Pythagoras, Ptolemy and the arithmetic tradition

Divisions of the Tetrachord 14-07-16 11:01 FA Divisions of the Tetrachord yh John Chalmers + on (Published by Frog Peak Music) Edited by Larry Polansky and Carter Scholz Designed by Carter Scholz Editor's Introduction (Larry Polansky), Forward (Lou Harrison), Preface (John Chalmers), Acknowledgements Chapter 1: The tetrachordin experimental music —7 Chapter 2: Pythagoras, Ptolemy. and the arithmetic tradition ® Chapter 3: Aristoxenos and the geometrization of musical space Chapter 4: The construction of new genera Chapter 5: Classification, characterization, and analysis of tetrachords Chapter 6: Scales, modes, and systems Chapterr 8: Schlesinger’S harmonia TE n's diaphonic cycles, and other similar constructs Chapter 9: The Catalog of tetrachords A note on this website: This is a set of scans of the original Frog Peak publication from the early 1990s. No editing has been done, nor any attempt to @pdate the book. (We will post the color cover soon). Larry Polansky http:/feamusic.dartmouth.edu/-larry/published_articles/divisions_of_the_tetrachord/ Pagina 1 van1

Divisions of the Tetrachord 14-07-16 11:07 Divisions of the Tetrachord John Chalmers (Published by Frog Peak Music) Edited by Larry Polansky and Carter Scholz Designed by Carter Scholz Editor's Introduction (Larry Polansky), Forward (Lou Harrison), Preface (John Chalmers), Acknowledgements Chapter 1: The tetrachord in experimental music Chapter 2: Pythagoras, Ptolemy, and the arithmetic tradition Chapter 3: Aristoxenos and the geometrization of musical space Chapter 4: The construction of new genera Chapter 5: Classification, characterization, and analysis of tetrachords Chapter 6: Scales, modes, and systems Chapter 7: Harmonization of tetrachordal scales Chapter 8: Schlesinger's harmoniai, Wilson's diaphonic cycles, and other similar constructs Chapter 9: The Catalog of tetrachords A note on this website: This is a set of scans of the original Frog Peak publication from the early 1990s. No editing has been done, nor any attempt to update the book. (We will post the color cover soon). Larry Polansky http://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/ Pagina 1 van 1

EDITOR’S INTRODUCTION WHEN I was 4 young student in California, Lou Harrison suggested thatI send one of my first pieces, Piano Study #5 (for JPR) to a Dr. Chalmers, who might publish it in his journal Xenbarmonikon. Flattered and fascinated, I did, and John did, and thus began what is now my twenty year friendship with this polyglot fungus researcher tuning guru science fiction devotee and general everything expert. Lou first showed me the box of papers, already called Divisions of the Tetrachord, in 1975. I liked the idea of this grand, obsessive project, and felt that it needed to be available in a way that was, like John himself, out of the ordinary. When Jody Diamond, Alexis Alrich, and I founded Frog Peak Music (A Composers’ Collective) in the early 80s, Divisions (along with Tenney’s then unpublished Meta + Hodos) was in my mind as one of the publishing collective’s main reasons for existing, and for calling itself a publisher of “speculative theory.” The publication of this book has been a long and arduous process. Revised manuscripts traveled with me from California to Java and Sumatra (John requested we bring him a sample of the local fungi), and finally to our new home in New Hampshire. The process of writing, editing, and publishing it has taken nearly fifteen years, and spanned various writing technologies. (When John first started using a word processor, and for the first time his many correspondents could actually read his long complicated letters, my wife and I were a bit sad—we had enjoyed reading his completely illegible writing aloud as a kind of sound poetry).

May people have contributed to the publication of this book, all voluntering their valuable time. David Doty (editor of 1/1, The Journal of the Jus hunation Network) and Daniel J. Wolf (who took over publication of Ynkmonikon for several issues in the 1980s) both made a tremendous editorial contribution to style and content. Jarrad Powell, Joel Mandeham, David Rothenberg (especially for chapter five) and Jody Diamond mateviluable suggestions. Lauren Pratt, who is to copy editing what John Chiers is to tetrachords, saw countless errors that were not there until shepointed them out. Carter Scholz, the one person I know who can give John Chalmers a run for his money in the area of polymathematics, began asthe book’s designer, and by virtue of his immeasurable contributions, became its co-editor. john Chalmers’s Divisions of the Tetrachord is a fanatic work. It is not a book that everyone will read or understand. It is a book that needs to exist, Larry POLANSKY lebanon, New Hampshire 1992

FOREWORD NEARLY TWENTY YEARS AGO John Chalmers and I had a number of very fruitful conversations. Well acquainted with the work of Harry Partch and also of younger musical theoreticians, Erv Wilson among them, John brought an immense amount of historical and scientific knowledge to our happy meetings. In turn, William Colvig and I brought the substance of professional musical life and the building of musical instruments. At that time I had rhapsodic plans for a “Mode Roem,” possibly for UNESCO, in which would be assembled some great world-book of notated modes, their preferred tunings and both ethnic and geographic provenance, along with such history of them as we might have. I had supposed a roomful of drawers, each holding an octave metallophone of a mode, and somewhere a harp or psaltery of some further octaves’ compass on which one might try out wider musical beauties of the mode under study. I even wrote out such a proposal in Esperanto and distributed it in an international ethnomusicology conference in Tokyo in 1961. However, a little later Mr. Colvig began to build extremely accurate monochords on which we could study anything at all, and we rushed, ina kind of ecstasy, to try everything at once. Bill and I designed and built a “transfer harp,” wirestrung and with two tuning systems, both gross and fine, Although innocently and quickly designed and built, its form, we discovered, is that of what the Chinese call a “standing harp”— the plate is parallel to the strings. We already owned a Lyon and Healey troubador harp, and, with these and with the addition of one or two other incidental

instruments, a bowed psaltery, drones, and small percussion, Richard Dee and Tin one rapturous weekend tuned and recorded improvisations in a fair number of modes from planetary history, especially from the classical civilizations and Islam. A little later, our friend Larry London, a professional clarinetist with wide intellectual interests and a composer of wide-ranging inquiry, made two improved versions of our original “transfer harp” and he actually revived what literature tells us is the way Irish bards played their own wirestrung harps, stopping off strings as you go. He has composed and plays a beautiful repertory of pieces and suites (each in a single mode) for his harps. I continue to want to hear hirn in some handsome small marble hall that reminds of Alexandria, Athens, or Rome. Thus, the “Mode Room,” about which I am still asked, turned into anyone’s room, with a good monochord and some kind of transfer instrument, But the great book of modes? Knowing that the tetrachord is the module with which several major civilizations assemble modes, John and I had begun to wonder about how many usable tetrachords there might be. We decided that the ratio 81/80 is the “flip-over” point and the limit of musical use, although not of theoretical use. This is the interval that everyone constantly shifts around when singing or playing major and minor diatonic modes, for it is the difference between a major major second (9/8) and a minor major second (10/9) and the distribution of these two kinds of seconds determines the modal characters. Thus our choice. John immediately began a program, and began to list results. I think that he used a computer and he soon had quite a list. From his wide reading he also gave attributions as historically documented formations turned up. It was enthralling, and this was indeed the “Great Book”— to my mind the most important work of musical theory since Europe’s Renaissance, and probably since the Roman Empire. But it has taken many years to mature. Not only is John a busy scientist and teacher, but he has wished to bring advanced mathematical thought to the work and enjoys lattice thinking and speculation, often fruitful. He tried a few written introductions which I in turn tried to make intelligible to advanced musicians, who, I thought, might see in his work a marvelous extension of humanist enquiry. Always he found my effort lacking to his needs. He often employed a style of scientese as opaque

to me as his handwriting is illegible. About the latter there is near universal agreement—John himself jestingly joins in this. In the last very few years all of us have finally had translations into English of Boethius, Ptolemy, and others—all for the first time in our language. For decades before this John worked from the Greek and other languages. This, too, was formidable. Few studies have stimulated me as has John Chalmers’s Divisions of the Tetrachord. It is a great work by any standards, and I rejoice. Lou Harrison

PREFACE THIS BOOK IS WRITTEN to assist the discovery of new musical resources, not to reconstruct the lost musical culture of ancient Greece. I began writing it as an annotated catalog of tetrachords while I was a post-doctoral fellow in the Department of Genetics at the University of California, Berkeley in the early 1970s. Much earlier, I had become fascinated with tuning theory while in high school as a consequence of an unintelligible and incorrect explanation of the 12-tone equal temperament in a music appreciation class. My curiosity was aroused and I went to the library to read more about the subject. There I discovered Helmholtz’s On the Sensations of Tone with A. J. Ellis’s annotations and appendices, which included discussions of non-12-tone equal temperaments and long lists of just intervals and historical scales. Later, the same teacher played the 1936 Havana recording of Julián Carrillo’s Preludio a Colón to our class, ostensibly to demonstrate the sorry condition of modern music, but I found the piece to be one of almost supernatural beauty, and virtually the only interesting music presented the entire semester. During the next summer vacation, I made a crude monochord calibrated to 19-tone equal temperament, and later some pan pipes in the 5- and 9tone equal systems. Otherwise, my interest in microtonal music remained more or less dormant for lack of stimulation until as a sophomore at Stanford I attended its overseas campus in Stuttgart. Music appreciation happened to be one of the required courses and Stockhausen was invited to address the class and play tapes of “elektronische Musik,” an art-form totally unknown to me at the time. This experience rekindled my interest in

music theory and upon my return to California, I tried to sign up for courses in experimental music. This proved impossible to do, but I did find Harry Partch’s book and a recording of the complete Oedipus in the Music Library. Thus I began to study microtonal tuning systems. My roommates were astonished whenI drove nails into my desk, strung guitar strings between them, and cut up a broom handle for bridges, but they put up with the resulting sounds more or less gracefully. During my first year of graduate school in biology at UCSD, I came across the article by Tillman Schafer and Jim Piehl on 19-tone instruments (Schafer and Piehl 1947). Through Schafer, who still lived in San Diego at that time, I met Ivor Darreg and Ervin Wilson. Later Harry Partch joined the UCSD music faculty and taught a class which I audited in 1967-68, About this time also, I began collaborating with Ervin Wilson on the generation of equal temperament and just intonation tables at the UCSD computer center (Chalmers 1974, 1982). After finishing my Ph.D. I received a post-doctoral fellowship from the National Institutes of Health to do research at the University of Washington in Seattle and from there I moved to Berkeley to the Department of Genetics to continue attempting to study cytoplasmic or non-Mendelian genetics in the mold Nezrospora crassa. A visit by John Grayson provided an opportunity to drive down to Aptos and meet Lou Harrison. I mentioned to Lou that I had begun a list of tetrachords in an old laboratory notebook and he asked me for a copy. I photocopied the pages for him and mailed them immediately. Lou urged me to expand my notes into a book about tetrachords, but alas, a number of moves and the demands of a career as both an industrial and academic biologist competed with the task. While working for Merck Sharp & Dohme in New Jersey before moving to Houston in the mid19708, I wrote a first and rather tentative draft. I also managed to find the time to edit and publish Xenbarmonikon, An Informal Fournal ofExperimental Music, while certain harmonic ideas gestated, but I had to suspend publication in 1979. Happily, it was resurrected in 1986 by Daniel Wolf and I resumed the editorship late in 1989. In the winter of 1980, I was invited to the Villa Serbelloni on Lake Como by the Rockefeller Foundation to work on the book and I completed another draft there. Finally, through the efforts of Larry Polansky and David Rosenboom, I was able to spend the summer of 1986 at Mills College

working on the manuscript. It was at Mills also that I discovered that the Macintosh computer has four voices with excellent pitch resolution and is easily programmed in BASIC to produce sound. This unexpected opportunity allowed me to generate and hear a large number of the tetrachords and to test some of my theories, resulting in a significant increase in the size of the Catalog and much of the material in chapter 7. After returning to Houston to work for a while as a consultant for a biotechnology firm, I moved back to Berkeley in the fall of 1987 so thatI could devote the necessary time to completing the book. With time out to do some consulting, learn the HMSL music composition and performing language developed at Mills College, and work as a fungal geneticist once again at the University of California, the book was finally completed. A few words on the organization of this work are appropriate. The first three chapters are concerned with tetrachordal theory from both classical Graeco-Roman and to a lesser extent medieval Islamic perspectives. The former body of theory and speculation have been discussed in extenso by numerous authorities since the revival of scholarship in the West, but the latter has not, as yet, received the attention it deserves from experimentally minded music theorists. After considerable thought, I have decided to retain the Greek nomenclature, though not the Greek notation. Most importantly, it is used in all the primary and secondary sources I have consulted; readers desiring to do further research on tetrachords will have become familiar with the standard vocabulary as a result of exposure to it in this book. Secondly, the Greek names of the modes differ from the ecclesiastical ones used in most counterpoint classes. To avoid confusion, it is helpful to employ a consistent and unambiguous system, which the Greek terminology provides. Since many of the musical concepts are novel and the English equivalents of a number of the terms have very different meanings in traditional music theory, the Greek terminology is used throughout. For example, in Greek theory, the adjective enharmonic refers to a type of tetrachord containing a step the size of a major third, with or without the well-known microtones. In the liturgical music theory of the Greek Orthodox church, also called Byzantine (Savas 1965; Athanasopoulos 1950), it refers to varieties of diatonic and chromatic tunings, while in traditional European theory, it refers to two differently written notes with the same pitch. Where

modern terms are familiar and unambiguous, and for concepts not part of ancient Greek music theory, I have used the appropriate contemporary technical vocabulary. Finally, I think the Greek names add a certain mystique or glamour to the subject. I find the sense of historical continuity across two and a half millen— nia exhiliarating—four or more millennia if the Babylonian data on the diatonic scale are correct (Duchesne-Guillemin 1963; Kilmer 1960). Harry Partch must have felt similarly when he began to construct the musical system he called monophony (Partch [1949] 1974). Science, including experimental musicology, is a cumulative enterprise; it is essential to know where we have been, as we set out on new paths. Revolutions do not occur in vacuo. The contents of the historical chapters form the background for the new material introduced in chapters 4 through 7. It is in these chapters that nearly all claims for originality and applicability to contemporary composition reside. In particular, chapters 5, 6, and 7 are intended to be of assistance to composers searching for new materia musica. Chapter 8 deals with the heterodox, though fascinating, speculations of Kathleen Schlesinger and some extrapolations from her work. While I do not believe that her theories are descriptive of Greek music at any period, they may serve as the basis for a coherent approach to scale construction independent of their historical validity. While not intended as a comprehensive treatise on musical scale construction, for which several additional volumes at least as large as this would be required, this work may serve as a layman’s guide to the tetrachord and to scales built from tetrachordal modules. With this in mind, a glossary has been provided which consists of technical terms in English pertaining to intonation theory and Greek nomenclature as far as it is relevant to the material and concepts presented in the text. Terms explained in the glossary are italicized at their first appearance in the text. The catalogs of tetrachords in chapter 9 are both the origin of the book and its justification—the first eight chapters could be considered as an extended commentary on these lists.

ACKNOWLEDGMENTS PorTIONS OF CHAPTER 5 and an earlier version of chapter 6 originally appeared in the journal Xenbarmonikon (Chalmers 1975; 1989). A much shorter draft of the book was written at the Centro Culturale Della Fondazione Rockefeller at Bellagio, Italy while I was a Scholar-inResidence in 1980. I would like to express my gratitude to Larry Polansky and David Rosenboom for arranging a summer residency for me at Mills College in 1986 to work on the manuscript, and for introducing me to the Macintosh as a word processor and acoustic workstation. Thanks are also due to Dr. Patricia St. Lawrence for the opportunity to come to Berkeley and work at the Department of Genetics during the academic years 1987-88 and 1988-89. Parts of this book are based on the unpublished work of Ervin M. Wilson who not only placed his notes at my disposal but also served as a teacher and critic in the early stages of the manuscript. Any errors or omissions in the presentation of his material are solely my fault. The same may be said of David Rothenberg, whose perception theories are a prominent part of chapter 5. Finally, it was Lou Harrison who suggested that I write a book on tetrachords in the first place and who has patiently awaited its completion.

The tetrachord in experimental music Why, IN THE LAST quarter of the twentieth century, would someone write a lengthy treatise on a musical topic usually considered of interest only to students of classical Greek civilization? Furthermore, why might a reader expect to gain any information of relevance to contemporary musical composition from sucha treatise? I hope to show that the subject of this book is of interest to composers of new music. The familiar tuning system of Western European music has been inherited, with minor modifications, from the Babylonians (Duchesne- Guillemin 1963). The tendency within the context of Western European “art music” to use intervals outside this system has been called microtonality, experimental intonation (Polansky 1987a), or xenharmonics (a term proposed by Ivor Darreg). Interest in and the use of microtonality, defined by scalar and harmonic resources other than the traditional 12-tone equal temperament, has recurred throughout history, notably in the Renaissance (Vicentino 1555) and most recently in the late nineteenth and early twentieth century. The converse of this definition is that music which can be performed in 12-tone equal temperament without significant loss of its identity is not truly microtonal. Moreover, the musics of many of the other cultures of the world are microtonal (in relation to 12-tone equal temperament) and European composers have frequently borrowed musical materials from other cultures and historical periods, such as the Ottoman Empire and ancient Greece. We owe our traditions of musical science to ancient Greece, and the theoretical concepts and materials of ancient Greek music are basic to an THE TETRACHORD IN EXPERIMENTAL MUSIC

understanding of microtonal music. Greek musical theory used the terrachord as a building block or module from which scales and systems could be constructed. A current revival of interest in microtonality, fueled by new musical developments and technological improvements in computers and synthesizers, makes the ancient tetrachord increasingly germane to contemporary composition, Contemporary microtonality Although 12-tone equal temperament became the standard tuning of Western music by the mid-nineteenth century (Helmholtz [1877] 1954), alternative tuning systems continued to find partisans. Of these systems, perhaps the most important was that of Bosanquet (Helmholtz [1877] 1954; Bosanquet 1876), who perfected the generalized keyboard upon which the fingering for musical patterns is invariant under transposition. He also championed the 53-tone equal temperament. Of nineteenthcentury theorists, Helmholtz and his translator and annotator A. J. Ellis (Helmholtz [1877] 1954) are outstanding for their attempts to revive the use of just intonation. The early twentieth century saw a renewed interest in quarter-tones (24tone equal temperament) and other equal divisions of the octave. The Mexican composer Julián Carrillo led a crusade for the equal divisions which preserved the whole tone (zero modulo 6 divisions) through 96-tone temperament or sixteenths of tones. Other microtonal, mostly quartertone, composers of note were Alois Hába (Czechoslovakia), Ivan Wyschnegradsky (France), and Mildred Couper (USA). The Soviet Union had numerous microtonal composers and theorists, including Georgy RimskyKorsakov, Leonid Sabaneev, Arseny Avraamov, E.K. Rosenov, A.S. Obolovets, and P.N. Renchitsky, before Stalin restrained revolutionary creativity under the doctrine of Socialist Realism (Carpenter 1983). Joseph Yasser (USA) urged the adoption of 19-tone equal temperament and Adriaan Fokker (Holland) revived the theories of his countryman, Christian Huygens, and promoted 31-tone equal temperament. More recently, Martin Vogel in Bonn and Franz Richter Herf in Salzburg have been active in various microtonal systems, the latter especially in 72-tone equal temperament. No discussion of alternative tunings is complete without mentioning Harry Partch, an American original who singlehandedly made extended CHAPTER 1

just intonation and home-built instruments not only acceptable, but virtually mandatory for musical experimenters at some stage in their careers. Composers influenced by him include Lou Harrison, Ben Johnston, James ‘Tenney, and younger composers such as Larry Polansky, Cris Forster, Dean Drummond, Jonathan Glasier, and the members of the Just Intonation Network. Ivor Darreg is an American composer working in California. He has been very actively involved with alternative tunings and new instrument design for more than five decades. Darreg has employed both non-r2-tone equal temperaments and various forms of just intonation in his music, theoretical writings, and instruments. More recently, he has begun to use MIDI synthesizers and has explored all the equal temperaments up to 53 tones per octave in a series of improvisations in collaboration with Brian McLaren. Ervin Wilson is one of the most prolific and innovative inventors of new musical materials extant and has been a major influence on me as well as a source for many tetrachords and theoretical ideas. He holds patents on two original generalized keyboard designs. Wilson has collaborated with Kraig Grady and other experimental musicians in the Los Angeles area. He also assisted Harry Partch with the second edition of Genesis of a Music by drawing some of the diagrams in the book. Some other North American microtonal composers are Ezra Sims, Easley Blackwood, Joel Mandelbaum, Brian McLaren, Arturo Salinas, Harold Seletsky, Paul Rapoport, William Schottstaedt, and Douglas Walker. While still very much a minority faction of the contemporary music community, microtonality is rapidly growing. Festivals dedicated to microtonal music have been held in recent years in Salzburg under the direction of Franz Richter Herf; in New York City, produced by Johnny Reinhard; and in San Antonio, Texas, organized by George Cisneros. Partch, Darreg, Wilson, Harrison, Forster, and William Colvig, among others, have designed and constructed new acoustic instruments for microtonal performance. Tunable electronic synthesizers are now available commercially and provide an an alternative to custom-built acoustic or electroacoustic equipment. A great deal of software, such as HMSL from Frog Peak Music, 7ICak by Robert Rich and Carter Scholz, and Antelope Engineering’s TuneUp, has been developed to control synthesizers microtonally via MIDI. THE TETRACHORD IN EXPERIMENTAL MUSIC

Good references for additional information on the history of microtonal systems are Helmholtz ([1877] 1954), Barbour (1951), Partch ([1949] 1974), and Mandelbaum (1961). Small press publications are a rich source and several journals devoted to music in alternative tunings have been published. The major ones are Xenbarmonikon, Interval, Pitch, and 1/1: The Journal of the Just Intonation Network. Finally, Musical Six-Six Bulletin, Leonardo: The InternationalJournal ofArts, Science, and Technology, Experimental Musical Instruments, and Musicworks have also contained articles about instruments in non-traditional tuning systems. The tetrachord in microtonal music Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world’s music, including that of Europe, the Near East, the Catholic and Orthodox churches, Iran, and India, are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the HYPATE PARHYPATE LICHANOS 1/1 MESE 4/3 L J F ai 3/2 PARAMESE 2/1 TRITE 1-1. The tetrachord. PARANETE NETE world’s music. The tetrachord is the interval of a perfect fourth, the diatessaron of the Greeks, divided into three subintervals by the interposition of two additional notes. The four notes, or strings, of the tetrachord were named hypate, parhypate, lichanos, and mese in ascending order from 1/1 to 4/3 in the first tetrachord of the central octave of the Greater Perfect System, the region of the scale of most concern to theorists. Ascending through the second tetrachord, they were called paramese, trite, paranete, and nete. (Chapter 6 discusses Greek scales and nomenclature.) Depending upon the spacing of these interposed tones, three primary genera may be distinguished: the diatonic, composed of tones and semitones; the chromatic, of semitones and a minor third; and the enharmonic, with a major third and two quarter-tones. Nuances or chroai (often translated “shades”) of these primary forms are further characterized by the exact tuning of these intervals. These four tones apparently sufficed for the recitation of Greek epic poetry, but soon afterwards another tetrachord was added to create a heptachord. As a feeling for the octave developed, the gamut was completed, CHAPTERI

and from this gamut various sections were later identified and given ancient tribal names (Dorian, Phrygian, et cetera). These octave species became the modes, two of which, the Lydian and Hypodorian, in the diatonic genus form the basis for the European tonal idiom. Although a formal nomenclature based on the position of the strings later developed, the four tetrachordal tones remained the basis for the Greek solfège: the syllables te, to, m, vo, (pronounced approximately teh, toe, tay, and tah in English) were sung in descending order to the notes of every genus and shade. The detailed history of the Greek tetrachordal scales is somewhat more complex than the sketchy outline given above. According to literary testimony supported at least in part by archaeology, the diatonic scale and its tuning by a cycle of perfect fifths, fourths, and octaves was brought from Egypt (or the Near East) by Pythagoras. In fact the entire r2-tone chromatic scale in this tuning is thought to have been known to the Babylonians by the second millennium sce and was apparently derived from earlier Sumerian precursors (Duchesne-Guillemin 1963, 1969; Kilmer 1960). Having arrived in Greece, this scale and its associated tuning doctrines were mingled with local musica] traditions, most probably pentatonic, to produce a plethora of scale-forms, melody-types and styles (see chapter 6). From a major-third pentatonic, the enharmonic genus can be derived by splitting the semitone (Winnington-Ingram 1928; Sachs 1943). The chromatic genera, whose use in tragedy dates from the late fifth century, may be relicts of various neutral and minor-third pentatonics, or conversely, descended from the earlier enharmonic by a process of “sweetening” whereby the pitch of the third tone was raised from a probable 256/243 to produce the more or less consonant intervals 5/4, 6/5, 7/6 and possibly 11/9 (Winnington-Ingram 1928). The resulting scales were rationalized by the number theory of Pythagoras (Crocker 1963, 1964, 1966) and later by the geometry of Euclid (Crocker 1966; Winnington-Ingram 1932, 1936) to create the body of theory called harmonics, which gradually took on existence as an independent intellectual endeavor divorced from musical practice. The acoustic means are now available, and the prevailing artistic ideology is sympathetic enough to end this separation between theory and practice. Many composers have made direct use of tetrachordal scales in recent compositions. Harry Partch used the pentatonic form of the enharmonic (16/15 - 5/4- 9/8 + 16/15 - 5/4) in the first of his Two Studies on Ancient Greek THE TETRACHORD IN EXPERIMENTAL MUSIC

Scales (1946) and the microtonal form in the second (in Archytas’s tuning, 28/27 - 36/35 - 5/4). Partch also employed this latter scale in The Dreamer that Remains, and in verse fifteen of Petals. His film score Windsong (1958) employs Ptolemy’s equable diatonic (diatonon homalon). Ivor Darreg’s On the Enharmonic Tetrachord from his collection Excursion into the Enharmonic, was composed in 1965 and published in Xenharmonikon 3 in 1975. Lou Harrison has used various tetrachords as motives in his “free style” piece A Phrase for Arion’s Leap (Xenharmonikon 3, 1975). An earlier piece, Suite (1949) was based on tetrachords in 12-tone equal temperament. Larry London published his Eight Pieces for Harp in Ditone Diatonic in Xenharmonikon 6 (1977) and his Four Pieces in Didymus’s Chromatic in Xenharmonikon 7+8 (1979). In 1984, he wrote a Suite for Harp whose four movements used Archytas’s enharmonic and a chromatic genus of J.M. Barbour. Gino Robair Forlin’s song in Spanish and Zapotec, Las Tortugas (1988), is based on the tetrachord 16/15 - 15/14 - 7/6. There are of course many other recent pieces less explicitly tetrachordal whose pitch structures could be analyzed in tetrachordal terms, but doing so would be a major project outside the scope of this book. Similarly, there is a vast amount of music from Islamic cultures, Hindustani, and Eastern Orthodox traditions which is also constructed from tetrachordal scales. These will not be discussed except briefly in terms of their component tetrachords. A psychological motivation for the consideration of tetrachords is provided by the classic study of George A. Miller, who suggested that musical scales, in common with other perceptual sets, should have five to nine elements for intuitive comprehension (Miller 1956). Scales with cardinalities in this range are easily generated from tetrachords (chapter 6) and the persistence of tetrachordal scales alongside the development of triad-based harmony may reflect this property. Tetrachords and their scale-like complexes and aggregates have an intellectual fascination all their own, a wealth of structure whose seductive intricacy I hope to convey in this book. CHAPTER I

Pythagoras, Ptolemy, and the arithmetic tradition GREEK MUSICAL TRADITION begins in the sixth century sce with the semi-legendary Pythagoras, who is credited with discovering that the frequency of a vibrating string is inversely proportional to its length. This discovery gave the Greeks a means to describe musical intervals by numbers, and to bring to acoustics the full power of their arithmetical science. While Pythagoras’s own writings on music are lost, his tuning doctrines were preserved by later writers such as Plato, in the Timaeus, and Ptolemy, in the Harmonics, The scale derived from the Timaeus is the so-called Pythagorean tuning of Western European theory, but it is most likely of Babylonian origin. Evidence is found not only in cuneiform inscriptions giving the tuning order, but apparently also as music in a diatonic major mode (DuchesneGuillemin 1963, 1969; Kilmer 1960; Kilmer et al. 1976). This scale may be tuned as a series of perfect fifths (or fourths) and octaves, having the ratios 1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1, though the Babylonians did not express musical intervals numerically. The next important theorist in the Greek arithmetic tradition is Archytas, a Pythagorean from the Greek colony of Tarentum in Italy. He lived about 390 BCE and was a notable mathematician as well. He explained the use of the arithmetic, geometric, and harmonic means as the basis of musical tuning (Makeig 1980) and he named the harmonic mean. In addition to his musical activities, he was renowned for having discovered a threedimensional construction for the extraction of the cube root of two. Archytas is the first theorist to give ratios for all three genera. His tunings are noteworthy for employing ratios involving the numbers 5 and 7 PYTHAGORAS, PTOLEMY, AND THE ARITHMETIC TRADITION

instead of being limited to the 2 and 3 of the orthodox Pythagoreans, for using the ratio 28/27 as the first interval (hypate to parhypate) in all three genera, and for employing the consonant major third, 5/4, rather than the harsher ditone 81/64, as the upper interval of the enharmonic genus. These tunings are shown in 2-1. | Other characteristics of Archytas’s tunings are the smaller second interval of the enharmonic (36/35 is less than 28/27) and the complex second interval of his chromatic genus. Archytas’s enharmonic is the most consonant tuning for the genus, especially when its first interval, 28/27, is combined with a tone 9/8 below the tonic to produce an interval of 7/6. This note, called byperhypate, is found not only in the harmoniai of Aristides Quintilianus (chapter 6), but also in the extant musical notation fragment from the first stasimon of Euripides’s Orestes. It also occurs below a chromatic pyknon in the second Delphic hymn (Winnington-Ingram 1936). This usage strongly suggests that the second note of the enharmonic and chromatic genera was not a grace note as has been suggested, but an independent degree of the scale (ibid.). Bacchios, a much later writer, calls the interval formed by the skip from hyperhypate to the second degree an ekbole (Steinmayer 1985), further affirming the historical correctness of Archytas’s tunings. The complexity of Archytas’s chromatic genus demands an explanation, as Ptolemy's soft chromatic (chroma malakon) 28/27 - 15/14. 6/5 would seem to be more consonant. Evidently the chromatic pyknon still spanned the 9/8 at the beginning of the fourth century, and the 32/27 was felt to be ARCHYTAS’S GENERA 2-1. Ptolemy’s catalog ofhistorical tetrachords, from the Harmonics (Wallis 1682). The genus 28/27 + 36/35 + 5/4 28/27 + 243/224 32/27 28/27: 8/3. 9/8 5/55. 22/21 - s/4 (31 +81 +386 cents) is also at- 63 + 49 + 386 63 + 141 + 294 63+231 +204 ENHARMONIC CHROMATIC DIATONIG ERATOSTHENES’S GENERA tributed to Ptolemy, Wallis says that this genus is in 40/39 + 39/38 - 19/15 44 + 45 + 409 ENHARMONIC all ofthe manuscripts, but is likely to be a later addi- 20/19: 19/18. 6/5 256/243 : 9/8 + 9/8 89 + 94 + 316 90 + 204 + 204 CHROMATIC DIATONIC tion. The statements ofAvicenna and Bryennios that 46/45 is the smallest melodic interval supports this view, DIDYMOS’$ GENERA 32/31 - 31/30. 5/4 55 + 57 + 386 ENHARMONIC 16/15-25/24 - 6/5 16/15: 10/9 . 9/8 112 +71 + 316 112 + 182 + 204 CHROMATIC DIATONIC CHAPTER 2

the proper tuning for the interval between the upper two tones. This may be in part because 32/27 makes a 4/3 with the disjunctive tone immediately following, but also because the melodic contrast between the 32/27 at the top of the tetrachord and the 7/6 with the hyperhypate below is notas great as the contrast between lower 7/6 and the upper 6/5 of Ptolemy’s tuning. Archytas’s diatonic is also found among Ptolemy’s own tunings (2-2) and appears in the /yra and kithara scales that Ptolemy claimed were in common practice in Alexandria in the second century ce. According to Winnington-Ingram (1932), itis even grudgingly admitted by Aristoxenos and thus would appear to have been the principal diatonic tuning from the fourth century BCE through the second cz, a period of some six centuries. Archytas’s genera represent a considerable departure from the austerity of the older Pythagorean forms: ENHARMONIC: 256/243 - 81/64 CHROMATIC: 256/243 - 2187/2048 « 32/27 DIATONIC: 256/243 -0/8 - 9/8 The enharmonic genus is shown as a trichord because the tuning of the enharmonic genus before Archytas is not precisely known. The semitone was initially undivided and may not have had a consistent division until the stylistic changes recorded in his tunings occurred. In other words, the incomposite ditone, not the incidental microtones, is the defining characteristic of the enharmonic genus. The chromatic tuning is actually that of the much later writer Gaudentius (Barbera 1978), but it is the most plausible of the Pythagorean chromatic tunings. The diatonic genus is the tuning associated with Pythagoras by all the authors from ancient times to the present (Winnington-Ingram 1932). 2-2. Ptolemy's own tunings. 46/45 24/23 5/4 28/27-15/14-6/5 22/21 12/11:7/6 21/20- 10/9 « 8/7 28/27 - 8/7 «9/8 256/243:9/8-9/8 16/15 + 9/8. 10/9 12/11: ı1/io-10/9 9 38+75+386 63+119+316 81+151+267 85+182+231 63+231 +204 go+204+204 112 +2044+ 182 151 + 165 + 182 ENHARMONIC SOFT CHROMATIG INTENSE CHROMATIC SOFT DIATONIC DIATONON TONIAION DIATONON DITONIAION INTENSE DIATONIG EQUABLE DIATONIC PYTHAGORAS, PTOLEMY, AND THE ARITHMETIC TRADITION

Ptolemy and his predecessors in Alexandria In addition to preserving Archytas’s tunings, Ptolemy (ca. 160 ce) also transmitted the tunings of Eratosthenes and Didymos, two of his predecessors at the library of Alexandria (2-1). Eratosthenes’s (third century BCE) enharmonic and chromatic genera appear to have been designed as simplifications of the Pythagorean prototypes. The use of 40/39 and 20/19 for the lowest interval presages the remarkable Tanbur of Baghdad of Al-Farabi with its subharmonic division by the modal determinant 40 (Ellis 1885; D’Erlanger 1935) and some of Kathleen Schlesinger’s speculations in The Greek Aulos (1939). Didymos’s enharmonic seems to be mere formalism; the enharmonic genus was extinct in music as opposed to theory by his time (first century BCE). His 1:1 linear division of the pyknon introduces the prime number 31 into the musical relationships and deletes the prime number 7, a change which is not an improvement harmonically, though it would be of less significance in a primarily melodic music. His chromatic, on the other hand, is the most consonant non-septimal tuning and suggests further development of the musical styles which used the chromatic genus. Didymos’s diatonic is a permutation of Ptolemy’s intense diatonic (diatonon syntonon). It seems to be transitional between the Pythagorean (3-/imit) and tertian tunings. Ptolemy’s own tunings stand in marked contrast to those of his predecessors. In place of the more or less equal divisions of the pyknon in the genera of the earlier theorists, Ptolemy employs a roughly 1:2 melodic proportion. He also makes greater use of superparticular or epimore ratios than his forerunners; of his list, only the traditional Pythagorean diatonon ditoniaion contains epimteres, which are ratios of the form (72 + 7)/n where m> 1. The emphasis on superparticular ratios was a general characteristic of Greek musical theory (Crocker 1963; 1964). Only epimores were accepted even as successive consonances, and only the first epimores (2/1, 3/2, and 4/3) were permitted as simultaneous combinations. There is some empirical validity to these doctrines: there is no question that the first epimores are consonant and that this quality extends to the next group, 5/4 and 6/5, else tertian harmony would be impossible, Consonance of the septimal epimore 7/6 is a matter of contention. To my ear, it is consonant, as are the epimeres 7/4 and 7/5 and the inversions of the epimores 5/4 and 6/5 (8/5 and 5/3). Moreover, Ptolemy noticed that octave CHAPTER 2

compounds of consonances (which are not themselves epimores) were aurally consonant. It is clear, therefore, that it is not just the form of the ratio, but at least two factors, the size of the interval and the magnitude of the defining integers, that determines relative consonance. Nevertheless, there does seem to be some special quality of epimore ratios. I recall a visitto Lou Harrison during which he began to tune a harp to the tetrachordal scale 1/1 27/25 6/5 4/3 3/2 81/50 9/5 2/1. He immediately became aware of the non-superparticular ratio 27/25 by perceiving the lack of resonance in the instrument. A complete list of all possible tetrachordal divisions containing only superparticular ratios has been compiled by I. E. Hofmann (Vogel 1975). Although the majority of these tetrachords had been discovered by earlier 2-3. Hofmann 's list ofcompletely superparticular divisions, This table has been recomposed after theorists, there were some previously unknown divisions containing very small intervals. The complete set is given in 2-3 and individual entries also appear in the Miscellaneous listing of the Catalog. The equable diatonic has puzzled scholars for years as it appears to be an academic exercise in musical arithmetic. Ptolemy’s own remarks rebut this interpretation as he describes the scale as sounding rather strange or foreign and rustic (Eevikotepov pev nog kat aypotkotepov, Winnington- Hofmann from Vogel (1975). See Main Catalogfor farther information.(s) has also been attributed to Ingram 1932). Even a cursory look at ancient and modern Islamic scales Tartini, but probably should be credited to Pachymeres, a thirteenth-century Byzantine author. a similar scale and very cleverly rationalized it according to the tenets of Greek theory. Such scales with 3/4-tone intervals may be related to from the Near East suggests that, on the contrary, Ptolemy may have heard I. 2§6/255+ 17/16. 5/4 NEW ENHARMONIC 14. 28/27. 15/14 : 6/5 PTOLEMY’S SOFT CHROMATIC 2. 3 136/135 - 18/17- 54 96/95 - 19/18 - 5/4 NEW ENHARMONIC WILSON’S ENHARMONIC 15. 16. 16/15 25/24: 6/4 20/19 - 19/18 - 6/5 DIDYMOS’S CHROMATIC ERATOSTHENES’S CHROMATIC 4. 76/75: 20/19: 5/4 AUTHOR'S ENHARMONIC 17. 64/63 - 9/8 + 7/6 BARBOUR AVICENNA 5. 64/63 + 21/20 + 5/4 SERRE'S ENHARMONIC 18. 36/35 : 10/0 + 7/6 6. 56/55 22/21 5/4 PSEUDO-PTOLEMAIC ENHARMONIC 19. 22/21: 12/11 - 7/6 PTOLEMY’S INTENSE CHROMATIC 7. 46/45 - 24/23 5/4 PTOLEMY’S ENHARMONIC 20. 16/15 : 15/14 : 7/6 AL-FARABI 8. 9. 40439: 26/25 - 5/4 28/27: 36/35 : 5/4 AVICENNA’S ENHARMONIC ARCHYTAS’S ENHARMONIC 21. 49/48 - 8/7 - 8/7 22. 28/27. 8/7 9/8 10. 32/31: 3130: 54 DIDYMOS’S ENHARMONIC 23. 21/20 : 10/9 - 8/7 PTOLEMY’S SOFT DIATONIG II. 100/99 - 11/10:6/ 12. 55/54: 12/11 - 6/5 NEW CHROMATIC BARBOUR 24. 25. 14/13: 13/12 + 8/7 16/15:19/18-10/9 AVICENNA PTOLEMY’S INTENSE DIATONIC 13. 40/39: 13/12 : 6/5 BARBOUR 26. 12/11: 11/10 - 10/9 PTOLEMY’S EQUABLE DIATONIC II AL-FARABI ARCHYTAS’S DIATONIC PYTHAGORAS, PTOLEMY, AND THE ARITHMETIC TRADITION

Aristoxenos’s hemiolic chromatic and may descend from neutral third pentatonics such as Winnington-Ingram’s reconstruction of the spondeion or libation mode (Winnington-Ingram 1928 and chapter 6), if Sachs’s ideas on the origin of the genera have any validity (Sachs 1943). In any case, the scale is a beautiful sequence of intervals and has been used successfully by 2-4. Genesis ofthe enharmonicpykna by katapyknoboth Harry Partch (Windsong, Daphne ofthe Dunes) and Lou Harrison, the sis. In principle, allpyknotic divisions can be generlatter in an improvisation in the early 197os. ated by this process, although very high multipliers Ptolemy returned to the use of the number seven in his chromatic and soft diatonic genera and introduced ratios of eleven in his intense chromatic and equable diatonic. These tetrachords appear to be in agreement with the may be necessary in some cases, The ones shown are merely illustrative. See the Catalogsfor the complete list, (1x) The basicform is the enbarmonic trichord, or major thirdpentatonic, often ascribed to Olympos, (2x) Didymos's enbarmonion, a “weak”form. (3x) Ptolemy's enbarmonion, a “strong” forms. To comply with Greek melodic canons, it was reordered as 46/45 - 24/23 + 5/4. (gx) Serre’ enharmonic, sometimes attributed to Tartini, and discussed by Perrett (1926, 26). Pachymeres may be the earliest source. (5x) Author's enharmonic, also on Hofmann’s list ofsuperparticular divisions, (6x) Wilson's enharmonic, also on Hofmann’s list of superparticular divisions. musical reality of the era, as most of the scales described as contemporary tunings for the lyra and kithara have septimal intervals (6-4). Ptolemy’s intense diatonic is the basis for Western European just intonation. The Lydian or C mode of the scale produced by this genus is the European major scale, but the minor mode is generated by the intervallic retrograde of this tetrachord, 10/9 + 9/8 - 16/15. This scale is not identical to the Hypodorian or A mode of 12-tone equally tempered, meantone, and Pythagorean intonations. (For further discussion of this topic, see chapters 6 and 7.) The numerical technique employed by Eratosthenes, Didymos, and Ptolemy to define the majority of their tetrachords is called linear division and may be identified with the process known in Greek as katapyknosis. Katapyknosis consists of the division, or rather the filling-in, of a musical INDEX NUMBERS IX 16 2X 32 31 3x 48 47 63 62 Is 30 PYKNA 16/15 32/31-31/30 46 45 24/23 + 46/45 61 60 64/63-21/20 4x 64 5x 6x 80 79 78 77 76 75 20/19-76/75 96 95 94 93 92 91 go 96/95-19/18 interval by multiplying its numerator and denominator bya set of integers of increasing magnitude. The resulting series of integers between the extreme terms generates a new set of intervals of increasingly smaller span as the multiplier grows larger. These intervals form a series of microtones which are then recombined to produce the desired melodic division, usually composed of epimore ratios. The process may be seen in 2-4 where it is applied to the enharmonic pyknotic interval 16:15. By extension, the pyknon may also be termed the katapyknosis (Emmanuel 1921). It consists of three notes, the barypyknon, or lowest note, the mesopyknon, or middle note, and the oxypyknon, or highest. The harmoniai of Kathleen Schlesinger are the result of applying katapyknosis to the entire octave, 2:1, and then to certain of the ensuing intervals. In chapter q it is applied to the fourth to generate indexed genera, The divisions of Eratosthenes and Didymos comprise mainly 1:1 divi- CHAPTER 2

sions of the pyknon while those of Ptolemy favor the 1:2 proportion, although in some instances the sub-intervals must be reordered so that the melodic proportions are the canonical order; small, medium and large. This 2-5. Ptolemy’s interpretation of Aristoxenos’s principle was also enunciated by Aristoxenos, but violated by Archytas, Didymos, and Ptolemy himself in his diatonic tunings. Amore direct method of calculating the divisions is to use the following genera. formulae (Winnington-Ingram 1932; Barbera 1978) where x/y is the interval to be linearly divided: ENHARMONIC 44 + 45 + 409 40/39 è 39/38 + 19/15 1/1 1/2 2/1 SOFT CHROMATIC 30/29 - 29/28 + 56/45 59 + 60 + 379 HEMIOLIC CHROMATIC 80/77 - 77/74: 37/30 66 + 69 + 363 INTENSE CHROMATIC 20/19 - 19/18 : 6/5 The final set of tetrachords given by Ptolemy are his interpretations of the genera of Aristoxenos (2-5). Unfortunately, he seems to have com- 89 + 142 + 267 pletely misunderstood Aristoxenos’s geometric approach and translated his “parts” into aliquot parts of a string of 120 units. Two of the resulting tetrachords are identical to Eratosthenes’s enharmonic and chromatic genera, INTENSE DIATONIC 89 + 192 +217 20/19 : 19/17 + 17/15 Finer divisions may be defined analogously; if a/b is the desired proportion and x/ the interval, then (a +5) -x/(bx+ay)-(bx+ay)/(a+b)-y=x/y. 89 + 04 + 316 SOFT DIATONIC 20/19 + 38/35 - 7/6 2x/(x+y) + y)/2y=ax/y, 3x/(2x + y) - (2x + y)/3y=x/y, Zelle +2) (+ 29)/3y=x/y. but the others are rather far from Aristoxenos’s intent. The Ptolemaic version of the hemiolic chromatic is actually a good approximation to Aristoxenos’s soft chromatic. Aristoxenos’s theories will be discussed in detail in chapter 3. The late Roman writers After Ptolemy’s recension of classical tuning lore, a few minor writers such as Gaudentius (fourth century ce) continued to provide tuning information in numbers rather than the fractional tones of the Aristoxenian school. Gaudentius’s diatonic has the familiar ditone or Pythagorean tuning, as does his intense chromatic (chroma syntonon), 256/243 + 2187/2048 - 32/27 (Barbera 1978). The last classical scholar in the ancient arithmetic tradition was the philosopher Boethius (sixth century ce) who added some novel tetrachords and also hopelessly muddled the nomenclature of the modes for succeeding generations of Europeans. Boethius’s tuning for the tetrachords in the three principal genera are below: 512/499 : 499/486 : 81/64 ENHARMONIC: CHROMATIC: 256/243 + 81/76 - 19/16 DIATONIC: 256/243 : 9/8 - 9/8 PYTHAGORAS, PTOLEMY, AND THE ARITHMETIC TRADITION

These unusual tunings are best thought of as a simplification of the Pythagorean forms, as the limma (256/243) is the enharmonic pyknon and the lowest interval of both the chromatic and diatonic genera. The enharmonic uses the 1:1 division formula to divide the 256/243, and the 19/16 is virtually the same size as the Pythagorean minor third, 32/27. The medieval Islamic theorists With the exception of Byzantine writers such as Pachymeres, who for the most part repeated classical doctrines, the next group of creative authors are the medieval Islamic writers, Al-Farabi (950 ce), Ibn Sina or Avicenna (1037 ce) and Safiyu-d-Din (1276 ce). These theorists attempted to rationalize the very diverse musics of the Islamic cultural area within the Greek theoretical framework. In addition to an extended Pythagorean cycle of seventeen tones, genera of divided fifths and a forty-fold division of the the string (Tanbur of Baghdad) in Al-Farabi, several new theoretical techniques are found. Al-Farabi analogizes from the 256/243 + 9/8 + 9/8 of the Pythagorean tuning and proposes reduplicated genera such as 49/48 - 8/7 -8/7 and 27/25 - 10/9 : 10/9. Avicenna lists other reduplicated tetrachords with intervals of approximately 3/4 of a tone and smaller (see the Catalog for these genera). The resemblance of these to Ptolemy’s equable diatonic seems more than fortuitous and further supports the notion that three-quarter-tone intervals were in actual use in Near Eastern music by Roman times (second century cE). These tetrachords may also bear a genetic relationship to neutral-third pentatonics and to Aristoxenos’s hemiolic chromatic and soft diatonic genera as well as Ptolemy’s intense chromatic. Surprisingly, I have been unable to trace the apparently missing reduplicated genus, 11/10 - 11/10: 400/363 (165 + 165 + 168 cents) that is a virtually equally-tempered division of the 4/3. Lou Harrison has pointed out that tetrachords such as this and the equable diatonic yield scales which approximate the 7-tone equal temperament, an idealization of tuning systems which are widely distributed in sub-Saharan Africa and Southeast Asia. Other theoretical advances of the Islamic theorists include the use of various arrangements of the intervals of the tetrachords. Safiyu-d-Din listed all six permutations of the tetrachords in his compendious tables, although his work was probably based on Aristoxenos’s discussion of the permutations of the tetrachords that occur in the different octave species. Iq CHAPTER 2

At least for expository purposes, the Islamic theorists favored arrangements with the pyknon uppermost and with the whole tone, when present, at the bottom. This format may be related to the technique of measurement termed messel, from the Arabic a/-mithal, in which the shorter of two string lengths is taken as the unit, yielding numbers in the reverse order of the Greek theorists (Apel 1955, 441-442.). The so-called neo-chromatic tetrachord (Gevaert 1875) with the augmented second in the central position is quite prominent and is also found in some of the later Greek musical fragments and in Byzantine chant (Winnington-Ingram 1936) as the palace mode. It is found in the Hungarian minor and Gypsy scales, but, alas, it has become a common musical cliché, the “snake-charmer’s scale” of the background music for exotic Oriental settings on television and in the movies. The present After the medieval Islamic writers, there are relatively few theorists expressing any great interest in tetrachords until the nineteenth and twentieth centuries. Notable among the persons attracted to this branch of music theory were Helmholtz ([1877] 1954) and Vogel (1963, 1967, 1975) in Germany; A. J. Ellis (1885), Wilfrid Perrett (1926, 1928, 1931, 1934), R. P. Winnington-Ingram (1928, 1932) and Kathleen Schlesinger (1933) in Britain; Thorvald Kornerup (1934) in Denmark; and Harry Partch (1949) and Ervin Wilson in the United States. The contributions of these scholars and discoverers are listed in the Catalog along with those of many other workers in the arithmetic tradition. After two and a half millennia, the fascination of the tetrachord has still not vanished. Chapter 4 will deal with the extension of arithmetical techniques to the problem of creating or discovering new tetrachordal genera. PYTHAGORAS, PTOLEMY, AND THE ARITHMETIC TRADITION

Aristoxenos and the geometrization of musical space ARISTOXENOS WAS FROM the Greek colony of Tarentum in Italy, the home of the famous musician and mathematician Archytas. In the early part of his life, he was associated with the Pythagoreans, but in his later years he moved to Athens where he studied under Aristotle and absorbed the new logic and geometry then being developed (Barbera 1980; Crocker 1966; Litchfield 1988). He was the son of the noted musician Spintharos, who taught him the conservative musical tradition still practiced in the Greek colonies, if not in Athens itself (Barbera 1978). The geometry of music The new musical theory that Aristoxenos created about 320 Bce differed radically from that of the Pythagorean arithmeticians. Instead of measuring intervals with discrete ratios, Aristoxenos used continuously variable quantities. Musical notes had ranges and tolerances and were modeledas loci in a continuous linear space. Rather than ascribing the consonance of the octave, fifth, and fourth to the superparticular nature of their ratios, he took their magnitude and consonance as given. Since these intervals could be slightly mistuned and still perceived as categorically invariant, he decided that even the principal consonances of the scale had a narrow, but still acceptable range of variation. Thus, the ancient and bitter controversy over the allegedly unscientific and erroneous nature of his demonstration that the perfect fourth consists of two and one half tones is really inconsequential. Aristoxenos defined the whole tone as the difference between the two fundamental intervals of the fourth and the fifth, the only consonances smaller than the octave. The octave was found to consist of a fourth anda ARISTOXENOS AND THE GEOMETRIZATON OF MUSICAL SPACE

fifth, two fourths plus a tone, or six tones. The intervals smaller than the fourth could have any magnitude in principle since they were dissonances and not precisely definable by the unaided ear, but certain sizes were traditional and distinguished the genera known to every musician. These conventional intervals could be measured in terms of fractional tones by the ear alone because musical function, not numerical precision, was the criterion, The tetrachords that Aristoxenos claimed were well-known are shown in 3-1. Aristoxenos described his genera in units of twelfths of a tone (Macran 1902), but later theorists, notably Cleonides, translated these units into a cipher consisting of 30 parts (moria) to the fourth (Barbera 1978). The enharmonic genus consisted of a pyknon divided into two 3-part micro- 3-1. The genera ofAristoxenos. The descriptions of tones or dieses and a ditone of 24 parts to complete the perfect fourth. Next tones have been converted to cents, assuming 500 come three shades of the chromatic with dieses of 4, 4.5, and 6 parts and upper intervals of 22, 21, and 18 parts respectively. The set was finished with Aristoxenos (Macran 1902) in terms of twelfths of cents to the equally temperedfourth, The interpretation ofAristoxenos’sfractional tones as thirty parts to the fourth is after the second century theorist Cleonides. two diatonic tunings, a soft diatonic (6 + 9 + 15 parts), and the intense diatonic (6+ 12 +12 parts). The former resembles a chromatic genus, but the latter is similar to our modern conception of the diatonic and probably ENHARMONIG o o 50 100 67 INTENSE CHROMATIC 500 o 100 6+6+ 18 PARTS 1/4+ 1/4+ 2 TONES 50 + 50 + 400 GENTS 1/2 + 1/2 + 1 1/2 TONES 100 + 100 + 300 CENTS SOFT CHROMATIC SOFT DIATONIC 133 4+4+ 22 PARTS 1/3 + 1/3 + 1 5/6 TONES 67 + 67 + 333 CENTS 500 o 100 250 6 +9 +I5 PARTS 1/2 + 3/4+ 1 1/4 TONES IOO + 150 + 250 CENTS HEMIOLIC CHROMATIC o 75 200 3+3+24 PARTS 150 4.5 + 4.5 + 21 PARTS 500 500 INTENSE DIATONIC 500 o 100 300 6+12+ 12 PARTS 3/8 + 3/8+ 1 3/4 TONES 1/2 + 1 + I TONES 75 + 75 + 350 CENTS 100+ 200+ 200 CENTS CHAPTER 3

represents the Pythagorean form. Two such 30-part tetrachords and a whole tone of twelve parts completed an octave of 72 parts. Several properties of the Aristoxenian tetrachords are immediately 3-2. Other genera mentioned by Aristoxenos. apparent. The enharmonic and three chromatic genera have small intervals UNNAMED CHROMATIC o 67 with similar sizes, as if the boundary between the enharmonic and chrosoo matic genus was not yet fixed. The two chromatics between the syntonic 500 chromatic and the enharmonic may represent developments of neutral-third pentatonics mentioned in chapter 2. The pyknon is always divided equally except in the two diatonic genera whose first intervals (half tones) are the same as that of the syntonic chromatic. Thus Aristoxenos is saying that the first interval must be less than or equal to the second, in agreement with Ptolemy’s views nearly five 200 4+8 + 18 PARTS 1/3 +2/3 + 1 1/2 TONES 67 + 133 + 300 CENTS DIATONIC WITH SOFT CHROMATIC DIESIS lo 67 300 4+14+ 12 PARTS hundred years later. +1 1/6 + I TONES The tetrachords of 3-2 are even more interesting, The first, an approved 67 + 233 + 200 CENTS but unnamed chromatic genus, not only has the 1:2 division of the pyknon, 1/3 but more importantly, is extremely close to Archytas’s chromatic tuning DIATONIC WITH HEMIOLIC CHROMATIC DIESIS lo 75 300 4.5 +13.5 +21 PARTS soo Archytas’s enharmonic is missing, though Aristoxenos seems to allude to it in his polemics against raising the second string and thus narrowing the 3/8 +1 1/8 + 1 TONES largest interval (ibid.). These facts clearly show that Aristoxenos understood 75 +225 + 200 CENTS REJECTED CHROMATIC (o 100 150 Ó + 3 + 21 PARTS 1/2 + 1/4+ 1 3/4 TONES 100 + SO + 350 CENTS (Winnington-Ingram 1932). The diatonic with soft chromatic diesis is a very good approximation to Archytas’s diatonic as well (ibid.). Only 500 the music of his time. The last two tetrachords in 3-2 were considered unmusical because the second interval is larger than the first. Winnington-Ingram (1932) has suggested that Aristoxenos could have denoted Archytas’s enharmonic tuning as 4 + 3 + 23 parts (67 + 50 + 383), a tuning which suffers from the same defect as the two rejected ones. A general prejudice against intervals containing an odd number of parts may have caused Aristoxenos to disallow UNMELODIC CHROMATIC o 75 133 4.5 + 3.5 + 22 PARTS 3/8 +7/24+1 5/6 TONES 75 + 58 + 367 CENTS 500 tetrachords such as 5 + II + 14, 5 + 9 + 16 (ibid.), and 5+6+ 19 (Macran 1902). The alleged discovery of equal temperament Because a literal interpretation of Aristoxenos’s parts implies equal tem- peraments of either 72 or 144 tones per octave to accommodate the hemiolic chromatic and related genera, many writers have credited him with the discovery of the traditional western European 12-tone intonation. This conclusion would appear to be an exaggeration, at the least. There is ARISTOXENOS AND THE GEOMETRIZATON OF MUSICAL SPACE

no evidence whatsoever in any of Aristoxenos’s surviving writings or from any of the later authors in his tradition that equal temperament was intended (Litchfield 1988). Greek mathematicians would have had no difficulty computing the string lengths for tempered scales, especially since only two computations for each tetrachord would be necessary, and only a few more for the complete octave scale. Methods for the extraction of the square and cube roots of two were long known, and Archytas, the subject of a biography by Aristoxenos, was renowned for having discovered a three-dimensional construction for the cube root of two, a necessary step for dividing the octave into the 12, 24, 36, 72, Or 144 geometric means as required by Aristoxenos’s tetrachords (Heath [1921] 1981, 1:246-249). Although irrationals were a source of great worry to Pythagorean mathematicians, by Ptolemy’s time various mechanical instruments such as the mesolabium had been invented for extracting roots and constructing geometric means (ibid., 2:104). Yet neither Ptolemy nor any other writer mentions equal temperament. Ptolemy, in fact, utterly missed Aristoxenos’s point and misinterpreted these abstract, logarithmic parts as aliquot segments of a real string of 120 units with 60 units at the octave, 80 at the fifth, and go at the fourth. His upper tetrachord had only twenty parts, necessitating the use of complicated fractional string lengths to express the actually simple relations in the upper tetrachords of the octave scales. There are two obvious explanations for this situation. First, Aristoxenos was opposed to numeration, holding that the trained ear of the musician was sufficiently accurate. Second, Greek music was mostly monophonic, with heterophonic rather than harmonic textures. Although modulations and chromaticism did exist, they would not have demanded the paratactical pitches of a tempered gamut (Polansky 1987). There was no pressing need for equal temperament, and if it was discovered, the fact was not recorded (for a contrary view, see McClain 1978). Later writers and Greek notation Although most of the later theorists continued the geometric approach taken by Aristoxenos, they added little to our knowledge of Greek music theory with few exceptions. Cleonides introduced the cipher of thirty parts to the fourth. Bacchios gave the names of some intervals of three and five dieses which were alleged to be features of the ancient style, and Aristides CHAPTER 3

Quintilianus offered a purported list of the ancient harmoniai mentioned 3-3, Two medieval Islamic forms. These two medieval Islamic tetrachords are Aristoxenian approximations to Ptolemy's eguable diatonic. The Arabs also listed Aristoxenos’s other tetrachords in their treatises, by Plato in the Timaeus. One exception was Alypius, a late author who provided invaluable information on Greek musical notation. His tables of keys or tonoi were deciphered independently in the middle of the nineteenth century by Bellermann (1847) and Fortlage (1847), and made it possible for the few extant fragments of Greek music to be transcribed into modern notation and understood. Unfortunately, Greek notation lacked both the numerical precision of the tuning theories, and the clarity of the system of genera and modes (chapter 6). Additionally, there are unresolved questions concerning the choice of alternative, but theoretically equivalent, spellings of certain passages. Contemplation of these problems led Kathleen Schlesinger to the heterodox theories propounded in The Greek Aulos. Others have simply noted that the notation and its nomenclature seem to have evolved away from the music they served until it became an NEUTRAL DIATONIC o 200 350 12 +9 + 9 PARTS 1 + 3/4 + 3/4 TONES 200 + 150 + 150 CENTS 500 Medieval Islamic theorists As the Roman empire decayed, the locus of musical science moved from EQUAL DIATONIG o 167 334 10+ 10+ IO PARTS 5/6 + 5/6 + 5/6 TONES 167 + 167 + 166 CENTS academic subject far removed from musical needs (Henderson 1957). For these reasons, little will be said about notation; knowledge of it is not necessary to understand Greek music theory nor to apply Greek theory to present-day composition. 500 Alexandria to Byzantium and to the new civilization of Islam. Aristoxenos’s geometric tradition was appropriated by both the Greek Orthodox church to describe its liturgical modes. Aristoxenian doctrines were also included in the Islamic treatises, although arithmetic techniques were generally employed. The tetrachords of 3-3 were used by Al-Farabi to express 3/4-tone scales similar to Ptolemy’s equable diatonic in Aristoxenian terms. If one subtracts 10 + 10 + 10 parts from Ptolemy’s string of 120 units, one obtains the series 120 110 100 go, which are precisely the string lengths for the equable diatonic (12/11-11/10: 10/9). It would appear that the nearly equal tetrachord 11/10 - 11/10 - 400/363 was not intended. The tetrachord 12+9+9 yields the permutation 120 108 99 go, or 10/9- 12/11 + 11/10. This latter tuning is similar to others of Al-Farabi and Avicenna consisting of a tone followed by two 3/4-tone intervals. Other tetrachords of this type are listed in the Catalog. ARISTOXENOS AND THE GEOMETRIZATON OF MUSICAL SPACE

Eastern Orthodox liturgical music The intonation of the liturgical music of the Byzantine and Slavonic Orthodox churches is a complex problem and different contemporary authorities offer quite different tunings for the various scales and modes (echoi). One of the complications is that until recently a system of 28 parts to the fourth, implying a 68-note octave (28 + 12 + 28 = 68 parts), was in use along with the Aristoxenian 30 + 12 + 30 parts (Tiby 1938). Another problem is that the nomenclature underwent a change; the term enharmonic was applied to both a neo-chromatic and a diatonic genus, and chromatic was associated with the neo-chromatic forms. Finally, many of the modes are composed of two types of tetrachord, and both chromaticism and modulation are commonly employed in melodies. Given these complexities, only the component tetrachords extracted from the scales are listed in 3-4. The format of this table differs from that of 3-1 through 3-3 in that the diagrams have been omitted and partially replaced by the ratios of plausible arithmetic forms. The four tetrachords from Tiby which utilize a system of 28 parts to the fourth are removed to the Tempered section of the Catalog. 3-4. Byzantine and Greek Orthodox tetrachords. Athanasopoulas’s enbarmonic and diatonic genera consist ofvarious permutations of 6+12 +12, ie. 12 +6+12. Xenakis permits permutations of the 12 + 11+7 and 6+12+12 genera. A closer, but nonsuperparticular, approximation to Xenakis's intense chromatic would be 22/21 -6/5 + 35/33. PARTS CENTS RATIOS 9+15+6 6+18+6 6+12+12 12+12+6 ATHANASOPOULOS (1950) I50+250+100 — 100+300+100 — : 100+200+200 — 200+200+100 — GENUS CHROMATIC CHROMATIC DIATONIC ENHARMONIC SAVAS (1965) 8+14+8 10+8+12 8+12+I0 I2+12+6 8+16+6 6+20+4 133+233+133 167+133+200 133+200+167 200+200+100 133+267+100 100+333 +67 — — — — — — 7+16+7 5+19+6 12+11+7 6+12+12 117+266+117 8343174100 200+183+117 I00+200+200 XENAKIS (1971) 16/15 -7/6- 15/14 SOFT CHROMATIC 256/243 6/5. 135/128 INTENSE CHROMATIC 9/8- 10/9. 16/15 DIATONIC 256/243 -9/8-9/8 ENHARMONIC 22 CHAPTER 3 CHROMATIC DIATONIC BARYS DIATONIC ENHARMONIG BARYS ENHARMONIC PALACE MODE (NENANO)

The tetrachords of Athanasopoulos (1950) are clearly Aristoxenian in origin and inspiration, despite being reordered. One of his chromatics is Aristoxenos’s soft diatonic and the other is Aristoxenos’s intense chromatic. The rest of his tetrachords are permutations of Aristoxenos’s intense diatonic. Savas’s genera (Savas 1965) may reflect an Arabic or Persian influence, as diatonics with intervals between 133 and 167 cents are reminiscent of Al-Farabi’s and Avicenna’s tunings (chapter 2 and the Catalog). They may plausibly represent 12/11 and 11/10 so that his diatonic tunings are intended to approximate a reordered Ptolemy’s equable diatonic. His chromatic resembles 14/13 - 8/7: 13/12 and his Barys enharmonic, 15/14: 7/6» 16/15. Savas’s ordinary enharmonic may stand for either Ptolemy’s intense diatonic (10/9 : 9/8 - 16/15) or the Pythagorean version (256/243 : 9/8 - 9/8). The palace mode could be 15/14 : 6/5 + 28/27 (Ptolemy’s intense chromatic). The above discussion assumes that some form of just intonation is intended. The tunings of the experimental composer Iannis Xenakis (1971) are clearly designed to show the continuity of the Greek Orthodox liturgical tradition with that of Ptolemy and the other ancient arithmeticians, though they are expressed in Aristoxenian terms. This continuity is debatable; internal evidence suggests that the plainchant of the Roman Catholic church is derived from Jewish cantillation rather than Graeco-Roman secular music (Idelsohn 1921). It is hard to see how the music of the Eastern church could have had an entirely different origin, given its location and common early history. A case for evolution from a common substratum of Near Eastern music informed by classical Greek theory and influenced by the Hellenized Persians and Arabs could be made and this might give the appearance of direct descent. The robustness of the geometric approach of Aristoxenos is still evident today after 2300 years. The musicologist James Murray Barbour, a strong advocate of equal temperament, proposed 2 + 14 + 14 and 8 + 8 + 14 as Aristoxenian representations of 49/48 - 8/7 .8/7 and 14/13 - 13/12 - 8/7in his 10) 1953 book on the history of musical scales, Tuning and Temperament. With Xenakis’s endorsement, Aristoxenian principles have become part of the world of international, or transnational, contemporary experimental music. In the next chapter the power of the Aristoxenian approach to generate new musical materials will be demonstrated. ARISTOXENOS AND THE GEOMETRIZATON OF MUSICAL SPACE

4 The construction of new genera THIS CHAPTER Is concerned with the construction of new genera in addition to those collated from the texts of the numerous classical, medieval, and recent writers. The new tetrachords are a very heterogeneous group, since they were generated by the author over a period of years using a number of different processes as new methods were learned or discovered. Including historical tetrachords, the tabulated genera in the catalogs number 723, of which 476 belong in the Main Catalog, 16 in the reduplicated section, 101 under miscellaneous, 98 in the tempered list, and 32 in the semi-tempered category. The genera in the Main Catalog are classified according to the size of their largest or characteristic interval (CI) in decreasing order from 13/10 (454 cents) to 10/9 (182 cents). There are 73 CIs acquired from diverse historical and theoretical sources (4-1). Sources are documented in the catalogs. The theoretical procedures for obtaining the new genera are described in this chapter and the next. New genera derived by linear division The first of the new genera are those whose Cls are relatively simple non-superparticular ratios such as 11/9, 14/11, and 16/13. These ratios were drawn initially from sources such as Harry Partch’s 43-tone, 11-limit just intonation gamut, but it was discovered later that some of these Cls are to be found in historical sources as well. The second group is composed of intervals such as 37/30, which were used sporadically by historical writers. To these ratios may be added their 4/3’s and 3/2’s complements, e.g. 27/22 THE CONSTRUCTION OF NEW GENERA

4-1. Characteristic intervals (CIs) ofnew genera injust intonation, The CI is the largest interval ofthe tetrachord and the pyknon or apyknon is the difference between the CI and the fourth. Because many ofthe new genera have historically known CIs, all ofthe Cls in the Main Catalog are listed in this table. The CIs ofthe reduplicated, miscellaneous, tempered, and semi-tempered lists are not included in this table, HYPERENHARMONIC GENERA The term byperenbarmonic is originally from EIO EII 34/27 113/90 18/17 120/113 399 + 99 394+104 C21 C22 20/17 27/23 17/15 92/81 281 +217 278 +220 n whose Cl is greater and refers to genera Wilso E12 64/51 17/16 393 + 105 C23 75/64 256/225 275+223 EI3 5/4 16/15 386+ 112 c24 7/6 8/7 267+231 genusis Wilson's 56/55- 55/54- 9/7. See chapter 5 E14 8192/6561 2187/2048 384 + 114 G25 136/117 39/34 261 +238 for classification schemes. PYKNON ci EIS E16 379 + 119 376 + 122 026 c27 36/31 80/69 31/27 23/10 259 +239 256+242 028 no 3833 254+ 244 52/45 15173 250724 than 425 cents. Theprototypical byperenbarmonic CENTS 56/45 41/33 15/14 44/41 HI Onlie 40/39 454+44 CHROMATIC GENERA H2 35/27 36/35 449 +49 of the chromatic genera range from 375 to The Cls 250 cents. 029 DIATONIC GENERA H3 22/17 34/33 446 + 52 H4 128/99 33/32 445 +53 cı 36/29 29/27 374 +124 The Cls of the diatonic genera range from 250 to HS 31/24 32/31 443 +55 c2 26/21 14/13 370 + 128 166 cents. In the diatonic genera, a pyknon does not H6 m7 #8 Ho ro art 40/31 58/45 9/7 104/81 50/39 32/25 31/30 30/29 28/27 27/26 26/25 25/24 441 +57 439 + 59 + 63 435 433 +65 430 +68 427 +71 C3 c4 CS cé c7 c8 21/17 100/81 130 37/3 16/13 27/22 11/9 68/63 27/25 of 40/37 13/12 88/81 12/11 366 + 132 365 +133 63 +1 + 135 303 359+ 139 355 + 143 347 + ISI exist. 15/13 DI 8/2 D2 30/23 23/20 D3 31/27 D4 39/34 DS ENHARMONIC GENERA co 39/32 128/117 342 +156 ns genera range from 375 The Cls of the enbarmonie 10425 cents, 424+ 73 24/23 23/18 EI cıo CII cI2 28/23 17/14 40/33 23/21 56/51 11/10 341 +157 336 + 162 333 + 165 421+77 cı3 20/24 32/29 52/45 22/1 9 80/69 36/31 136/117 248 + 250 242 +256 5 242 +256 239 +259 238+261 8/7 7/6 231 + 267 D7 D8 Dg 256/225 25/22 92/81 75/64 88/75 27/23 223 +275 221 +277 220 +278 328 + 170 DIO 76/67 67/57 218 +280 E2 88/69 23/22 E3 _ 50/41 160/153 421 +77 ciq 6/5 10/9 316 + 182 DIT 17/15 20/17 217 +281 14/11 22/21 418 +81 Cis 25/21 28/25 302 + 196 DI2 112/99 33/28 298 + 201 294 + 204 293+205 289 + 209 284 +214 DI3 44/39 DI4 152/135 DIS 9/8 DI6 160/143 DI7 10/9 13/11 45/38 32/27 143/120 214 + 284 209 + 289 205 + 293 + 294 204 194+304 6/5 182 + 316 E4 Es E6 E7 E8 EQ 80/63 33/26 10/15 81/64 24/19 21/20 104/99 20/19 256/243 19/18 414 +84 413+85 409 + 89 408+ 90 404 +94 cı6 cı7 cı8 cıg c2o 19/16 32/27 45/38 13/11 33/28 64/57 9/8 152/135 44/39 CHAPTER 4

is the 3/2’s complement of 11/9 and 52/45 the 4/3’s complement of 15/13. Various genera were then constructed by dividing the pykna or apykna by linear division into two or three parts to produce 1:1, 1:2, and 2:1 divisions. Both the 1:2 and 2:1 divisions were made to locate genera composed mainly of superparticular ratios. Even Ptolemy occasionally had to reorder the 4-2. Indexed genera. The terms4 and 3 which represent the 1/1 and 4/3 ofthefinal tetrachord are multiplied by the index. The lefthand sets of tetrachords are those generated by selecting and recombining the successive intervals resultingfrom the additional terms after the multiplication. The righthand sets oftetrachords have been reduced to lowest terms and ordered with the CI uppermost. intervals resulting from triple division before recombining two of them to produce the two intervals of the pyknon (2-2 and 2-4). More complex divisions were found either by inspection or by katapyknosis with larger multipliers, Indexed genera One useful technique, originated by Ervin Wilson, is a variation of the katapyknotic process. In 4-2 this technique is applied to the 4/3 rather than MULTIPLIER: 4 TERMS: 16 15 14 13 X2 16/15: 15/14- 14/12 16/15-15/13: 13/12 16/14: 14/13 - 13/12 MULTIPLIER: 5 TERMS: 20 19 20/19- 19/18- 18/15 20/19: 19/17 17/15 20/19- 19/16: 16/15 20/18: 18/17: 17/15 20/18 - 18/16: 16/15 20/17: 17/16: 16/15 MULTIPLIER: 6 16/15 - 15/14- 7/6 16/15 - 13/12 > 15/13 14/13 . 13/12 «8/7 18 17 16 15 20/19: 19/18 :6/5 20/19: 19/17: 17/15 20/19: 16/15 - 19/16 18/17. 10/9- 17/15 : 9/8 16/15- 10/9 17/16: 16/15 : 20/17 TERMS: 24 23 22 21201918 24/13: 23/22 ‚22/18 24/23: 23/21 - 21/18 24/23 +23/20: 20/18 24/23: 23/19: 19/18* 24/22 -22/21 - 21/18 24/22 - 22/20: 20/18 24/22 + 22/19- 19/18 24/21 «21/20- 20/18 24/21 - 21/19: 19/18 24/10: 20/19 + 19/18 * see Catalog number 536. 24/23- 23/22: 11/9 24/23: 23/21 - 7/6 - 23/20 24/23. 10/9 24/23: 19/18 . 23/19 22/21: 12/11 7/6 12/11: 11/10 + 10/9 19/18- 12/11: 22/19 21/20: 10/9- 8/7 19/18: 21/19 - 8/7 » 6/5 20/19: 19/18 to the pyknon (as it wasin 2-4). The 1/1 and 4/3 of the undivided tetrachord are expressed as 3 and 4, and are multiplied by a succession of numbers of increasing magnitude, The new terms resulting from such a multiplication and all the intermediate numbers define a set of successive intervals which may be sequentially recombined to yield the three intervals of tetrachords. I have termed the multiplier, the index, and the resulting genera indexed genera. The intermediate terms are a sequence of arithmetic means between the extremes. The major shortcoming of this procedure is that the number of genera grows rapidly with the index. There are 120 genera of index 17, and not all of these are worth cataloguing, since other genera of similar melodic contours and simpler ratios are already known and tabulated. The technique is still of interest, however, to generate sets of tetrachords with common numerical relations for algorithmic composition. Pentachordal families Archytas’s genera were devised so that they made the interval 7/6 between their common first interval, 28/27, and the note a 9/8 below the first note of the tetrachord (Erickson 1965; Winnington-Ingram 1932; see also 6-1). Other first intervals (x) may be chosen so that in combination with the 9/8 they generate harmonically and melodically interesting intervals. These intervals may be termed pentachordal intervals (PI) as they are part of a pentachordal, rather than a tetrachordal tonal sequence. Three such groups or families of tetrachords are given in 4-3 along with their initial and pentachordal intervals. THE CONSTRUCTION OF NEW GENERA

The 28/27 family is an expansion of Archytas’s set of genera. The 40/39 family fits quite well into 24-tone equal temperament because of the reasonably close approximation of many of the ratios of 13 to quarter-tone intervals. The 15/13 is another plausible tuning for the interval of five 4-3. Pentachordal intervals andfamilies. These tetrachords are defined by two parameters: the dieses which was reputed to be a feature of the oldest scales (chapter 6; Bacchios, 320 CE in Steinmayer 1985). The 16/15 family contains the most consonant tunings of the chromatic and diatonic genera. The pentachordal intervals of 4-3 are the mediants (“thirds”) of the triads pentachordal interval, 9x/8, and the characteristic interval, which determines the genus. An initial interval x results in a pentachordal interval (PI) of 9x/8. These pentachordalfamilies are the most which generate the tritriadic scales of chapter 7, where they are discussed in important tritriadic genera ofchapter 7. The initials are thefirst intervals ofthe tetrachords. greater detail. In general, all tetrachords containing a medial 9/8 may function as generators of tritriadic scales. DISJUNCTIVE TONES / 8/9 1/1 gnome INTERVALS x y 4/3 3/2 32 3ylı 2/1 ZO PENTACHORDAL INTERVALS x= 40/39, PI=15/13 ENHARMONIC 40/39 + 39/38 - 19/15 ERATOSTHENES 40/39 + 26/25 - 5/4 AVICENNA CHROMATIG 40/39 + 13/12 - 6/5 40/39 - 39/35 + 7/6 40/39 + 11/10 + 13/11 DIATONIC 40/39 : 52/45 + 9/8 BARBOUR INITIAL 16/15 28/27 13/12 112/99 44/39 68/63 256/243 136/117 80/68 92/81 PI 6/5 7/6 39/32 14/11 33/26 17/14 32/27 17/13 30/23 23/18 INITIAL 10/9 12/11 128/117 40/39 104/99 64/57 9/8 7/6 56/45 184/171 x=28/27, Pla 7/6 ENHARMONIC ARCHYTAS 28/27 è 36/35 « 5/4 CHROMATIG ARCHYTAS 28/27 - 243/224+ 32/27 PTOLEMY 28/27: 15/14 : 6/5 27/26 : 26/21 MAIN CATALOG 28/27 « DIATONIC ARCHYTAS 28/27: 8/7 + 9/8 MAIN CATALOG 28/27« 39/35 + 15/13 40/39 : 91/80 - 8/7 28 CHAPTER 4 PI 5/4 27/22 16/13 15/13 13/11 24/19 81/64 21/16 23/20 57/46 INITIAL 8/7 88/81 22/21 52/45 56/51 19/18 52/51 64/63 24/23 76/69 PI 9/7 11/9 33/28 13/10 21/17 19/16 39/34 8/7 27/23 23/19 x=16/15, PI=6/5 CHROMATIC DIDYMOS 16/15 - 25/24 + 6/5 AL-FARABI 16/15: 15/14- 7/6 16/15- 20/19 + 19/16 KORNERUP DIATONIC PTOLEMY 16/15 - 9/8. 10/9 16/15: 13/12 - 15/13 MAIN CATALOG

Mean tetrachords The mathematician and musician Archytas may have been the first to recognize the importance of the arithmetic, harmonic, and geometric means to music. He was credited with renaming the mean formerly called the “subcontrary” as the harmonic mean because it produced more pleasing melodic divisions than the arithmetic mean (Heath [1921] 1981; Erickson 4-4. Means: formulae and equivalent expressions from Heath 1921, 1:85-87, exceptfor the logarithmtic, ratio, and root mean square means. Number 12 is the framework ofthe scale when a = 12 andb = 6. The tetrachords generated by number 17 are extremely close numerically to the counterlogarithmicmean tetrachords ofthe other kinds. They also resemble the subcontraries to the geometric means. 1, ARITHMETIC (a-b)/(b-c) =a/a=b/bmec/c a+c=2b 1965). His own tunings were constructed by the application of only the harmonic and arithmetic means, but there were actually nine other means known to Greek mathematicians and which might be used to construct tetrachords (Heath [1921] 1981). To this set of twelve may be added the root mean square or quadratic mean and four of my own invention whose definitions are given along with the historical ones in 4-4. The logarithmic mean divides an interval into two parts, the ratio of whose widths is the inverse of the ratio of the extremes of the interval, For example, the logarithmic mean divides the 2/1 into two 10. UNNAMED (SAME AS FIBONACCI SERIES) (a-c)\/(a-b) =b/e a=b+c 2, GEOMETRIC 11, UNNAMED (a-b)/(b-c)=atb=b/e ac=b? (a-dHa-b)=a/b a? =2ab-be 3. HARMONIG I2. MUSICAL PROPORTION a:(a+b)/2 = 2ab/(a +b):b (a-b)/(b-c)=ale 1/a+1/c=2/b b=2ac/(a+c) 4. SUBCONTRARY TO HARMONIC (@-c)/(b-c)=c/a (a? +2)/(a+c) =b 5. FIRST SUBCONTRARY TO GEOMETRIC (a-b)/b-g=cAb a=b+c-2/b 6. SECOND SUBCONTRARY TO GEOMETRIC (a-b)/b-c)=b/a c=a+b-a°/b 7. UNNAMED (a-b-)=ate È=zac-ab 13. LOGARITHMIC MEAN logb=(clogaraloge)/(a +) (ba)= (cb)? 14. COUNTER-LOGARITHMIC MEAN logb=(aloga+clogc)/(a+c) (ba)a=(wb)c 15. RATIO MEAN (a-)/0-c)mxly ca(bx-4))/(x-3) 16, SECOND RATIO MEAN (a-c}/(a-b)=x/y c=(ay-ax+bx)/y 8, UNNAMED 17. ROOT MEAN SQUARE (a-c/(a-bj=ak a? +2 =4(b+#c) bav((a2+00)/2) Pm(a+c*)/2 9. UNNAMED (a-)/b-o)=b/e P+mc(s +b) THE CONSTRUCTION OF NEW GENERA

4-5. Generating tetrachords with means. intervals of 400 and 800 cents in the proportion of 1:2 (0, 400, and 1200 cents). The counter-logarithmic mean effects the same division in the opposite order, i.e., 800 and 400 cents (0, 800, and 1200 cents). MEAN TETRACHORDS OF THE FIRST KIND 8/9 1/1 4/3 HYPERH. H.MESON PARHYPATE LICHANOS L ji 3/2 MESE PARAMESE + 8 of 4-4, differing only in that the ratio of the difference of the extremes to the difference between the mean and one of the extremes is dependent upon the parameter x. There are still other types of mean, but these seventeen are sufficient to FIRST MEAN 1 mal | The two ratio means, numbers 15 and 16, are variations of numbers 7 and generate a considerable number of tetrachords (4-6-8) and may be of further utility in the algorithmic generation of melodies. The most obvious procedures for generating tetrachords from these means are shown in 4-5. Mean tetrachords of the first kind are constructed by first calculating the lichanos as the mean between 1/1 and 4/3, or equivalently between 4 = 4 and and c = 3. The next step is the computation SECOND MEAN Lichanos is defined as the appropriate mean between Aypate meson (1/1) and mese (4/3). Parhypate is then computed as the identical mean between lichanos and hypate. of parhypate as the same mean between 1/1 and the just calculated lichanos MEAN TETRACHORDS OF THE SECOND KIND 8/9 w/t 4/3 3/2 HYPERH, H.MESON PARHYPATE LICHANOS MESE PARAMESE (4-6). Tetrachords of the second kind have the mean operations performed in reverse order (4-7). Tetrachords of the third kind are found by taking the means between 1/1 and 3/2 and between 8/9 and 4/3 (4-8); the smaller is defined as parhypate; the larger becomes the lichanos. FIRST MEAN The construction of sets of genera analogous to those of Archytas, which are composed of a mean between 8/9 and 4/3 and its “subcontrary” or “counter”-mean between 8/9 and 32/27 (Erickson 1965; Winnington- SECOND MEAN Parhypate is defined as the appropriate mean between hypate meson (t/t) and mese (4/3). Lichanos is then computed as the identical mean between parbypate and mese. MEAN TETRACHORDS OF THE THIRD KIND 8/9 1/1 4/3 HYPERH, H.MESON PARHYPATE LICHANOS | 3/2 MESE PARAMESE FIRST MEAN 12/11 - 11/10: 10/0, 10/9 - 11/10 : 12/11. The geometric mean equivalent is the new genus 166.667 + 166.667 + 166.667 cents (see the discussion of tempered tetrachords below). N L Ingram 1932), is left for future investigations as it involves deep questions about the integration of intervals into musical systems. Multiple means may be defined for the arithmetic, harmonic, and geometric means. The insertion of two arithmetic or harmonic means into the 4/3 results in Ptolemy’s equable diatonic and its intervallic retrograde, 4 SECOND MEAN Lichanos is defined as the appropriate mean between bypate meson (1/1) and paramese (3/2). Parbypate is then computed as the identical mean between mese (4/3) and byperkypate (8/9). CHAPTER 4

4-6, Mean tetrachords ofthefirst kind. The lichanoi are the means between 1/1 and 4/3; the barbypatai are the means between 1/1 and the lichanoi. I, ARITHMETIC t/t 2, 5 . FIRST SUBCONTRARY TO GEOMETRIC 6, SECOND SUBCONTRARY TO GEOMETRIC 7. UNNAMED 8 UNNAMED 9. UNNAMED 10, FIBONACCI SERIES IL, UNNAMED 12, MUSICAL PROPORTION LO 1.07457 1.15470 1.33333 vi 16/15 8/7 4/3 t/t 533/483 25/21 4/3 1.0 109429 1.18046 1.33333 LO 1.09185 1.17704 1.33333 1/1 6/5 5/4 4/3 ui 157/156 13/12 4/3 1.0 1.21677 1.26376 1.33333 NO SOLUTION ir 256/255 16/15 4/3 1/1 8/7 7/6 4/3 13. LOGARITHMIC MEAN LO 1.05956 1.33333 1.05956 : 1.06763 - 1.17867 14. COUNTER-LOGARITHMIC MEAN IS, RATIO MEAN (X/Y = 4/3) 16, SECOND RATIO MEAN (X/Y = 4/3) 17. ROOT MEAN SQUARE 10 1/1 1/1 LO 1.09301 1.17867 1.33333 19/16 5/4 4/3 157/156 13/12 4/3 1.09290 1.17851 1.33333 109301 + 1.07837 + 1.13122 19/16 - 20/19 - 16/15 157/156 - 169/157 + 16/13 1.09291 « 1.078328: 1.13137 GEOMETRIC 3 ‚ HARMONIC 4 . SUBCONTRARY TO HARMONIC 13/12 7/6 4/3 1.13122 13/12 + 14/13 - 8/7 1.07457‘ 1.07457: 1.15470 16/15 15/14- 7/6 533/483 + 575/533 - 28/25 1.09429 : 1.07874 : 1.12950 1.09185 - 1.07803 : 1.13278 6/5 «25/24+ 16/15 157/156 - 169/157 : 16/13 1.21677 « 1.03862 - 1.05505 256/255 - 17/16 : 5/4 8/7 - 49/48 - 8/7 139+128 + 231 125 +125 +249 II2 +119 + 267 I7I+I3I +196 156+13I +211 152 +130 +216 316+ 71 +112 II +128 + 359 340+66+93 7 + 105 + 386 231+ 364231 100+ 113 +285 154+ 131 +213 208+89 +112 11 +128 +359 154+131+214 4-7. Mean terrachords ofthe second kind, The parhypatai are the means between 1/1 and 4/3; the lichanoi are the means between the parbypatai and 4/3. I. ARITHMETIC 2. GEOMETRIC 3 4 HARMONIC 5 FIRST SUBCONTRARY TO GEOMETRIC SUBCONTRARY TO HARMONIC 6. SECOND SUBCONTRARY TO GEOMETRIC 7. UNNAMED 8. UNNAMED 9 UNNAMED 10, FIBONACCI SERIES II. UNNAMED 12. MUSICAL PROPORTION 13. LOGARITHMIC MEAN 14. COUNTER-LOGARITHMIC MEAN 15. RATIO MEAN (x/v=4/3) 16. RATIO MEAN (x/v=4/3) 17. ROOT MEAN SQUARE 1/1 LO 1/1 1/1 Lo LO 1/1 1/1 7/6 5/4 4/3 1.15470 1.24081 1.33333 8/7 16/13 4/3 25/21 1409/1113 4/3 118046 1:25937 1.33333 1.17704 1.25748 1.33333 5/4 85/64 4/3 13/12 217/192 4/3 LO 126376 1.3299 1.33333 NO SOLUTION 1/1 16/15 10/9 4/3 1/1 8/7 7/6 4/3 LO 113122 1.21987 1.33333 10 117867 1.25839 1.33333 ri 5/4 21/16 4/3 t/t 13/12 55/48 4/3 1.0 117851 1.22583 1.33333 31 7/6 15/14 16/15 267 +IIg+I1z 115470: 1.07457 - 1.07457 8/7 14/13 - 13/12 25/21 > 1409/1325 : 1484/1409 1.18046 : 1.06685 : 1.05873 1.17704: 1.06833 : 1.06032 §/4+17/16+ 256/255 13/12 + 217/208 - 256/217 1.26376- 1.05321 - 1.00260 249 +125 +125 231 +128 +139 302 + 106 +90 287+112 +09 282+114+ 101 386+105+7 139+ 73 +286 405+88+4 16/15 + 25/24+ 6/5 8/7 + 49/48 + 8/7 1.13122 + 1.07837 - 1.09301 1.17867 : 1.06763 : 1.05956 5/4: 21/20 - 64/63 112+71+316 231+36+23 213+131+154 285+ 113 + 100 386+ 84 +27 139 + 07 +262 284+ 113 + 100 13/12 + 55/52 + 64/55 1.17851 : 1.067708 - 1.059625 THE CONSTRUCTION OF NEW GENERA

Summation tetrachords Closely related to these applications of the various means is a simple nique which generates certain historically known tetrachords as v some unusual divisions. Wilson has called this freshman sums, and h plied it in many different musical contexts (Wilson 1974, 1986, 1989 numerators and denominators of two ratios are summed separat obtain a new fraction of intermediate size (Lloyd and Boyle 1978 example, the freshman sum of 1/1 and 4/3 is 5/4 , and the sum of 5. 1/1 is 6/5. These ratios define the tetrachord 1/1 6/5 5/4 4/3. Similar “sum” of 5/4 and 4/3 is 9/7, and these ratios delineate the 1/1 5/4 9 tetrachord, The former is a permutation of Didymos’s chromatic gen the latter is the inversion of Archytas’s enharmonic. If one emp multiplier/index as in 4-2 and expresses the 1/1 as 2/2, 3/3...,, ani set of graded tetrachords may be generated. The most important a a saa +H Atm FF LS M » H HH ee © 2 SAAN Dw PH 4-8. Mean tetrachords ofthe third kind. The lichanoi ofthese tetrachords are the means between 1/1 and 3/2; the parbypatai are the means berween 8/9 and 4/3, These tetracbords are also tritriadic genera, ARITHMETIC GEOMETRIC HARMONIC SUBCONTRARY TO HARMONIC FIRST SUBCONTRARY TO GEOMETRIC SECOND SUBCONTRARY TO GEOMETRIC UNNAMED UNNAMED UNNAMED . FIBONACCI SERIES + UNNAMED . MUSICAL PROPORTION . LOGARITHMIC MEAN . COUNTER-LOGARITHMIC MEAN + RATIO MEAN (X/y = 2/1) » RATIO MEAN (X/¥ = 2/1) . ROOT MEAN SQUARE teresting ones are tabulated in 4-9. Similarly, the multiplier may be applied to the 4/3 rather than the yield 8/6, 12/9... . The resulting tetrachords fall into the enharmor hyperenharmonic classes and very quickly comprise intervals too sr be musically useful. A few of the earlier members are listed in 4-10. 1/1 10/9 5/4 4/3 1.0 1.08866 1.22474 133333 1/1 16/15 6/5 4/3 1/1 52/45 13/12 4/3 LO 1.13847 1.28078 1.33333 LO 1.12950 1.27069 1.33333 NO SOLUTION 1/1 28/27 7/6 4/3 NO SOLUTION NO SOLUTION NO SOLUTION NOT DEFINED LO 1.04540 1.17608 1.33333 LO 1.13371 1.27542 1.33333 1/1 10/9 5/4 4/3 1/1 10/9 5/4 4/3 LO 1.1331 1.27475 133333 32 CHAPTER 4 10/9 - 9/8 : 16/15 1.08866 . 1.125 : 1.08866 16/15 - 9/8» 10/9 1.13847 - 1.125 - 1.04Io 1.1295 + 1.125+ 1.0493 182 + 204 + II 147 + 204 + 14 112 + 204+ 18 250+ 204 + 44 225 + 204+ 70 211 + 204+ 83 28/27 -9/8 - 8/7 63 + 2044+ 231 1.0454 1.125+ 1.1337 77 +204+21 217 + 204 + 77 182+ 204+ 11 182+ 204+ 11 216+ 204+ 78 52/45 + 9/8 40/39 1.1337 1.125 + 1.0454 10/9 - 9/8 - 16/15 + 9/8 - 16/15 1.1331 «1.125 - 1.04595

TETRACHORD 1/1 6/5 5/4 4/3 RATIOS 6/5 + 25/24- 16/15 SOURCE DIDYMOS 4-9. Summation tetrachords of the first type. 2. 1/1 5/4 9/7 4/3 5/4: 36/35 + 28/27 ARCHYTAS Unreduced ratios bave been retained to clarify the generating process. 3. 4. 5. 6 7. 8. 9. 10. 11. 12, 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 2/2 8/7 6/5 4/3 2/2 6/5 10/8 4/3 3/3 10/9 7/6 4/3 3/3 7/6 11/9 4/3 4/4 12/11 8/7 4/3 4/4 8/7 12/10 4/3 5/5 14/13 9/8 4/3 5/5 9/8 13/11 4/3 6/6 16/15 10/9 4/3 6/6 10/9 14/12 4/3 7/7 18/17 11/10 4/3 7/7 11/10 15/13 4/3 8/8 20/19 12/11 4/3 8/8 12/11 16/14 4/3 9/9 22/21 13/12 4/3 9/9 13/12 17/15 443 10/10 24/23 14/13 4/3 10/10 14/13 18/16 4/3 II/IT 26/25 15/14 4/3 11/11 15/14 19/17 4/3 12/12 28/27 16/15 4/3 12/12 16/15 20/18 4/3 8/7: 21/20+ 10/9 6/5-25/24: 16/15 10/9 + 21/20 + 8/7 9/6- 22/21 - 12/11 12/11 - 22/21 7/6 8/7. 21/20. 10/9 14/13 - 117/112 » 32/27 9/8 - 104/99 - 44/39 16/15-25/24 6/5 10/9- 21/20 + 7/6 18/17 : 187/180 : 40/33 11/10: 150/143 : 52/45 20/19 - 57/55 11/9 12/11 + 22/21 : 7/6 22/21 + 91/88 - 16/13 13/12 - 68/65 : 20/17 24/23 161/156 : 26/21 14/13-117/I12 - 32/27 26/25- 375/364- 56/45 15/14 266/255 - 68/57 28/27: 36/35: 5/4 16/15- 25/24 -6/5 PTOLEMY DIDYMOS PTOLEMY PTOLEMY PTOLEMY PTOLEMY MISC. CAT. MAIN CAT. DIDYMOS PTOLEMY MISC, CAT, MISC. CAT. MAIN CAT, PTOLEMY MISC. CAT. MAIN CAT, MISC. CAT. MISC. CAT. MISC. CAT. MISC. CAT. ARCHYTAS DIDYMOS I. TETRACHORD r/ı 10/8 9/7 8/6 RATIOS 5/4 36/35 « 28/27 SOURCE ARCHYTAS 4-10. Summation tetrachords of the second type. 2. 1/1 9/7 17/13 8/6 9/7-119/117: 52/51 MISC. CAT. Unreduced ratios bave been retained to clarify the generating process. 3. 4. 5. 6. 7. 8. 9. 10. 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 14/11 + 143/140 : 40/39 13/10 + 250/247 : 76/75 9/7: 119/117 - 52/51 17/13 429/425 100/99 22/17: 357/352 : 64/63 2r/16-656/651. 124/123 13/10: 250/247 76/75 25/19 -931/925 148/147 MISC, CAT, MISC. CAT. MISC. CAT, MISC. CAT. MISC. CAT. MISC. CAT, MISC. CAT, Misc. CAT. 33 THE CONSTRUCTION OF NEW GENERA 14/11 13/10 12/9 13/10 25/19 12/9 18/14 17/13 16/12 17/13 33/25 16/12 22/17 21/16 20/15 21/16 41/31 20/15 26/20 25/19 24/18 25/19 49/37 24/18

PARTS CENTS APPROXIMATION PTOLEMAIC INTERPRETA 80/79- 79/78 - 13/10 ENHARMONIC genera with 4-11, Neo-Aristoxenian constant CI, 1.5+1.5+27 25+25+450 80/79 : 79/78 - 13/10 1+2+27 17+ 33 +450 120/119 - 119/117+ 13/10 120/119 - 119/117- 13/1 56/55: 55/54 9/7 44/43 43/42 14/11 55/54: 36/35 - 14/11 60/59: 59/57: 19/15 40/39 - 39/38 19/15 56/55 22/21. 5/4 60/59 - 59/58. 58/45 48/47 - 47/46 : 23/18 - 23/18 60/59- 118/115 60/59 » 59/57 + 19/15 40/39 - 38/39 - 19/15 60/59- 118/113 - 113/9c 2+2+26 33 +33 +433 2.5+2.5+25 2+3+25 2+4+24 3+3+24 2+5+23 42+42+417 33 +50+417 33 + 67 +400 50+ 50 + 400 33+83+383 3+4+23 50 + 67 +383 36/35 28/27 - 5/4 40/39- 117/113 - 113/9¢ 3.5+3.5+23 584584383 32/31 : 31/30 - 5/4 240/233 + 233/226. 113, CHROMATIG 2+6+22 8/3+16/3+22 3+5 +22 4+4+ 22 2+7+21 3+6+21 33 + 100+ 367 44+89+367 50+ 83 + 367 67 + 67 + 367 33 + 117 + 350 50 + 100+ 350 51/50: 18/17: 100/81 40/39 - 21/20: 26/21 34/33 - 22/21 + 21/17 28/27 + 27/26 - 26/21 56/55» 15/14 11/9 34/33 18/17: 11/9 60/59 - 59/56 : 56/45 45/44 - 22/21 + 56/45 40/39+ 117/112 - 56/45 30/29 - 29/28 - 56/45 60/59- 118/111 - 37/30 40/39« 39/37 - 37/30 4+5+21 67+ 83 +350 28/27. 22/21: 27/22 30/29- 116/111 - 37/30 4.5+4.5+21 2+10+18 3+9+18 4+8+18 4.5+7.5+18 5+7+18 6+6+18 75+75+350 33 + 167 + 300 50+ 150+ 300 67 + 133 +300 754125 +300 83 + 117 + 300 100+100+300 24/23 « 23/22 + 11/9 45/44 11/10: 32/27 33/32 12/11 » 32/27 28/27 + 243/224: 32/27 25/24: 27/25 + 32/27 21/20+ 15/14+ 32/27 256/243 - 2187/2048 - 32/27 80/77 - 77/74: 37/30 60/59 - 59/54 6/5 40/39 + 13/12 + 6/5 30/29 - 29/27 - 6/5 80/77: 77/72 + 6/5 24/23 - 115/108 - 6/5 20/19 - 19/18 - 6/5 33 + 217 + 250 50+200+250 67 + 183 +250 83 + 167 +250 100+217+250 117 + 217 +250 125 125 +250 33 +267+200 50+ 250 +200 67+ 233 + 200 75+225 +200 83+ 217 +200 100+200+200 117+183+200 113 +167+200 45/44 : 44/39 + 52/45 34/33 - 19/17: 22/19 27/26: 10/9 + 52/45 104/99+ 11/10 - 15/13 19/18 + 12/119 22/19 104/97 + 97/909 15/13 15/14 + 14/13 + 52/45 64/63 + 7/6: 9/8 40/39 - 52/45 + 9/8 28/27 - 8/7 + 9/8 24/23-92/81: 9/8 22/21: 112/90 - 9/8 256/243 9/8 - 9/8 16/15: 10/9-9/8 320/297: 11/10 : 9/8 60/59 - 118/105 - 7/6 40/39 - 39/35 : 7/6 30/29- 116/105 : 7/6 24/23-23/21:7/6 20/19 - 38/35 : 7/6 120/113 - 113/105 : 7/6 16/15 - 15/14 : 7/6 60/59: 59/51. 17/15 40/39- 39/34- 17/15 30/29 - 58/51 - 17/15 80/77-77/68. 17/15 24/23 - 115/102 - 17/15 20/19- 19/17 17/15 120/113 - 113/102 - 17/1 15/14 56/51. 17/15 DIATONIG 2+13 +15 3+12 +15 4+11+15 5 +10 +15 6+9+15 7+8+15 75+75+15 2+16+12 3+15+12 4+14+ 12 4.5+13.5+12 5+13+12 6+12+12 34 CHAPTER 4

Neo-Aristoxenian tetrachords with Ptolemaic interpretations While Aristoxenos may have been documenting contemporary practice, even a cursory look at his tables suggests that many plausible neo- Aristoxenian genera could be constructed to “fill in the gaps” in his set. The most obvious missing genera are a diatonic with enharmonic diesis, 3 + 15 +12 (50 + 250+ 200cents), a parachromatic, 5 + § + 20(83 + 83 +334 cents), and a new soft diatonic, 7.5+7.5+15 (125 +125+250 Cents). Although Aristoxenos favored genera with 1:1 divisions of the pyknon, Ptolemy and the Islamic writers preferred the 1:2 relation. More complex divisions, of course, are also possible. 4-11 lists a number of neoAristoxenian genera in which the CI is held constant and the pyknotic division is varied. With the exception of the first five genera which represent byperenharmonic forms and three which are a closer approximation of the enharmonic (383 cents, rather than 400 cents), only Aristoxenos’s CIs are used, For each tempered genus an approximation in just intonation is selected from a genus in the Main Catalog. Furthermore, an approximation in terms of fractional parts of a string of 120 units of length, analogous to Ptolemy’s interpretation of Aristoxenos’s genera, is also provided. While these Ptolemaic interpretations are occasionally quite close to the ideal tempered forms, they often deviate substantially. One should note, however, that the Ptolemaic approximations are more accurate for the smaller intervals than the larger. Intervals whose sizes fall between one third and one half of the perfect fourth may be be repeated within the tetrachord, leaving a remainder less than themselves. These are termed reduplicated genera and a representative set of such neo-Aristoxenian tetrachords with reduplication is shown in 4-12. APPROXIMATION 4-12. Neo-Aristoxenian genera with PARTS CENTS reduplication. 2+14+14 4+13+13 6+12+12 8+11+11 IO+10+10 49/48 8/7 - 8/7 344233 +233 67 +217 +217 300/289- 17/15-17/15 1004200+200 256/243:9/8-9/8 133+183+183 = 27/25+10/9+ 10/9 16641674167 Ir/io-II/10- 400/363 35 THE CONSTRUCTION OF NEW GENERA PTOLEMAIC INTERPRETATION 60/59 : 59/52 - 52/45 30/29: 116/103- 103/90 20/19+ 19/17+ 17/15 16/14: 112/101 12/11: 11/10. 10/9

4-13. Neo-Aristoxenian genera with constant pyknoticproportions. I:I PYKNON 1.5+1.5+27 2+2+26 2.5+2.5 +25 3+3+24 3.5 +3.5 +23 4+4+22 4.5+4.5+2E 5+5+20 CENTS APPROXIMATION 25+25+450 80/79 + 79/78 . 13/10 33 +33 + 433 42 + 42 + 417 50 + 50 + 400 56/55: 55/54: 9/7 44/43: 43/42 + 14/11 40/39* 39/38: 19/15 32/31 - 31/30 - 5/4 58 + 58 + 383 67 +67 + 367 75 +75+ 350 83 + 83 + 334 55 +5-5 +19 92 +92+ 317 6+6+18 100 + 100 + 300 108 + 108 + 283 117+ 117 + 267 125 +125 +250 6.5+6.5+17 7+7+16 75 +75 +15 8+8+14 8.5+8.5+13 9+9+12 9.5 +9.5 +II Io+ IO + IO 1:2 PYKNON I+2+27 4/3 +8/3 +26 5/3 +10/3 +25 2+4+24 7/3 +14/3 +23 8/3 + 16/3+ 22 3+6+21 10/3 +20/3 + 20 11/3 + 22/3 + 19 4+8+18 13/3 +26/3 +17 14/3+ 28/3 + 16 5 +10+ 15 16/3 + 32/3 + 14 17/3 + 34/3 +13 6+12+12 36 133 + 133 +234 142 + 142 +217 150+ 150+ 200 158 +158+ 183 166 + 166+ 167 17+33+450 22+44+433 28+56+417 33 +67 +400 39 +78+ 383 44 + 89 + 367 50+ 100+ 350 56+ III + 333 61 +122 + 317 67 +133 + 300 72 + 144+ 283 78+ 156 + 267 83 +167 +250 89+ 178 + 233 94+ 189+ 217 IOO + 200 + 200 CHAPTER 4 28/27 27/26- 26/21 24/23 ‚23/22: 11/9 22/21 «21/20: 40/33 20/19 - 19/18 - 6/5 18/17 - 17/16: 32/27 17/16- 16/15 + 20/17 16/15 - 15/14 - 7/6 15/14- 14/13 + 52/45 14/13 -13/12 7/6 40/37: 37/34- 17/15 64/59- 59/54: 9/8 12/11: 11/10. 10/9 11/10: 11/10: 400/363 120/119 - 119/117 : 13/10 84/83 : 83/81 - 9/7 64/63 + 33/32 . 14/11 57/56 - 28/27: 24/19 46/45 - 24/23 "5/4 40/39 - 21/20: 26/21 34/33: 18/17 - 11/9 33/32: 16/15 - 40/33 28/27: 15/14 6/5 27/26: 13/12 + 32/27 51/49» 49/45 + 20/17 22/21. 12/11. 7/6 104/99: 11/10: 15/13 21/20- 10/9 - 8/7 20/19 + 19/17 + 20/17 256/243 + 9/8 + 9/8 PTOLEMAIC INTERPRETA 80/79 « 79/78. 13/10 60/59 : 59/58. 58/45 48/47 + 47/46 : 23/18 40/39- 39/38 - 19/15 240/233 - 233/226 -113/ 30/29 - 29/28. 56/45 80/77 - 77/74: 37/30 24/23 - 23/22 11/9 240/229 : 229/218 - 109/ 20/19 + 19/18. 6/5 240/227 + 227/214+ 107/ 120/113 - 113/106 + 53/4: 16/15 - 15/14 : 7/6 15/14: 14/13 - 52/45 240/223 - 223/206 - 103/ 40/37: 37/34 17/15 240/221 + 221/202 + 101/1 12/11 + 11/10 + 10/9 120/119- 119/117+ 13/1 90/89 - 89/87 : 58/45 72/71 + 71/69 : 23/18 60/59 - 59/57 19/15 360/353 + 353/339 113/ 45/44 : 22/21 56/45 40/39 - 39/37 + 37/30 36/35 - 35/33 «11/9 360/349 + 349/327 109/ 30/29 : 29/27 + 6/5 360/347 - 347/321 - 107/ 180/173 - 173/159 - 53/4. 24/23 + 23/22 + 7/6 45/43 © 43/39 « 52/45 360/343 : 343/309+ 103/ 20/19 + 19/17 - 17/15

Finally, in 4-13, the pyknotic proportions are kept constant at either 1:1 or 1:2 and the Cls are allowed to vary. These neo-Aristoxenian tetrachords may be approximated in just intonation or realized in equal temperaments whose cardinalities are zero modulo 12. The zero modulo 12 temperaments provide opportunities to simulate many of the other genera in the Catalogs as their fourths are only two cents from 4/3 and other intervals of just intonation are often closely approximated. One may also use them to discover or invent new neoAristoxenian tetrachords. To articulate a single part difference, a temperament of 72 tones per octave is required. The 1/2 parts in the hemiolic chromatic and several other genera normally demand 144 tones unless all the intervals including the disjunctive tone have a common factor. In this case, the 48-tone system suffices. For the 1:2 pykna which employ 1/3 parts, 216-tone temperament is necessary unless the numbers of parts share common factors. These data are summarized in 4-14. 4-14. Aristoxenian realizations. Theframework is the number of “parts” in the two tetrachords and the disjunctive tone. The corresponding equal temperament is the sum ofthe parts ofthe framework, The articulated genera are those that may be played in the corresponding equal temperaments. The scheme of 144 parts was used by Avicenna and Al-Farabi (D'Erlanger 1930). FRAMEWORK ET ARTICULATED GENERA 525 Io 4 Io 15 6 15 12 24 36 Diatonicand syntonic chromatic. Enbarmonic, syntonic and soft diatonics, syntonic chromatic. Syntonic diatonic, syntonic and soft chromatics, unnamed. 20 8 20 48 25 10 25 30 12 30 60 72 35 14 35 40 16 40 84 96 45 18 45 so 20 50 55 22 55 60 24 60 90 36 go 108 120 132 144 216 Hemiolic chromatic, soft and syntonic diatonics, syntonic chromatic, diatonic with hemiolic chromatic dieses. See 24-tone ET. Syntonic diatonic and chromatic. AI previous genera except bemiolic chromatic and genera with hemiolic chromatic dieses (see 24-tone ET). Syntonic diatonic and chromatic, Enharmonic, syntonic diatonic, soft diatonic, syntonic and bemiolie chromatic. See 24-tone ET. See 36-tone ET. See 24-tone ET. See 12-tone ET. All genera except 1:2 pykna with 1/3 parts. All genera defined in text, Chromatic, diatonic with soft chromatic dieses. THE CONSTRUCTION OF NEW GENERA

Semi-tempered tetrachords The computation of the mean tetrachords also generates a number of genera containing irrational intervals involving square roots. These tetrachords contain both tempered intervals as well as at least one in just intonation, the 4/3, and may therefore be called semi-tempered. There also are the semitempered tetrachords resulting froma literal interpretation of the late classical theorists Nichomachos and Thrasyllus (Barbera 1978). The first of these is Nichomachos’s enharmonic, defined verbally as a ditone with an equally divided Jima and mathematically as V(2 56/243): V(256/243) -81/64 (45 + 45 + 408 cents). The second is Thrasyllus's chromatic, described analogously as having a Pythagorean trihernitone or minor third and a whole tone pyknon. Literally, this genus would be V(9/8) : (9/8) - 32/27 (102 + 102 + 294 cents), but it is possible that Thrasyllus meant the standard Pythagorean tuning in which the pyknon consists of a limma plus an apotome, i.e., 256/243 - 2187/2048 -32/27 (90+114+ 294 cents). Other semi-tempered forms result from Barbera’s assumption that Aristoxenos may have intended that the perfect fourth of ratio 4/3 be divided 4-15. Semi-tempered Aristoxenian tetrachords, These tetrachords are literal interpretations of Aristoxenos's genera under Barbera’s assumption that Aristoxenos meant to divide the perfectfourth of ratio 4/3 into 30 equal parts. geometrically into thirty parts. Barbera (1978) offers this literal version of the enharmonic: 10V(4/3). 19V(4/3) - !°V(65536/6561), or 50+ 50 + 398 cents, where 65536/6561 is (4/3). It is an easy problem to find analogous interpretations of the remainder of Aristoxenos’s genera. These and a few closely related genera from 3-1-3 have been tabulated in 4-15. PARTS ROOTS CENTS 3+3+24 4/3 1/10 . 4/3 110 . ars 2. 444422 3 4st qg eet EIS 5 ns 4/320. 4/33/20, 4/37/10 50 + 50 + 398 66 + 66 + 365 ENHARMONIC HEMIOLIC CHROMATIC 4. 6+6+18 a” «4/35 «4/335 75 +75 + 349 100 + 100 + 299 INTENSE CHROMATIG 5. 649415 aa . 4/33/10. 4/32 100 + 149 + 250 SOFT DIATONIC 6. 4139. gl3?!5 «4/325 | 100+ 199 + 199 INTENSE DIATONIC 66 + 232 + 199 DIATONIC WITH SOFT CHROMATIC DIESES 6412412 7. rigen AUS 43705 «4/35 GENUS 8 45+135+i2 4/3 20. 4/3920, 4/32/5 9. 4+8+18 4732/55 . 43/5 . 4/335 10. 6+3 +21 11. 4.64 3.5422 4/3 «4/310. 473710 75+224+199 66+133 +299 100+ SO + 349 4/320. 4/57/60, 4/3101S 75+58 +365 12. lo+io+io 13. 12+9+9 4/3. 4/313. 4/33 a/325 « 4/33/10. 4733710 166 + 166 + 166 200+ 149 + 149 38 SOFT CHROMATIG DIATONIC WITH HEMIOLIC CHROMATIC DIESES UNNAMED REJECTED REJECTED SEMI-TEMPERED EQUABLE DIATONIG ISLAMIC DIATONIC CHAPTER 4

Equal divisions of the 4/3 The semi-tempered tetrachords suggest that equally tempered divisions of the 4/3 would be worth exploring. Such scales would be analogous to the equal temperaments of the octave except that the interval of equivalence is the 4/3 rather than the 2/1. Scales of this type are very rare, though they have been reported to exist in contemporary Greek Orthodox liturgical music (Xenakis 1971). A possible ancestor of such scales is the ancient Lesser Perfect System, which consisted of a chain of the three tetrachords hypaton, meson, and synemmenon. In theory, all three tetrachords were identical, but this was not an absolute requirement, and in fact, in Ptolemy’s mixed tunings, they would not have been the same. (See chapter 6 for the derivations of the various scales and systems, and chapter 5 for the analysis of their properties.) The most interesting equal divisions of the 4/3 resemble the equal temperaments described in the next section and in 4-14 and 4-17. The melodic possibilities of these scales should be quite rich, because in those divisions with more than three degrees to the 4/3 not only can several tetrachordal genera be constructed, but various permutations of these genera are also possible. The harmonic properties, however, may be very different from those of the octave divisions as the 2/1 may not be approximated closely enough for octave equivalence to be retained. Moreover, depending upon the division, other intervals such as the 3/2 or 3/1 may or may not be acceptably consonant, The equal divisions of the 4/3 which correspond to equal octaval temperaments are described in 4-16. A few supplementary divisions such as the one of 11 degrees have been added since they reasonably approximate harmonically important intervals. For reasons of space, only a very limited number of intervals was examined and tabulated. To gain an adequate understanding of these tunings, the whole gamut should be examined over a span of at least eight 4/3’s. Additionally, the nearest approximations to the octave and the number of degrees per 2/1 are listed. This information allows one to decide whether the tuning is equivalent to an octave division, or whether it essentially lacks octave equivalence. Composition in scales without octave equivalence is a relatively unexplored area, although the THE CONSTRUCTION OF NEW GENERA

DEGREES PER 4/3 CENTS/DEGREE DEGREES/OCTAVE CENTS/OCTAVE 166.0 4 5 6 OCTAVE DIVISION OTHER CONSONANT INTERVA 7.228 1162.1 70 GOLDEN RATIO (PHI) = 5 124.5 9.638 1245.1 10 (+) 7/1= 27 99.61 12.05 1195.3 12 (—) 5/1= 28 83.01 14.46 1162.1 14) 7/5=7 7 71.16 16.86 1209.5 17 (+) — 8 62.26 19.27 1182.9 19 (>) 7/1=54 9 55.34 21.68 1217.4 22 (+) 5/3 = 16, 6/1 = 56 10 49.80 24.09 1195.3 24 (-) 3/2 = 14, 5/1 = 56 II 45.28 26.50 1222.5 27 (+) 13 38.31 31.32 1187.6 310) 3/1 = 42, 4/1 = 531 5/2 = 35, 6/1 = 81, 7/1 = 88, 8/1 = 94 14 35-57 33-73 1209.5 34 (+) 7/2 = 61 15 33.20 36.14 1195.3 36 (—) 5/1 = 84, PHI = 25 17 29.30 40.96 1201.2 41 (+) 3/2 = 24, 7/2 = 74 20 24.90 48.19 1195.3 48 (-) s/ı = 112, 7/4 = 39 22 22.64 53.01 1199.8 53 (-) 3/2 = 31, 5/3 = 39 25 19.92 60.24 1195.3 60 (—) 5/1 = 140, 7/1 = 169 28 17.79 67.46 1191.8 67 (—) 3/1 = 107, 4/1= 135 30 16.605 72.28 1195.3 72 (-) 7/1 = 203, 7/5 = 35 35 14.23 84.33 1195.3 84 (-) 7/4 = 68, 7/5 = 41 40 12.45 96.38 1195.3 96 (-) 6/1 = 249, §/3 = 71 45 11.07 108.4 1195.3 108 (-) 3/1 = 172, 4/1 = 217 50 9.961 120.5 1195.3 120 (-) 3/1 = 191, 4/1 = 241 55 9.055 132.5 1204.4 133 (+) 7/4= 107, PHI = 92, 3/1 = 21 60 8.301 144.6 1203.6 145 (+) 3/1 = 229, 4/1 = 289 go 5.534 216.8 1200.8 217 (+) 3/2 = 127 4-16, Equal divisions ofthe 4/3. These are equal temperaments ofthe 4/3 rather than the 2/1. “Degrees/octave” is the number ofdegrees ofthe division corresponding to the 2/1 or octave. For many ofthese divisions, the octave no longer functions as an interval ofequivalence. “Cents/octave” is the cent value ofthe approximations to the 2/1. “Octave division” is the closest whole number ofdegrees to the 2/1. (-) indicates that the octave is compressed and less than 1200 cents. (+) means that it isstretched and larger than 1200 cents. “Consonant intervals” are the degrees in good approximations to the intervals listed. All divisions ofthe 4/3 have good approximations to the 10/1 as (4/3)? + the skbisma equals 10/1. Divisons that are multiples of 3 also bave good approximations to the 11/1. 17 is a slightly stretched 41-tone equal temperament. 22 is audibly equivalent to 53-tone equal temperament. 28 is analogous to the division ofthefourth into 28 parts according to Tiby’s theory ofGreek Orthodox liturgical music (Tiby 1938). 30 is analogous to Aristoxenos’s basic system. 55 is analogous to 13 2-tone equal temperament. 60 isanalogous to 144-tone equal temperament. 90 is analogous to 2 16-tone equal temperament, The Golden Ratio or Phi is (1+ 5)/2, approximately 1.618. CHAPTER 4

composer and theorist Brian McLaren has recently written a number of pieces in non-octaval scales mostly of his own invention (McLaren, personal communication, 1991). Xenakis has also mentioned chains of fifths consisting of tetrachords and disjunctive tones (Xenakis 1971). These suggest analogous divisions of the 3/2, including both those with good approximations to the 4/3 and those without. Similarly, there are divisions in which octave equivalence is retained and those in which it is not. An example of one with both good fourths and octaves is the seventh root of 3/2, which corresponds to a moderately stretched 12- tone equal temperament of the octave (Kolinsky 1959). Tetrachords in non-zero modulo 12 equal temperaments Tetrachords may also be defined in non-zero modulo 12 equal temperaments. For some combinations of genus and tuning the melodic and harmonic distortions will be negligible, but for others the mappings may distort the characteristic melodic shapes unacceptably. As an illustration, the three primary genera, the enharmonic, the syntonic chromatic, and the 4-17. Tetrachords in non-zero modulo 12 equal temperaments. These genera are defined in ETs where the perfectfourth does not equal 2 1/2 “whole tones.” The framework is the number of “parts” in the twofourths and the disjunctive tone. More than one framework is plausible in some temperaments without goodfourths or with more than 17 notes. The corresponding equal temperament ts the sum of the parts oftheframework, The genera in a generalized, non-specific sense may be approximated in these equal temperaments, “Diatonic/chromatic” means that there isno melodic distinction between these genera. The chromatic pykna in 9-, 10-, and rr- tone ET consist oftwo small intervals and one large, while the disjunction may larger or smaller than the CI. Genera indifferently enharmonic and chromatic occur around 19 tones per octave and neoAristoxenianforms may be realizable in many ofthe ETs. FRAMEWORK 313 3 2 3 4 1 4 4 2 4 4 3 4 535 626 636 727 737 7 4 7 (8 2 8) 8 3 8 4 8 9 3 9, 8 5 8 949 959, 10 3 Io 13 5 13 14 6 14 17 7 17 22 Q 22 41 ET 7 8 IO II I3 Iq I5 16 17 18 19 20 21 22 23 31 34 41 53 GENERA DIATONIC/CHROMATIC DIATONIC/CHROMATIC CHROMATIC CHROMATIC CHROMATIC DIATONIC, CHROMATIC DIATONIC, CHROMATIC DIATONIG, CHROMATIG DIATONIC, CHROMATIC DIATONIC, CHROMATIC DIATONIC, CHROMATIC (ALL THREE) DIATONIC, CHROMATIC ALL THREE ALL THREE ALL THREE ALL THREE ALL THREE ALL THREE ALL THREE ALL THREE THE CONSTRUCTION OF NEW GENERA

4-18. Augmented and diminished tetrachords. These tetrachords are closely related to those in 8-5 and 8-15. For tetrachords with perfectfourths incorporating the diminishedfourths as intervals, see the Main and Miscellaneous Catalogs. A few additional intervals of similar size bave been used as Cls in 4-1, but not divided due to their complexity. The last three intervals are technically diminished fiftbs, but theyfunction asaugmentedfourths in certain afthe barmoniai ofchapter 8. RATIOS 14/11 23/18 32/25 9/7 31/14 22/17 13/10 30/3 17/13 21/16 29/22 31/23 23/17 19/14 15/11 26/19 11/8 40/29 18/13 25/18 32/23 7/5 1024/729 45/32 24/17 17/12 44/31 10/7 CENTS EXAMPLES 418 19/13 13/22. 12/11 424 23/22-11/10- 10/9 427 32/31: 31/30 - 6/5 435 443 446 454 460 464 471 478 18/17-17/16-8/7 31/30-10/9- 9/8 11/10- 10/9 - 18/17 13/12-12/11 - 13/10 15/14: 7/6 : 24/23 17/16. 8/7. 14/13 21/20-10/9-9/8 29/28. 7/6- 12/11 517 31/30. 5/4 - 24/23 523 529 537 543 551 23/22-11/9- 18/17 19/18-6/5-15/14 ıs/ıq- W/6- 12/11 26/15 - s/4- 20/19 II/10-10/9-9/8 557 87. 7/6 - 30/29 563 9/8 - B/7- 14/13 569 5/4-20/19- 19/18 572 _ 16/15 5/4 + 24/23 583 14/13-13/12- 6/5 588 256/243 . 8/7. 7/6 590 16/15 - 10/9 - 6/5 597 6/5-10/9-18/17 603 17/16-8/7- 7/6 606 11/10- 5/4- 32/31 617 10/9-9/8-8/7 diatonic, will be mapped into the 12-, 19-, 22-, and 24-tone equal temperament (ET) below: ET FOURIH ENHARMONIC CHROMATIC DIATONIC 12 5° — 1+1+3 I+2+2 Ig 8° I+1+6 2+2+4 2+3+3 22 9° 1+1+7 24245 I+4+4 24 10° 1+1+8 2+2+6 2+4+4 The enharmonic is not articulated in 12-tone ET, or at least not distinguishable from the chromatic except as a semitonal-major third pentatonic. In 19-tone ET, the soft chromatic is identical to the enharmonic and the syntonic chromatic is close to a diatonic genus like 125 + 125 + 250 cents. The enharmonic is certainly usable in 22-tone ET but the diatonic is deformed, with a quarter-tone taking the place of the semitone. These distortions, however, are mild compared to the 9-tone equal temperament in which not only are the diatonic and chromatic genera equivalent as 1 + 1 + 2 degrees, but the semitone at two units is larger than the whole tone. Whether these intervallic transmogrifications are musically useful remains to be tested. There are, however, many fascinating musical resources in these non- 12-tone tunings. As Ivor Darreg has pointed out, each of the equal temperaments has its own particular mood which suffuses any scale mapped into it (Darreg 1975). For this reason the effects resulting from transferring between tuning systems may be of considerable interest. Because of the large number of systems to be covered, the mappings of the primary tetrachordal genera into the non-zero modulo 12 equal temperaments are summarized in 4-17. The tetrachordal framework and primary articulated genera in the equal temperaments of low cardinality or which are reasonable approximations to just intonation are shown in this figure. Augmented and diminished tetrachords The modified or altered tetrachords found in some of the non-zero modulo 12 equal temperaments of 4-17 suggest that tetrachords based on augmented and diminished fourths might be musically interesting. This supposition has historical and theoretical support. The basic scales (thats) of some Indian ragas have both augmented and perfect fourths (Sachs 1943), and the octaval barmoniai of Kathleen Schlesinger contain fourths of di- CHAPTER 4

magnitudes (Schlesinger 1939; and chapter 8). Wilson has exploited the fact that any scale generable by a chain of melodic fourths must incorporate fourths of at least two magnitudes (Wilson 1986; 1987; and chapter 6). His work implies that scales may be produced from chains of fourths of any type, but that their sizes and order must be carefully selected to ensure that the resulting scales are recognizably tetrachordal. A number of altered fourths are available for experimentation. 4-18 lists those which commonly arise in conventional theory and in the extended theory of Schlesinger’s harmoniai described in chapter 8. Scales may be constructed by combining these tetrachords with each other or with normal ones and with correspondingly altered disjunctive tones to complete the octaves. Alternatively, the methods described in chapter 6 to generate non-heptatonic scales may be employed. THE CONSTRUCTION OF NEW GENERA

Classification, characterization, and analysis of tetrachords THIS CHAPTER CONTAINS a complex mixture of topics regarding the description or characterization of tetrachords. Some of the concepts are chiefly applicable to single tetrachords, while others refer to pairs of tetrachords or the complete tetrachordal space. The most interesting of the newer methods, those of Rothenberg and Polansky, are most usefully applied to the scales and scale-like aggregates described in detail in chapter 6. Moreover, Polansky’s methods may be applied to parameters other than pitch height. The application of these techniques to tetrachords may serve as an model for their use in broader areas of experimental intonation. The first part of the chapter is concerned with the historical approach to classification and with two analyses based on traditional concepts. These concepts include classification by the size of the largest, and usually uppermost, incomposite interval and subclassification by the relative sizes of the two smallest intervals. A new and somewhat more refined classification scheme based on these historical concepts is proposed at the end of this section. These concepts and relationships are displayed graphically in order that they may become more intuitively understood. A thorough understanding of the melodic properties of tetrachords is a prerequsite for effective composition with tetrachordally derived scales. Of particular interest are those tetrachords which lie near the border of two categories. Depending upon their treatment, they may be perceived as belonging to either the diatonic or chromatic genera, or, in other cases depending on the CIs, to either the enharmonic or chromatic. An example is the intense chromatic or soft CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

diatonic types, where the interval near 250 cents may be perceived as either a large whole tone or a small minor third. This type of ambiguity may be made compositionally significant in a piece employing many different tetrachords. The middle portion of the chapter deals with various types of harmonic and melodic distance functions between tetrachords having different intervals or intervallic arrangements. Included in this section is a discussion of the statistical properties of tetrachords, including various means (geometric mean, harmonic mean, and root mean square; see chapter 4) and statistical measures of central tendency (mean deviation, standard deviation, and variance). Both tabular and graphical representations are used; the tabular is useful to produce a feeling for the actual values of the parameters. These concepts should be helpful in organizing modulations between various tetrachords and tetrachordal scales. For example, one could cut the solid figures generated by the various means over the whole tetrachordal space by various planes at different angles to the axes. The intersections of the surfaces with the planes or the interiors of the bounded portions of the figures of intersection define sets of tetrachords. Planes parallel to the bases define tetrachordal sets with invariant values of the means, and oblique planes describe sets with limited parametric ranges. Similarly, lines (geodesics) on the surfaces of the statistical measures delineate other tetrachordal sets, These techniques are similar to that employed by Thomas Miley in his compositions Z-View and Distance Music, in which the inter. sections of spheres and planes defined sets of intervals (Miley 1989). The distance functions are likewise pertinent both to manual and algorithmic composition, James Tenney has used harmonic and melodic distance functions in Changes: Sixty-four Studiesfor Six Harps, a cycle of pieces in 11-limit just intonation, Polansky’s morphological metrics are among the most powerful of the distance functions. Polansky has used morphological metrics in a number of recent compositions, although he has not yet applied them to sets of tunings (Polansky, 1991, personal communication). His compositions employing morphological metrics to date are 17 Simple Melodies of the Same Length (1987), Distance Musics I-VI (1 987), Duet (1989), Three Studies (1989) and Bedhaya Sadra/Bedhaya Guthrie (1988-1991). In the absence of any published measurements known to the author of the perceptual differences between tetrachordal genera and tetrachordal permutations, the question ofwhich of the distance functions better models CHAPTER 5

perception is unanswerable. There may be a number of interesting research problems in the psychology of music in this area. The chapter concludes with a discussion Rothenberg’s concept of propriety as it applies to tetrachords and heptatonic scales derived from tetrachords. Rothenberg has used propriety and other concepts derived from his theoretical work on perception in his own compositions, i.e., Inbarmonic Figurations (Reinhard 1987). Historical classification The ancient Greek theorists classified tetrachords into three genera according to the position of the third note from the bottom. This note was called lichanos (“indicator”) in the hypaton and meson tetrachords and paranete in the diezeugmenon, hyperbolaion, and synemmenon tetrachords (chapter 6). The interval made by this note and the uppermost tone of the tetrachord may be called the characteristic interval (CI), as its width defines the genus, though actually it has no historical name. If the lichanos was a semitone from the lowest note, making the CI a major third with the 4/3, the genus was termed enharmonic. A lichanos roughly a whole tone from the 1/1 produced a minor third CI and created a chromatic genus. Finally, a lichanos a minor third from the bottom and a whole tone from the top defined a diatonic tetrachord. The Islamic theorists (e.g., Safiyu-d-Din, 1276; see D’Erlanger 1938) modified this classification so that it comprised only two main categories translatable as “soft” and “firm.” (D’Erlanger 1930; 1935) The soft genera comprised the enharmonic and chromatic, those in which the largest interval is greater than the sum of the two smaller ones, or equivalently, is greater than one half of the perfect fourth. The firm genera consisted of the diatonic, including a subclass of reduplicated forms containing repeated whole tone intervals. These main genera were further subdivided according to whether the pykna were linearly divided into approximately equal (1:1) or unequal (1:2) parts. The 1:1 divisions were termed “weak” and the 1:2 divisions, “strong.” These theorists added many new tunings to the corpus of known tetrachords and also tabulated the intervallic permutations of the genera. This led to compendious tables which may or may not have reflected actual musical practice, CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

Crocker’s tetrachordal comparisons Richard L. Crocker (1963, 1964, 1966) analyzed the most important of the ancient Greek tetrachords (see chapters 2 and 3) in terms of the relative magnitudes of their intervals. Crocker was interested in the relation of the older Pythagorean tuning to the innovations of Archytas and Aristoxenos. He stressed the particular emphasis placed on the position of the lichanos by Archytas who employed 28/27 as the first interval (parhypate to 1/1) in all three genera. In Pythagorean tuning, the chromatic and diatonic parhypatai are a limma (256/243, 90 cents) above hypate, while the enharmonic division is not certain. The evidence suggests a limmatic pyknon, but it may not have been consistently divided much prior to the time of Archytas (Winnington-Ingram 1928). Archytas’s divisions are in marked contrast to the genera of Aristoxenos, who allowed both lichanos and parhypate to vary within considerable ranges. With Archytas the parhypatai are fixed and all the distinction between the genera is carried by the lichanoi. These relations can be seen most clearly in 5-1, 5-2, and 5-3. These figures have been redrawn from those in Crocker (1966). This type of comparison has been extended to the genera of Didymos, Eratosthenes and Ptolemy in 5-4, 5-5, and 5-6. The genera of Didymos and Eratosthenes resemble those of Aristoxenos with their pykna divided in rough equality. Ptolemy’s divisions are quite different. For Aristoxenos, Didymos, and Eratosthenes, the ratio of the intervals of the pyknon are roughly 1:1, except in the diatonic genera. Ptolemy, however, uses approximately a 2:1 relationship. Barbera’s rate of change function C. André Barbera (1978) examined these relations in more detail. He was especially interested in the relations between the change in the position of the lichanoi compared to the change in the position of the parhypatai as one moved from the enharmonic through the chromatic to the diatonic genera. Accordingly, he defined a function over pairs of genera which compared the change in the location of the lichanoi to the change in that of the parhypatai. His function is (lichanos; — lichanos:) / (parhypatez — parhypat e) where the corresponding notes of two tetrachords are subscripted. This function is meaningful only when computed on a series of related genera CHAPTER 5

5-1. Archytas’s genera. These genera have a constant 28/27 as theirparhypate. 5-3. Aristoxenos’s genera, expressed in Cleonides’s o 63 cents, 112 63 498 ENHARMONIC so 100 204 63 9/8 204 500 3+3+24 PARTS 498 DIATONIC 8/9 28/27 o 5/4 CHROMATIC 243/224 32/27 28/27 (o) parts rather than ratios. One part equals 16.667 ENHARMONIG SOFT CHROMATIC 133 67 498 500 4+4+ 22 PARTS HEMIOLIC CHROMATIG 75 150 4.5 + 4.5 +21 PARTS 500 5-2. Pythagorean genera. These genera are traditionally attributed to Pythagoras, but in fact are of INTENSE CHROMATIC 100 200 6 +6 + 18 PARTS 500 Babylonian origin (Duchesne-Guillemin 19 63, 1969). The division ofthe enbarmonic pyknon is not known, but severalplausible tunings are listed in the Main Catalog. SOFT DIATONIC 100 250 6 +9 + 15 PARTS soo > ? _—— o ? ENHARMONIG Too 81/64 go INTENSE DIATONIG 50 3 6+12 +12 PARTS 3 co 498 CHROMATIC 256/243 2187/2048 32/27 (o) 90 498 DIATONIC 9/8 256/243 o 204 go 294 9/8 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

5-4. Didymos’s genera, Didymos’s chromatic is probably the mast consonant tuningfor the 6/5 genus. His diatonic differsfrom Ptolemy's only in the order ofthe 9/8 and 10/9. 5-6, Ptolemy’s genera. Only Ptolemy’s own genera are shown, Ptolemy's tonic diatonic is the same as Archytas’s diatonic. His ditone diatonic is the Pythagorean diatonic, ENHARMONIC 5/4 32/31 31/30 o 55 112 498 ENHARMONIG 46/45 24/23 CHROMATIC 16/15 25/24 o 112 o 113 183 28/27 SOFT CHROMATIC 15/14 6/5 498 o 10/9 112 63 182 294 498 o INTENSE CHROMATIC 12/11 7/6 81 232 85 267 5-5. Eratosthenes’s genera. Eratosthenes’s diatonic 16/15 (o) ENHARMONIC 19/15 9/8 II2 12/11 498 (e CHROMATIC e] 19/8 89 498 DIATONIC 9/8 256/243 o 6/5 183 go 294 498 INTENSE DIATONIG is the same as Ptolemy’s ditone diatonic. o 44 89 498 SOFT DIATONIC 10/9 8/7 21/20 o 20/19 498 9/8 22/21 40/39 39/38 498 6/5 DIATONIC 16/15 5/4 o 38 9/8 498 50 CHAPTER 5 10/9 316 498 EQUABLE DIATONIC 11/10 10/9

5-7. Barbera function applied to Aristoxenos’s and Ptolemy’s genera. such as Aristoxenos’s enharmonic and his chromatics or on the corresponding ones of Ptolemy. The extent to which such calculations give consistent values is a measure of the relatedness of the tetrachordal sets. SOFT CHR./ENH. SOFT CHR./ENH. In 5-7, the results of such calculations are shown. The value for Aristoxenos’s non-diatonic genera is 2.0. Ptolemy’s genera yield values near 3.0, and the discrepancies are due to his use of superparticular ratios and just intonation rather than equal temperament. The proportion of the Ptolemaic to the Aristoxenian values is near 1.4. HEM, CHR./SOFT CHR, INT. CHR./SOFT CHR. These facts suggest that both theorists conceived their tetrachords as INT, CHR./HEM. CHR. INT. DIA./INT. CHR. internally related sets, not as isolated tunings. Presumably, the increase from 2.0 to about 3 of this parameter reflects a change in musical taste in the nearly soo years elapsed between Aristoxenos and Ptolemy. 5-8. Ratio of lichanos to parbypate in Aristoxenos’s and Ptolemy’s genera, 2.0 ENHARMONIC ki ee 2.047 nnn 1.474 messa 2.0 cement 2.880 SOFT CHROMATIC INTENSE CHROMATIC i Both ancient theorists presented additional genera not used in this computation. Some, such as Aristoxenos’s hemiolic chromatic or Ptolemy’s equable diatonic, had no counterpart in the other set. Ptolemy’s soft diatonic appears to be only a variation or inflection of his intense (syntonic) chromatic, His remaining two diatonics, the tonic and ditonic, were of historical origin and not of his invention. The same is true of Aristoxenos’s intense diatonic which seems clearly intended to represent the archaic ditone or Pythagorean diatonic. A comparison of the corresponding members of these two authors’ sets of tetrachords by a simpler function is also illuminating. If one plots the ratio of lichanos to parhypate or, equivalently, the first interval versus the sum of the first two, it is evident that Aristoxenos preferred an equal division of the pyknon and Ptolemy an unequal 1:2 relation. These preferences are shown by the data in 5-8, where the lichanos/parhypate ratio is 2.0 for SOFT DIATONIC INTENSE DIATONIC Aristoxenos’s tetrachords and about 3.0 for Ptolemy’s non-diatonic genera. HEMIOLIC CHROMATIC One may wonder whether Ptolemy’s tetrachords are theoretical innovations or whether they faithfully reflect the music practice of second TONIC DIATONIC È DITONE DIATONIC EE EQUABLE DIATONIC X B ARISTOXENOS # PTOLEMY # RATIO (PTOLEMY/ARISTOXENOS) century Alexandria. The divisions of Didymos and Eratosthenes, authors who lived between the time of Aristoxenos and Ptolemy, resemble Aristoxenos’s, and there are strong reasons to assume that Aristoxenos is a trustworthy authority on the music of his period (chapter 3). The lyra and kithara scales he reports as being in use by contemporary musicians would seem to indicate that the unequally divided pyknon was a musical reality (chapter 6). Ptolemy’s enharmonic does seem to be a speculative CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

construct as the enharmonic genus was extinct by the third century BCE (Winnington-Ingram 1932). His equable diatonic, however, resembles modern Islamic scales and certain Greek orthodox liturgical tetrachords (chapter 3). These historical studies are important not only for what they reveal about ancient musical thought but also because they are precedents for organizing groups of tetrachords into structurally related sets. The use of 5-9. Neo-Aristoxenian classification, a+ b+c = 500 constant or contrasting pyknotic/apyknotic proportions can be musically cents. This classification is based on the size ofthe largest or characteristic interval (CD); the equal division ofthe pyknon (a+b) is only illustrative and other divisions exist. The byperenbarmonic genera bave Cls between the major third and thefourth and pyknotic intervals ofcommatic size. The enbarmonic genera contain Cls approximating major thirds, The significant. Modulation of genus (neraßoAe kata yevoo) from diatonic to chromatic genera rangefrom the soft chromatic to the soft diatonic ofAristoxenas or the intense chromatic of ofPtolerry. The diatonic are all those genera without pykna, i.e., whose largest interval is less than 250 cents. HYPERENHARMONIC Sto <a+b$ 3/17 234234454 10 37.5+37.5 +425 cents Bo/79- 79/78-13/10 to 50/49+49/48 32/25 ENHARMONIC zij <a+bsch 37.5+37.5+425 to 62,5+62.5+375 cents 48/47 47/46-23/18 to 30/29-29/28 56/45 CHROMATIC d3<at+hse 62.5+62.5+375 fo 125+125+250 cents 29/28 -28/27-36/29 to 15/14: 14/13-52/45 DIATONIG e<a+bSıc 125+125+250 #0 167+167+167 cents 104/97:97/90- 15/13 to 11/10 11/10: 400/363 chromatic or enharmonic and back was a significant stylistic feature of ancient music according to the theorists. Several illustrations of this technique are found among the surviving fragments of Greek music (Winnington-Ingram 1936). Neo-Aristoxenian classification The large number of new tetrachordal divisions generated by the methods of chapter 4 indicates a need for new classification tools. A conveniently simple scheme is the neo-Aristoxenian classification which assumes a tempered fourth of 500 cents and categorizes tetrachords into four classes according to the sizes of their CIs. For tetrachords in just intonation, the fourth has 498.045 cents, and the boundaries between categories will be slightly adjusted. The essential feature of this scheme is the geometrical approach of chapter three. Those new genera whose CIs fall between a major third and perfect fourth may be denoted byperenbarmonic after Ervin Wilson (personal communication) who first applied it to the 56/55 : 55/54 : 9/7 genus. The hyperenharmonic Cls range from roughly 450 cents down to 425 cents. The next class is the enharmonic with Cls ranging from 425 to 375 cents, a span of 5o cents, The widest division is the chromatic, from 375 cents to 250 cents as it includes CIs whose widths vary from the neutral thirds of approximately 360-350 cents (16/13, 11/9, 27/22) through the minor and subminor thirds (6/5, 7/6) to the “half-augmented seconds” (15/13, 52/45) near 250 cents. Beyond this limit, a pyknon no longer exists and the genera are diatonic. This neo-Aristoxenian classification is summarized in 5-9. The limits of the categories are illustrated with representative tetrachords in just intonation, CHAPTER 5

These four main classes may be further subdivided according to the proportions of the two intervals which divide the pyknon, or apyknon in the case of the diatonic genera. Because of the large number of possible divisions, it is clearer and easier to display the various subgenera graphically than to try to name them individually. Thus a number of representative 5-10. Plot of characteristic intervals versus tetrachords from the Main Catalog have been plotted in 5-10-12 to illustrate the most important types. In 5-10, the first interval, as defined by the position of the note parhypate, parbypatai. The four notes of the illustrative meson has been plotted against the characteristic interval. For most of the histetrachord in ascending order ofpitch are hypate, torical tetrachords of chapters 2 and 3, this is equivalent to plotting the parkypate, lichanos, and mese. The CI is the interval between lichanos and mese. smallest versus the largest intervals or the first against the third. The exceptions, of course, are Archytas’s enharmonic and diatonic and Didymos’s chromatic. 5-11 shows the position of the third note, lichanos, graphed against the second, parhypate. This is equivalent to comparing the size of the whole pyknon (or apyknon) to its first interval. This particular display recalls the Greek classification by the position of the lichanoi and the differentiation into shades or chroai by the position of the parhypatai. The first interval is plotted against the second in 5-12. In this graph, 5-11. Plot oflichanoi versus parbypatai. 5-12. First intervalplotted against second intervals ofmajor tetrachordal genera. The tetrachords plotted here are 50 + 50 + 400, 100 + 100 + 300, 100 +150 however, all of the permutations of this set of typical tetrachords are also +250, 100 +200 + 200, and 166.67 + 166.67 + plotted. This type of plot reveals the inequality of intervallic size between 166.67 cents in all oftheir intervallic permutations. The permutations ofthe soft diatonic genus delineate genera and distinguishes between permutations when the tetrachords are not in the standard Greek ascending order of smallest, medium, and the region ofRothenberg-proper diatonic scales. large. 5-10. 5-11. 400 - 5-12. 400 » HYPERENHARMONIC 1 ENHARMONIC d SOFT DIATONIC È INTENSE DIATONIG INTENSE CHROMATIC 4 ENHARMONIG . CHROMATIC 4 400 DIATONIG x 4 w oO < 5= 4 a DIATONIC ' mn 100 PARHYPATE 200 EQUAL DIATONIC 4 CHROMATIC ENHARMONIC HYPERENHARMONIC T 100 PARHYPATE 53 2Q à “ 1 200 Oo TT a T T 1 200 FIRST INTERVAL CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

Intervallic inequality functions More quantitative measures of intervallic inequality are seen in 5-13. The first measure is the ratio of the logarithms of the largest interval to that of the smallest. In practice, cents or logarithms to any base may be used, This 5-13. Intervallic inequalityfunctions onjust and tempered tetrachords. ratio measures the extremes of intervallic inequality. The second measure is the ratio of the largest to the middle-sized interval. For tetrachords with reduplicated intervals, i.e., 256/243 - 9/8 - 9/8 or 16/15 - 16/15 : 75/64, the CI/MIN CI/MID MID/MIN RATIOS HYPERENHARMONIC 56/55 - 55/54 - 9/7 middle-sized interval is the reduplicated one, and this function is equal to one of the other two functions. The third measure is the ratio of the mid- 13.95 13-70 1.018 dle-sized interval to the smallest. This function often indicates the relative ENHARMONIC 28/27 : 36/35 - 5/4 7.921 66.136 1.291 32/31 + 31/30 + 5/4 7.028 6.805 1.033 sizes of the two intervals of the pyknon and distinguishes subgenera with 46/45 + 24/23+5/4 10.15 5.243 1.936 CHROMATIC 20/19 » 19/18 - 6/5 3.554 3.372 18/27 + 15/14 6/5 5.013 2.642 26/25 -25/24-16/13 5.294 5.086 39/38 - 19/18 + 16/13 7.994 3.840 24/23 + 23/22 11/9 4.715 4.514 34/33 «18/17 11/9 6.722 3511 16/15 - 15/14» 7/6 2.389 2.234 22/21 + 12/11 < 7/6 3.314 1.772 1.054 1.897 1.041 2.081 1.044 1.915 1.069 1.870 DIATONIC 14/13 - 13/12 - 8/7 1.802 21/20 + 10/9 - 8/7 2.737 28/27. 9/8 - 8/7 3.672 16/15 : 10/9 + 9/8 1.825 256/243 :9/8:9/8 2.260 1.668 1.267 1.133 1.118 1.000 1.080 2.159 3.239 1.633 2.260 12/11 + II/10:10/9 1IOS 1.095 1.211 TEMPERED TETRACHORDS 50 + 50 + 400 8.00 8.00 66.67 + 133.33 + 300 4.50 2.25 TOO + 100 + 300 300 3.00 100 + 150 + 250 2.50 1.67 109 + 200 + 200 2,00 1.00 166.67 + 166.67 + 166.67 1.00 1.00 1.00 2.00 I.00 1,50 2,00 1.00 the same CI, These functions measure the degree of inequality of the three intervals and may be defined for tetrachords in equal temperament as well as in just intonation. All of these functions are invariant under permutation of intervallic order. Harmonic complexity functions In addition to being classified by intervallic size, tetrachords may also be characterized by their harmonic properties. Although harmony in the sense of chords and chordal sequences is discussed in detail in chapter 7, it is appropriate in this chapter to discuss the harmonic properties of the tetrachordal intervals in terms of the prime numbers which define them. The simplest harmonic function which may be defined on a tetrachord or over a set of tetrachords is the largest prime function. The value of this function is that of the largest prime number greater than 2 in the numerators or denominators of three ratios defining the tetrachord. The tetrachord (or any other set of intervals) is said to have an #-limit or be an n-limit construct when 7 is the largest prime number in the defining ratio(s), irrespective of its exponent and the exponent’s sign. : One limitation of the n-limit function is that it uses only a small part of the information in the tetrachordal intervals. As a result, numerous genera with different melodic properties have the same #-limit. However, this one-dimensional descriptor is often used by composers of music in just intonation (David Doty, personal communication). For example, the following diverse set of tetrachords all contain 5 as their largest prime number: 25/24 + 128/125 - 5/4, 256/243 - 81/80 - 5/4, 16/15 + 25/24 : 6/5, 256/243 : 54 CHAPTER 5

5-14. Harmonic complexity and simplicityfunctions on tetrachords injust intonation, (1) Cl complexity: the sum ofthe primefactors ofthe largest interval. (2) Pyknotic complexity: thejoint complexity ofthe two intervals ofthe pyknon. (3) Average complexity: the arithmetic mean ofthe Cl andpyknotic complexities. (4) Total complexity: thejoint complexity ofthe entire tetrachord. (5, Harmonic simplicity: x over the sum ofthe primefactors greater than 2 ofthe ratio defining the CI. It bas been normalized by dividing by 0.2, as the maximum value of the unscaledfunction is 0,2, corresponding to 5/4 whose Wilson's complexity is 5. RATIOS 1 2 3 4 5 HYPERENHARMONIC 56/55-55/54-9/7 13 32 22.5 32 3846 ENHARMONIC 28/27-36/35:5/4 32/31 - 31/30: 5/4 5 5 21 39 13 22 21 1.000 39 1.000 46/45-24/23:5/4 5 34 19.5 34 1.000 CHROMATIC 20/19-19/18-6/5 8 30 19 28/27-15/14:6/5 8 21 14.5 26/25-25/24:16/13 13 26 19.5 39/38-19/18-16/13 13 38 25.5 30 21 26 38 .6250 .6250 .3846 .3846 40 2941 24/23:13/21-11/0 17 34/33-18/17-11/0 17 34 25.5 34 -2941 37 27 16/15:15/14:7/6 22/21-12/11:7/6 10 15 10 21 12.5 15.5 15 .5000 2I .5000 DIATONIG 14/13-13/12-8/7 7 23 IS 23 21/20: 10/9-8/7 7 18 12.5 18 28/27-9/8.8/7 7 16 11.5 16 16/15 -10/9-9/8 6 11 85 11 256/243-9/8.9/8 6 15 ros 15 12/11-11/10+10/0 I II IO IS 22 7143 7143 .7143 8333 8333 4545 135/128 - 6/5, 16/15 : 75/64 : 16/15, 10/9 + 10/9 : 27/25, and 16/15 : 9/8 - 10/9. Similarly, all the Pythagorean tunings in the Catalog are at the 3-limit. The second limitation of the largest prime number function when applied to the whole tetrachord is that it does does not distinguish between intervals which may be of differing harmonic importance to the composer. Primary distinctions between genera are determined by the sizes of their characteristic intervals. Genera with similarly sized CIs may have quite different musical effects due to the different degrees of consonance of these intervals. Similar effects are seen with the pyknotic intervals as well, particularly those due to the first interval which combines with mese or the added note, hyperhypate, to form an interval characteristic of the oldest Greek styles (Winnington-Ingram 1936 and chapter 6). In these cases, the largest prime function must be applied to the individual intervals and not just to the tetrachord as a whole. For these reasons, other indices of harmonic complexity have been developed which utilize more of the information latent in the tetrachordal intervals. These indices have been computed on a representative set of tetrachords and their component intervals, The first of the indices is Wilson’s complexity function which for single intervals may be defined as the sum of their prime factors (greater than 2) times the absolute values of their exponents. For example, the complexities of 3/2 and 4/3 are both 3 and those of 6/5 and 5/3 are both 8 (3 + 5). Similarly, the intervals 9/7 and 14/9 both have complexities of 13 (3 + 3 + 7). The complexities of the Cls of some important genera are tabulated in 5-14. Wilson’s complexity function may also be applied to sets of intervals by finding the modified least common multiple of the prime factors (with all the exponents made positive). The pyknon of Archytas’s enharmonic consists of the intervals 28/27 and 36/35. The first ratio may be expressed as 7 + 33 and the second as 3? + 5 + 7. The modified least common multiple of this set is 33 - 5 - 7 and the Wilson’s complexity is 21 (3 + 3 + 3 + 5 +7). The average complexity, which is the arithmetic mean of the complexities of the CI and the pyknon, and the total complexity, which is the joint complexity of all three intervals, are also shown in 5-14. In most cases the latter index equals the pyknotic complexity. An alternative index which may be more convenient in some cases is the harmonic simplicity, which is the reciprocal of the complexity. This function CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

5-15. Euclidean distances between genera injust intonation. The upper set ofnumbers is the distance may be normalized, as it is in 5-14, by dividing its values by 5, which is the maximum simplicity of a CI or tetrachord (because 5/4 is the simplest interval smaller than 4/3). calculated on the largest versus the smallest intervals ofthe tetrachords. The lower set iscomputedfrom the first and second intervals. The Euclidean distance is the square root of the sum of the squares of the Euclidean distances between tetrachords differences between corresponding intervals, Values The methods described in chapter 4 and in the compilations of the historical art in cents, authors provide many tetrachords with diverse melodic characteristics. To bring some order to these resources, some measure of the perceptual distance between different genera or between different permutations of the same genus is desirable. While a useful measure of the distance between 5-16. Euclidean distances between temperedgenera. genera may be obtained from the differences between the characteristic The 1:2 chromatic is the “strong” form corresponding to the intense chromatic ofAristoxenas. The equal diatonic is 166.67 + 166.67 + 166.67 intervals, this measure does not distinguish between the subgenera (i.e, the cents. the Euclidean distances between genera on a plot of the CI versus the 28/27 -15/14+ 6/5 28/27 - 36/35 « 5/4 72.09 70.67 28/27 - 15/14 6/5 1:1 and 1:2 divisions of the pyknon). A more precise measure is afforded by 25/24-16/15+6/5 73-99 22/21-12/t1+ 7/5 123.59 16/15:9/8.10/9 192.96 103.37 162.62 145.59 7.71 10,91 51.84 35.81 121.91 97.54 159.50 98.81 49.76 40.14 119.04 100.91 155.39 96.09 70.26 61.73 109.77 71.56 22/21 - 12/11 - 7/6 16/15 + 9/8: 10/9 44.45 55-02 1:2 CHROMATIC È CHROMATIC 7 + 133 + 300) INTENSE CHROMATIC 227.94 63.43 25/24: 16/15 :6/5 ENHARMONIC (50 + 50 + 400) 12/11 - 11/10. 10/9 101.36 84.89 INTENSE CHROMATIC SOFT DIATONIC INTENSE DIATONIC EQUAL DIATONIG 111.80 70,71 158.11 111.80 206.16 158.11 260.8 164.00 33-33 60.09 105.41 166.67 47.14 37.27 (zoo + 100 + 300) 74.54 50.0 100.0 50.0 100.0 SOFT DIATONIC (100+ 150 + 250) (o) 200 INTENSE DIATONIC (roa + 200 + 200) 105.41 149.07 94.28 pu ; 74:54 CHAPTER 5

smallest interval or of the first versus the second interval. The distances are calculated according to the Pythagorean relation: the distance is defined as the square root of the sum of the squares of the dif- 5-17. Euctidean distances between permutations of Archytas’s enharmonic genus. Thefunction tabulated is the distance calculated on the plot ofthefirst by the second interval ofthe tetrachord, The other distancefunction, computedfrom the graph ofthe ferences of the coordinates. The Euclidean distance is Y[(CH — CH)? + (parhypate2-—paryhypate:)] in the first case and V[(frst interval, -first interval)? greatest versus the least interval, is always zero + (second interval» _ second interval1)?] in the second. It is convenient to convert the ratios into cents for these calculations. The distances between berween permutations ofthe same genus. some representative tetrachords in just intonation are tabulated in 5-15 and some in equal temperament with similar melodic contours in 5-16. One may also use the second Euclidean distance function to distinguish between permutations of tetrachords as shown in 5-17 and 5-18. 5-18. Euclidean distances between permutations of tempered genera. 28/17: 5/4- 36/35 28/27 - 36/35 + 5/4 337.54 28/27 : 5/4: 36/35 36/35 « 5/4 - 28/27 36/35-28/27.5/4 5/4 28/27 - 36/35 5/4: 36/35 28/27 337-84 20.07 323.66 323.35 14.19 323.66 323.55 457.29 467.43 337.54 467.43 155.39 337.84 14.19 36/35 + 5/4« 28/27 36/35+ 28/27 - 5/4 5/4: 28/27 : 36/35 ENHARMONIC 50 + 50 + 400 50 + 400 + 50 350.0 50 + 400 + 50 INTENSE CHROMATIC] 100 + 100 + 300 494.97 100 + 300 + 100 200.0 100 + 300 + 100 200 + 100+ 200 100 + 200 + 200 141.42 100 + 100 + 200 100+ 150+ 250 100 + 250 + 150 150+ 100 + 250 300 + 100 + 100 200.0 282.84 INTENSE DIATONIC SOFT DIATONIG 400 + 50 + 50 350.0 200 +200 + 100 100.0 100.0 100+ 250+ 150 100.0 150+ 100+ 250 70.71 158.11 150 + 250 + 100 250+ 100+ 150 150+ 250+ 100 11181 250 + 100 + 150 158.11 250+ 150+ 100 150.0 50.0 212.13 180.28 150.0 100.0 180.28 111.80 141,42 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

Minkowskian distances between tetrachords The closely related Minkowski metric or city block distance function is shown in 5-19 and 5-20 for the same sets of tetrachords. The two functions shown here are defined as the sum of the absolute values of the differences between corresponding intervals. For the upper set of numbers, the function is (1 CI: — Cl + | parhypate2 —paryhypate; |) and for the lower set, (| first interval; ~ first interval; | + {second interval, — second interval; |). These computations 5-19. Minkowski or “city block” distances between have also been done in cents throughout for ease of comparison. The distances between permutations may also be compared by means genera injust intonation. of the second distance function (5-21 and 5-22). 28/27 15/14 6/5 28/27 + 36/35 « 5/4 84.86 70.67 28/29: 15/14 + 6/5 25/24-16/15-6/5 22/2r-12/11-7/6 16/15-9/8-10/g 12/11 - T1/t0- 10/9 92.57 70.67 151.21 119.44 245.36 203.91 305.78 203.91 7.71 66.35 160.50 220.91 15.42 48.77 133.24 133.24 58.64 152.79 213.20 48.77 133-24 133.24 94.16 109.77 84.47 84.47 25/24 16/15 : 6/5 22/21: 12/11 - 7/6 77.81 16/15: 9/8. 10/9 60.41 3-20. Minkowski or “city block” distances between tempered genera. I:2 CHROMATIC ENHARMONIC (50 + 50 + 400) 1:2 CHROMATIC (67 + 133 + 300) 116.67 100.0 INTENSE CHROMATIC SOFT DIATONIC INTENSE DIATONIC EQUAL DIATONIC 150.0 100.0 200.0 150.0 250.0 200.0 350.0 233.33 33.33 66.67 83.33 50.0 133.33 100.0 233.33 200.0 INTENSE CHROMATIC 50.0 100.0 200.0 (100 + 100 + 300) 50.0 100.0 133.33 SOFT DIATONIC 50.0 150. (100 + 150 + 250) 50,0 83.33 INTENSE DIATONIC 100.0 (100 + 200 + 200) CHAPTER 5

5-21. Minkowski or “city block” distances between permutations of Archytas’s enbarmonic genus. 28/27 - 5/4: 36/35 28/27+ 36/35 « 5/4 36/35 + 5/4: 28/27 337-54 28/27 - 15/14 + 6/5 36/35-28/27-5/4 5/4-28/27-36/35 3/4 - 36/35 28/27 351.73 28.38 337.54 323.35 14.19 337.54 646.71 660.90 323.35 660.90 675.09 337.54 351.73 14.19 25/24 - 16/15 : 6/5 22/21 12/11. 7/6 16/15 + 9/8 - 10/9 5-22. Minkowski or “city block” distances between permutations of tempered genera. ENHARMONIC 50 + 400 + 50 go + 50 + 400 350.0 100 + 250 + 150 400 + 50 + 50 350.0 700.0 INTENSE CHROMATIC IOO + 300 + 100 300 + 100 + 100 100 + 100 + 300 100 + 300 + 100 200.0 200.0 INTENSE DIATONIG 200 + 100 + 200 200 +200 + 100 100 + 200 + 200 200.0 100.0 400.0 200 + 100 + 200 100.0 SOFT DIATONIC 100 + 250+ 150 150 + 100 + 250 150 +250 + 100 250+100+150 250+I50+1I00 100+ 150+ 250 100.0 100.0 200.0 150.0 50.0 200.0 300.0 150.0 250.0 150.0 100.0 250.0 150.0 200.0 50.0 100 + 250+ 150 150+ 100 + 250 150 + 250 + 100 250+ 100 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

5-23. Tenney pitch and harmonic distance funcions on the intervals of tetrachords in just intonation. Tenney’s pitch and harmonic distance functions The composer James Tenney has developed two functions to compare intervals (Tenney 1984), and has used these functions in composition, SMALL 56/55 - 55/54 9/7 .0078 28/27» 36/35 - 5/4 .0122 3.100 32/31 - 31/30 + 5/4 „0138 2.997 .0096 46/45 » 24/13: 5/4 3-489 3.156 particularly in Changes: Sixty-four Studies for Six Harps. The first function is the pitch-distance function defined as the base-2 logarithm of a/b where a and bare the numerator and denominator respectively of the interval in an extended just intonation. This function is equivalent to Ellis’s cents which are 1200 times the base-2 logarithm. The second function is his harmonic distance, defined as the logarithm of 4 - b. This distance function is a special use of the Minkowski metric in a tonal space where the units along each of the axes are the logarithms of prime numbers. Thus the pitch distance of the interval 9/7 is log (9/7) and the harmonic distance is 2 - log 20/19: 19/18. 6/5 0223 2.580 28/29. 15/14 - 6/5 „0158 2.878 26/25: 25/24: 16/13 «0170 distances for each of the three intervals. This has been done for the set of representative tetrachords in 5-23. The upper set of numbers is the pitch (3) + log (7). These functions may be used to characterize tetrachords by computing 2.813 distances; the lower, the harmonic distances. Alternatively, one could also 39/38 - 19/18. 16/13 .0113 3.171 apply it to the notes of the tetrachord after fixing the tonic and calculating the notes from the successive intervals. 24/23: 23/22. 11/9 0185 2.742 By a slight extension of the definition, the pitch distance function may 34/33» 18/17: 11/9 .0130 3.050 also be applied to tempered intervals. The pitch distance is the tempered interval expressed as a logarithm. For intervals expressed in cents, the 16/15: 15/14 - 7/6 .0280 2.380 formula is pitch distance = cents / 1200 log (2); other logarithmic measures could be used. This function will be most interesting for intervals which 22/11: 12/11. 7/6 .0202 2.664 tance function is not well defined for tempered intervals unless they closely 14/13: 13/12. 8/7 „0322 approximate just intervals. 2.260 21/20: 10/9 - 8/7 .0212 2.623 28/27 - 9/8. 8/7 1.580 2.879 16/15: 10/9 : 9/8 ‚0280 2.380 256/243 + 9/8. 9/8 0226 4-794 .0378 2.121 12/11 «11/10 + 10/9 are close approximations to those in just intonation. The harmonic dis- The Tenney functions also may be used to measure the distance between tetrachords. The pitch distance between the Cls of two genera is the logarithm of the quotient of their ratios; i.e., the pitch distance between 5/4, the CI of the enharmonic, and 6/5, the CI of the intense chromatic, is the logarithm of 25/24. The harmonic distance is the logarithm of 3/2, the product of 5/4 and 6/5. The pitch distance and harmonic distance functions on the CIs distinguish genera quite well, though obviously not permutations of the genera. The Tenney distance functions between representative set of tetrachords in just intonation are shown in 5-24. One could also apply the CHAPTER 5

Tenney distance functions on the pyknotic intervals to distinguish subgenera with the same CI, The distances between tetrachords in equal temperament may also be measured by the Tenney functions. The pitch distance of the Cls is simply the difference in cents or tempered degrees, The harmonic distance is the sum of the CIs. Data on representative tempered tetrachords are shown in 5-25. 5-24. Tenney pitch and harmonic distances between genera in just intonation. 28/27 - 36/35 - 5/4 28/27 15/14 6/5 26/24:16/15- 6/5 22/21 12/11 7/6 16/15-9/8-10/9 .0177 „1761 ‚0177 „1761 10270 „1638 0458 1481 0512 1427 0.0 ‚1584 .0122 „1461 .0280 1303 .0334 .1249 .0122 «1461 .0280 „1303 «0334 „1249 .0158 .1181 0212 „II2I 28/27 - 15/14 + 6/5 25/24- 16/15 + 6/5 22/21 - 12/11 «7/6 16/15 - 9/8. 10/9 12/11- 11/10 - 10/9 .0054 .0969 5-25. Tenney pitch and harmonic distances between tempered genera. I:2 CHROMATIC ENHARMONIC 50 + 50 + 400 1:2 CHROMATIC 67 + £33 + 300 100.0 700.0 INTENSE CHROMATIC SOFT DIATONIC INTENSE DIATONIC EQUAL DIATONIC 100.0 700.0 150.0 650.0 200.0 600.0 233.33 566.67 0.0 600.0 50.0 550,0 100.0 500.0 133.33 466.67 50.0 550.0 100.0 500.0 133.33 466.67 50.0 450.0 83.33 416.67 INTENSE CHROMATIC 100 + 100 + 300 SOFT DIATONIC 100 + 150 + 250 INTENSE DIATONIC 100+ 200+ 200 33-33 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

factor of 2 - E(4cf), where hef is the highest common factor, must be subtracted from the denominator of the formula. 5-27. Barlow's specific barmonicityfunction on tetrachords and tetrachordal scales. The specific barmonicityfunction is the square ofthe number of tones in the scale divided by sum ofthe reciprocals of the harmonicities ofthe combinatorial intervals (Barlow 1987) without regard to sign. For the tetrachord, the number oftones is 4, n° = 16, and there are six combinatorial intervals (see 5-28). The specific harmonicity ofthe Dorian mode is defined as above save that n = 8 (including the octave), n° = 64, and there are 28 intervals (n + (n-1)/2). Barlow’s harmonicity function is applied to set of tetrachords in just intonation in 5-26. The harmonicities of the three intervals are computed separately. The harmonicity of 4/3 is the constant —0.2143. The harmonicities of the pykna are also included to complete the characterization of the tetrachords. In the case of the general tetrachord 4 - 5 + c, where c= 4/346, there are four ratios, 1/1, 4,4 - b, and 4/3. Then: (n — 1)/2 = 6 combinatorial intervals are a, ab, 4/3, b, 4/3a, and 4/3ab. For example, Archytas’s enharmonic, 28/27 36/35 - 5/4, yields the tones 1/1, 28/27, 16/15, and 4/3. The combinatorial intervals are 28/27, 16/15, 4/3, 36/35, 9/7, and 5/4 the six non-redundant differences between the four tones of the tetrachord. The definition of RATIOS TETRACHORD DORIAN these intervals for equally tempered tetrachords is shown as the Polansky set in 5-48. In just intonation, the sums and differences become products I. 56/55: 55/54 - 9/7 „1063 0973 and quotients and the zero and soo cents are replaced by 1/1 and 4/3 2. 3. 28/27: 36/35 5/4 32/31: 31/30 - 5/4 „1859 0724 4. 46/45-24/23: 5/4 ‚0885 0815 5. 6. 9. 8 9. 10. It, 12. 13. 20/19 19/18 - 6/5 28/27-15/14: 6/5 26/25 25/24: 16/13 39/38. 19/18. 16/13 24/23. 23/22. 11/9 34/33: 18/17 - 11/9 16/15: 15/14: 7/6 22/21-12/11 7/6 14/13: 13/12 : 8/7 .1042 .1911 „1062 ‚0719 0767 .0848 „2170 „1375 .1247 .0677 respectively. For scales and other sets of ratios, Barlow defined a third function, termed specific harmonicity. The specific harmonicity of a set of ratios is the square of the number of tones divided by the sum of the absolute values of the reciprocals of the harmonicities of the combinatorial intervals (Barlow 1987). For the tetrachord, 7 = 4 and n? = 16. The specific harmonicities are .0698 presented in 5-27-29 for various sets of tetrachords. .0807 „1879 „1274 „1143 14. 21/20: 10/9 - 8/7 .1739 „1627 15. 16. 17. 28/27-9/8- 8/7 16/15 10/9 + 9/8 256/243 - 9/8 - 9/8 „2101 2658 .2212 Similarly, the specific harmonicities of scales generated from tetrachords may be computed. In the case of heptatonic scales, there are eight tones including the octave (2/1) and 28 combinatorial relations, which are defined analogously to the six of the tetrachord. The specific harmonicities of the same set of tetrachords as in 5-26 are given in 5-27. The specific har- 2363 18. 12/11. II/10- 10/9 „1609 19. 20. 11/10: 11/10:400/363 16/15-25/24: 6/5 .0829 2374 „1633 ‚0660 „0946 „1721 0998 „1885 „2025 1437 0797 .2133 monicities of both the tetrachords and a representative heptatonic scale are included in this table. The Dorian mode was selected for simplicity, but other scales could have been used as well (see chapter 6 for a detailed discussion of scale construction from tetrachords). It is the scale composed of an ascending tetrachord, a 9/8 tone, and an identical tetrachord which completes the octave. Abstractly, the tones are 1/1 a ab 4/3 3/2 34/2 3ab/2 2/1, where a. b. 4/3ab is the generalized tetrachord in just intonation. The set of combinatorial intervals is a, ab, 4/3, 3/2, 34/2, 3ab/2, 2/1, b, 4/34, 3/24, 3/2, 30/2, 2/a, 4/3ab, 3/24b, 3/26, CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

3/2, 2/ab, 9/8, 04/8, gab/8, 3/2, a, ab, 4/3, b, 4/34, 4/3ab. The repeated 5-28. Barlow's specificharmonicityfunction on the permutations of Ptolemy's intense diatonic genus. An Pw ym RATIOS TETRACHORD 16/15 + 9/8 - 10/9 16/15 : 10/9 : 9/8 9/8. 10/9 - 16/15 9/8 : 16/15 : 10/9 10/9+ 16/15 : 9/8 10/9 + 9/8 - 16/15 DORIAN 2794 2567 „2658 „2658 „2586 „2586 2794 „2363 2535 „2407 2398 2486 intervals are a consequence of the modular structure of tetrachordal scales. As can be seen from 5-27, the specific harmonicity function distinguishes different tetrachords and their derived scales quite well. 5-28 shows the results of an attempt to use this function to distinguish permutations of tetrachords from each other. Although the specific harmonicity function does not differentiate between intervallic retrogrades (4 - b- cversusc-b. 4) of single tetrachords, it is quite effective when applied to the corresponding heptatonic scales. Finally, since the specific harmonicity function is basically a theoretical measure of consonance, it would be interesting to use it to determine the most consonant tunings or shades (chroai) of the various genera. Accordingly, a number of tetrachords whose intervals had relatively “digestible” prime factors were examined. The results are tabulated in 5-29. It is clear that while the diatonic genera are generally more consonant than chromatic and they in turn are more harmonious than the enharmonic, there is considerable overlap between genera and permutations. In particular, the most consonant chromatic genera are more consonant than many of the diatonic tunings. 5-29. The most consonant genera according to Barlow’s specific barmonicityfunction. RATIOS TETRACHORD DORIAN IA. IB. 2A. 2B. 3A. 38. ENHARMONIG 256/243 - 81/80 5/4 .1878 5/4-81/80.256/243 .1878 28/27 36/35 5/44 .1859 5/4 - 36/35 - 28/27 „1859 25/24 128/125 - 5/4 .1806 5/4 128/125 «25/24 .1806 IA. 16/15 - 25/24 : 6/5 2374 .2133 6/5 + 25/24 + 16/15 16/15: 75/64 - 16/15 34. 10/9 - 81/80 - 32/27 38. 32/27-81/80. 10/9 25/24 27/25-32/27 2374 .2317 .2290 .2290 .1926 «2145 „2008 „2046 „2035 1745 ‚1669 ‚1715 „1633 „1667 .1550 .1556 CHROMATIC IB. Ga. 9/8 - 64/63 : 7/69 63, 7/6: 64/63 : 9/8 7A. 10/9 + 36/35 + 7/6 73. 7/6: 36/35 - 10/9 .2137 2137 .2032 .2032 .1937 .1903 „1783 1797 IA. 1B, 24. 28, 34, 38. 4A. 43. 5. DIATONIC 9/8 + 28/27 : 8/7 2176 8/7. 28/27 - 9/8 .2176 10/9 : 21/20: 8/7 .2104 8/7: 21/20: 10/0 .2104 16/15 - 9/8 - 10/9 .2794 10/9 + 9/8 - 16/15 2794 256/243 : 9/8 - 9/8 „2212 0/8-9/8- 256/243 „2212 10/9. 27/25 : 10/9 2251 „2027 .I914 .1888 „1856 .2567 2486 „2025 „2105 CHAPTER 5

Euler’s gradus suavitatis function A function somewhat similar to Wilson’s, Tenney’s, and Barlow’s functions is Euler's gradus suavitatis (GS) or degree of harmoniousness, consonance, or pleasantness (Euler 1739 [1960]; Helmholtz [1877] 1954). Like the other functions, the GS is defined on the prime factors of ratios, scales, or chords. Unlike Barlow’s functions, the GS is very easy to compute. The GS of a prime number or of the ratio of a prime number relative to 1 is the prime number itself, i.e, the GS of 3/1 is 3. The GS of a composite number is the sum of the GSs of the prime factors minus one less than the number of factors. The GS of a ratio is found by first converting it to a section of the harmonic series and then computing the least common multiple of the terms, The GS of the least common multiple is the GS of the ratio. Sets of ratios such as chords and scales may be converted to sections of the harmonic series by multiplying each element by the lowest common denominator. For example, the harmonic series form of the major triad 5-30. Euler's gradus suavitatisfunction on tetrachords injust intonation, (1) isa byperenbarmonic genus, (2)-(4) are enbarmonic, (5)-(12) and (20) are chromatic, and (13)-(19) are diatonic. The tetrachords are in their standard form with the small intervals at the base and the largest interval at the top. See 5-32 and 5-33 for other permutations ofthe tetrachord. I. 2. 3 4. 5. RATIOS 56/55. 55/54 : 9/7 28/27. 36/35 « 5/4 32/31. 31/30 - 5/4 46/45 24/23 + 5/4 20/19. 19/18 - 6/5 INTERVAL A 24 I5 36 32 25 INTERVAL B 22 17 38 28 24 CI II 7 7 7 8 PYKNON 15 (28/27) 11 (16/15) 11 (16/15) ı1 (16/15) 10 (10/9) 6. 7. 8. 9. 10. II. 12. 13. 14. 15. 16. 17. 28/27- 15/14 + 6/5 15 14 8 10 (10/9) 26/25-25/24: 16/13 39/38-19/18. 16/13 24/23: 23/22: 11/9 34/33 - 18/17 - 11/9 16/15: 15/14 7/6 22/21 + 12/11 7/6 14/13: 13/12 + 8/7 21/20 10/9 - 8/7 28/27-9/8- 8/7 16/15: 10/9 : 9/8 256/243 -9/8 - 9/8 22 34 28 30 II 20 20 15 15 II 19 14 24 34 22 14 15 17 10 8 Io 8 17 17 15 15 10 10 Io Io 10 8 8 17 (13/12) 17 (13/12) 15 (12/11) 15 (12/1) 10 (8/7) 10 (8/7) 10 (7/6) 10 (7/6) 10 (7/6) 12 (32/27) 12 (32/27) 18. 12/11 - 11/10 - 10/9 15 16 10 8 (6/5) 19. 11/10-11/10:400/363 16 16 35 31 (121/100) 20. 16/15 + 25/24 - 6/5 11 14 8 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

5-31. Euler's gradus suavitatisfunction on tetrachords and tetrachordal scales. (1) is a byperenharmonic genus, (2)-(4) are enbarmonic, (5)(12) and (20) are chromatic, and (13)-(19) are diatonic. The harmonic series representation of the Dorian mode of 16/15 - 9/8 - 10/9 is 30:32:3 6:40: 45:48:54:60. Its least common multiple is 4320 and is GS is 16. 1/1 5/4 3/2 is 4:5:6. The least common multiple of this series is 60 and the GS of the major scale thus is 9. The GSs of the component intervals of the usual set of tetrachords are shown in 5-30. The GS of 1/1 is 1 and that of 4/3 is 5. In 5-31, the GSs of both the tetrachords and the Dorian mode generated from each tetrachord are tabulated. The GSs of the Dorian mode are 3 more than the GSs of the corresponding tetrachords, reflecting the structure of the mode which has the identical series of intervals repeated at the perfect fifth. The GS seems not to be particularly useful for distinguishing perRATIOS TETRACHORD 1. 56/55-55/54-9/7 30 2. 3 28/27-36/35: 5/4 32/31. 31/30. 5/4 21 42 4. 46/45-24/23: 5/4 35 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. zo. 20/19+ 19/18. 6/5 28/27-15/14: 6/5 26/25 +25/24- 16/13 39/38-19/18. 16/13 24/23 : 23/22» 11/9 34/33: 18/17: 11/9 16/15: 15/14. 7/6 22/21-12/11: 7/6 14/13:13/12-8/7 21/20: 10/9 + 8/7 28/27: 9/8 -8/7 16/15 + 10/9+9/8 256/243 : 9/8 - 9/8 12/11-11/10: 10/9 11/10. 11/10:400/363 16/15 25/24: 6/5 29 19 27 39 40 33 17 22 24 19 16 16 19 21 35 17 DORIAN 33 24 45 38 32 22 30 42 43 36 20 25 27 23 19 19 22 24 38 20 mutations of tetrachords, as evidenced by 5-32. It is noteworthy that the most harmonious arrangements of Ptolemy’s intense diatonic are those which generate the major and natural minor modes (see the section on tritriadic scales in chapter 7). As with Barlow’s functions, the GS ranks the enharmonic the least harmonious of the major genera, though the most consonant tunings and arrangement overlap with those of the chromatic (5-33). Similarly, the most harmonious chromatic tunings approach those of the diatonic. Interestingly, however, the most harmonious enharmonic tuning is 28/27 « 5/4 - 36/35 and its retrograde which have the largest interval medially. The same is true for the chromatic 16/15 - 6/5 - 25/24. Of the diatonic forms, the two arrangements of Ptolemy’s intense diatonic with the 9/8 medial are the most consonant. Although the GS is an interesting and potentially useful function, it does have one weakness. Because the ratios defining small deviations from ideally consonant intervals contain either large primes or large composites, the GS of slightly mistuned consonances can become arbitrarily large. Thus the GS would predict slightly mistuned consonances to be extremely dissonant, a prediction not consistent with observation. 5-32. Euler's gradus suavitarisfunction on the permutations of Ptolemy's intense diatonic genus. (1) is the prime form. (2) is the order given by Didymos. I. 2. 3 4. 5. 6. RATIOS TETRACHORD 16/15 +9/8 + 10/9 13 16/15-10/9- 9/8 16 9/8-10/9- 16/15 16 9/8-16/15- 10/9 16 10/9. 16/15 : 9/8 16 10/9. 9/8. 16/15 13 66 CHAPTER 5 DORIAN 16 19 19

RATIOS TETRACHORD DORIAN ENHARMONIC 5 5243 on do” 5/4 23 26 This failure, however, is a feature shared by the other simple theories of consonance based upon the prime factorization of intervals. Helmholtz’s beat theory (Helmholtz [1877] 1954) and the semi-empirical “critical band” n 7 Bi, ° a és i tT 24 theories of Plomp and Levelt (1965) and Kameoka and Kuriyagawa (19692, se. M 24 22 1969b) avoid predicting infinite dissonance for mistuned consonances, but ja . 36/35:bi 28/27: s/4 25/24: 128/125 « 5/4 2 25 are more complex and difficult to use. The prime factor theories are ade- 14. CHROMATIC 16/15 + 25/24. 6/5 17 20 IB. 25/24: 16/15 : 6/5 18 21 ie. 16/15 6/5 > 25/24 16 19 Statistical measures on tetrachordal space «Le . > ‘ . . quate for theoretical work and for choosing between ideally tuned musical structures. so. 2. 16/15- 75/64 - 16/15 17 20 The concepts of the degree of intervallic inequality and of the perceptual 34. 38. 10/9: 81/80. 32/27 32/27-81/80: 10/9 18 18 21 21 differences between tetrachords may be clarified by computing some of the 4a. 25/24 27/25 + 32/27 20 23 48. 32/27: 27/25: 25/24 20 33 ga. 16/15-15/14- 7/6 17 20 sn. 6a. 16/15-7/6:15/14 9/8 - 64/63 : 7/6 19 19 22 22 standard statistical measures on a set of representative tetrachords. The arithmetic mean of the three intervals is 500/3 or 166.667 cents in equal temperament or 3V(4/3) in just intonation. The mean deviation, standard deviation, and variance are calculated according to the usual formulae for entire populations with x = 3. These data are shown in 5-34 for some rep- 68. 64/63 - 9/8 - 7/6 7a. 10/9: 7/6 lo 36/35 116. «36155 17 18 i 20 resentative tetrachords in just intonation and in 5-35 for a corresponding set Be 107970 ? 3983 21ni in qequalmperament. temperaent. While e not notdistinguishi t ons, thesethese fun funcdistinguishing permutations, tions differentiate between genera quite well, although the degree to which 7e. 36/35: 10/0- 7/6 20 23 IA. OrATONIG 9/8. 28/27 + 8/7 18 21 19 ratic mean) may be calculated in a similar fashion. Like the other statistical in. 8/7-9/8-28/27 16 . | . . the mathematical differences correlate with the perceptual is not known. The CE geometric mean, harmonic mean, and root mean square (or quadmeans . na | q q . 24. 10/9: 21/20: 8/7 18 21 measures above, these are non-linear functions of the relative sizes of the 28. 21/20:10/9:8/7 19 22 intervals and they have considerable ability to discriminate between the 34 38. 16/15 9/8. 10/9 10/9: 9/8 - 16/15 13 13 16 16 various genera. The relevant data are shown in 5-36 and 5-37. Several properties of these functions are apparent: for a given degree of 9%? intervallic asymmetry, the root mean square will show the greatest value, dA. 5. 256/243 9/8 9/8 10/9: 27/25: 10/9 17 20 5-33. The most consonant genera according to Euler’s gradus suavitatisfunction. These ratios are the most consonant permutations ofthe most consonant tunings ofeach ofthe genera. In cases where the most consonant permutation according to Barlow'sfunctions is different from the one(s) according to Euler’s, both are given. The gradus suavitatis ofa set of ratios is the GS oftheir least common multiple after the set has been transformed into a harmonic series. CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

5-34. Mean deviations, standard deviations, and variances ofthe intervals oftetracbords injust intonation. The arithmetic mean bas the constant value 166.67 cents (500/3) for all genera. In just intonation its value isthe cube root of 4/3. The standard deviation and variance are computed with n=3. 5-35. Mean deviations, standard deviations, and variances ofthe intervals oftempered tetrachords. 28/27 + 36/35 - 5/4 28/27: 15/14 6/5 25/14: 16/15 - 6/5 22/21 + 12/11. 7/6 16/15 : 9/8 - 10/9 12/11: 11/10+ 10/9 ENHARMONIC (50 + 50 +400) 1:2 CHROMATIC 5-36. Geometric mean, harmonic mean, and root root of'a-b-(500 —a —b); the harmonic mean is 3/2 (1/i), where 1/i = 1/2, 1/b, and 1/(500 -a-b); the root mean square is N(E(?)/3), where i? = a2, b?, (soo-a-b)2. 28/27: 36/35 + 5/4 28/27. 15/14: 6/5 25/24: 16/15 - 6/5 22/21 - 12/11: 7/6 16/15 + 9/8: 10/9 12/11. 11/10+ 10/9 ENHARMONIC 5-37. Geometric mean, barmonic mean, and root mean square of tempered tetrachords. STANDARD DEV. VARIANCE 146.87 99.75 99.75 67.24 36.19 10.93 155.88 108.29 107.12 76.84 39.38 12.99 24299.31 11725.73 11474.97 5904.95 1550.44 168.70 MEAN DEV. STANDARD DEV. VARIANCE 155.56 164.99 27222.22 88.89 98.13 9629.62 88.89 94.28 8888.89 55.56 62.36 3888.89 44.44 47-14 2222.22 0.0 0.0 0.0 (67 + 133 + 300) INTENSE CHROMATIC (100 + 100 + 300) SOFT DIATONIC (zoo + 150 + 250) INTENSE DIATONIG (100 + 200 + 200) EQUAL DIATONIC mean square ofthe intervals oftetrachords injust intonation. For n = 3, the geometric mean is the cube MEAN DEV. GEOMETRIC HARMONIC RMS 105.86 133.40 135.58 147.90 160.77 165.51 76.97 109.40 114.21 131.57 155.15 165.01 227.73 198.21 197.58 182.94 170.62 166.52 GEOMETRIC HARMONIC RMS 100.0 70.59 234.52 138.79 116.38 193.41 144.23 128.57 191.41 155.36 145.16 177.95 158.74 150.0 173.21 166.67 166.67 166.67 (50 + 50 + 400) 1:2 CHROMATIC (67 + 133 + 300) INTENSE CHROMATIC (100 + 100 + 300) SOFT DIATONIC (100 + 150 + 250) INTENSE DIATONIG (100 + 200 + 200) EQUAL DIATONIG 68 CHAPTER 5

the geometric the next, and the harmonic the least, except for the arithmetic mean, which is insensitive to this parameter. The set of all possible tetrachords instead of just representative examples or selected pairs may be studied by computing these standard statistical measures over the whole of tetrachordal space. This space may be defined by magnitudes of the first and second intervals (parhypate to hypate and lichanos to parhypate) as the third interval (mese to lichanos) is completely determined by the values of the first two. This idea may be made clearer by plotting a simple linear function such as the third tetrachordal interval itself versus the first and second intervals. The third interval may be defined as soo — x — y, where x is the lowest interval and y the second lowest. The domain of this function is defined by the inequalities o $ x S 500 cents, o £ y S 500 cents, and x + y S 500 cents. 5-38 depicts the “third interval function” from two angles. Its values range from o to 500 cents. The arithmetic, geometric, harmonic, and root mean square functions 5-38. The third intervalfunction, seen frontally and obliquely. The three intervals are parbypate to hypate, lichanos to parhypate, and mese to lichanos. They always sum 500 cents (3/2 injust intonation). are shown in 5-39 through 5-41. The arithmetic mean is a plane of constant height at 166.667 cents for all values of the three intervals. The geometric and harmonic means have dome and arch shapes respectively, while the root mean square somewhat resembles the roof of a pagoda. The shapes of these latter means may be clearer in the contour plots in the lower portions of the figures. One may conclude that the arithmetic mean obscures the apparent distance between genera, the geometric mean reveals it, the harmonic mean maximizes it, and the root mean square exaggerates it. This conclusion is illustrated in 5-43 where a cross-section through the plot is made where the second interval has the value 166.667 cents and the first interval varies from THIRD INTERVAL THIRD INTERVAL SECOND INTERVAL FIRST INTERVAL SECOND INTERVAL FIRST INTERVAL CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

5-39. Arithmetic mean ofthe three tetrachordal intervals. The arithmetic mean bas the constant value of 166.67 cents. The domain ofthisfunction is the xand y axes (0 <x< 500), (0 <y < 500), andthe © to 333.333 cents, The means are all equal when all three intervals of the tetrachord are 166.667 cents. The analogous representation is applied to the mean deviation, standard deviation, and variance, which are shown in 5-44-46. The variance has liney = 500 —x, where xand y are the first and been divided by roo so that it may be plotted on the same scale as the other second intervals ofthe tetrachord. The third interval may also approach zero, statistical functions. These functions have a minimum value of zero when all three intervals of the tetrachord are 166.667 cents each. This is seen most clearly in the cross-section plot of 5-47. ARITHMETIC MEAN Based on its properties with respect to the four means and three statistical measures, the equally tempered division of the fourth appears to be a most interesting genus. Itis the point where the three means are equal and where the statistical functions have their minima. FIRST INTERVAL SECOND INTERVAL 5-40. Geometric mean ofthe three tetrachordal intervals, 5-41. Harmonic mean of the three tetrachordal intervals. GEOMETRIC MEAN HARMONIC MEAN 5-42. Root mean square of the three terrachordal intervals, ROOT MEAN SQUARE RER N GENOA ij if dé ge in fl f (it FIRST INTERVAL SECOND INTERVAL SECOND INTERVAL 500 FIRST INTERVAL SECOND INTERVAL 500 500 î È SECOND INTERVAL È A 2 8 8 a FIRST INTERVAL FIRST INTERVAL u © 500 FIRST INTERVAL 70 CHAPTER 5 FIRST INTERVAL

three tetrrachordal intervals when the second 5-47. Cross-section ofthe mean deviation, standard deviation, and variance ofthe three tetrachordal interval equals 166.67 cents. intervals when the second interval equals 166.67 5-43. Cross-sections ofthe various means ofthe cents, 300 | 200 VARIANCE/100 STAND. DEV. RMS ARITHMETIC | 4 MEAN DEV. GEOMETRIC HARMONIC 400 Ô 300 FIRST INTERVAL FIRST INTERVAL 5-44. Mean deviation ofthe three tetrachordal 5-45. Standard deviations ofthe three tetra- 5-46. Variance ofthe three tetrachordal intervals. chordal intervals, intervals. MEAN DEVIATION STANDARD DEVIATION te DIG tte A CH ra FIRST INTERVAL SECOND INTERVAL 500 FIRST INTERVAL SECOND INTERVAL FIRST INTERVAL SECOND INTERVAL FIRST INTERVAL SECOND INTERVAL SECOND INTERVAL di o 500 SECOND INTERVAL 500 N o FIRST INTERVAL 71 FIRST INTERVAL CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

Polansky’s morphological metrics A more sophisticated approach with potentially greater power to discriminate between musical structures has been taken by Larry Polansky (1987b). While designed to handle larger and more abstract sets of ele5-48. Interval sets ofthe abstract tetrachord, o a atb soo. Injust intonation the abstract tetrachord may be written 1/1 a ab 4/3 ora a atb 498 cents, and the intervals adjusted accordingly. ments than tetrachords, i.e., the type of scale and scale-like aggregates discussed in chapters 6 and 7, and even sets of timbral, temporal, or rhythmic information, Polansky’s morphological metrics may be applied to smaller formations as well. Morphological metrics are distance functions computed on the notes or intervals between the notes of an ordered musical structure. A morphological metric is termed linear or combinatorial according to the number SUCCESSIVE INTERVALS 4 ath b 500 500-4-bh POLANSKY SET a ath ath b 500 mum combinatorial length for a morphology of length L is the binomial coefficient (L?-L)/2, notated as Ly. . The simplest of Polansky’s metrics is the ordered linear absolute mag- DIFFERENCE SET a+b b soo-a-b b-a 500-a26 500 — 3h relationships between component parts. A strictly linear interval set as well as two of the possible combinatorial interval sets derived from an abstract, generalized tetrachord are shown in 5-48. For a strictly linear interval set of a morphology (or scale) of length L, there are L - 1 intervals. The maxigoo 500 — 4 500-a—b a of elements or intervals used in the computations: the more intervals or elements used in the computation, the more combinatorial the metric. In other words, combinatorial metrics tend to take into account more of the nitude (OLAM) metric which is the average of the absolute value of dif500 ferences between corresponding members of two tetrachords. In the case of two tetrachords spanning perfect fourths of 500 cents, this function reduces to the sum of the absolute values of the differences between the two parhypatai and the two lichanoi divided by four. Given two tetrachords a; +b1+ 500-41—-h1 and a2 +b2+ 500-42—b2, the equation is: L x | e1j_e2; | IL, ind where L = 4 and én;= (0, #1, 41 + bi, 500) cents and (0, 42, 42 + b2, 500) cents. When not divided by L, this metric is identical to the Minkowski or “city block” metric previously discussed. Note that the OLAM metric does not take intervals into account, so it looks at L rather than L ~ 1 values. A simpler formula, ( laa ai | and |a2+b, - a; —b; | }/ 2, would be defensible in this context as zero and 500 cents are constant for all tetrachords of this type. If the tetrachords are built above different tonics or their CHAPTER 5

fourths spanned different magnitudes, i.e., 500 and 498 or 583, etc., the first equation must be used. The next simplest applicable metric is the ordered linear intervallic magnitude (OLIM) metric which is the average of the absolute values of the difference between the three intervals which define the tetrachords. In the case of the two tetrachords above, the intervals are 41,51, 500—a@; ~4; and 47, bz, 500-42-b;, The equation for this metric function is: L 5-49. Ordered linear absolute magnitude (upper) | E( | C1;-1j-] | |e2,~e2;_,1)| /(L-1), L-1 =3, and ordered linear intervallic magnitude (lower) metrics on tetrachords injust intonation. i2 where i ranges from 2 through L, since intervals are being computed. In 5-49, these two simple metrics are applied to a group of representative tetrachords in just intonation. The melodically similar tempered cases are 5-50. Ordered linear absolute magnitude (upper) shown in 5-50, Permutations of genera are analyzed in 5-51 and 5-52. The and ordered linear intervallic magnitude (lower) OLAM metric distinguishes between these genera quite well; the OLIM metrics on tempered genera. less so, but patterns are suggested which data on a larger set of tetrachords 28/27-15/14-6/5 28/27: 36/35 - 5/4 25/24-16/15:6/5 z22/21-12/11-+7/6 16/15-9/8-10/g 17.67 19.60 34-25 63.17 72.90 47-11 47-11 79.63 135.94 135.94 1.93 5.14 16.59 32.51 45.50 88.83 55.23 88.83 14.66 32.51 43-57 88.83 53.30 88.83 28.92 5631 38.64 56.31 28/27: 15/14 + 6/5 25/24: 16/15 - 6/5 22/21 - 12/11 + 7/6 16/15 : 9/8 - 10/9 9.73 25-94 1:2 CHROMATIC ENHARMONIG (50 + 50 + 400) 1:2 CHROMATIC (67 + 133 + 300) 12/11 - x1/10 + 10/9 29.17 66.67 INTENSE CHROMATIC SOFT DIATONIC INTENSE DIATONIG EQUAL DIATONIC 37-50 66.67 50.0 100,0 62.50 133.33 87.50 155.56 8.33 22.22 20.83 33-33 33.33 66.67 58.33 88.89 8.333 33-33 25.0 66.67 50.0 88.89 12.50 33-33 37.50 55.56 INTENSE CHROMATIC (100+ 100+ 300) SOFT DIATONIC (100+ 150 + 250) INTENSE DIATONIC (100 + 200 + 200) 25.0 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

5-51. Ordered linear absolute magnitude (upper) and ordered linear intervallic magnitude (lower) metrics onArchytas’s enbarmonic genus. 28/27 -5/4+36/35 28/27 : 36/35 « 5/4 84.39 225.03 28/27 + 5/4 + 36/35 36/5: 5/4+ 28/27 36/35 28/27 5/4 5/4 + 28/27-36/35 5/4. 36/35 - 28/27 84.39 225.03 3.55 9.46 165.22 225.03 161.68 215,57 7.10 9.46 87.93 225.03 80.83 215.57 84.38 225.03 80.83 215.57 87.93 225.03 84.39 225.03 225.03 225.03 165.22 225.03 36/35 « 5/4: 28/27 36/35 28/27. 5/4 5/4 + 28/27 + 36/35 3-55 9.46 5-52. Ordered linear absolute magnitude (upper) and ordered linear intervallic magnitude (lower) metrics on permuted tempered tetrachords. ENHARMONIG 50 + 400 + 50 400 + 50 + 50 50+ 50 + 400 87.50 2333 175.0 233.3 87.50 233.3 300 + 100 + 100 50 + 400 + 50 INTENSE CHROMATIC | IOO + 300 + IOO 100 + 100 + 300 50.0 133.3 100 + 300 + 100 INTENSE DIATONIC | 200 +100+ 200 100 + 200 + 200 25.0 66.67 200 + 100 + 200 SOFT DIATONIC 100 + 150 + 250 100 + 250 + 150 150 + 100 + 250 100.0 133.3 50.0 133.3 200 + 200 + 100 50.0 66.67 25.0 66.67 100 + 250 + 150 25.0 66.67 150+ 100 + 250 150 + 250 + IOO 250 + 100 + 150 250 + 150 + 100 12.50 33-33 50.0 100.0 62.50 100.0 75.0 100.0 37-50 100.0 25.0 33-33 37-50 100.0 50.0 100.0 37-50 100.0 50.0 66.67 62.50 100.0 37.50 100.0 25.0 66.67 12.50 150 +250+ 100 250 74 CHAPTER 5

may reveal, In particular, the OLIM metric fails to distinguish between permutations of tempered tetrachords, In theory, morphological metrics on combinatorial interval sets have greater discriminatory power than metrics on linear sets. Two sets of combinatorial intervals were derived from the simple successive intervals of 5-48. The first set, the Polansky set, is that described by Polansky (1987b). The second set, the difference set, was constructed from iterated differences of differences (Polansky, personal correspondence). The ordered combinatorial intervallic magnitude (OCIM) metric is the average of the absolute value of the differences between corresponding elements of the musical structure. Its definition is: Lel Lj EE lAlersen;aj) Alea 02:44) |/ Im, ful im where L,, = the number of intervals in the set (the binomial coefficient, described above). To apply it to other combinatorial interval sets, it must be appropriately modified to something like: L E |Zo) \/ Ln im? where In; are the elements of a set like the difference set of 5-48. As can be seen in 5-53 and 5-54, the OCIM metric calculated on the two sets of intervals from these tetrachords discriminates between genera very well. Both sets of intervals are roughly equivalent with this metric. Permutations are studied in 5-55 and 5-56, On neither interval set does the OCIM metric distinguish permutations completely. 5-53. Ordered combinatorial intervallic magnitude metric on the Polansky (upper) and difference (lower) interval setsfrom tetrachords injust intonation. 28/27. 15/14 > 6/5 28/27. 36/35 : 5/4 28/27 - 15/14 :6/5 35.34 94.23 25/24 16/15 6/5 22/21 12/11 9/6 16/15-9/8-10/g ra2/rr- 11/10: 10/9 36.62 86.52 62.65 141.68 110.08 223.11 116.57 184.20 3.86 27.31 47.45 74.75 128.88 81.23 104.01 10,28 26.03 25/24 - 16/15 «6/5 55.16 22/21- 12/11: 7/6 16/15: 9/8. 10/9 15.36 79-94 136.59 106.58 47-43 81.43 53-92 61.10 19.45 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

5-54. Ordered combinatorial intervallic magnitude metric on the Polansky (upper) and difference (lower) interval setsfrom tempered tetrachords. 1:2 CHROMATIC ENHARMONIC (50 + 50 + 400) 52.78 116.67 1:2 CHROMATIC (67 + 133 + 300) INTENSE CHROMATIC SOFT DIATONIC INTENSE DIATONIC EQUAL DIATONIC 58.33 83.33 83.33 150.0 108.33 216.67 136.11 194.44 16.67 44.44 30.56 38.89 55.56 100.0 83.33 100.0 25.0 66.67 50.0 136.33 77-78 III.II 25.0 66.67 52.78 61.11 INTENSE CHROMATIC (100+ 100+ 300) SOFT DIATONIC (100 + 150 + 250) INTENSE DIATONIC (100 + 200 + 200) 38.39 55.56 5-55. Ordered combinatorial intervallic magnitude metric on Polansky (upper) and difference (lower) interval sets on permutations of Archytas’s enbarmanic genus. 28/27 + 36/35 - 5/4 28/27 + 5/4 - 36/35 28/27 + 5/4+36/35 36/35:5/4-28/27 168.77 450.06 168.77 450.06 36/35-28/27-5/4 7.10 18.92 sig 28/27- 36/35 222.66 229.76 5/4: 36/35 + 28/27 215,57 215.57 9.46 171.14 161.68 168.77 9.46 435.87 431.14 450.06 161.68 171.14 168.77 431.14 435.87 450.06 225.03 225.03 222.66 229.76 36/35 : 5/4 28/27 36/35 + 28/27 5/4 5/4 28/27 - 36/35 7.10 CHAPTER 5

5-56. Ordered combinatorial intervallic magnitude metric on the Polansky (upper) and difference (lower) interval setsfrom permuted tempered tetrachords. ENHARMONIC 50 + 50 + 400 50 + 400 + 50 400 + 50 + 50 175.0 233.33 466.67 233.33 50 + 400 + 50 175.0 466.67 INTENSE CHROMATIC | 100 + 300 + 100 300 + 100 + 100 100+ 100 + 300 133.33 133.33 100.0 266.67 100 + 300 + 100 100.0 266.67 INTENSE DIATONIC 200 + I00 + 200 100 + 200 + 200 50.0 133.33 200 + 100 + 200 SOFT DIATONIC 100+ 150 +250 100 +250 + 150 200 + 200 + 100 66.67 66.67 50.0 133.33 100+ 250+ 150 150 + 100 + 250 150+ 250+ 100 250+ 100 + 150 250 +150 +100 50.0 25.0 83.33 91.67 100.0 133.33 66.67 150,0 116,67 100.0 75.0 200.0 33.33 33.33 75.0 200,0 83.33 150.0 75.0 200.0 66.67 66.67 91.67 116.67 75.0 200.0 50.0 133.33 150+ 100 + 250 150 + 250 + IOO 250 + 100 + 150 25.0 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRAGHORDS

Unordered counterparts of the ordered metrics are also defined. Although the unordered linear absolute or intervallic magnitude metrics are of little use in this context, the unordered combinatorial intervallic magnitude (UCIM) metric is rather interesting when computed on these two interval sets, For the Polansky interval set, the metric is: LI Lj Lel Lj |E XA (€1;,€1;4)/Lm-£ EA(ez, 024)! Lm |, Lm= 6. jal inl jal inl This function is the absolute value of the difference between the averages of the corresponding intervals. For the difference set, the formula becomes: L L ind im | E (1j)/Ly-E(b;)/Lm|,Lm=6, where the I,; are the elements of the set. 5-57 and 5-58 show the data for the same group of tetrachords as before. Genera are fairly well discriminated by this metric, especially when calculated on the Polansky interval set, but not as well with the difference set intervals, Neither are particularly successful for distinguishing permutations with this metric (5-59 and 5-60). 5-57. Unordered combinatorial intervallic magnitude metric on the Polansky (upper) and difference (lower) interval setsfrom tetrachords injust intonation. 28/27 15/14 6/5 28/27 + 36/35 « 5/4 28/27 + 15/14 + 6/5 11.78 47.11 25/2441 6/15 + 6/5 10.49 44:54 1.29 2.57 25/24: 16/15 - 6/5 22/21 -12/11:7/6 16,98 25.86 19.37 119.68 106.71 5.20 26.65 14.08 72.57 7.59 59.60 6.48 15.36 8.88 29.23 75.14 62.17 16/15 + 9/8 - 10/9 1 nen 12/11-11/10- 10/9 73.77 22/21 - 12/11 + 7/6 78 16/15:9/8- 10/9 CHAPTER 5 8.88 2.39 45-91

5-58. Unordered combinatorial intervallic magnitude metric on the Polansky (upper) and difference (lower) interval setsfrom tempered tetrachords. 1:2 CHROMATIC ENHARMONIG 50+50+400 13.889 61.11 INTENSE CHROMATIC 8.333 50.0 SOFT DIATONIC 16.67 83.33 INTENSE DIATONIC 25.0 116.67 EQUAL DIATONIC 19.44 116.67 1:2 CHROMATIC 5.556 2.778 II.II 5.556 67+133 + 300 11.11 22.22 55.56 55.56 8.333 33-33 16.67 66.67 II.II 66.67 INTENSE CHROMATIC 100 + 100 + 300 SOFT DIATONIC 8.333 100 + 150 + 250 33-33 INTENSE DIATONIC 100 + 200 + 200 2.778 33-33 5.556 0.0 5-59. Unordered combinatorial intervallic magnitude metric on Polansky (upper) and difference (lower) interval sets on permutations ofArchytas’s enharmonic genus. 28/27. 5/4-36/35 28/27 : 36/35 - 5/4 28/27 + 5/4+ 36/35 56.26 225.03 36/35 5/4: 28/27 56.26 220.30 0,0 4.73 36/35 28/27 5/4 36/35 + 28/27: 5/4 3/4: 36/35 - 28/27 2.36 2.36 0.0 4-73 53.89 117.24 53.89 107.78 56.26 220.30 36/35 « 5/4 + 28/27 5/4: 28/27-36/35 107.78 117.24 53.89 53.89 56.26 215.57 103.05 112.51 0,0 112.51 222,66 103.05 2.36 9.46 5/4+ 28/27 - 36/35 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

5-60. Unordered combinatorial intervallic magnitude metric on the Polansky (upper) and difference (lower) interval sets from permuted tempered tetrachords, ENHARMONIC 50+ 50 + 400 50 + 400 + 50 58.33 233-33 50 + 400+ 50 100 + 200 + 200 33-33 0.0 66,67 33-33 66.67 200 + 100 + 200 16.67 33-33 200 + 100+ 200 SOFT DIATONIC 100 + 150 +250 100 +250+ 150 300 + 100 + 100 133.33 100+ 300+ 100 INTENSE DIATONIC 0.0 116.67 58.33 116.67 INTENSE CHROMATIC | 100 + 300 + 100 100 + 106 + 300 400 + 50 + 50 100 + 250 + 150 16.67 66.67 200 + 200 + 100 0.0 33-33 16.67 66.67 150 + 100 + 250 150 + 250 + 100 250+ 100+ I50 250 + 150 + 100 8.333 16.67 16.67 83.33 8.333 16.67 0.0 50.0 25.0 83.33 0.0 16.67 25.0 50.0 16.67 16.67 25.0 100.0 0.0 33.33 8.333 66.67 25,0 66.67 16.67 33-33 150+ 100 + 250 150 +250 + IOO 250+ 100 + 150 8.333 CHAPTER 5

In addition to absolute and intervallic metrics, directional metrics are also defined. Directional metrics measure only the contours of musical structures, i.e., whether the differences between successive elements are positive, negative or zero. Although these metrics are perhaps the most interesting of all, they are generally inapplicable to tetrachords because tetrachords are sets of four monotonically increasing pitches whose dif5-61. Ordered (upper) and unordered (lower) ferences are always positive (or negative if the tetrachord is presented in “fore ocean ta date mes on Ifference sets from tetrachoras tn just intonation. J descending order). Directional metrics, however, are very applicable to : melodies constructed from the notes of tetrachords or from tetrachordally derived scales such as those of chapter 6. The intervals of the tetrachordal difference set, however, are not necessarily monotonic and therefore combinatorial directional metrics may be computed on these intervals. Two such metrics were calculated for the same set of tetrachords and permutations used above, the ordered 5-62. Ordered (upper) and unordered (lower) combinatorial interval direction metrics on difference setsfrom tempered genera. 28/27 + 36/35 : 5/4 28/27 15/14:6/5 25/24. 16/15: 6/$ 1667 3333 1667 3333 0.0 0.0 28/27. 15/14 + 6/5 25/24: 16/15 : 6/5 22/21+ 12/11 7/6 16/15-9/8. 10/9 12/11: 11/10 - 10/9 „1667 3333 0.0 0.0 „5000 3333 -3333 .6667 „1667 3333 0.0 0,0 0.0 0.0 .3333 ‚6667 o,o 0.0 „3333 0,0 667 0,0 22/21 12/11. 7/6 16/15 «9/8 - 10/9 „5000 3333 1:2 CHROMATIC ENHARMONIC (50 + 50 + 400) 1:2 CHROMATIC (67 + 133 + 300) INTENSE CHROMATIC SOFT DIATONIC INTENSE DIATONIC EQUAL DIATONIC „1667 0.0 3333 „1667 9.0 „5000 3333 3333 3333 ‚6667 1667 3333 0.0 0.0 3333 6667 „5000 1,00 1667 3333 ‚5000 3333 3333 ‚6667 3333 ‚6667 +5000 1.00 INTENSE CHROMATIG (100 + 100 + 300) SOFT DIATONIC (100 + 150 + 250) INTENSE DIATONIC „3333 (100 + 200 + 200) +3333 CLASSIFICATION, GHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

combinatorial intervallic directional (OCID) metric and its unordered counterpart, the unordered combinatorial intervallic directional (UCID) metric. The OCID metric is the average of the differences of the signs of corresponding intervals. The sign (sgn) of an interval is -1, 0, or +1 according to whether the interval is decreasing, constant or increasing. The difference (diff) is 1 when the signs are dissimilar, otherwise the difference is zero. The definition of the OCID metric on the difference set is: L E dif (sgn (11) 86m (Ja)! Im Lm=6. The UCID metric is the average of the absolute values of the numbers of intervals with each sign. The definition of UCID on the difference set is: L L |#ei-f#ey Dl Lips, Lm = 6, im2 where #e,” = the number of intervals in the matrix such thatv = sgn (J,,); i.e., v=[-1,0, 1]. The data from these computations are shown in 5-61 and 5-62. Similar results were obtained with tetrachordal permutations (5-63 and 5-64). 5-63. Ordered (upper) and unordered (lower) combinatorial interval direction metrics on difference setsfrom permutations ofArchytas’s enbarmonic genus. 28/27 36/35 - 5/4 28/27 « 5/4 + 36/35 28/27: 5/4 36/35 36/35 + 5/4 28/27 36/35-28/27-5/4 5/4.- 28/27-36/35 ‚5000 3333 „5000 1667 „1667 0.0 -3333 3333 3333 0.0 0.0 0.0 +3333 ‚6667 3333 0.0 5000 3333 3333 „6667 3333 0.0 ‚5000 3333 3333 6667 1667 3333 36/35 « 5/4 28/27 36/35 + 28/27 « 5/4 5/4 28/27: 36/35 S/a- 36/35 « 28/27 „1667 CHAPTER 5

5-64. Ordered (upper) and unordered (lower) combinatorial interval direction metrics on difference setsfrom permuted tempered tetrachords, ENHARMONIC SO + 50 + 400 50 + 400 + 50 400 + 50+ 50 „5000 3333 3333 5000 3333 ‚6667 50 +400 + 50 INTENSE CHROMATIC 100 + 300 + 100 300 + 100 + TOO .5000 .6667 3333 3333 .5000 ‘3333 INTENSE DIATONIC 200 + 100 + 200 200 + 200 + 100 TOO + 200 + 200 .5000 3333 3333 3333 100 + 100 + 300 100 + 300 + 100 „5000 200 + I00 + 200 „6667 SOFT DIATONIC 100 + 150 + 250 100 +250 + 150 150 +250+ I00 250+ 100+ 150 250 + 150+ 100 1667 3333 5000 3333 3333 0.0 0.0 .1667 5000 3333 3333 0.0 100 + 250 + 150 150 + 100+ 250 3333 1667 3333 6667 3333 6667 3333 5000 3333 0.0 0.0 ‚5000 3333 150+ 100 + 250 150 +250+100 ‚6667 -1667 ‘3333 250+ 100+ 150 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

Rothenberg propriety David Rothenberg has developed criteria derived from the application of concepts from artificial intelligence to the perception of pitch (Rothenberg 1969, 1975, 1978; Chalmers 1975, 1986b). In Rothenberg’s own words (personal communication); “These concepts relate the intervallic structure of scales to the perceptibility of various musical relations in music using these scales. Only the relative sizes of the intervals between scale tones, not the precise sizes of these intervals are pertinent.” These concepts are applicable to scales of any cardinality whether or not the intervals repeat at some interval of equivalence. In practice, most scales repeat at the octave, though cycles of tetrachords and pentachords are found in Greek Orthodox liturgical music (Xenakis 1971; Savas 1965). To apply Rothenberg’s concepts, the first step is to construct a difference matrix from the successive intervals of an n-tone scale. The columns of the matrix are the intervals measured from each note to every other one of the scale, The rows ty of the matrix are the sets of adjacent intervals measured from successive tones, These intervals are defined conventionally: the row of seconds (t,) comprises the differences between adjacent notes; the row of thirds (tz) consists of the differences between every other note; etc., up to the interval of equivalence (t,). Row ty contains the original scale. A number of functions may be calculated on this matrix. The most basic of these is propriety. A scale is strictly proper if for all rows every interval in row ty-.1is less than every interval in row t,. If the largest interval in any row tn-1 is at most equal to the smallest interval in row t,, the scale is termed proper. These equal intervals are considered ambiguous as their perception depends upon their context. A familiar example is the tritone (F-B in the C major mode in 12-tone equal temperament), which may be perceived as either a fourth or a fifth. Scales with overlapping interval classes, i.e., those with intervals in rows tn-1 larger than those in rows tn, are improper. These contradictory intervals tend to confound one’s perception of the scale as a musical entity, and improper scales tend to be perceived as collections of principal and ornamental tones. Improper scales may contain ambiguous intervals as well. 5-65 illustrates these concepts with certain tetrachordal heptatonic scales in the 12- and 24-tone equal temperaments. The first example is the intense diatonic of Aristoxenos. The scale is proper and the tritone is ambiguous. The second scale is Aristoxenos’s soft diatonic which is also CHAPTER 5

5-65. Rothenberg difference matrices. The row index is tn. Max Gy) is the largest entry in row tn. Min (tq) is the smallest enry in row ty. The intense diatonic tetrachord is 1 +2 +2 degrees or 6+12+12 parts. The soft diatonic derivesfrom 2 +3 + 5 or 6+9 +15 parts. The neutral diatonic is 3 +4 +3 degrees, a permutation of 9 +9 + 12 parts. The intense chromatic is 1 +1 +3 degrees. The enharmonic tetrachord is 1+1+8 degrees. Intervals in parentheses are ambiguous; those in square brackets are contradictory. proper, but replete with ambiguous intervals. A composer using this scale might prefer to fix the tonic with drone or restrict modulation so as to avoid exposing the ambiguous intervals. The next scale is patterned after certain common Islamic scales employing modally neutral intervals. It is strictly proper, a feature it shares with the more familiar five-note black key scale in 12-tone equal temperament. The final two examples, Aristoxenos’s intense chromatic and his enharmonic, are improper. The majority of the intervals of these scales are either ambiguous or contradictory. These scales are most likely to be heard and used as pentatonic sets with alternate tones or inflections. Because the major (o 400 700 cents, 4:5:6 in just intonation), minor (o 300 700 cents, 10:12:15), subminor (0 250 700 cents, 6:7:9), and supramajor (o 450 700 cents, 14:18:21) triads are strictly proper, they can serve to ty INTENSE DIATONIG IN 12-TONE ET: PROPER o 2 4 6 7 9 II 12/0 I 2 2 2 I 2 2 MAX (£3) = MIN (t4) =6 to ti INTENSE CHROMATIC IN 12-TONE ET; IMPROPER 0 1 2 5 7 8 9 12/0 it 3] (2) 1 1 [3] Max (fy) > MIN (#2) nn t3 4 ts 3 5 7 8 4 4 © 5 7 7 9 9 3 5 7 8 3 5 ta tz 4 ts Ll 4 [sl 3 [2] 4 4 MAX G2)> MIN (f3) 5 (6 (6) ld] 5 5 5 7 7 7 7 6) (6) BI 8 8 10 8 [7] 9 m te IO II IO IO IO II IO t6 9 IT II 9 IO II II t7 12 12 12 I2 I2 I2 12 ta 12 I2 I2 I2 I2 I2 I2 to SOFT DIATONIC IN 24-TONE ET: PROPER o 2 5 IO 14 16 19 24/0 to ENHARMONIC IN 24-TONE ET: IMPROPER o I 2 IO Iq 15 16 24/0 8 4 5 7 9 3 5 7 9 tr 2 3 (5) 4 2 3 (5) Max (ti) = MIN (2) ty 101 [B] 4 1 1 [8] MAX (ti) > MIN (2) t2 (65) 8 (9) 6 (5) 8 (5) Max (¢2) = MIN (£3) t2 tl Ur] 5 [2] 9 9 t3 10 (12)11 (9) 10 10 10 MAX (f3)= MIN (f4) ta 10 [13] [13] [6] 10 10 10 MAX (£3) > MIN (ta) te ts 14 16 ts (19) 22 t7 24 ta ts te t7 14 14 14 14 15 15 22 15 16 23 23 16 24 24 24 24 14 14 14 (12) 13 (15) MAX (fg) = MIN (ts) 17 (19) 16 (15) 18 (19) ETc. 21 (19) 20 22 21 24 24 24 24 24 24 9 Max (2 > MIN (#3) [rr] [rr] [18] [12]ı9 22 20 23 23 24 24 24 NEUTRAL DIATONIC IN 24-TONE ET: STRICTLY PROPER to O 3 7 IO 14 17 21 iz 3 4 3 4 3 4 3 7 7 6 27 7 7 7 t3 ta ty ts f7 II 14 18 21 24 IO 14 17 20 24 II IO 10 IO 14 13 I4 13 17 17 17 17 21 20 2I 20 24 24 24 24 IO I4 17 21 24 24/0 MAX (En-1) < MIN (En) CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

as sets of principal tones for improper scales. The various sets of principal tones would be used as the main carriers of melodies, while the auxiliary 5-66. Propriety limits oftetrachords. The differences are in cents and an underlying zero modulo 12 equal temperament is assumed, The results for just intonation are virtually identical except that thefourth of 498.045 cents and a whole tone of203.91 cents replace the soo- and 200-cent intervals in the computations. tones would be used as ornaments. This topic deserves more extended discussion than is appropriate here and Rothenberg’s original papers should be consulted (Rothenberg 1969, 1975, 1978). The fact that the minor and septimal minor triads are strictly proper may explain certain musically significant cadential formulae in the Dorian modes of the enharmonic and chromatic genera. These consist of a downward leap from the octave to the lowered submediant (trite), then ROWS ti DIFFERENCE MATRIX a b 500-4-b 12 a+b soo-a soo-h ty 500 500 500 CONSTRAINTS: 0 < 4 < 250; 0 < b < 250; 250<4 +h< soo, VERTICES: 0, 250; 250, 0} 250, 250. down to the subdominant (mese) before ending up on the dominant (paramese). This formula may be repeated a fifth lower, beginning with a leap from the subdominant (mese) to the lowered supertonic (parhypate) and then down to the subtonic (hyperhypate) before ending on hypate (chapters 6 and 7). Minor triads are outlined in the chromatic genus and septimal minor triads in the enharmonic. The latter chords contain the important interval of five dieses called eklysis by the Greek theorists, and in fact, the jump from parhypate to hyperhypate is seen in the Orestes fragment (Winnington-Ingram 1936). The upper submediants (lichanos and paranete) may be substituted in both genera; the major triad appearing in the 5-67. Propriety limitsfor isolated tetrachords and conjunct chains oftetrachords. chromatic genus is also strictly proper. As has been seen above, the propriety criterion separates those scales derived from chromatic and enharmonic tetrachords from those generated by diatonic genera. As will be seen later, the situation is somewhat more complex; under certain conditions, some diatonic tetrachords yield only improper scales, while some chromatic genera can combine with diatonic tetrachords to generate proper mixed heptatonic scales. Propriety may be computed for abstract classes of scales or subscalar 300 modules rather than for specific instances by replacing one or more of the intervals by variables. If the three subintervals of the tetrachord are written 300 as a, b, and 500 - 4 —b (a, b, and 45/3a in just intonation), one can calculate the Rothenberg difference matrix and determine the propriety limits for isolated tetrachords or conjunct chains where the interval of equivalence is the fourth. Such chains were present in the earlier stages of classical Greek music and are still extant in contemporary Greek Orthodox liturgical music (chapter 6 and Xenakis 1971). The computation is performed by solving the inequalities formed by setting each of the elements of rows # less than each of those in rows ty +1 86 CHAPTER 5

In practice, the work may be minimized because only the elements in the first (2 + 1) / 2 rows of an n-tone scale need be considered. One may also 5-68. Propriety limits ofpentachords. ignore relations that are tautological when all the intervals are positive. The result is a set of constraints on the sizes of intervals a and b, shown ROWS DIFFERENCE MATRIX ab 500-4-b 200 12 4+b 500-4 700-a-b 20048 in 5-66. Tetrachords and conjunct chains of tetrachords spanning perfect fourths, are strictly proper when intervals 4 and è satisfy these constraints. The tetrachords and chains are proper when their intervals equal the ex- #3 trema of the constraints. For values outside these limits, the tetrachords and 14 500 _700 700-4 7oo-b 700 700 200 +a + 700 GONSTRAINTS: 0 < 4 € 250; 0 <b < 250; 250 <a . +b < 500; 24 +b < 700; a + 2b < 700; b-a< 200; 300 < 24 +b. | . conjunct chains are improper. Because the three intervals a, è, and 500 — 4 — è add to a constant value, there are only two degrees of freedom. Therefore, the domain over which VERTICES! 250, 0; 50, 200; 33.3, 233.3; 100, tetrachords are proper may be displayed graphically in two dimensions. 300; 233.3, 233.3; 250, 200. The region in the 4 - è plane within which tetrachords are strictly proper is shown in 5-67. The vertices define an area in the 4 - b plane within which the 5-69. Propriety limitsfor isolated pentachords and constraints are satisfied. Points on the edges of the triangular region correspond to proper tetrachords. The two points on the axes are also proper conjunct chains of pentachords. as trichords, which are degenerate tetrachords with only three notes. Similarly, the propriety limits for pentachords consisting of a tetrachord and an annexed disjunctive tone (200 cents or 9/8) may be determined. The 300 difference matrix is shown in 5-68. As all circular permutations of a scale have the same value for propriety, it is immaterial whether the disjunctive tone is added at the top or bottom of the tetrachord. The region satisfying the propriety constraints for isolated pentachords and pentachordal chains is shown in 5-68. Similar calculations may be carried out for complete heptatonic scales consisting of two identical tetrachords and a disjunctive tone. This tone 300 5-70. Propriety limitsfor heptatonic scales with identical tetrachords. 5-71. Propriety limitsfor heptatonie scales with identical tetrachords. a a+b b 500-4 500-4-b 700-4-b 200 200+4 a ath b 500-4 soo-a-b 500-6 soo 700 700-4 700 700-5 700 200+4+b 700 500 500+4 500 500+5 500 1000-4-b 300 |. i : CONSTRAINTS: 100 <4 < 250; 100<b<250;250<4+b<400, VERTICES: 100,150; 100,250; 150,100; 150,250; 250,150; 250,100. CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

may be placed between the tetrachords or at either end to complete the octave (chapter 6). The results of the calculations are given in 5-70. The region of propriety is shown in 5-71. Complete tetrachordal space An alternative mode of graphic representation may be clearer. Physical chemists have long been accustomed to plotting phase diagrams for three component mixtures on equilateral triangle graphs. The three altitudes are interpreted as the fractions of each component in the whole mixture. There A : INN I DLS 4 AME are only two degrees of freedom as the sum of the composition fractions must equal unity. The data from 5-66, 5-68, and 5-70 have been replotted in 5-72-73. 5-72. Propriety limitsfor tetrachords and tetrachordal chains. These limits arefor chains ofconjunct tetrachords such as arefound in Greek Orthodox liturgical music (Xenakis 1971). 5-72 shows the range over which the intervals 4, b, and soo — 4 - b may vary and still result in proper tetrachords. Pentachords are shown in 5-73 and heptatonic scales in 5-74. The advantage of the triangular graph over the conventional rectangular type is most evident with the heptatonic scales of 5-74. All points in the interior of the semi-regular hexagonal region correspond to strictly proper scales, while the edges are sets of intervals that define scales that are merely cma-b 500 b (o) a (e) a € b 500 (o) 500 5-73. Propriety limitsforpentachords and pentachordal chains, 5-74. Proper heptatonic scales. CHAPTER 5

proper. The three triangular spaces lying between the long sides of the hexagon and the edge of the space contain diatonic genera which yield 500 improper heptatonic scales. In certain cases to be discussed later, some of these tetrachords may be combined with other genera to produce proper mixed scales, The six vertices of the central hexagon in 5-74 are the six permutations ofthe soft diatonic genus of Aristoxenos, 100 + 150 + 250 cents. The center of overall symmetry is the equal diatonic genus, 166.667 + 166.667 + 166.667 cents. The intersection of the altitudes of the triangle and the midpoints of the long sides of the hexagon are the three permutations of the 500 o 500 intense diatonic, 100 + 200 + 200 cents, while the intersections with the midpoints of the short sides define the arrangements of the neo- Aristoxenian genus, 125 + 125 + 250 cents. This genus lies on the border of the chromatic and diatonic genera, but sounds chromatic because of the 5-75. Non-diatonic genera. equal division of the pyknon. The non-diatonic or pyknotic genera are portrayed in 5-75. The empty border around the filled regions delimits the commatic (25 cents) and subcommatic intervals. The small triangular regions in dark color near the vertices are the hyperenharmonic genera whose smallest intervals fall between 25 and so cents in this classification (see the neo-Aristoxenian classification above for more refined limits on the boundaries between the hyperenharmonic, enharmonic, and chromatic genera). Next are the trapc=a-b 500 ezoidal enharmonic and chromatic zones which flank the unmarked central diatonic area. The enharmonic zone contains pyknotic intervals from 50 to 100 cents and the chromatic from roo to 125 cents. These data are summarized in 5-76. The diatonic tetrachords generating proper and strictly proper scales map into the central zone, The three triangular zones flanking the central region along the long sides of the hexagon are diatonic tetrachords which contain one of the small hyperenharmonic, enharmonic, or chromatic intervals. These diatonic genera 500 o 5-76. Complete tetrachordal space. 500 yield improper scales. As in 5-75, the chromatic tetrachords lie in the large trapezoidal regions, with the enharmonic and hyperenharmonic beyond. The outer belts of the chromatic zones depict genera with enharmonic and hyperenharmonic intervals. Similarly, the enharmonic regions are divided into realms of pure enharmonic and enharmonic mixed with hyperenharmonic intervals. CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

Propriety of mixed scales The computation of the propriety limits for heptatonic scales containing dissimilar tetrachords is a more complex problem. Since there are now four degrees of freedom, two for each of the tetrachords, the graphical methods used for the single tetrachord case are of limited use. It is possible, however, to consider the upper and lower tetrachords separately and to calculate absolute limits on the intervals of each. If 4, b, and 500 ~ 4 — bare assigned to the intervals of the lower tetrachord and c, d, and soo —c—d to the upper, one can compute the range of values for a and over which it is possible to find an upper tetrachord with which a proper scale can be generated. Similar computations may be done for ¢ and d, These results of these calculations are tabulated in 5-77 and are graphed in 5-78 and 5-79. These graphs use only those relations which are solely functions of 4 and b or c and d. Triangular plots of the same data are depicted in 5-80 and 5-81. The union of the the upper and lower tetrachord regions corresponds to the pentachordal limits of 5-68 and 5-73, and their intersection is the proper diatonic region of 5-74. The upper and lower tetrachord regions are also the intervallic retrogrades of each other as propriety is unaffected by retrogression or circular permutation of the intervals. The solution to the general case of finding the limits for mixed tetra5-77. Propriety limitsfor beptatonic scales with mixed tetrachords. (Only thefirstfour rows are shown.) a atb 500 700 b 500-4 700-4 700-a+c 500-4-h 700-4-b 700-4-b+c 700-4-b+c+d chordal scales must satisfy all the inequalities that relate a, 4, c, and d. It is difficult to display this four-dimensional solution space in two dimensions. One can, however, choose tetrachords from the lower or upper absolute 200 200 +¢ 200+¢+d 700 € c+d 500 500+4 d 500-6 500-c+4 500-c+4+b 500-6-d 500-c-d+4 soo-c-d+a+b 1000-c-d CONSTRAINTS ON @ AND b: 0 <4 <250;250 <a +b<500;24 +b<700;4 + 2h <700. VERTICES: 100, 150; 100, 300; 250, 200; 250, 0; 233.3, 233.3. CONSTRAINTS ON CAND dic < 250; 250 <¢ +d< 400; d~¢< 200; 300 <2¢ +d. VERTICES: 50, 200; 33.3, 233.3; 100, 300; 250,150; 250, 0. MUTUAL CONSTRAINTS ON 4, b, €, AND d:4<c+d;b<c+dc<a+bid<a+b;c<24;4+c<500;h+c<500j4+d<500;h-6<200;26-4< 300;4-c<100;c+d-8<300;a+b+c<7o0; 2¢+d-a<so00;¢+2d—a<500;4+b4+d< 700; 24 + 2h-¢ < 700; 4+b—-c~d < 100; 300 <a +c+d;c+d<2a+b;200 <24 +2b—-¢-d; 2¢+d~a—b < 300; 24-¢—d < $00; 200<20+b-Gc+b+d-a< 500; 500 <a+b+c+d!300< 20 +2d-a;24+b-27-d<200. 90 CHAPTER 5

propriety regions of 5-80 and 5-81 and find companion tetrachords which produce proper heptatonic scales when joined to them by a disjunctive tone. These computations are performed in the same way as in 5-70 and 5-77, except that the variables in one of the two tetrachords are replaced by the cents values of the intervals. The result of the calculations will be a T INTERVAL è 300 |- range of values for the companion tetrachord. INTERVAL 4 300 The three permutations of the intense diatonic genus in 12-tone equal temperament (100 + 200 + 200 cents, 200 + 100 + 200 cents, and 200 + 200 5-78. Absolute propriety limitsfor lower tetrachords. + 100 cents) as well as the neochromatic form of the syntonic chromatic (100 + 300 + 100 cents) were selected as lower tetrachords. The propriety limits for the upper companion tetrachords were then computed. These Points in the interiors of the regions yield strictly proper scales, while those on the peripheries produce scales that are merely proper. The neochromatic tetrachord has only a one-dimensional solution space; the uppermost point corresponds to a mode of the harmonic minor scale. LI INTERVAL è Ww le] (e) ï results are shown in 5-82. Similar calculations were performed for an additional 23 tetrachords and INTERVAL 4 300 the results are tabulated in 5-83. In agreement with previous results (5-74 and 5-78), no proper scales could be formed from lower tetrachords whose 5-79. Absolute propriety limitsfor upper tetrachords. first intervals were microtones. c=a-b 500 IN N Vo? / NANNY AZ AVAVAVI a 500 i N [N AN AVAN P____AZIZ NZVISA VW \VAVAVARA c o b 500 a 500 5-80. Absolute propriety limitsfor lower tetrachords. € (e) b 500 5-81. Absolute propriety limitsfor upper tetrachords. CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

5-82. Propriety rangesfor upper companion tetrachords: limits for the tetrachords (a) 100 + 200 + 200 cents, (b) 200 + 100 + 200 cents, (¢) 200 + 200 + 100 cents, (d) zoo + 300 +100 cents. subsequently calculated to yield scales that are the intervallic retrogrades or octave inversions of those above. A number of interesting conclusions may be drawn from these data. Proper heptatonic tetrachordal scales containing microtones are only possible under certain conditions. The microtonal intervals may be present in either the upper or lower tetrachord provided they are not in the extreme 300 | positions, i.e., not intervals 4 or 500-c-d. Proper hexatonic scales also exist when tetrachordal intervals} or d equal T INTERVAL b Upper tetrachords may also be chosen and lower companion ranges INTERVAL 4 300 INTERVAL 4 300 zero and 4 and care 250 cents. These scales may be analysed as containing a tetrachord, a disjunctive tone, and a trichord. The tetrachordal genera which appear as vertices of the propriety regions are of great interest. In particular, the equal division 166.667 + 166.667 + 166.667 accepts as upper companions both chromatic and improper diatonic genera, including some with subcommatic intervals, Other new tetrachords occurring as vertices are the improper diatonic genera 33.333 + 233-333 + 233.333; this is very close to Al-Farabi’s 49/48 - 8/7 . 8/ 7, and 50 + 250 + 200, which is approximated rather well by 40/39 « 52/45 9/8, Work of other investigators Several other investigators have independently developed descriptors functionally identical to Rothenberg’s strict propriety. Gerald Balzano has used the notion of “coherence” in his work on microtonal analogs of the diatonic scale in 12-tone equal temperament (Balzano 1980). Though not INTERVAL 4 300 300 L = tetrachordal, Balzano’s scales are homologous to the tritriadic scales discussed in chapter 7. Ervin Wilson (personal communication) has applied the term constant structure to scales in which each instance of a given interval subtends the same number of subintervals, but not necessarily subintervals of the same magnitude or order. This property is also equivalent to propriety. È N 1 INTERVAL 4 1 1 CHAPTER 5

LOWER TETRACHORD VERTICES 5-83. Proper mixed tetrachord scales, in cents. These I. 100200200 50, 200; 50, 250; 200, 200; 200, 50 tetrachords can combine with a disjunctive tone and 2. 200 100 200 100, 150; 100, 300; 200, 200; 200, 50 any tetrachord in the region defined by the vertices to yieldproper or strictly proper scales. The retrogrades of these tetrachords can also serve as the upper tetra- 3, 4. 5, 200200 100 100 300 100 100150 250 100, 200; 100, 300; 250, 150; 250, 50 100, 200; 200, 100 50, 250; 50, 200; 150, 150; 150, 100 chords of proper scales, The third interval of each tet- 6. 100250 150 100, 150; 100, 250; 200, 150; 200, 50 rachord may be found by subtracting the sum of the two tabulated intervals from 500 cents. The neochromatic tetrachord number4 is the upper tetrachord of the harmonic minor mode. Its region of propriety is reduced toa line rather than an area in the tetrachordal intervalplane. Tetrachords 11, 12, and 26 cannot form proper scales with any upper tetrachord. 7. 8. 9. IO. II. 12, 13. Iq. 15. 16. 17. 18, 19. 20. 21. 150100250 150150 100 250100 150 250 150 100 50 250 200 50 200 250 200 50 250 200250 50 250 50 200 250 200 50 125 125 250 125 250 125 250125 125 150 150 200 150 200 160 50, 200; 50, 250; 150, 150; 150, 100 100, 275; 100, 200; 150, 250; 225,175; 225, 75 150, 150; 150, 250; 250, 150; 250, 50 150, 150; 150, 250; 250, 150; 250, 50 NO PROPER SCALES NO PROPER SCALES 100, 150; 100, 200; 150, 150; 150, 100 200, 150; 200, 200; 250, 150; 250, 100 150, 150; 150, 250; 200, 200; 200, 100 200, 150; 200, 200; 250, 150; 250, 100 50, 200; 50, 250; 150, 150; 150, 100 87.5, 187.5; 87.5, 287.5; 212.5, 162.5; 212.5, 62.5 150, 150} 150, 250; 250, 150; 250, 50 50, 200; 50, 250; 200, 200; 200, 50 75, 175; 75, 2253 83.3, 283.3; 150, 2503 225, 175; 225,25 100, 150; 100, 300; 250, 150; 150,0 87.5, 187, 53 87.5, 237.5; 200, 125; 200, 75 100, 175; 100, 250; 212.5, 137.55 212.5, 62.5 : 22. 200150 150 23. 100275 125 24. 125 275 100 25. 233.33 233.33 33.33 233.33, 133-335 233.33, 166.67 26. 27. 33.33 233.33 233.33 166.7 166,7 166.7 NO PROPER SCALES 66,67, 183.33; 66.67, 266.67; 88.89, 288.89; 133.33, 266.67; 233.33, 166.67; 233.33, 16.67 CLASSIFICATION, CHARACTERIZATION, AND ANALYSIS OF TETRACHORDS

6 Scales, modes, and systems THE FORMATION OF heptatonic scales from tetrachords was mentioned briefly in chapters 1 and 5. In the present chapter, scale construction will be examined at greater length—in particular, the formation of nontraditional and non-heptatonic scales from tetrachordal modules. Before introducing this new material, however, a brief review of the salient features of the Greek theoretical system is necessary as an introduction to scale construction. The hierarchy of scalar formations The ancient Greek theorists recognized a hierarchy of increasingly large scalar formations: tetrachord, pentachord, hexachord, heptachord, octachord, and system. The canonical forms of each of these scalar formations may be seen in 6-1. The smaller formations were finally absorbed into the Perfect Immutable System which with its fifteen pitch keys or tonoi was the highest structural level of the Greek theoretical doctrine. As the tetrachordal level has been introduced in earlier chapters, the discussion will focus on the pentachord and larger structures. | The pentachord Pentachords may be considered as tetrachords with disjunctive tones added at either extremity. They divide the perfect fifth into four subintervals and occur in several forms in the various modes of heptatonic scales. The two forms of greatest theoretical importance are described in 6-1. While of relatively minor musical prominence, the pentachord has considerable pedagogical value in explaining how certain tunings and scales may have arisen. SCALES, MODES, AND SYSTEMS

For example, Archytas’s complex septimal tuning system can be best understood by considering not just the three species of tetrachord, but the pentachords formed with the note a whole tone below. These would be the note hyperhypate for the meson tetrachord and mese for the diezeugmenon (Winnington-Ingram 1932; Erickson 1965). By the use of the harmonic mean between hyperhypate (8/0) and mese (4/3), Archytas defined his enharmonic lichanos as 16/15. His tuning for the note parhypate (28/27) in all three genera was placed as the arithmetic mean between the 8/9 and 32/27, the diatonic lichanos. This construction may be seen in 6-2, The notes D F G and A form the harmonic series 6:7:8:9 and the notes D G Aa minor triad, 10:12:15. The 7/6 which the hyperhypate (D) makes with parhypate (F) is found in all three of his genera and is duplicated a fifth higher between mese (A) and trite (C). This interval was very important in Greek theory and had its own name, ekbole (Steinmayer 1985). It occurs in the Dorian harmonia shown in 6-q and in the fragments of surviving Greek music. As this interval has the value of 7/6 only in Archytas’s tunings and those others of the 7/6 pentachordal family (chapter 4), it is interesting to consider analogous pentachords with the 28/27 replaced by other intervals. 6-2 also depicts such a system, employing a more Aristoxenian 1/4-tone interval, 40/39, which was used by the theorists Eratosthenes, Avicenna, and Barbour in their genera (See the Main Catalog and 4-3). This system has a number of interesting harmonic and melodic intervals and could be played very well in 24-tone equal temperament. 6-1. The hierarchy of scalarformations, The tetrachord may be any ofthe those listed in chapter 9. The interval ofequivalence is the 4/3. The two canonicalforms ofthe pentachord are given. Other forms occur in the various modes ofbeptatonic scales of different genera and may have the 9/8 interpolated between the tetrachordal intervals. Miscellaneous pentachordal structures According to Xenakis, chains of conjunct tetrachords and pentachords (trochos) are used in the liturgical music of the Greek Orthodox church With the addition ofthe octave 2/1, the beptachord FORM NOTES becomes the Mixolydian mode ofthe complete beptatonic or octachordal scale. Ifthe 8/9 is added below the 1/1 the scale becomes the Hypodorian mode transposed downwards by a whole tone (9/8). The next bighest structural level is that ofa system which TETRACHORD PENTACHORD I! 2: HEXACHORD I: 2: HEPTACHORD OCTACHORD 1/1 ab 4/3 1/1 4 b 4/3 3/2 8/9 1/1 ab 453 I/t 4 d 4/3 3/2 3/2 1/1 4 b 4/3 3/2 34/2 1/1 ab 4/3 44/3 4bl3 16/9 1/1 2 b 4/3 3/2 34/2 36/2 2/1 contains al} the lower ones. The octachord is the beptatonic Dorian mode. CHAPTER 6

(Xenakis 1971, and chapters 2 and 5). These chains exhibit cyclic permutation of their constituent intervals. Most importantly, they are examples of those rare musical systems in which the octave is not the modulus or interval of equivalence. Additionally, more traditional heptatonic modes (echoi), some of which appear to have genetic continuity with classic Greek theory, if not practice, 6-2. Pentachordal systems. are employed. These may be analyzed either as composed of two tetrachords or as as combinations of tetrachord and pentachords. A number of ARCHYTAS’S SYSTEM D 8/0 E ii F Gh 28/27 16/15 & 9/8 6/5 A 4/3 9/7 7/6 8/7 E 1/1 40/39 SYSTEM F Gh G G 40/39 16/15 10/9 32/27 6/5 15/13 5/4 13/10 5/4 15/13 Some irregular species of Greek and Islamic origin are also listed in the Islamic scales were apparently modes used in actual music. 8- or g-tone pseudo-tetrachordal octave scales may be formed by combining these with A 4/3 appropriate fifths or fourths. The hexachord, heptachord, and gapped scales The hexachord and heptachord generally appear as transitional forms between the single tetrachord and the complete heptatonic scale or oc- 6/5 52/45 tetrachords from these modes are listed in the Catalogs. chapter 8 along with Kathleen Schlesinger’s harmoniai to which they bear some resemblance. These divide the fourth into four parts and the fifth into five. The Greek forms are merely didactic patterns taken from Aristoxenos and interpreted by Kathleen Schlesinger as support for her theories, while 5/4 7/6 D 8/9 G 32/27 tachord. The hexachord appears as a stage in the evolution of the enharmonic genus from a semitonal pentatonic scale similar to that of the modern Japanese koto to the complete heptatonic octave. This 5-note scale is often called the enharmonic of Olympos (6-3) after the legendary musician who was credited with its discovery by Plutarch (Perrett 1926), This and other pentatonic scales may be construed as two trichords combined with a whole tone to complete the octave. The two intervals of the trichord may be a semitone with a major third, a whole tone with a minor third, or any other combination of two intervals whose sum equals a perfect fourth. At some point the semitone in the lower trichord was divided into two dieses. This produced the spondeion or libation mode which consisted of a lower enharmonic tetrachord combined by disjunction with an upper trichord consisting of a semitone and a major third (6-3). This hexachord or hexatonic scale evolved into the spondeiakos or spondeiazon tropos. Eventually the semitone in the upper trichord was also split and a hep- SCALES, MODES, AND SYSTEMS

6-3. Gapped or irregular scales, The notation used here reproduces that of the references. The plus sign indicates a tone 1/g-tone higher than normal. Unless otherwise noted, no particular tuning is assumed, but either Pythagorean or Archytas’s supplemented as required with undecimal ratios would be appropriate historically. tatonic scale in the enharmonic genus resulted. This transformation may have been completed about the time of Plato, who writes as if he distrusted these innovations. In later times, the ancient pentatonic and hexatonic melodic patterns were retained in compositions for voice and accompaniment (Winnington-Ingram 1936). In principle, a hexachord can be obtained from a heptatonic scale in four ways by omitting one tone in either tetrachord. 6-3 lists the versions found in the literature. In these cases, the omitted note is the sixth degree, though Pentatonic forms the second version which lacks the seventh instead is a plausible inter- ENHARMONIC OF OLYMPOS e fa be (e) SPONDEION (WINNINGTON-INGRAM 1928) e f a b c+ or e f+ a b c+ 1/1 12/11 4/3 3/2 18/11 (2/1) Some controversy, however, exists in the literature about the tuning of these early gapped or transilient scales. The arguments over the relative b merits of enharmonic or diatonic tunings were discussed by WinningtonIngram (1928) whose scales and notation are reproduced in 6-3. Notable SPONDEION (MOUNTFORD 1923) 28/27 4/3 3/2 18/11 (2/1) are his and Mountford’s undecimal or 11-limit tunings for the pentatonic forms. Winnington-Ingram’s undecimal neutral third pentatonic could be Hexatonic forms SPONDEIAKOS 07 SPONDEIAZON TROPOS (WINNINGTON-INGRAM 1928) e e+ f a be the progenitor of the hemiolic chromatic genus (75 + 75 + 350 cents) and diatonics similar to the equable diatonic such as 150 + 150 + 200 cents. Henderson (1942) has also offered two quite different non-standard in- SPONDEION (HENDERSON 1942) f a b dé et ore e+ f a 1/1 with b+ d' & c' in the accompaniment DIATONIC OF WEIL & REINACH (WINNINGTON-INGRAM 1928) efgabd with by c & e' in the accompaniment considerations. The hypothetical diatonic versions of these scales according to the Reinach provide a conventional diatonic form (Winnington-Ingram 1928). DIATONIC OF GREIF (WINNINGTON-INGRAM 1928) de fa b c# (d) SCHLESINGER (1939, 183) ıvıo 11/9 11/8 11/7 1/6 terpretations of the enharmonic pentatonic based on etymological suggestions of several scholars are listed in this table as well. Weil and GAPPED SCALE OF TERPANDER & NICOMACHOS (HELMHOLTZ 1877, 266) e f g abd (e) wt pretation in some cases. Schlesinger’s version is based on her theories which are described in detail in chapter 8. The version of Greif appears to be derived from the Lesser Perfect or Conjunct System with the addition of a tone below the tonic as seen in the Dorian harmonia of 6-4 (ibid.). It should be compared with the ancient non-octaval heptachord which may also be formally derived from the conjunct system (6-1). The medieval diatonic hexachord of Guido D’Arezzo, cd e fg a c', may (2/1) be included with these scales too, although it is much later in time. In just intonation, it is usually considered to have the ratios 1/1 9/8 5/4 4/3 3/2 Heptatonic form 5/3, derived from the Lydian mode of Ptolemy’s syntonic diatonic instead CONJUNCT HEPTACHORD of the Pythagorean 1/1 9/8 81/64 4/3 3/2 27/16. In the septimal diatonic tuning of Archytas it would have the ratios 1/1 8/7 9/7 4/3 32/21 12/7. cf g a b CHAPTER 6

The octachord or complete heptatonic scale The union of a tetrachord and a pentachord creates an octachord or com- 6-4. The oldest harmoniai in three genera. Dorian ENHARMONIC de f-ggabc-d'pe' CHROMATIC defgabcdje' DIATONIC defgabcd'e Phrygian ENHARMONIG def-gabc-dyd' CHROMATIC defgabcd'd' DIATONIC defgabcd' Lydian ENHARMONIC f-gabc-dye'fCHROMATIC fgabcdy,e'f DIATONIC fgabed'e'f Mixolydian ENHARMONIC Bc-dydef-gb GHROMATIC Bcddefgb DIATONIC Bcdef(g)(a)b Syntonolydian ENHARMONIC BC-dyeg CHROMATIC BC deg DIATONIC cdefg 2ND DIATONIC BCdeg Ionian (lastian) ENHARMONIC BC-dyega CHROMATIC BCdega DIATONIC cefga 2NDDIATONIG BCdega plete heptatonic scale. There is evidence, however, that initially two diatonic tetrachords were combined by conjunction, with a shared note between them, to form a 7-note scale less than an octave in span (6-1). The later addition of a whole tone at the top, bottom, or middle separating the two tetrachords, completed the octave gamut. Traces of this early heptachord may be seen in the construction of the Lesser Perfect System and in the irregular scales of 6-3 and 6-4. Similarly, two enharmonic tetrachords were joined by disjunction with the 9/8 tone between them to create the Dorian harmonia to which a lower tone was added (6-4). An alternative genesis would connect two pentachords whose extra tones were at their bases to produce the 9-tone Dorian harmonia to which other tones might accrete. By analogy, both the enharmonic and diatonic proto-scales converged to the same multi-octave structures later called by the name of system. In the fifth century sce the wide ditone or major third of the enharmonic genus was gradually narrowed to a minor or subminor third by a process termed “sweetening.” Eventually, this process resulted in the chromatic genus which was raised to the same status as the diatonic and enharmonic genera. The Greater and Lesser Perfect Systems However the early evolution of the Greek musical system actually occurred, the result came to be schematized as the Perfect Immutable System. Its construction was as follows: two identical tetrachords of any genus and a disjunctive tone (9/8) formed a central heptatonic scale which became the core of the system. Another identical tetrachord was then added by conjunction at both ends of the scale and disjunctive tone was patched on at the bottom of the whole array. A fifth tetrachord, synemmenon, was inserted conjunctly into the middle of the system to recall the ancient heptachord and to facilitate commonly occurring modulations at the fourth. This supernumerary tetrachord was also a useful pedagogical device to illustrate unusual intervals (Erickson 1965; Steinmayer 1985). The final results consisted of sets of five tetrachords linked by conjunction and disjunction into arrays of fifteen notes spanning two octaves. These systems, in turn, could be transposed into numerous pitch keys or tonoi, at intervals roughly a semitone apart according to the later authors. SCALES, MODES, AND SYSTEMS

The subset of four alternately conjunct and disjunct tetrachords (hypaton, meson, diezeugmenon, and hyperbolaion) was termed the greater perfect (or complete) system (syotena tehetov pertov). The three conjunct tetrachords (hypaton, meson, and synemmenon), was called the Lesser Perfect (or Complete) System (oyotnpo. teActov eLattov or ekacoov). Their union was called variously the Changeless System or the Perfect Immutable System (ovompa teAeiov aneraßoAov) by different authors. The Perfect Immutable System By the fourth century Boe, the Greek theorists had analyzed the scales or harmoniai of their music into sections of this theoretical two octave gamut. This 15-note span is conventionally transcribed into our notation as lying between A and a’, The Perfect Immutable System could be tuned to each of the three genera, and while in theory all five of the tetrachords must be the same, in practice mixed tetrachords and considerable chromaticism occurred. Not only was the diatonic lichanos meson (D in the Dorian or E mode) added, but other extrascalar notes led to successions of more than two semitones (Winnington-Ingram 1936). 6-5 depicts the Perfect Immutable System in its theoretical form and in its two most historically important intonations. The fixed notes (hestotes) of the Perfect Immutable System were proslambanomenos, hypate hypaton, hypate meson, mese, paramese, nete diezeugmenon, nete hyperbolaion, and nete synemmon. The moveable tones (Kivovpevol) were the parhypatai, the lichanoi, the tritai, and the paranetai of each genus. Lichanos hypaton, also called hyperhypate, a diatonic note a whole tone (9/8 in Archytas’s and most other just tunings) below the tonic, was added to the Dorian octave species in the chromatic and enharmonic genera in the harmoniai of Aristides Quintilianus, certain planetary scales, and the Euripides fragment (ibid.). Erickson (1965) and Vogel (1963, 1975) have shown that a number of interesting tetrachords occur in the region where the synemmenon tetrachord overlaps with the diezeugmenon tetrachord in Archytas’s system. These include the later and historically important 16/15 : 9/8 . 10/9 (Ptolemy’s syntonic diatonic), 16/15 + 10/9 : 9/8 (Didymos’s diatonic), the three permutations of the Pythagorean diatonic, 256/243 - 9/8 - 9/8, (90 + 204 + 204 cents), the Pythagorean chromatic 32/27 - 2187/2048 - 256/243 (294 + CHAPTER 6

114 + go cents), and Avicenna’s chromatic 7/6 - 36/35 - 10/9 (267 +49 + 182 cents). Some unusual divisions such as 28/27 - 81/70- 10/9 (63 + 253 + 182 6-5. The Perfect Immutable System in the diatonic, chromatic, and enharmonic genera, tuned according to Archytas’s and Pythagorean tuning. The transcription is in the natural key to avoid accidentals and the mistaken late shift ofemphasisfrom Dorian to Hypolydian (Henderson 1957). The - andy indicate that these are different pitches in the enharmonic genus, Erickson (1965) proposes 64/45 as an alternative tuningfor trite synemmenon. cents), 28/27 - 2187/1792 - 256/243 (63 + 345 + go cents), 16/15 - 35/32 - 8/7 (112 +155 + 231 cents), 16/15: 1215/1024: 256/243 (112 + 296+ go cents), 7/6 . 81/80 : 9/8 (267 + 22 + 204 cents), 32/27 - 81/80 - 10/9 (294 + 22 + 182 cents), 28/27 -64/63 -81/64(63 +22 +408 cents), 6/5 - 135/128 +2 56/243 (316 +92 + gocents), and 256/243 -81/80- 5/4 (go + 22 + 386 cents) are also found here. Notable are the intervals of 253 cents, another possible tuning for the ekbole, the neutral third of 345 cents, the three-quarter tone 35/32 (155 cents), and the minor whole tone 10/9. The alternate tunings 16/15 and 28/27 for the first interval of the synemmenon tetrachord may have been used in order to obtain the spondeiasmos, an interval of three dieses approximating 150 cents, mentioned by Bacchios (Steinmayer 1985; Winnington-Ingram 193 2). These intervals would measure 35/32 (155 cents) as the difference between 14/9 and 64/45, or 243/224 (141 cents) as the difference between 112/81 and 3/2. The in- TRANSCRIPTION DIA. CHR. ENH. ARCHYTAS DIA. CHR. PROSLAMBANOMENOS A A A 2/3 2/3 HYPATE HYPATON B B B PARHYPATE HYPATON C C C- 3/4 7/9 LICHANOS HYPATON D D Dy HYPATE MESON E E E PARHYPATE MESON F F F- LICHANOS MESON MESE G a G a Ge a PARAMESE TRITE DIEZEUGMENON b c b c b c- ENH. PYTHAGOREAN DIA. CHR. ENH. 2/3 2/3 2/3 2/3 3/4 7/9 3/4 7/9 3/4 64/81 3/4 64/81 3/4 384/499 8/9 1/1 28/27 32/27 27/32 1/1 28/27 9/8 4/5 1/1 28/27 16/15 8/9 1/1 256/243 32/27 27/32 1/1 256/243 9/8 64/81 1/1 512/499 256/243 4/3 4/3 4/3 4/3 4/3 4/3 3/2 14/9 16/9 3/2 14/9 27/16 3/2 14/9 8/5 3/2 128/81 16/9 3/2 128/81 17/16 3/2 768/499 128/81 PARANETE DIEZEUGMENON d d dy NETE DIEZEUGMENON e e e TRITE HYPERBOLAION PARANETE HYPERBOLAION f g f & fE 2/1 56/27 2/1 56/17 2/1 56/27 2/1 512/243 2/1 512/243 2/1 1024/499 64/27 9/4 32/15 64/27 9/4 512/243 NETE HYPERBOLAION a a’ a' 8/3 8/3 8/3 TRITE SYNEMMENON (28/27) bi b b- G Gy 112/81 64/45 8/3 1024/7209 128/81 8/3 2048/1497 c 112/81 3/2 8/3 1024/729 PARANETE SYNEMMENON 112/81 128/81 3/2 1024/729 NETE SYNEMMENON d d D 16/9 16/9 r6/9 16/9 16/9 SCALES, MODES, AND SYSTEMS

3/2. The interval of three dieses also appears in Archytas’s chromatic as the difference between the 28/27 and the 9/8. In many cases the scales containing these tetrachords would be mixed, but deliberately mixed scales were not unknown. 6-6 lists some varieties of mixed scales recorded by Ptolemy in the second century cr. The scales actually employed in Greek music are a matter of some confusion because of the paucity of extant musical examples and the variety of theoretical works from different traditions written over a period of several centuries (fourth century BCE to fourth century ce). In the theoretical treatises, the seven octave species or circular permutations of the basic heptatonic scale are singled out and given names derived from early tribal groups. These scales are notated in all three genera in 6-7. Their intervals and notes are in shown in ratios for both Archytas’s and Pythagorean tuning in 6-8 and 6-9. 6-10 gives the diatonic form in Ptolemy’s syntonic diatonic (16/15 : 9/8 - 10/9), and 6-11 gives the retrograde of this genus (10/9 : 9/8 16/15). The Lydian mode in the former tuning is the standard just intonation of the major scale, and the latter is that of the natural minor mode 6-6. Scales in common use according to Ptolemy. In the text, the names ofthe tunings are always given in pluralform. (1), not the ditonic or Pythagorean, appears to have been the standard diatonic. On the kithara, in the Hypodorian mode it was called tritat; in the Phrygian, bypertropa. (2a) is given in twoforms in differentplaces in the Harmonics; the intense chromatic (1:84), where it is mistranslated as “di- (see chapter 7). For the Pythagorean tuning of the enharmonic, I have used Boethius’s much later arithmetic division of the pyknon, as the actual tuning prior to Archytas is not known. Since the division of the semitone in both tetra- I, STEREA, A LYRA TUNING: TONIC DIATONIC t/t 28/27 32/27 4/3 3/2 14/9 16/9 2/1 atonic chromatic,” and the soft chromatic (2:208). The tables (2:178) use the intense chromatic; the soft chromaticfits the sense ofthe name better. On the kithara, (2b) in the Hypodorian mode is called tropoi or tropikoi. In the Dorian mode on the kithara, (3) is calledparypatai, (4) is in the Hypophrygian mode. (5), inthe Dorian mode, is given variously as either pure tonic diatonic ora mixture oftonic diatonic and intense and is also referred to as metabolika. (6) is from Avicenna (D'Erlanger 1935, 2:239), who sometimes approximated complex ratios like 72/65 with superparticulars of similar magnitude such as 2. MALAKA, A LYRA TUNING: SOFT OR INTENSE CHROMATIC AND TONIC DIATONIC A. I/t 28/27 10/9 4/3 3/2 14/9 16/9 2/1 B.I/t 22/21 8/7 4/3 3/2 14/9 16/9 2/1 3: METABOLIKA, ANOTHER LYRA TUNING: SOFT DIATONIC AND TONIC DIATONIC 1/1 21/20 7/6 4/3 3/2 14/9 16/9 2/1 4. IASTI-AIOLIKA, A KITHARA TUNING: TONIC DIATONIC AND DITONIC DIATONIC 1/1 28/27 32/27 4/3 3/2 27/16 16/9 2/1 5. IASTIA OR LYDIA, KITHARA TUNINGS: INTENSE DIATONIC AND TONIC DIATONIC 1/1 28/27 32/27 4/3 3/2 8/5 9/5 2/1 22/21, but the exact ratio is clearfrom the context. 6, A MEDIEVAL ISLAMIC SCALE OF ZALZAL FOR COMPARISON 1/1 9/8 81/64 4/3 40/27 130/81 16/9 2/1 CHAPTER 6

chords was completed only near end of the fourth century sce, the division may not have been standardized and was most likely done by ear during the course of the melody (Winnington-Ingram 1928), in which case the approximate equality of the dieses in Boethius’s tuning probably captures the flavor of the scale adequately. Euler’s eighteenth-century tuning (Euler [1739] 1960, and Catalog number 79) is similar and considerably simpler. An impractical, if purely Pythagorean, solution (number 81) as well as some other approximations are given in the Main Catalog. Although these scales are analogous to the “white key” modes, the latter are named out of order due to a misunderstanding in early medieval times. TONIC NAME MESE HYPERMIXOLYDIAN, HYPERPHRYGIAN, LOCRIAN MIXOLYDIAN, HYPERDORIAN LYDIAN PHRYGIAN DORIAN HYPOLYDIAN HYPOPHRYGIAN, IONIAN HYPODORIAN, AEOLIAN mous OH OVS ERR Me ERS HYPERMIXOLYDIAN, HYPERPHRYGIAN, LOCRIAN MIXOLYDIAN, HYPERDORIAN LYDIAN PHRYGIAN DORIAN HYPOLYDIAN HYPOPHRYGIAN, IONIAN HYPODORIAN, AEOLIAN Enharmonic 103 SCALES, MODES, AND SYSTEMS oF Fe OMY Chromatic m 6-7. The octave species in all three genera, The traditional names are given first and alternate ones subsequently. The Hypermixolydian was denounced by Ptolemy as otiose and by the city ofArgos as illegal (Winnington-Ingram 1936), This transcription uses the natural key for clarity. Late theorists mistakenly built the system and notation about the F mode (Hypolydian) rather than the correct E mode (Dorian) (Henderson 1957). Although the Dorian, Phrygian, and Lydian modes bave distinctive tetrachordalforms, these forms were never named after their parent modes by any of the Greek theorists. In the chromatic and enharmonic genera the tonics of the species are transformed. An alternative nomenclature for the enharmonic tetrachord is E E+ FA. The mese kata thesin is four scale degrees above the tonic with which it usually makes an interval of a perfect fourth. © FrHpows Diatonic HYPERMIXOLYDIAN, HYPERPHRYGIAN, LOCRIAN MIXOLYDIAN, HYPERDORIAN LYDIAN PHRYGIAN DORIAN HYPOLYDIAN HYPOPHRYGIAN, IONIAN HYPODORIAN, AEOLIAN

Diatonic (28/27 » 8/7 - 9/8) HYPOLYDIAN (F — f) 1/1 MIXOLYDIAN (B — b) 28/27 32/27 4/3 112/81 128/81 16/9 2/1 28/27 + 8/7 : 9/8 . 28/27 - 8/7 + 9/8 : 9/8 243/224 9/7 81/56 3/2 729/448 27/14 2/1 243/224: 32/27-9/8-28/27- 243/224 - 32/27: 28/27 1/1 ri LYDIAN (C —c) 8/7 9/7 4/3 32/21 12/7 27/14 2 8/7 + 9/8 » 28/27 - 8/7 + 9/8 - 9/8 - 28/27 32/27 4/3 112/81 3/2 16/9 448/243 21/1 32/27 + 9/8 + 28/27 - 243/224 - 32/27 + 28/27 | 243/224 Mi PHRYGIAN (D —d) 9/8 7/6 445 3/2 27/6 7/4 di 9/8 + 28/27 + 8/7 - 9/8 - 9/8 : 28/27 + 8/7 ur 1/1 DORIAN (E e) 28/27 32/27 4/3 3/2 4/9 16/9 2/1 28/27 + 8/7 > 9/8 - 9/8 + 28/27 « 8/7 - 9/8 1/1 MIXOLYDIAN (B — b) 28/17 16/15 4/3 112/81 64/45 16/9 2/1 28/27 - 36/35 + 5/4 » 28/27 - 36/35 > 5/4 - 9/8 ri HYPOLYDIAN (F — f) 8/7 9/7 81/56 3/2 12/7 27/14 2/1 8/7 « 9/8 + 9/8 - 28/27 - 8/7 - 9/8 + 28/27 ur LYDIAN (C- —c-) 36/35 9/7 4/3 48/35 12/7 27/14 alt 36/35 + 5/4 > 28/27 + 36/35 + 5/4 + 9/8 : 28/27 ir 9/8 I/I PHRYGIAN (Dy, — dy) sla 35/27 43 5/3 15/8 35/18 2/1 5/4 + 28/27 + 36/35 + 5/4 + 9/8 - 28/27 - 36/35 1/1 DORIAN (E — e) 28/27 16/15 43 3/2 14/9 B/5 z/ 28/27 « 36/35 + 5/4 + 9/8 - 28/27 - 36/35 « 5/4 MIXOLYDIAN (B — b) ir 28/27 9/8 4/3 112/81 3/2 16/9 2/1 28/27 : 243/224 + 32/27 + 28/27 + 243/224 - 32/27 > 9/8 I/I HYPOLYDIAN (F- — f-) 36/35 9/7 81/56 3/2 54/35 27/14 2/ 36/35 + 5/4 + 9/8 + 28/27 - 36/35 - 5/4 > 28/27 LYDIAN (C — c) ir 243/224 9/7 4/3 81/56 12/7 27/14 2/1 243/224 : 32/27 + 28/27 » 243/224 + 32/27 + 9/8 - 28/27 1/ı PHRYGIAN (Dj — dh) ı/ı HYPOPHRYGIAN (G ~ g) 81/64 21/16 3/2 27/16 7/4 HYPOPHRYGIAN (Gy — gi) HYPODORIAN ih 2/1 HYPODORIAN (A — a) 9/8 76 4h 3/2 14/9 16/9 2/1 9/8 - 28/27 + 8/7 « 9/8 + 28/27 - 8/7 - 9/8 3/2 14/9 27/16 2/1 HYPOPHRYGIAN (Gy — gy) 32/27 896/729 4/3 128/81 16/9 448/243 2/1 32/27 + 28/27 + 243/224 : 32/27 : 9/8 - 28/27 - 243/224 DORIAN (E — e) 28/27 9/8 4/3 3/2 14/9 27/16 2/1 28/27 - 243/224 - 32/27 + 9/8 - 28/27 > 243/224 : 32/27 6-8, The intervals of the octave species in all three genera in Archytas’s tuning. 104 81/64 Enharmonic (28/27 : 36/35 : 5/4) Chromatic (28/27 « 243/224 : 32/27) 1/1 7/6 9/8 - 28/27: 243/224: 32/27 : 28/27 - 243/224 - 32/27 9/8 « 9/8 + 28/27 - 8/7 « 9/8 - 28/27 - 8/7 1/1 9/8 CHAPTER 6 5/4 45/32 35/24 3/2 15/8 35/18 alt 5/4 « 9/8 : 28/27 + 36/35 + 5/4 + 28/27 > 36/35 HYPODORIAN (A — a) 9/8 7/6 6/5 3/2 14/9 9/8 + 28/27 + 36/35 + 5/4 + 28/27 : 36/35 + 5/4

Diatonic (256/243 « 9/8 - 9/8) MIXOLYDIAN (B — b) 256/243 32/27 4/3 1024/9729 128/81 16/9 2/1 256/243 + 9/8 > 9/8 + 256/243 : 9/8 - 9/8 - 9/8 LYDIAN (C —c) 1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1 9/8 + 9/8 : 256/243 + 9/8 - 9/8 + 9/8 : 256/243 DORIAN (E- e) it 256/243 9/8 4/3 3/2 128/81 27/16 2/1 256/243 : 2187/2048 - 32/27 - 9/8 : 256/243 - 2187/2048 - 32/27 HYPOLYDIAN (F — f) 1/1 2187/2048 81/64 729/512 3/2 6561/4096 243/128 2/1 2187/2048: 32/27 : 9/8 » 256/243 - 2187/2048 . 32/27 - 256/243 HYPOPHRYGIAN (Gj— gy) PHRYGIAN (D —d) ı/ı 9/8 32/27 44 3/2 29/16 16/9 2/1 9/8 + 256/243 - 9/8 + 09/8 : 9/8 - 256/243 - 9/8 I/T 32/27 4/3 729/512 3/2 16/9 4096/2187 2/1 32/27 - 9/8 - 256/243 - 2187/2048 - 32/27 | 256/243 + 2187/2048 DORIAN (E-e) rr 9/8 32/27 81/64 3/2 128/81 27/16 2/1 9/8. 256/243 2187/2048 : 32/27 - 256/243 2187/2048 - 32/27 1/1 256/243 3227 4/3 3/2 128/81 169 2/1 256/243 - 9/8 - 9/8 - 9/8 - 256/243 + 9/8 - 9/8 HYPOLYDIAN (F ~ f) 243/128 2/1 ir 9/8 81/64 729/512 3/2 27/16 9/8 + 9/8 + o/8 : 256/243 + 9/8 + 9/8 : 256/243 ir HYPODORIAN (A —a) HYPOPHRYGIAN (G ~ g) 9/8 81/64 4/3 3/2 27/16 16/9 2/1 9/8 - 9/8 . 256/243 + 9/8 - 9/8 - 256/243 - 9/8 HYPODORIAN (À —a) 1/1 9/8 32/27 4/3 3/2 128/81 16/9 2/1 9/8 : 256/243 : 9/8 : 9/8 - 256/243 : 9/8 : 0/8 Enharmonic (512/499 : 499/486 - 81/64) MIXOLYDIAN (B —b) 1/1 512/499 256/243 4/3 2048/1497 1024/729 16/9 2/1 512/499 : 499/486 « 81/64 + 512/499 : 499/486 - 81/64 : 9/8 LYDIAN(C--c-) 1/1 499/486 499/384 4/3 998/729 499/288 499/256 2/1 499/486 : 81/64 - 512/499 : 499/486 : 81/64 - 9/8 - 512/499 PHRYGIAN (Dy, — dy) 1/1 81/64 648/499 4/3 27/16 243/128 972/499 2/1 81/64 + §12/499 - 499/486 « 81/64 - 9/8 - 512/499 - 499/486 Chromatic (256. 2187/2028. 32/27) DORIAN (E—e) MIXOLYDIAN (B — b) 1/1 256/243 9/8 4/3 1024/729 3/2 16/9 21/1 256/243 - 2187/2048. 32/27 - 256/243 + 2187/2048 - 32/27 + 9/8 ir 512/499 256/243 4/3 3/2 768/499 128/81 24 512/499 : 499/486 : 81/64 : 9/8. 512/499 « 499/486 - 81/64 LYDIAN (C —c) 1/1 499/486 499/384 1497/1024 3/2 499/324 499/256 2/1 1/1 2187/2048 81/64 4/3 729/512 27/16 243/128 2/1 2187/2048 + 32/27 : 256/243 : 2187/2048 - 32/27 : 9/8 - 256/243 PHRYGIAN (D, - di) 1/1 32/27 8192/6561 4/3 128/81 16/9 4096/2187 2/1 32/27 - 256/243 + 2187/2048 + 32/27 - 9/8 - 256/243 : 2187/2048 HYPOLYDIAN (F- — f-) 499/486 » 81/64 : 9/8 - 512/499 : 499/486 » 81/64 : 512/499 HYPOPHRYGIAN (Gy— giù) 1/1 81/64 729/512 729/499 3/2 243/128 972/499 2/1 81/64 - 9/8 - 512/499 : 499/486 - 81/64 : 512/499 : 499/486 HYPODORIAN (A — a) 1/1 9/8 576/499 32/27 3/2 768/499 € 128/81 2A 9/8 : 512/499 - 499/486 . 81/64 - 512/499 : 499/486 - 81/64 6-9. The intervals of the octave species in Pythagorean tuning. The tuning of the preArchytas enbarmonic is not known, but at first it had undivided sernitones, obtaining the pyknon later. Boethius’s tuning is used here. SCALES, MODES, AND SYSTEMS

6-10. The intervals ofthe octave species ofPtolemy's intense diatonic genus. Seefigures 6-3 and 6-6for names ofnotes. The diatonic tetrachord is 16/15 9/8 + 10/9. The Lydian mode in this tuning is the major mode injust intonation. The Hypodorian or A mode is not the minor made as thefourth degree is 29/20 instead of 4/3. Although they are conventionally presented as sections of the two octave gamut, they were actually retunings of the central octave so that the sequences of intervals corresponding to the cyclic modes fell on the notes of the Perfect Immutable System (hypate meson to nete diezeugemenon, e to e!). These abstract sequences of intervals are shown in 6-12. Thus, in the Dorian tonos, the interval sequence of the Dorian mode filled the central octave; in the Phrygian, the Phrygian sequence was central and the Dorian, 6-11. The intervals ofthe octave species ofthe Prolemy's intense diatonic genus, reversed, The diatonic tetrachord is 10/9 : 9/8 « 16/15. The Lydian or C mode in this tuning is the minor mode injust intonation. The Dorian or E mode is not the major mode as the second degree is 10/9 instead of9/8. This scale transposed to C isJohn Redfield’s tuningfor the major scale (Redfield 1928). a tone higher. In the Hypolydian tonos, the initial A, proslambanomenos, was raised a semitone, as was its octave, mese, the supposed tonal center of the whole system. From the original set of seven pitch keys (tonoi), a later set of thirteen or fifteen theoretical keys at more or less arbitrary semitonal intervals developed, irrespective of genus (Crocker 1966; Winnington-Ingram 1936). In Roman times, the theorists moved the entire system up a semitone so MIXOLYDIAN (B — b) irr 16/15 6/5 4/3 64/45 8/5 16/9 2/1 16/15 - 9/8 - 10/9 - 16/15 + 9/8 + 10/9 - 9/8 1/1 MIXOLYDIAN (B — b) 109 5/4 4/3 40/27 55 16/9 2/1 10/9 « 9/8 - 16/15 - 10/9 + 9/8 - 16/15 - 9/8 LYDIAN (C — c) 2/1 1/1 9/8 5/4 aa 3/2 553 15/8 9/8 - 10/9 + 16/15 - 9/8 + 10/9 « 9/8 + 16/15 1/1 LYDIAN (C — c) 9/8 6/5 aA 3/2 8/5 9/5 2/1 9/8 + 16/15 + 10/9 - 9/8 + 16/15 + 9/8 » 10/9 PHRYGIAN (D - d) 1/1 10/9 32/27 als 40/27 5/3 16/9 2/1 10/9 + 16/15 + 9/8 + 10/9 + 9/8 - 16/15 - 9/8 PHRYGIAN (D — d) ir 16/15 32/27 4/3 64/45 8/5 16/9 2/1 16/15 + 10/9 - 9/8 - 16/15 : 9/8 - 10/9 - 9/8 DORIAN (E — e) 16/15 6/5 4/3 3/2 Ba 09% 2/1 16/15 + 9/8 - 10/9 - 9/8 - 16/15 : 9/8 - 10/9 1/1 HYPOLYDIAN (F — f) 1/1 9/8 5/4 45/32 3/2 27/16 15/8 2/1 9/8 - 10/9 «9/8. 16/15 + 9/8 - 10/9 - 16/15 ri 1/1 1/1 DORIAN (E. —e) HYPOPHRYGIAN (G - g) 10/9 sa 4/3 3/12 5/3 16/9 2/1 10/9 + 9/8 + 16/15 - 9/8 - 10/9 - 16/15 - 9/8 109 sla a 342 5a 18/8 2/1 10/9 : 9/8 » 16/15 « 9/8 - 10/9 - 9/8 - 16/15 HYPOLYDIAN (F - f) 9/8 6/5 27/20 3/2 27/6 9/5 2/1 9/8 - 16/15 - 9/8. 10/9 - 9/8 - 16/15 « 10/9 HYPOPHRYGIAN (G — g) ui 16/15 6/5 4/3 3/2 8/5 16/9 2/1 16/15 » 9/8 . 10/9 - 9/8 - 16/15 - 10/9 - 9/8 HYPODORIAN (A — a) ir 9/8 6/5 27/20 3/2 8/5 9/5 2/1 9/8 + 16/15 - 9/8 + 10/9 - 16/15 - 9/8 + 10/9 1/1 106 HYPODORIAN (A — a) 9/8 sla 45/32 3/2 5/3 15/8 9/8 . 10/9 - 9/8 © 16/15 - 10/9 - 9/8 « 16/15 CHAPTER 6

6-12. Interval sequences of the octave species ofthe that the central octave began on either E or F in modern notation. In this abstract tetrachord a-b-c.a-b-c=4/3(c=4/3ab) injust intonation or a+b +500-a-b with the disjunctive tone equaling 200 cents in the zero modulo 12 equal temperaments. In the Main Catalog, c is equal to the CI. final form, however, the central octave had the interval sequence of the Hypolydian mode rather than the Dorian. The modal retunings could also be considered as transpositions of the entire Perfect Immutable System. The order of the keys ran in the opposite direction to that of the hornonymous octave species and the octave species could be described either by the positions of their interval sequences in MIXOLYDIAN HYPOLYDIAN a.b.cva-b.c.g/8 b.c-.g/B-a.b.c-a LYDIAN HYPOPHRYGIAN b:c'a.b:c-0/8.4 c-9/8.a.b.cva.b PHRYGIAN c-a-b-¢.9/8-a-b HYPODORIAN Q/Ba-b-coa-bee DORIAN a:b.c-g/d.a-b.c 6-13. Vogel's transcription ofthe Greek notations. Only the upper octave from mese to nete hyperbolaion is shown. Vogel’s German notation has been transcribed into the American form. His notes bave been transposed up an octave, and those marked with a bar in the original are given a + here. 512/405 (406 cents) replaces 81/64 (408 cents), in Vogel's tuning. In the upper halfofthe scale, 2048/1215 replaces 27/16. relation to the untransposed Dorian or by the relative pitch of the entire Perfect Immutable System. This duality is reflected in the two nomenclatures employed by Ptolemy, the “onomasia kata thesin” (by position) and “onomasia kata dynamin” (by function). The thetic nomenclature in the natural key is used in the tables of this chapter and chapter 8 as it is the same for all tonoi. The dynamic refers all notes to the Dorian tonos for which the thetic and dynamic nomenclatures are identical. NOTE RATIO NOTATION MESE TRITE SYNEMMENON PARANETE SYNEMMENON I/x 28/27 16/15 (ENHARMONIC) A Bi Bi+ PARANETE SYNEMMENON, PARAMESE 9/8 (CHROMATIC) B TRITE DIEZEUGMENON 7/6 C- PARANETE SYNEMMENON 32/27 (DIATONIC) C PARANETE DIEZEUGMENON 6/5 (ENHARMONIC) C+ ‘NETE SYNEMMENON NETE DIEZEUGMENON TRITE HYPERBOLAION PARANETE HYPERBOLAION PARANETE HYPERBOLAION 896/729 D,- 512/405 (CHROMATIC) 4/3 (DIATONIC) 112/81 Dy+ D E- 64/45 B 3/2 14/9 8/5 (ENHARMONIC) 128/81 E FF+ F 3584/2187 G- 2048/1215 (caromatIc) 16/9 (DIATONIC) G, 448/243 Ap 256/135 NETE HYPERBOLAION 107 2/1 SCALES, MODES, AND SYSTEMS

The Greeks named the modes from their keynotes as octave species of the Perfect Immutable System, while the medieval theorists named them 6-14. Unusual tetrachords in Vogel's transcription. CENTS RATIOS 64/63 «81/80 - 35/27 27 +22 + 449 81/80 - 2240/2187 + 9/7 22 + 41 +435 49 + 41 + 408 36/35 - 2240/2187 + 81/84 36/35 - 256/243 - 315/256 49 + 90 + 359 27 +112 + 359 64/63 : 16/15 : 315/256 64/63 : 2187/2048 - 896/729 27+ 114+ 357 896/729 + 36/35 + 135/128 357 +49 +92 28/27 + 256/243 : 2187/1792 63 +90 + 345 16/15 + 2240/2187 - 2187/1792 112 + 41 + 345 28/27 - 128/105 - 135/128 63+343+92 6/5 + 35/32 + 64/63 316+155 +27 6/5 : 2240/2187 + 243/224 316 +41 +141 7168/6561 : 36/35 - 1215/1024 153 +49+ 296 16/15 - 1215/1014 + 256/243 112+296+90 28/27 . 1024/945 + 1215/1024 63 + 139+ 296 267+ 139 + 92 7/6: 1024/945 : 135/128 28/27 : 81/70 + 10/9 63 +253 +182 81/70. 2240/2187 : 9/8 153 + 4I +204 81/70 : 256/243 - 35/32 253 +90+155 135/128 . 7168/6561 - 81/70 92+153+253 16/15 + 280/243 - 243/224 112+245+ I41 36/35 + 9/8 - 280/243 49 + 204+ 245 8/7 + 81/Bo - 280/243 231+22+245 9/8 : 7168/6561 : 243/224 204+153+1I4I 9/8 : 4096/3645 : 135/128 204+ 202 + 92 155 +139 + 204 35/32 + 1024/0945 «9/8 4096/3645 + 35/32 + 243/224 202 +155 + 141 in order of their transpositions (Sachs 1943). The two concepts became confused by the time of Boethius. For this reason the names of the ecclesiastical modes are different from those of ancient Greece. In more recent periods, other ecclesiastical nomenclatures were developed. Greek alphabetic notations In addition to the thetic and dynamic nomenclatures, which were really tablatures derived from the names of the strings of the kithara or similar instrument, there were two alphabetical cipher notations, the vocal and the instrumental, These were recorded for the each of the tonoi in all three genera by the theorist Alypius. The independent elucidation of Alypius’s tables by Bellermann (1847) and Fortlage (18477) have permitted scholars to transcribe the few extant fragments of Greek music into modern notation. Vogel (1963, 1967) has translated these cipher notations into a tuning system based on Archytas’s and Pythagoras’s genera (6-4). This set of tones includes a number of unusual tetrachords, most of which occur in several permutations (6-13). Some of these are good approximations to the neo- Aristoxenian types: 50 + 100 + 350 cents, §0 + 150 + 300 cents, 50 + 250 + 200 cents, and 150+150+200 cents of chapter 4. The Greek notations, however, were not entirely without ambiguity, and some uncertainly exists over the meaning of certain presumed “enharmonic” equivalences, i.e. two notes of the same pitch written differently. Kathleen Schlesinger developed her somewhat fantastic theories, detailed in chapter 8, in part from deliberations on the apparent anomalies of these notations, Concise descriptions of the notational systems may be found in Sachs (1943) and Henderson (1957). The oldest harmoniai or modes Although the melodic canons laid down by Aristoxenos (330 BCE) stated that the smallest interval the melody could move from the pyknon was a whole tone and that notes four or five positions apart must make either perfect fourths or fifths, both literary evidence and the surviving fragments attest to mixed scales and chromaticism (Winnington-Ingram 1936), as mentioned previously, A late writer, Aristides Quintilianus, gave a list of what he said were the scales approved by Plato in the Republic. These scales CHAPTER 6

are in the enharmonic genus and depart quite strongly from the conventional octave species of 6-7. Since it is known that both diatonic and chromatic scales of the same name existed, it is tempting to try to reconstruct them. 6-4 contains Aristides’s enharmonic harmoniai, Henderson’s (1942) diatonic versions, and my own chromatic and diatonic forms. The chromatic versions are based on Winnington-Ingram’s indication that there is literary evidence for certain chromatic versions (1936). The diatonic harmoniai are from Henderson (1942), except in the cases of the Syntonolydian and Iastian where I have supplied a second diatonic which I feel better preserves the melodic contours. In the enharmonic and chromatic forms of some of the harmoniai, it has been necessary to use both a d and either a d, or dy because of the non-heptatonic nature of these scales. C and F are synonyms for dy and gy. The appropriate tunings for these scales are those of Archytas (Mountford 1923) and Pythagoras. These scales are very important evidence for the use of extrascalar tones (diatonic lichanos meson, called hyperhypate) and scalar gaps, which were alluded to by Aristoxenos as an indispensable ingredient in determining the ethos of the mode. Furthermore, one of the fragments, a portion of the first stationary chorus of Euripides’s Orestes, uses hyperhypate and the enharmonic in such a way as to prove that the middle tone of the pyknon (mesopyknon) was not merely a grace note, but a full member of the scale (Winnington-Ingram 1936). Ptolemy’s mixed scales Still more remote from the conventional theory are the mixed scales listed by Ptolemy in the Harmonics. These scales are ones that he said were in common use by players of the lyra and kithara in Alexandria in the second century ce (6-6). These scales bear some resemblance to modern Islamic modes containing 3/4-tone intervals, as does Ptolemy’s equable diatonic, 12/11» 11/10 + 10/9. They offer important support and evidence for the combination of tetrachords of varying genera and species to generate new musical materials. Permutation of intervals Although traditional techniques can generate a wealth of interesting material for musical exploration, the Greek writers suggested only a small _ fraction of the possibilities inherent in the permutations and combinations of tetrachords. While Aristoxenos mentioned the varying arrangements of SCALES, MODES, AND SYSTEMS

6-15. Permutations of sequentialfourths, See Wilson 1986forfurther details. This example begins with the Dorian mode ofthe standard ascending formfor clarity and consistency with other sections of this treatise, The sizes ofthefourths rangefrom 6/5 (316 cents) to 35/24 (653 cents). Interval 7 in the original sequence isa fixedfourth, The pair ofpermutedfourths are in boldface. The last tetrachord is Archytas's diatonic. zauzwern 1/1 ORIGINAL SCALE 28/27 16/15 4/3 3/2 14/9 8/5 2/1 28/27 36/35: 5/4 9/8 + 28/27 - 36/35 5/4 FOURTHS 1/1 to 4/3 4/3 to 8/5 8/5 to 16/15 16/15t014/9 14/9 t0 28/27 28/27 to 3/2 3/2 to 2/1 SIZE 43 6/5 4h 35/24 443 81/56 4h the intervals of the tetrachord in the different octave species, the Islamic theorists, such as Safıyu-d-Din, gave lengthy tables of all the permutational forms of tetrachords with two and three different intervals. However, the construction of 5-, 6-, and 7-tone scales from permuted tetrachords and trichords (gapped tetrachords) has been studied most thoroughly by the composer Lou Harrison (1975). Harrison constructed scales from all the permutations of the tetrachords and trichords and allowed different permutations in the upper and lower parts of the scale. In chapter 5, the melodic properties of scales constructed of either identical or dissimilar tetrachords, irrespective of permutational order, are analyzed according to the perception theories of David Rothenberg (1969, 1975, 1978; also Chalmers 1975). Wilson's permutations and modulations Perhaps the most sophisticated use to date of tetrachordal interval permutation in a generative sense is Ervin Wilson’s derivation of certain North Indian thats (raga-scales) and their analogs (Wilson 1986a; 1987). In “The Marwa Permutations” (1986a), Wilson’s procedure is to permute the order of the sequential fourths of heptatonic scales constructed from two iden- ORIGINAL SEQUENGE I 2 3 4/3 6/5 4/3 35/24 4/3 81/56 (4/3) 04 5 6 7 PERMUTED SEQUENCE I 3 204 4/3 4/3 6/5 35/24 4/3 5 6 7 81/56 (4/3) NEW SCALE 1/1 28/27 16/15 4/3 3/2 14/9 16/9 2/1 28/27 - 36/35 : 5/4 + 9/8 - 28/27 - 8/7 - 9/8 tical tetrachords. These sequential fourths are computed in the usual manner by starting with the lowest note of one of the modes and counting three melodic steps upwards. The process is continued until the cycle is complete and one is back to the original tone. The resulting seven fourths are the same as the adjacent fourths of the difference matrices of chapter 5, but in a different order. In abstract terms, if the intervals of the tetrachord are @-b/a. 4/3b, the scale is 1/1 ab 4/3 3/2 34/2 34/2, and 2/1. The sequential fourths from 1/1 are thus 4/3, 3/24, 34/25, 9b/8, 4/3, 4/3, and 4/3. Itis clear that these fourths must be of at least two different sizes even in Pythagorean intonation, While holding the position of one fourth constant to avoid generating cyclic permutations or modes, pairs of fourths are exchanged to create new sequences of intervals in general not obtainable by the traditional modal operations. Both the choice of the positionally fixed fourth and the arrangement of the tetrachordal intervals affect the spectrum of scales obtainable from a given genus. 6-15 illustrates this process with the enharmonic genus of Archytas. The exchange of the second and third fourths converts the upper tetrachord into CHAPTER 6

lamic Archytas’s diatonic and yields a mixed scale, half enharmonic and half diional atonic. Further application of this principle produces additional scales until t, the the original sequence is restored. Each of these scales could be modally sand (cyclically) permuted as well. y the Il the perither r, are 969, 6-16, Modulations by sequentialfourths. This example begins with the Dorian modefor consistency with other sections ofthis treatise. The sizes ofthe fourths rangefrom 6/5 (316 cents) to 35/24 (653 cents). In the original sequence the exceptionalfourth is in boldface. In the rotated sequence the scale has been modally permuted to separate the exceptional fourth (in boldface)from the rest. In thefirst modulated sequence the 6/5 (in boldface) has been interpolated betweenfourths 7 and 1 ofthe original perThe rder series, In the second modulated sequence the 6/5 (in boldface) has been interpolated betweenfourths 3 and Wilson derives a number of the thats of North Indian ragas by operating on various arrangements of the tetrachords 256/243 - 9/8 - 9/8, 16/15 9/8 - 10/9, 28/27 + 8/7 -9/8, 16/15 + 135/128 - 32/27, and 10/9: 10/9 - 27/25. He then generates analogs of these scales from other tetrachords, including those with undecimal intervals. In his 1987 paper, Wilson described a complementary technique of modulation (“The Purvi Modulations”). This technique makes use of the fact that at least one of the fourths differs greatly in size from the rest. The exceptional fourth may be abstracted from the linear fourth sequence and interpolated between successive pairs to generate derived scales. At the end of seven such interpolations, the linear sequence is cyclically permuted by one position and the process of interpolation continued. After 42 steps the 4 ofthe original series. The new tetrachord is Archytas’s diatonic. den- ORIGINAL SCALE ting 1/1 28/27 16/15 4/3 3/2 14/9 8/5 2/1 28/27 - 36/35 + 5/4: 0/8. 28/27 - 36/35 : 5/4 le is ter ord ear an RAR Y PE sual FOURTHS SIZE 1/1 TO 4/3 4/3 To 8/5 8/5 To 16/15 16/15 TO 14/9 14/9 To 18/27 28/27 To 3/2 3/2 To 2/1 4h 6/5 4/3 35/24 4/3 81/56 4/3 ng ew dal I 4/3 2 6/5 4/3 81/56 4/3 4 MODULATED SEQUENCE I 2 6/5 3 4 5 6 7 4/3 35/24 4/3 81/56 4/3 1/1 9/8 I 4/3 7/6 6/5 3/2 14/9 8/5 MODULATED SEQUENGE 2 3 4 5 6 7 4/3 35/24 4/3 81/56 4/3 3 4 § 6 7 4/3 35/24 4/3 81/56 4/3 I 4/3 3 2 7 I 4/3 6/5 35/24 4/3 81/56 4/3 4 5 6 4/3 NEW SCALE 2 1/1 9/8 7/6 4/3 3/2 14/0 8/55 2/1 9/8 + 28/27: 8/7 + 9/8 - 28/27 - 36/35 : 5/4 2 6/5 NEW SCALE tt 5/4 35/27 4/3 5/3 2/1 9/8. 28/27 : 36/35: 5/4 28/27 - 36/35 + 5/4 ROTATED SEQUENCE he 35/24 NEW SCALE I ORIGINAL SEQUENCE ir to THE LINEAR SEQUENCE OF FOURTHS 4/3 15/8 35/18 2/1 5/4 + 28/27 - 36/35 - 5/4 - 9/8 > 28/27 - 36/35 SCALES, MODES, AND SYSTEMS

original scale is restored, but transposed to a new and remote key. Wilson also provides an alternate derivation which better brings out the transpositional character of the process. In this case the linear sequence of non-exceptional fourths is tandemly duplicated to form a series of indefinite extent. Successive overlapping 6-unit segments of this series are appended with the exceptional fourth to form octave scales. After seven operations, the sequence repeats with a new mode of the original scale. The process is illustrated in 6-16. Non-traditional scale forms In the remainder of this chapter, some non-traditional approaches to scale construction from tetrachordal modules will be presented. These approaches are presented as alternatives to the historical modes and other types of scales which were discussed in the earlier parts of this chapter. The first group of non-standard tetrachordal scales is generated by combining a given tetrachord with an identical one transposed by one of its own structural intervals or the inversion of one of these intervals (6-17). This process yields 7-tone scales, including three of the traditional modes if the interval is 4/3, 3/2, or with a slight stretching of the concept, 9/8 and 6-17, Complexes ofone tetrachordalform. 3/2 together. The other tetrachordal complexes, however, are quite different from the historical modes. I. TRANSPOSITION BY 4 1/1 4 b 24 ab 4/3 44/3 2/1 7. TRANSPOSITION BY 9/8 & 3/2, HYPODORIAN 1/1 9/8 94/8 gb/8 3/2 34/2 36/2 2/1 2. TRANSPOSITION BY è 1/1 ab ab 2b 4/3 4b/3 2/1 8. TRANSPOSITION BY 4/35 1/14 b4/3b 4/3 galzb 16/96 2/1 3. TRANSPOSITION BY 4/3, MIXOLYDIAN 1/1 4b 4/3 44/3 4b/; 16/0 2/1 9. TRANSPOSITION BY 4/34 1/1 a b 4/34 4/3 4b/3a 16/94 2/1 4. TRANSPOSITION BY 3/2, DORIAN 1/14 b 4/3 3/2 3a/2 3b/2 2/1 1/1 a2/bab qa/3b 4/3 afb 2/1 5. TRANSPOSITION BY 2/b 1/14 b 4/3 2/b alb 4/3b 2/1 1/1 b/a ab bıla 4/3 ablza ılı 10. TRANSPOSITION BY 4/b I1. TRANSPOSITION BY b/a 6. TRANSPOSITION BY 2/4 1/14 bah ala bla 4/3a 2/1 CHAPTER 6

6-18 provides examples of the resulting scales when the generating tetrachord is Archytas’s enharmonic, 28/27 - 36/35 - 5/4. In this case interval a equals 28/27 and bis 16/15 (28/27-36/35). As some of these tetrachordal complexes have large gaps, one might try combining two of them, one built upwards from 1/1 and the other downwards from 2/1 to create a more even scale, though there are precedents for such gapped scales, i.e., the Mixolydian harmonia (6-4). While the normal ascending or prime form of the tetrachord—the one whose intervals are in the order of smallest, medium and largest—is used to demonstrate the technique, any of the six permutations would serve equally well. In fact, Archytas’s enharmonic and diatonic genera are not strictly of this form as 28/ 27 is larger than 36/35 and 8/7 is wider than 9/8. The next class of tetrachordal complexes are those composed of a tetrachord and its inverted form. 6-19 lists some simple examples of this ap6-18. Complexes ofthe prime form ofArchytas’s enbarmonie. proach; 6-20 lists the resulting notes in Archytas’s enharmonic tuning. These scales have six, seven, or eight tones. I. TRANSPOSITION BY4 1/1 28/27 16/15 784/729 448/405 4/3 112/81 2/1 o 63 112 126175 498 561 1200 7. TRANSPOSITION BY 9/8 & 3/2, HYPODORIAN 1/1 9/8 7/6 6/5 3/2 14/9 8/5 2/1 0 204 267 316 702 765 814 1200 2. TRANSPOSITION BY b 1/1 28/27 16/15 448/405 256/225 4/3 64/32 2/1 063 112 175 223 498 610 1200 1/1 28/27 16/15 5/4 35/27 4/3 5/3 2/1 063 112 386 449 498 884 1200 8. TRANSPOSITION BY 4/36 9. TRANSPOSITION BY 4/34 1/1 28/27 16/15 9/7 4/3 48/35 12/7 2/1 3. TRANSPOSITION BY 4/3 MIXOLYDIAN 1/1 28/27 16/15 4/3 112/81 64/45 16/9 24 063 112 498 561 610 996 1200 063 112 435 498 547 933 1200 10, TRANSPOSITION BY 4/b 1/1 245/243 28/27 16/15 35/27 4/3 35/18 2/1 o 14.63 112 449 498 1151 1200 4. TRANSPOSITION BY 3/2, DORIAN t/t 28/27 16/15 4/3 3/2 14/9 8/5 2/1 063112 498 702 765 814 1200 LI. TRANSPOSITION BY b/a 1/1 36/35 28/27 16/15 192/175 4/3 48/35 2/1 o 49 63 112 161 498 561 1200 5. TRANSPOSITION BY 2/b 1/1 28/27 16/15 5/4 4/3 15/8 35/18 2/1 o 63 112 386 498 1088 1151 1200 6. TRANSPOSITION BY 2/4 1/1 36/35 28/27 16/15 9/7 4/3 27/14 2/1 o 49 63 112 435 498 1137 1200 SCALES, MODES, AND SYSTEMS

I, TRANSPOSITION AND INVERSION BY 4, 6 TONES, A HEXANY 1/1 ab 4a/3b 4/3 4a/3 2/1 2. TRANSPOSITION AND INVERSION BY È, 6 TONES, A HEXANY 1/1 4 b 4/3 4b/3a 4b/3 2/1 6-19. Simple complexes ofprime and inverted forms. Two versions ofthe pseudo- (¥-) Hypodorian mode are shown to illustrate the effect ofreversing the placement ofthe prime and invertedforms. The two scales are not modes ofeach other. 3: TRANSPOSITION AND INVERSION BY 4/3, 7 TONES, W-MIXOLYDIAN 1/1 a b 4/3 16/9b 16/ga 16/0 2/1 4. TRANSPOSITION AND INVERSION BY 3/2, 7 TONES, W-DORIAN 1/1 4 b 4/3 3/2 2/b 2/a 2/1 5. TRANSPOSITION AND INVERSION BY 2/b, 8 TONES, AN OCTONY 1/1 ab 4/3 2/b 4/36 4/3ab 4/3b 2/1 6. TRANSPOSITION AND INVERSION BY 2/4, 8 TONES, AN OCTONY 1/1 ab 4/3 2/a 4/3a? 4/3ab 4/34 2/1 7: TRANSPOSITION AND INVERSION BY 9/8 & 3/2, 7 TONES, W-HYPODORIAN I 1/1 9/8 3/2b 3/24 3/2 3a/2 3b/2 2/1 8. TRANSPOSITION AND INVERSION BY 9/8 & 3/2, 7 TONES, W-HYPODORIAN 2 1/1 9/8 04/8 gb/B 3/2 2/b 2/a 2/1 9. TRANSPOSITION AND INVERSION BY I/I, 6 TONES, A HEXANY 1/14b4/3b4/3a 4/3 2/1 10. TRANSPOSITION AND INVERSION BY 4/35, 8 TONES, AN OCTONY 1/1 a b 4/3b 4/3 16/0b? 16/gab 16/9b 2/1 LI. TRANSPOSITION AND INVERSION BY 4/34, 8 TONES, AN OCTONY 1/1 ab 4/34 4/3 16/94b 16/94? 16/94 2/1 12, TETRACHORDAL HEXANY, 6 TONES, A-MODE 1/1 b/a b 4/3a 4/3 4bl3a 2/1 13. EULER'S GENUS MUSICUM, 8 TONES, AN OCTONY 1/1 ab ab 4/3 qal3 4b/3 4ab/3 2/1 14. TRANSPOSITION AND INVERSION BY B/A, 8 TONES, AN OCTONY 1/1 b/a a b 4/34 4/3 4b/3a2 4b/3a 2/1 15. TRANSPOSITION AND INVERSION BY A/B, 8 TONES, AN OCTONY 1/1 4b 4a/3b? 4/3b galzb 4/3 a/b zlı CHAPTER 6

The 7-tone scales are analogous to the traditional Greek modes, whose 6-20. Simple complexes of the prime and inverted forms ofArchytas’s enharmonic, in ratios and cents. Two versions of the Y-hypodorian mode are shown to illustrate the effect of reversing the placement of the prime and invertedforms. The two scales are not modes ofeach other. names are appropriated with a prefixed W (for pseudo) to indicate their relationship to the prototypes. Although these 7-tone scales were produced by pairing a tetrachord with its inversion, in principle any two dissimilar permutations would yield a heptatonic scale. This degree of flexibility is not true of the 6- and 8-tone types for which the pairing of prime and inverted forms is mandatory. I, TRANSPOSITION AND INVERSION BY 4, 6 TONES, A HEXANY 1/1 28/27 16/15 35/27 4/3 112/81 2/1 o 63 112 449 498 561 1200 9. TRANSPOSITION AND INVERSION BY I/1, 6 TONES, A HEXANY 1/1 28/27 16/15 5/4 9/7 4/3 2/1 © 63 112 386 435 498 1200 2, TRANSPOSITION AND INVERSION BY #, 6 TONES, A HEXANY 10. TRANSPOSITION AND INVERSION BY 4/35, 8 TONES, AN OCTONY 1/1 28/27 16/15 5/4 4/3 25/16 45/28 5/3 2/1 o 63 112 386 498 773 821 884 1200 1/1 28/27 16/15 4/3 48/35 64/45 2/1 0 63 112 498 547 610 1200 3. TRANSPOSITION AND INVERSION BY 4/3, 7 TONES, W-MIXOLYDIAN 1/1 28/27 16/15 4/3 5/3 12/7 16/9 2/1 o 63 112 498 884 933 996 1200 4. TRANSPOSITION AND INVERSION BY 3/2, 7 TONES, W-DORIAN 1/1 28/27 16/15 4/3 3/2 15/8 27/14 2/1 o 63 112 498 702 1088 1137 1200 12, TETRACHORDAL HEXANY, 6 TONES, A-MODE 5. TRANSPOSITION AND INVERSION BY 2/b, 8 TONES, AN OCTONY 1/1 28/27 16/15 75/64 135/112 5/4 4/3 15/8 2/1 063 112 275 323 386 498 1088 1100 6. TRANSPOSITION AND INVERSION BY 2/4, 8 TONES, AN OCTONY 1/1 28/27 16/15 135/112 243/196 9/7 4/3 27/14 2/1 o 63 112 323 372 435 498 1137 1200 7. TRANSPOSITION AND INVERSION BY 9/8 & 3/2, 7 TONES, W-HYPODORIAN I 1/1 9/8 45/32 81/56 3/2 14/9 8/5 2/1 0 204 590 639 702 765 814 1200 8. TRANSPOSITION AND INVERSION BY 9/8 & 3/2, 7 TONES, W-HYPODORIAN 2 1/1 9/8 7/6 6/5 3/2 15/8 27/14 2/1 0 204 267 316 702 1088 1137 1200 IIS II. TRANSPOSITION AND INVERSION BY 4/34, 8 TONES, AN OCTONY 1/1 28/27 16/15 9/7 4/3 45/28 81/49 12/7 2/1 o 63 112 435 498 821 870 933 1200 1/1 36/35 16/15 9/7 4/3 48/35 2/1 O 49 112 435 498 547 1200 13. EULER'S GENUS MUSICUM, 8 TONES, AN OCTONY 1/1 28/27 16/15 448/405 4/3 112/81 64/45 1792/1215 2/1 o 63 112 175 498 561 610 673 1200 14. TRANSPOSITION AND INVERSION BY b/a, 8 TONES, AN OCTONY 1/1 36/35 28/27 16/15 9/7 324/245 4/3 48/35 2/1 0 49 63 112 435 484 498 561 1200 15. TRANSPOSITION AND INVERSION BY 4/b, 8 TONES, AN OCTONY 1/1 28/27 16/15 175/144 5/4 35/27 4/3 35/18 2/1 o 63 112 338 386 449 498 1151 1200 SCALES, MODES, AND SYSTEMS

6-21. Ther 3 5 7 tetradic hexany. Thefactor 1 may be omittedfrom the three tones which contain it. This diagram was invented by Ervin Wilson and represents the six tones ofthe bexany mapped over the six vertices ofthe regular octabedron (Wilson 1989). Each triangularface is an essential consonant chord ofthe hexany harmontic system and every pair of tones separated by a principal diagonal is a dissonance. The keynote is 3.5. Tetrachordal hexanies ‘The 6-tone complexes are of greater theoretical interest than either the seven or 8-tone scales. Because of their quasi-symmetrical melodic structure, which is a circular permutation of the interval sequence c ba è cd (a, b, c, and d not necessarily different intervals), they are members of a class of scales discovered by Ervin Wilson and termed combination product sets (Wilson 1989; Chalmers and Wilson 1982; Wilson, personal communication). The same structure results if interval 4 is replaced with interval d and intervals b and care exchanged. A combination product set of six tones is called a hexany by Wilson. The notes of the hexany are the melodic expansion of the intervals of a I+3 3°5 generating tetrad or tetrachord. They are obtained by forming the six binary products of the four elements of the generator. If these four elements are labelled x, y, z, and w, the resulting notes are x - y, x: 2,7 + W, y+ 2, y-w, and w :z. In the case where the generator is the dominant seventh tetrad, 1/1 5/4 3/2 7/4, written in factor form as 1 3 5 7, the resulting hexany is that of 6-21, where it has been mapped over the vertices of a regular octahedron. This diagram has been named a “hexagram” by Wilson. 1:7 3°7 reg NOTES AND INTERVALS OF HEXANY 1/1 7/6 7/6 c 8/7 b a 7/5 21/20 a 8/5 8/7 b 28/15 7/6 £ 15/14 d 6-22. Consonant chords of the 1 3 5 7 hexany, 2/1 It is convenient to choose one of these tones and transpose the scale so that it starts on this note. The note 3 - 5 has been selected in 6-21. This note, however, should not be considered as the tonic of the scale; the combination product sets are harmonically symmetrical, polytonal sets with virtual or implicit tonics which are not tones of the scale. Although the hexany is partitionable into a set of rooted triads (see below), the global 1/1 for the whole set is not a note of the scale. In this sense, combination product sets are a type of atonal or non-centric musical structure in just intonation. The four elements of the generator are related to the melodic intervals asx=1/1,y=b,z=bh-c,andw=a-b?. c, although the actual tones may have to be transposed or circularly permuted to make this relationship clearer. CHORD HARMONIC SUBHARMONIC 135 137 157 357 17 15 13 13 35 1-5 13 37 1713 5717 15 57 37 35 116 CHAPTER 6 37 35 35 15 57

6-23. The tetrachordal bexany. Based on the generating tetrad 1/1 a b 4/3. After transposition by a, it is equivalent to complex 12 of 6-19 and 6-20. 1-44 The six tones of the hexany may be partitioned into four sets of three tones and their inversions. In the hexagram or octahedral representation, the 3-tone sets appear as triangular faces or facets. The triads of 6-21 are tabulated in 6-22. These chords are the essential consonant chords of the hexany, and all chords containing pairs of tones separated by diagonals are considered dissonant. Armed with thís background, one can now proceed to the generation of hexanies from tetrachords. Starting with the tetrachord 1/1 a b 4/3 (the generator of complex 12 in 6-19), the generative process and the relationships between the notes may be seen in 6-23. Archytas’s enharmonic (1/1 28/27 16/15 4/3; 28/27 36/35 5/4; a= 28/27, b = 16/15) is the specific generator (see also 6-20, complex 12). This hexany has been transposed so that the starting note 1-4 is 1/1. Tetrachordal hexanies are melodic developments of the basic intervals 4413 4- biz a:b tetrachord. Since this is basically a melodic development, the faces will be referred to as essential subsets rather than consonant chords, (For the same NOTES AND INTERVALS OF HEXANY 1/1 bla b 1/1 36/35 16/15 ala 9/7 rather than harmonic expansions of tetrads. The triangular faces of tetrachordal hexanies are 2-interval subsets of the three intervals of the original 4h 4/3 ab/za 48/35 2/1 2/1 36/35 28/27 135/112 28/27 36/35 35/24 € b 4 b c d reason, the terms harmonic and subharmonic are replaced by prime and inverted.) These hexanies may be partitioned into essential subsets as shown in 6-24. The generator of complex 1 of 6-19 and 6-20 (inversion and transposition by 4) is the permuted tetrachord 1/1 b/a b 4/3 (1/1 36/35 16/15 4/3; 36/35 « 28/27 : 5/4; a = 36/35, b = 16/15). The generators of complexes zand g are 1/1 b/ab 4b/3a (1/1 36/35 16/15 48/35; 36/35 + 28/27 « 9/7) and 6-24. Essential subsets ofthe hexamies based on the tetrachords 1/1 a b 4/3 and 1/1 28/27 16/15 4/3 (Archytas’s enbarmonic). For the sake ofclarity, thefactor 1 (1/1) bas been omittedfrom 1-a, 1-b, and 1-4/3. The + signs are also deleted. Both bexanies are given in their untransposed forms. SUBSET rab 1/1 0443 1/1 b 4/3 ab 453 PRIME 4/3 44/3 4b/3 b ab 45/3 a ab 44/3 a b 483 INVERTED abba 44/3 4/3 a 43/3 4/3 b 41/3 44/3 ab 1/1 28/27 16/15 1/1 28/27 4/3 1/1 16/15 4/3 28/27 16/15 4/3 4/3 112/81 64/45 448/405 16/15 28/27 112/81 4/3 28/27 64/45 4/3 16/15 64/45 112/81 448/405 IIY 16/15 448/405 64/45 28/27 448/405 112/81 28/27 16/15 4/3 SCALES, MODES, AND SYSTEMS

6-25. Ther 3 5 7 tetradic octony. This structure is also an Euler's genus (Fokker 1966; Euler 1739). 1/1 b/a b 4/34 (1/1 36/35 16/15 9/7; 36/35 - 28/27 - 135/112) respectively. In these hexanies, the tetrachordal generators are bounded by augmented and diminished fourths rather than 4/3’s, but the subset relations are analogous to those with perfect fourths. Tetrachordal Euler genera The 8-tone complexes represent a different type of scale which may be 57 3°5 called an interval symmetric set (Chalmers and Wilson 1982; Chalmers 1983). These scales have the melodic sequence de ba bc d e which is homologous to the ¢ è 4 è c d sequence of the hexany. However, these 8-tone scales lack some of the harmonic and structural symmetries that characterize the combination product sets. Wilson has pointed out that these sets are members of a large class of scales invented by Leonhard Euler in the eighteenth century and publicized by A. D. Fokker (Wilson, personal communication). While they have been given the generic name of octony in analogy with the hexany 1.7 and other combination product sets, the terms Euler genus or EulerFokker genus would seem to have priority as collective names (Fokker 1/1 1966; Rasch 1987). The generation of an octony from the 1 3 5 7 tetrad is shown in 6-25. In this representation, the eight tones have been mapped over the vertices of a cube. This diagram may be called an “octagram.” The octony may also 6-26. Essential chords of the 1 3 5 7 tetradic octony. CHORD FACE VERTEX PRIME 1/1 1-3 135 3-7 1/1 1-5 16 3:5 1/1 1:7 15 5-7 V1l3 15107 1-7 57 1/1 37 1-5 1/1 5735 1:3 3:5 3:7 1/1 DIAGONAL I/I 5:7 3:5 3-7 INVERTED 371535357 175737357 1:3 3:7 35 357 357353757 151335357 371317357 571715357 357151317 be partitioned into inversionally paired subsets, but the chords are generally more complex than those of hexanies derived from the same generator (6-26). Chords considered as the essential consonances of a harmonic system based on the octony appear not only as faces (face chords), but also as vertices with their three nearest neighbors connected by edges (vertex chords) or by face diagonals (vertex-diagonal chords) (Chalmers 1983). Essential dissonances are any chords containing a pair of tones separated by a principal diagonal of the cube. With the exception of the generator itself and its inversion, each of the 4-note chords consists of the union of a harmonic and subharmonic triad of the form 1/1 x y and x y x-y. An analogous chord in traditional theory is the major triad with the major seventh added, 1/1 5/4 3/2 15/8, which could be construed as a major triad on 1/1 fused with a minor triad on 5/4. As in the case of the hexany, octonies may be constructed from tetrachords and their inversions (6-27). The clearest example is complex 13 of CHAPTER 6

6-18 which is generated by the tetrachord 1/1 a è 4/3. Its subset structure 6-27. The tetrachordal octony. This 8-tone Euler genus is generatedfrom the generalized tetrachord ala a b 4/3. is shown in 6-28. The generating tetrachord and its inversion appear as face chords. The other chords are more complex intervallic sets. Like the hexany above, the octony should be viewed as a melodic rather than a harmonic development of the tetrachord. The other 8-tone complexes of 6-19 are also octonies. The complexes generated from Archytas’s enharmonic genus are listed in 6-20. qb/3 4/3 Tetrachordal diamonds aab/3 The next group of non-traditional tetrachordal scales is even more complex than the previous constructions. The first of these are based on the Partch diamond (Partch [1949] 1974) which is an interlocking matrix of harmonic NOTE AND INTERVALS OF OCTONY ı/ı ab 1/1 1/1 a 28/27 b ab 4/3 16/15 448/405 4/3 4433 112/81 4b/3 64/45 44b/3 1792/1215 2/1 24 28/27 + 36/35 28/27: 135/112 + 28/27 > 36/35 + 28/27 + 1215/896 d 6 b a b c d e 6-28. Essential subsets ofthe tetrachordal octonies 1/1 ab4/3 and 1/1 28/27 16/15 4/3 (Archytas’s enharmonic). The term essential subset rather SUBSET PRIME INVERTED FACE 1/1 4/3 4aha 1/1 4/3 46/36 gab/3 ab b 4b/; 4ab/3 ab a 44/3 w/t 4 b ab 4ab/3 4bl3 gals 4/3 1/1 4 b 4/3 4/3 1/1 44/3 42/3 a 4/3 46/3 4/3 b 1/1 40/3 44/3 ab 446/3 4b/y 40/3 ab 4b/3 ab qab/z ba qablzb 4b/3 ab 1/1 qablza ab qal3 1/1 qab/3 ab 4/3 1/1 4/3 112/81 28/27 1/1 4/3 64/45 16/15 1/1 28/27 16/15 448/405 t/t 28/27 16/15 4/3 4/3 1/1 112/81 64/45 112/81 28/27 4/3 16/15 64/45 4/3 16/15 28/27 1/1 64/45 112/81 448/405 1792/1215 448/405 16/15 64/45 1792/1215 448/405 28/27 112/81 1792/1215 64/45 112/81 4/3 1792/1215 64/45 112/81 448/405 448/405 1792/1215 16/15 28/27 64/45 448/405 1/1 1792/1215 448/405 112/81 1/1 1792/1215 1792/1215 28/27 16/15 4/3 VERTEX than consonant chord és employed as the tetrachordal octony isprimarily a melodic structure. DIAGONAL FACE VERTEX DIAGONAL SCALES, MODES, AND SYSTEMS

chords built on roots that are the elements of the corresponding subharmonic ones. An example of what is called a 5-/it diamond may be seen in 6-30. This example has been constructed from harmonic 1 3 5; major triads and subharmonic 1 3 5; or minor triads, The structure is referred to as having a 5-limit because the largest prime number appearing among its ratios is five. Diamonds, however, may be constructed from any chord or scale of any cardinality, magnitude, or limit. The simplest of the tetrachordal diamonds consists of ascending tetrachords erected on the notes of their inversions. Either the octave or the 4/3 (numbers 1 and 2 of 6-29) may be used as the interval of identity in the diamond. In the latter case, the resulting structure is one of the rare examples of musical scales in which the octave is not the interval of equivalence, The second group of diamond-like complexes employs entire heptatonic scales in place of triads or tetrachords as structural elements. Four examples are given, all derived from scales of the Dorian or Y-Dorian type in which. prime or inverted tetrachords appear in either or both positions relative to the central disjunctive tone (6-29, numbers 2, 4, 5; and 6-34). The primeprime and inverted-inverted diamonds have prime or inverted tetrachords in both halves of the generating scales. Because of the inversional symmetry 6-29. Tetrachordal diamonds. The octave modular tetrachordal diarnond in I, THIRTEEN TONE OCTAVE MODULAR DIAMOND 1/1 bla a b4/3b 4134 4h; 3/2 30/2 34/2 2/b ala a/b 2/1 Archytas's enharmonic tuning is shown 2. EIGHT TONE FOURTH MODULAR DIAMOND 1/1 a b 4/3b 4/3a galzb 4/3 45534 in 6-33. 3. PRIME-PRIME AND INVERTED-INVERTED HEPTATONIC DIAMONDS, 27 TONES t/t b/a a 8 9/8 04/8 ob/8 4/36 4/34 qal3b 4/3 4b/30 galy 3/26 4b/3 3/20 zalab 3/2 36/24 34/2 30/2 16/9b 16/94 16/0 2/b 2/a a/b 2/1 4. PRIME-INVERTED HEPTATONIC DIAMOND, 25 TONES 1/1 bla aba? ab 9/8 b? 4/3b 4/3a 4/3 gals 3/2b 4bl33/2a 3/2 3a/2 36/2 2/b? 16/9 2/ab 2/a? 2/b 2/a a/b 2/1 5. INVERTED-PRIME HEPTATONIC DIAMOND, 25 TONES 1/1 b/a a b 9/8 94/8 9b/8 94°/8 gab/8 4/3b ob*/8 4/344/3 3/2 34/2 16/962 35/2 16/gab 16/ga? 16/9b 16/94 16/92/b 2/a a/b 2/1 CHAPTER 6

Partch diamond, after “The Incipient Five-limit Tonality Diamond” (Partch [1949] 1974, 6 "32. Thiirteen-tone 110). Based on the 13 5 major triad 1/1 5/4 3/2 and its inversion, the subbarmonic 1 3 5 minor triad 2/1 di octave modular tetrachordal | 8/5 4/3. 6-31. Eight tonefourth modular diamond, Based on 6-33. Thirteen-tone octave modular tetrachordal the tetrachord 1/1 ab 4/3, with 4/3 as the interval diamond based on Archytas’s enharmonic genus. ofequivalence. D È E> Se SCALES, MODES, AND SYSTEMS

of the diamond, both scales are identical. The prime-inverted and inverted-prime diamonds are constructed from the corresponding tetrachordal forms and are non-equivalent scales, as in general, tetrachords are not inversionally symmetrical intervallic sequences. 6-35 and 6-36 show examples of these diamonds based on Archytas’s enharmonic genus and its inversion. 6-34. Tetrachordal beptatonic diamonds. These tables may be rotated 45 degrees clockwise to bring Stellated tetrachordal hexanies the diagonal of 2/1 's into vertical position and compared tofigures 6-30-33. The scale derivedfrom the primeform ofthe tetrachord is seen in the rightmost column and its inversion in the bottom row. The last of the non-traditional tetrachordal complexes to be discussed are two examples of stellated hexanies. Hexanies may bestellated by adding the eight tones which complete the partial tetrad or tetrachord on each face (Wilson 1989; Chalmers and Wilson 1982). The result is a complex of four PRIME-INVERTED PRIME-PRIME 2/1 bla bo gh/8 3/2 3b/2a 3b/2 2/1 bla 4/34 3/24 z/ab ılar 21/4 alb 2/17 a 94/8 zahb 3/2 34/2 a/b 2/1 4/35 3/2b z2/b2 z2/ab 2/b a/b la 2/1 9/8 yhb 3/18 3/2 34/2 3bla 2/1 9/8 3/2b 3/2a 3/2 16/9b 16/94 16/9 2/1 4b ala 4h 42/3 ab 16/9 2/1 4/36 4/34 4h 4/3 aba 453 3b/2 2/1 bla b ab ba ab 3b/2 2/1 bla b qalzb 4/3 gab 34/2 afb 21/1 a a2 ab 4a/3 3al2 afb 2/1 4 4/3b ala 4/3 3/2 zb 2/a t/t 4 b 4/3 3/2 z2/b ala 1 INVERTED-INVERTED INVERTED-PRIME 2/1 bla alzb 3/2a 3/2 3b/2a 21/4 2/1 bla b 95/8 gab/B 9b2/8 3b/2 alb 2/1 4/3b 3/26 zahb 3/2 a/b alb 2/1 a 94/8 ga2/8gab/8 34/2 34/2 3b/2 2/1 9/8 galb € ghia 3/2 2/b la 2/1 9/8 94/8 06/8 3/2 44/3 ab 16/9 2/1 a b 44 16/9b 16/ga 16/9 2/1 a b 4h 4/5 gba 16/ga 2a 2/1 bla 4/3a 16/gab 16/0942 16/94 2/4 2/1 ba 4/34 qalzb 4/3 16/06 2/b atb alı 4/3b 16/9h2 16/gab 16/9h z2/b alb 2/1 4/3b 4 b 4/3 gala 3bla vi 4/3b 3/2 34/2 3bla 1/1 3/2 122 CHAPTER 6

prime and four inverted tetrachords with a total of fourteen tones, though certain genera may produce degenerate complexes with fewer than 14 different notes. Wilson has variously termed these structures “mandalas” from their appearance in certain projections, and “tetradekanies” or “dekatesseranies” from their fourteen tones. Their topology is that of Kepler's stella octangula, an 8-pointed star-polyhedron (Coxeter 1973; Cundy and Rollett 1961). The prime form of the tetrachord 1/1 4 b 4/3 generates the hexany tones a, b, 4/3, 44/3, 4b/3 and ab (a = 1/1-a or 1-4, etc.). This hexany is equivalent 6-35. Tetrachordal diamonds based on Archytas's enharmonic, in ratios and cents. 1/1 le) 13-TONE OCTAVE MODULAR DIAMOND 14/9 3/2 4/3 9/7 765 702 498 435 5/4 386 16/15 112 28/27 63 36/35 49 35/18 1151 27/14 1137 15/8 1088 8/5 814 2/1 1200 8-TONE TETRACHORD MODULAR DIAMOND 1/1 28/27 16/15 o 63 112 5/4 9/7 35/27 4/3 48/35 386 435 449 498 547 = PRIME-PRIME AND INVERTED-INVERTED HEPTATONIG DIAMONDS, 27 TONES r/ı 28/27 36/35 o 49 63 45/32 590 64/45 610 81/56 639 1/1 (e) 36/35 49 28/27 63 45/32 590 64/45 610 81/56 639 36/35 1/1 (e) 49 9/7 435 4/3 498 3/2 702 4/3 48/35 112/81 449 498 547 561 9/8 7/6 6/5 112 204 267 316 386 435 5/3 884 12/7 0933 Wo 765 54/35 751 3/2 702 8/5 814 16/9 996 PRIME-INVERTED HEPTATONIC DIAMOND, 25 TONES 256/225 9/8 448/405 784/729 223 204 175 126 16/15 112 14/9 765 8/5 814 225/128 977 16/9 996 405/224 1025 729/392 1074 INVERTED-PRIME HEPTATONIC DIAMOND, 25 TONES 98/81 6/5 7/6 9/8 16/15 330 316 267 204 112 28/27 63 14/9 765 35/27 16/15 35/24 653 3/2 702 9/7 5/4 25/16 773 8/5 814 45/28 821 123 81/49 870 5/3 884 12/7 933 15/8 996 SCALES, MODES, AND SYSTEMS 27/14 1137 15/8 1088 4/3 498 9/7 435 5/4 386 15/8 1088 27/14 1137 35/18 IISI 35/18 1137 2/1 1200 112/81 561 2/1 1200 32/25 427 5/4 386 56/45 379 27/14 1088 35/18 IISI 2/1 IIsI

6-36. Tetrachordal heptatonic diamonds based on Archytas’s enbarmonic. The generating tetrachords are 1/1 5/4 9/7 4/3 and 1/1 28/27 16/15 4/3. PRIME-INVERTED PRIME~PRIME 2/t 3645 16/15 6/5 3/2 54/35 B/5 2/1 36/35 9/7 81/56 405/224 729/392 27/14 35/18 2/1 28/27 7/6 35/24 3/2 14/9 35/18 2/1 5/4 45/42 225/128 gos/224 15/8 15/8 27/14 2/1 9/8 45/32 81/56 3/2 14/9 8/5 2/1 9/8 45/32 81/56 3/2 54 12/7 16/9 2/1 5/4 9/7 4 112/81 64/45 16/9 2/1 5/4 9/7 4/3 4h 4835 64/45 8/5 2/1 36/35 16/15 448/405 256/225 64/45 8/5 2/1 36/35 16/15 35/27 4h 112/81 14/9 35/18 2/1 28/27 784/729 448/405 112/81 14/9 35/18 2/1 28/27 5/4 9/7 4/3 3/2 15/8 27/14 1/1 28/27 3/2 15/8 27/14 1/1 16/15 43 INVERTED-INVERTED INVERTED-PRIME 2/1 36/35 9/7 81/56 3/2 54/35 27/14 2/1 36/35 16/5 6/5 56/45 32/25 8/5 35/18 2h 5/4 45/32 35/24 3/2 15/8 35/18 2/1 28/27 7/6 98/81 56/45 14/9 14/9 8/5 2/1 9/8 7/6 6/5 3/2 15/8 27/14 2/1 9/8 7/6 6/5 3/2 112/81 64/45 16/9 2/1 28/27 16/5 4/3 54 12/9 16/9 2/1 28/27 16/15 4/3 4/3 48/35 12/7 27/14 2/1 36/35 9/7 45/28 81/49 12/7 27/14 2/1 36/35 9/7 35/17 4h 5/3 15/8 35/18 2/1 5/4 25/16 45/28 5/3 15/8 35/18 2/1 5/4 28/27 16/5 43 3/2 14/9 8/5 1/1 5/4 9/7 4/3 3/2 14/9 8/5 1/1 6-37. Stellated bexanies generated by the prime tetrachord 1/1 ab 4/3. The bexany notes are a, b, 4/3, ab, 4a/3, and #b/3. The 8 extra notes are (1/1)2=1/1, a, bf, 16/9, 3ab/2, gab/3, 4a/3b, and 4b/3a. The second stellated bexany is based on number 1 of figure 6-29. Instances ofeach are based on Archytas’s enharmonic. The first is generated by prime tetrachord 1/1 28/27 16/15 4/3. The bexany notes are 28/27, 16/15, 4/3, 448/405, 112/81, and 64/45. The second is based on (1) of 6-20. FIRST STELLATED TETRACHORDAL HEXANY be gal3b 4/3 qb/3a 3ab/2 16/9 2/1 44/3 45/3 4ab/3 1 28/27 16/15 784/729 448/405 256/225 35/27 4/3 48/35 112/81 64/45 1792/1215 224/135 16/9 2/1 63 112 126 175 223 561 610 673 877 996 1200 449 498 547 ~ 1/1 I 4 b al ab SECOND STELLATED TETRACHORDAL HEXANY I + ~~ 1/1 b/a ba? b bè/a be 4/34 4/3 4blza ga 36/35 1296/1225 16/15 192/175 256/225 0/7 4/3 48/35 112/81 98 II2 161 223 435 498 547 561 49 124 CHAPTER 6 qblz q4blza 3b/2a 16/9 2/1 64/45 610 256/175 288/175 16/9 2/1

to complex 12 of 6-19 when transposed so as to begin on the tone a. The stellated form of this hexany is the first of 6-37, while complex 1 of 6-19 yields the second of 6-37. The eight supplementary tones of the first stel- 6-38. (a) Essential tetrachords of the first stellated hexany. For the sake of clarity, the factor 1 (1/1) bas been omitted from 1 <a, 1-b, 1 + 4/3, etc. The - signs are also deleted, The boldfaced notes in each lated hexany are 1/1, 42, b?, 16/9, 44/3b, 4ab/3, zab/2, and 45/34. These notes may be deduced by inspection of 6-23, the tetrachordal hexany. The first four extra notes are the squares of the elements of the generator, 1/1, 42, b?, and 16/9 (x?, y?, 22, and w?) from 1/1 4 band 4/3. The remaining four notes chord are the starting notes of the prime and inverted tetrachords, 1/1 a b 4/3 and 4/3 4/3a are the mixed product-quotients needed by the subharmonic faces. These have the form x-y-z/w (3ab/2), x-y-w/z (44/36), x-2-w/y (4b/3a), and y-z-w/x (4ab/3). Two stellated hexanies based on Archytas’s enharmonic are shown in 6-37. The notes of the second type of stellated hexany of 6-30 are derived analogously by replacing 4 in the prime tetrachord with 2/4. The tetrachord 1/1 28/27 16/15 4/3 in the first type is thus replaced by 1/1 36/35 16/15 4/3. The essential tetrachords of the first stellated hexany are seen in 6-38, and those of the second may be found by analogy. The component tetra- 4/3b 1/1. PRIME w/t 4 È INVERTED 4h 4/3 44/3 4b/3 16/9 4/3 ab b ab br 4 40/3 4/3 gab a a rr 4 a 4bl3 ablza 4/3 40b/3 4b/3 4al3 b ab ab 44/3 b ab aa ab 1/1 b a 3ab/2 chords of the first stellated hexany derived from Archytas’s enharmonic are listed in 6-39. Those of the second kind may be derived by replacing the 28/27 of the first tetrachord with 36/35. The other tetrachordal hexanies of 6-18 also generate stellated hexanies, but their tetrachords are bounded by intervals other than 4/3. 6-39. Essential tetrachords ofthe 1/1 28/27 16/15 4/3 stellated hexany. 1/1 4/3 16/15 PRIME 16/15 28/27 64/45 112/81 448/405 256/225 4/3 16/9 64/45 4/3 448/405 112/81 INVERTED 5/4 9/7 28/27 16/15 35/27 4/3 28/27 784/729 448/405 112/81 64/45 48/35 4/3 16/15 1/1 28/27 16/5 43 1792/1215 64/45 112/81 448/405 125 v/ı SCALES, MODES, AND SYSTEMS

7 Harmonization of tetrachordal scales SCALES BASED ON tetrachords are found in the musics of a large part of the world. Although much of this music is primarily melodic and heterophonic, this is due neither to the intrinsic nature of tetrachords nor to the scales derived from them. Rather, it is a matter of style and tradition. Many, if not most, tetrachordal scales have harmonic implications even if these implications are contrary to the familiar rules of European tonal harmony. The melodies of the ancient Greeks were accompanied by more or less independent voices, but polyphony and harmony in their traditional senses appear to have been absent. “A feeling for the triad,” however, does appear in the later Greek musical fragments, but this may be a modern and not ancient perception (Winnington-Ingram 1936). The scales of North Indian music are also based on tetrachords (Sachs 1943; Wilson 1986a, 1987). In this music, drones emphasizing the tonic and usually the dominant of the scale are essential elements of performance. Their function may be to fix the tonic so that ambiguous intervals are not exposed (chapter 5 and Rothenberg 1969, 1978). Islamic music of the period of the great medieval theorists Al-Farabi, Safiyu-d-Din, and Avicenna (Ibn Sina) was likewise heterophonic rather than harmonic (Sachs 1943; D’Erlanger 1930, 1935, 1938). In recent times, however, some Islamic groups have adopted certain elements of tonal harmony into their music. Harmonizing tetrachordal scales Many tetrachordal scales are nevertheless suitable for harmonic music. The HARMONIZATION OF TETRACHORDAL SCALES

Lydian mode of Ptolemy’s intense diatonic genus is the just intonation of the major mode. The diatonic Arabo-Persian scale bbidjazi, is more con7-1. Endogenous harmonization of tetrachordal scales. The addition of the subtonic 9/8 below 1/1 to the enharmonic and chromatic genera where it was called byperbypate is attested both theoretically and musically (Winnington-Ingram 1936, 25). The dotted lines indicate the lower octave ofthe dominant of the triads on 4/3. (8/9) 1/1 a ab 4/3 3/2 34/1 zablı 2/1 (9/4) sonant than the 12-tone equal-tempered tuning of the major scale (Helmholtz [1877] 1954). Harry Partch pointed out that many of the other tetrachordal genera also have harmonic implications which may be exploited in the context of extended just intonation (Partch [1949] 1974). As an example, he offered Wilfrid Perrett's harmonization of a version of the enharmonic tetrachord. Partch added a repeat to Perrett’s progression and transposed it into his 43-tone scale (Partch [1949] 1974; Perrett 1926). Partch also challenged his readers to limit themselves to the notes of the scale. 7-1 depicts the triadic resources of a generalized tetrachordal scale in which both tetrachords are identical. The dark lines delimit triads which are available in all genera while the light ones indicate chords which may or may not be consonant in certain genera. The three sub-intervals of the tetrachord are denoted as a, b, and 4/3ab, resulting in the tones, 1/1, a, ab, and 4/3, duplicated on the 3/2. Because there is both musical and literary evidence for the customary addition of the 7-2. Endogenous harmonization ofArchytas's enharmonic. (8/9) 1/1 28/27 16/15 4/3 3/2 14/0 8/5 2/1 note hyperhypate a 9/8 whole tone below the tonic in the enharmonic and chromatic genera (Winnington-Ingram 1936, 25), it has been included. The inversion of this interval has also been added to allow the construction of a consonant dominant triad in some genera or permutations. The types of these triads depend upon the tuning of the tetrachord. In Archytas’s enharmonic genus, the triads on 4/3 and 8/9 will be septimal minor, 6:7:9. The triad on 4 (28/27) is the septimal major triad, 14:18:21. The triad on ab (16/15) is a major triad, 4:5:6, and the alternative triads on 4/3 and 8/9, are minor, 10:12:15, The tonal center appears not to be the 1/1, but rather the 4/3 or mese. These chords are shown in 7-2. The tonal functions of these triads are determined by the mode or circular permutation of the scale. The Lydian or C mode of Ptolemy’s intense diatonic, in its normal form, 16/15 : 9/8 : 10/0, is the familiar major mode with 4:5:6 triads on 1/1, 4/3, and 3/2. The reverse arrangement of this tetrachord, 10/9 : 9/8 : 16/15, generates the natural minor mode with 10:12:15 or subharmonic 4:5:6 triads on these degrees. This scale is not identical to the Hypodorian or A mode of the first scale because that scale has a 27/20 rather than a 4/3 as its fourth degree. The chordal matrices and tetrachordal forms of these scales are shown in 7-3. CHAPTER 7

The seven modes or octave species of the reversed tetrachord scale are the exact inversions of those of the major scale above. The C mode of this 7-3. The 4:5:6 triad and its derived tritriadic scale. scale is the diatonic scale of John Redfield (1928, 191-197). Redfield as- The tritriadic or matrixform is the C or Lydian mode ofthe tetrachordal scale. The tonic ofthe triad signed Hebraic names to these modes and termed the triads with the comma-enlarged fifth “Doric.” is denoted t or 1/1, the third or mediant, m and the Sift or dominant, d. The tetrachordalform is the E or Dorian mode ofthe tritriadic scale. The mode that is the inversion of the major scale may be harmonized SUBDOMINANT 4/3 5/3 2/1 TONIG 1/1 5/4 3/2 2/d m/d 2/1 1/1 md DOMINANT ddmd 3/2 15/8 9/8 with three triads built downwards from 2/1, 3/2, and 4/3. An otherwise obscure composer named Blainville wrote a short symphony in this scale and was ridiculed by Rousseau for doing so (Perrett 1931; Partch [1949] 1974). This kind of inverted harmony was called the phonic system by the nineteenth and early twentieth century theorist von Ottingen (Helmholtz [1877] 1954; Mandelbaum 1961) in contrast to the traditional tonic w/t 9/8 5/4 4/3 3/2 56/3 15/8 2/1 9/8 - 10/9 - 16/15 - 9/8 - 10/9 : 9/8 : 16/15 system. THE TETRACHORDAL FORM 1/3 16/15 6/5 4/3 3/2 Ba 9/5 2/1 16/15 : 9/8: 10/9 - 9/8 - 16/15 : 9/8. 10/9 Tritriadic scales The scales derived from tetrachords with 9/8 as their second interval may (16/15 + 9/8 : 10/9) roots 1/1, 4/3, and 3/2. They are harmonizable with analogs of the familiar THE 10:12:15 TRIAD & ITS DERIVED TRITRIADIC SCALE SUBDOMINANT 4/3 8/5 2/1 TONIC t/t 6/5 3/2 2/d mid 2/1 1/1 md DOMINANT d dm d? 1/1 9/8 3/2 9/5 9/8 6/5 4/3 3/2 8/5 o/5 2/1 9/8. 16/15 - 10/9 - 9/8 - 16/15 : 9/8- 10/9 THE TETRACHORDAL FORM 1/1 10/9 5/4 4/3 3/2 5/3 15/8 2/1 10/9 : 9/8 . 16/15 + 9/8 » 10/9 - 9/8 + 16/5 (10/9 : 9/8 - 16/15) be called tritriadics because they may be divided into three triads on the 1 Iv (1) v1 and 1 rv (vu) m vi (1) v 1 progressions (Chalmers 1979, 1986, 1987, 1988). In general, however, the vu and chords will be out of tune (Lewin 1982) and probably should be omitted in the progressions unless extra notes are employed. The composer Erling Wold, however, has made a case for a more adventurous utilization of available tonal resources (Wold 1988). Partch ([1949] 1974) has done so too in a discussion of a letter from Fox-Strangways concerning the alleged defects of just intonation and their effect on modulation. The three primary triads on 1/1, 4/3, and 3/2 are of the same type, but the triads on the third (mediant) and sixth (submediant) degrees are of the conjugate or 3/2’s complement type. For example, the primary triads of number 1a of 7-4 are major, while the mediant and submediant triads are minor. In number 1b, the modalities are just the reverse. In addition to the principle triads of these scales, triads on other degrees may also be usable. Similarly, in some tunings, seventh or other chords may be useful. Phonic or descending harmonizations are also possible in certain modes of tritriadic scales. Lewin, in fact, proposes what might be called both phonic major and minor harmonizations (Lewin 1982). HARMONIZATION OF TETRACHORDAL SCALES

The generalized triad is denoted as t:m:d, after Lewin (1982), where t is the tonic, 7 the mediant, and d the dominant. In principle, any tetrachord containing the interval 9/8 can be arranged as a tritriadic generator, but the majority of the resulting triads will be relatively discordant. If the mediant of a triad is denoted by m, then the tetrachord has the form 4/3m - 9/8 . 8m/9, where 4/37 - 87/9 = 32/27. The conjugate tritriadic scale is generated by the permutation 8/9 : 9/8 4/37. The magnitude of m may range from 9/8 to 4/3 and generate a seven tone tritriadic scale, though the Rothenberg propriety (chapter 5) of the scale and the consonance of the triads will depend of the value of #. Triads with perfect fifths (4 = 3/2) whose mediants (m) are greater than 32/27 and less than 81/64 generate strictly proper scales (chapter 5; Rothenberg 1969, 1975, 1978; Chalmers 1975). Strictly proper scales tend to be perceived as musical gestalts and are used in styles where motivic transposition is an important structural element. Improper scales, on the other hand, are usually employed as sets of principal and auxiliary or ornamental tones. 7-4. Tritriadic tetrachords. I standsfor “improper,” and SPfor “strictly proper” (Rothenberg 1969, 1975, 1978). Injust intonation, tritriadic scales are either strictly proper or improper. SEES TA. IB. 24 2B, 3A. 58. 6a. 6B. 7A TRIAD 4:5:6 10:12:15 6:7:9 14:18:21 18:22:27 22:27:33 26:32:39 32:39:48 22:28:33 28:33:42 10:13:15 26:30:39 22:26:33 MED. 5/4 6/5 7/6 9/7 11/9 27/22 16/13 39/32 14/11 33/28 13/10 15/13 13/11 CTS 386 316 267 435 347 355 359 342 418 284 454 248 289 TETRACHORD 16/15-9/8.10/9 10/9-9/8. 16/15 8/7. 9/8. 28/27 28/27.-9/8.8/7 12/11 - 9/8. 88/81 88/81.o/8. 12/11 13/12:-9/8-128/17 128/17-9/8 13/12 22/21-9/8- 112/99 112/99- 9/8 22/21 40/39. 9/8 - 52/45 52/45 + 9/8 - 40/39 44/39: 9/8: 104/99 75. 26:33:39 33/26 413 104/99 - 9/8 - 44/39 BA 56/51-9/8. 68/63 14:17:21 17/14 336 Only a limited number of acceptably consonant triads exist in just intonation and also generate useful tritriadic scales. The most important of these have been tabulated in 7-4. As indicated above, triads 1a and 1b generate the major and natural minor modes, and 2a and 2b generate the PROPRIETY SP SP I I SP sp SP sp I I 1 1 I I sp 130 CHAPTER 7 8B. 34:42:51 DA. 16:19:24 98. 38:48:57 IOA. 64:81:96 IoB. 54:64:81 IIA. 26:34:39 IIB. 34:39:51 124, 14:16:21 12B, 16:21:24 134. 20:23:30 138. 46:60:69 IgA. 18:23:27 I4B. 46:54:69 ISA, 38:46:57 158. 46:57:69 21/17 366 19/16 298 24/19 404 81/64 408 32/27 294 17/13 464 39/34 238 8/7 231 21/16 471 23/20 242 30/23 460 23/18 424 27/23 278 23/19 331 57/46 371 68/63 + 9/8 + 56/51 64/57-9/8.19/18 19/18 - 9/8 : 64/57 256/243-9/8:9/8 9/8 - 9/8 - 256/243 52/51-9/8-136/11 136/117-9/852/51 7/6-9/8- 64/63 64/63 -9/8-7/6 80/69-9/8- 46/45 656/45:9/8.80/69 24/23: 0/8 - 92/81 92/81 - 9/8 - 24/23 184/171 9/8. 76/69 76/69: 9/8 184/171 SP 1 1 1 I ı 1 I I 1 1 I

corresponding septimal minor and septimal major scales. The septimal 7-5. Mixed tritriadic scales. The triads are 4:5:6 and 6:7:9. (Poole 1850). Mixed scales may often be decomposed into two tetrachords and a disjunctive tone in more than one way. Farnsworth’s scale is a mode ofPoole’s. It may be construed as a tonic major triad, a dominant seventh chord, or a septimal minor triad (6:7:9) on the supertonic (Farnsworth 1958, 1969). POOLE’S “DOUBLE DIATONIC” OR “DICHORDAL SCALE” SUBDOMINANT 4/3 5/3 2/1 2/d x 2/1 TONIC 1/1 5/4 3/2 1/1md DOMINANT 3/2 7/4 9/8 dsd? w/t 98 5/4 4/3 3/2 5/3 7/4 2/1 9/8 . 10/9 - 16/15 : 9/8 - 10/9. 21/20. 8/7 ALTERNATE TETRACHORDAL FORM 1/1 10/9 7/6 4/3 3/2 5/3 16/9 2/1 10/9 » 21/20 : 8/7. 9/8 + 10/9+ 16/15 - 9/8 FARNSWORTH’S SCALE SUBDOMINANT 21/1627/162/1 d-s d3 2/4 TONIC 1/1 5/4 3/2 1/1md DOMINANT 3/2 15/8 8 21/16 ddamd?ds 1/1 9/8 5/4 21/16 3/2 27/16 15/8 2/1 9/8. 10/9 + 21/20- 8/7 - 9/8 - 10/9« 16/15 TETRACHORDAL FORM 1/1 9/8 5/4 4/3 3/2 5/3 7/4 2/1 9/8. 10/9 + 16/15 + 9/8 + 10/9 » 21/20 + 8/7 minor or subminor scale sounds rather soft and mysterious, but the septimal major is surprisingly harsh and discordant. Triads ga and gb are virtually equally tempered and sound very much like their 12-tone counterparts, The scales based on roa and rob are the Pythagorean tunings of the major and minor modes in which the thirds are the brilliant, if somewhat discordant, 81/64 and 32/27. Triads with undecimal, tridecimal, and septendecimal thirds (numbers 3a8b of 7-4) are less consonant than those discussed above. However, these triads are still relatively smooth and may be useful in certain contexts. Their tetrachords are also interesting melodically as they approximate certain medieval Islamic and neo-Aristoxenian genera (chapter 4). The tetrachords generated by the even less harmonious triads 24:31:36, 64:75:96, 34:40:51, 30:38:45, and 24:29:36 and their conjugates will be found in the Main Catalog. Scales with mixed triads Tritriadic scales may also be constructed from triads with different mediants, provided that d remains 3/2. An example where the tonic and subdominant triads are 4:5:6 and the dominant triad is 6:7:9 is shown in 7-5 (Helmholtz [1877] 1954, 474). The tetrachordal structure may be described as 9/8 - 8/9 - 4/3m (where m is the mediant of the tonic triad) for the lower tetrachord and 22/3 : s/x « 2/s (where x and s are the sixth and seventh of the scale) for the upper tetrachord. However, as 7-5 indicates, mixed tritriadics may often be divided into two tetrachords and a disjunctive tone is more than one way. Farnsworth’s scale, also shown in 7-5, is a mode of Poole’s Double Diatonic (Farnsworth 1969). It may be construed as a major triad on 1/1, a dominant seventh chord on 3/2, and a subminor triad (6:7:9) on 9/8. In chapter 5, the limits on the propriety of mixed modes are discussed. Ellis’s duodenes Composers may find the intrinsic harmonic resources of tetrachordal scales rather sparse, even with the addition of one or more historically motivated supplementary tones. Two simple remedies immediately come to mind, One is to enlarge the chain of chordal roots of tritriadic scales to encompass four or more triads. This procedure may tend to hide the tetrachords beneath a mass of chords, but by way of compensation, HARMONIZATION OF TETRACHORDAL SCALES

more tetrachords are created. The process may be seen in 7-6. The parent tritriadic scale contains five tetrachords, all of which are permutations of 16/15-9/8- 10/9 (112 + 204 + 182 cents). The new pentatriadic scale contains 42 tetrachords of six different genera. The second solution is to extend both the d and m axes to generate structures analogous to A. J. Ellis's duodenes, the twelve note “units of modulation” in his theory of just intonation in European tonal harmony (Helmholtz [1877] 1954). The duodene generated from the 4:5:6 triad and some analogs generated by other triads are illustrated in 7-7. These scales likewise consist of large numbers of tetrachords of diverse genera in a harmonic context. Perrett’s harmonizations Wilfrid Perrett, an English theorist, developed some highly imaginative, if controversial, ideas about Greek music and its early history. In Some Questions ofMusical Theory, Perrett harmonized a version of the enharmonic tetrachord (21/20 - 64/63 : 5/4) which he attributed to Tartini, but it is more likely that Pachymeres has priority. Perrett used familiar tonic, subdominant, and dominant chord progressions by adding tones, effectively embedding the tetrachord in a larger microchromatic gamut (Perrett 1926, 1928, 1931, 1934). It is this harmonization that Partch quoted in Genesis of THE 4:5:6 TRIAD AND A DERIVED PENTATRIADIC SCALE 7-6. Pentatriadic scales, À pentatriadic is an expansion ofa tritriadic by the addition of the subdominant of the subdominant and the dominant of the dominant. An alternative form has a third dominant in place of the second subdominant and is a mode of the scale above. 16/9 10/9 443 2/8 mid 2/d SUBDOMINANT TONIC 4/3 5/3 2/1 1/1 5/4 3/2 2/d mld 2/1 w/t md DOMINANT 3/2 15/8 9/8 9/8 45/32 27/16 d dem d2 d? md? di 1/1 10/9 9/8 5/4 4/3 45/32 3/2 6/3 27/16 16/9 15/8 2/1 10/9-81/80-10/9-16/15-135/128-16/15-10/9-81/80-256/243-135/128-16/15 TETRACHORDS IN SCALE RATIOS 1. 81/80 - 256/243 - 5/4 2. 256/243 - 135/128-6/5 3. 135/128- 16/15 - 32/27 4. 81/80 - 10/9 » 32/27 5. 16/15 - 9/8 + 10/9 6. 256/243 : 9/8. 9/8 132 CHAPTER 7 CENTS 22 + 90+ 396 90+92+316 92+112+294 22 +182 +294 II2 +204 + 182 90 + 204 + 204 NUMBER 3 3 8

4 Music (Partch [1949] 1974, 171). Perrett placed the tetrachord in the soprano voice and added sufficient extra tones in the lower registers to obtain the desired chord progression. 7-8 simplifies Partch’s presentation by leaving out the repeated chords under 16/15, 21/20, and 1/1 that follow the one under 4/3, and by transposing the pitches from 5/3 to 1/1. Perrett also devised harmonizations for a number of other tetrachords listed by Ptolemy. These harmonizations are shown in 7-9 where they have been transposed to 1/1 and tabulated in a standard format. Perrett also discovered a harmonization of Archytas’s enharmonic, 28/ 27 ‘36/5 « 5/4, a much more plausible and consonant tuning than the 21/ 20 : 64/63 « 5/4 he chose initially (Perrett 1928, 95). He expressed the solution in the 171-tone equal temperament and later translated it into a 7-7. Ellis’s duodenes. This table is based on Helmholtz [1877] 1954, 457-464. The axes have been reversedfrom the original in which the chain of3/2’s was vertical, Note the interlocking prime (major) and conjugate (minor) triads, The 4:5:6 duodene contains sq tetrachords ofdiverse genera, 10:12:15 is a conjugate duodene which should be compared with the one above ofwhich it is not a “mode.” It contains 48 tetrachords ofdifferent genera. 6:7:9 is a non-tertian duodene. It contains 62 tetrachords of various genera, TRADITIONAL DUODENE BASED ON THE 4:5:6 TRIAD 5/3 5/4 15/8 45/32 44 IX 3/2 9/8 16/15 8/5 6/5 9/5 DUODENE BASED ON THE 10:12:15 TRIAD 27/20 9/5 6/5 8/5 4/3 ı/ı 3/2 9/8 10/9 5/3 5/4 15/8 DUODENE BASED ON THE 6:7:9 TRIAD 7/6 7/4 I 3/2 9/7 12/7 14/9 443 8/7 7-8. Perrett’s harmonization ofPachymeres’s enharmonic. The numbers under the note ratios repre_ sent the harmonicfactors or Partch “Identities” ofthe chords. The uppermost voice contains the tones ofthe tetrachord. The ratios ofeach ofthe chordal components are shown below. Asterisks indicate the roots ofharmonic chords, “Otonalities” in Parich'snomenclature. The 28/15 does not occur in the Partch gamut, buta transposed version is available in Partch’s system starting on 1/1 = 5/3. The pitches of the tetrachord then become 5/3 7/4 16/9 and 10/9. 21/16 9/8 27/14 1/1 21/20 16/15 4/3 5 7 8 5 4 6 7 4 3 5 6 3 I I I I $=2/1 4=8/5 3= 6/5 7=21/20 6= 9/5 5=3/2 8 = 16/15 7= 28/15 6 = 8/5 5=4/3 4= 16/15 3 =8/5 1 8/5 1=6/5 1=16/15 1=16/15 8/5* 6/5 * 16/15 * HARMONIZATION OF TETRACHORDAL SCALES

17-limit just intonation (Perrett 1934, 158). This harmonization is shown as number 7 of 7-9. I have devised another harmonization, which is noteworthy in that the movement between the roots of last two chords of the cadence is by a 40/ 27 rather than a 3/2. This example is shown in 7-10. These harmonizations are rather simple, with few nonharmonic tones 7-9. Perrett’s other tetrachord harmonizations. The namesfor numbers 3 and 4 are Perrett’s; the tetrachord is actually Archytas’s diatonic and Ptolemy's tonic diatonic genus rearranged. In ascendingform, the tetrachord ofnumbers 1 and 6 is 28/27. 15/14 + 6/5, Ptolemy's soft chromatic. I. INVERTED PTOLEMY’S SOFT CHROMATIC ı/ı 6/5 9/7 4/3 5 4 5 6 9 7 7 6 3 I 4 I 5 2 5 I or passing chords. More sophisticated techniques including the use of subharmonic chords would seem appropriate. More complex treatment is obviously possible in larger microchromatic scales such as Partch’s 43-tone gamut. With the help of a computer, 4022 occurrences of tetrachords and 1301 heptatonic scales in which both tetrachords are identical have been found in this scale. Among these are the instances of the Ptolemaic sequence, Partch’s name for the major mode, and a number of other tetrachords from Ptolemy’s catalog. Smaller systems such as Perrett’s 19-tone scale have considerable tetrachordal resources; 269 tetrachords and 52 heptatonic tetrachordal scales occur in this gamut. 2. PTOLEMY’S SOFT CHROMATIG 1/1 6 28/27 7 10/9 5 4/3 6 5 6 4 5 4 I 5 I 3 I 4 I 3. PTOLEMY’S “SOFT DIATONIC,” 5. ARCHYTAS'S DIATONIC 1/1 6 5 28/27 14 12 32/27 16 12 4/3 16 12 4 9 9 8 2 4 6 5 6. INVERTED PTOLEMY’S SOFT CHROMATIC, REARRANGED 1/1 6 28/27 7 7/6 7 4/3 8 5 6 6 7 4 5 5 6 I I I I 4. PTOLEMY’S “SOFT DIATONIG,” REARRANGED, ALTERNATIVE CHORDS 1/1 6 28/27 7 7/6 5 ALTERNATIVE CHORDS 1/1 5 4 6/5 5 6 9/7 go 70 4/3 20 15 3 I 4 63 12 I 45 Io 7. ARCHYTAS’S ENHARMONIC 4/3 8 5 6 4 7 4 5 3 6 I I I I 1/1 8-16 5-10 28/27 12 10 16/15 28 24 4/3 6 5 3-7 2-4 7 4 17 10 CHAPTER 7

7-10. Another harmonization ofArchytas’s enharmonic. The root ofthe chord under 28/27 is go/ 27 a syntonic comma lower than 3/2. The septimal tetrad on 16/15 lacksa major third. 1/1 28/29 16/15 4/3 5 4 3 7 6 5 8 7 6 5 4 3 I I I I Many of these tetrachords closely approximate divisions based on higher harmonics or equal temperaments, such as those found in Aristoxenian theory. Because they are composed of secondary or multiple number ratios whose factors are limited to 11, their tones may be harmonized by comparatively simple harmonic or subharmonic chords in a tetradic or hexadic texture. Wilson’s expansions Perhaps the most innovative technique for harmonizing tetrachords is due to Ervin Wilson (personal communication, 1964). Wilson’s technique is based on sequences of chords of increasing intervallic span linked by a common tone. Wilson’s have the property that the successive differences between the chordal factors follow a consistent pattern. This pattern is termed the unit-proportion (ur). It controls both the rate of intervallic expansion and less directly the degree of consonance. For harmonic chords, it may be expressed as a string of signed, positive integers, i.e., the unitproportion of the major triad 4:5:6:8 is +1 +1 +2. Subharmonic unitproportions are written with prefixed — signs; the unit-proportion of the chord 8:6:5:4 is -2 -ı —1. Sequences of chords with identical unitproportions make up an expansion which progresses from a dense, relatively discordant chord through chords of decreasing tension to a stable consonance, usually a triad with the root doubled. Sequences of such chords may be used in many musical contexts, and somewhat similar chordal sequences have been explored by Fokker (1966, 1975). Wilson’s expansions are particularly attractive when applied to tetrachords and tetrachordal scales. The application of Wilson’s technique to tetrachordal scales is best seen by example. Wilson’s original examples were harmonizations of the inverted enharmonic genera, 1/1 5/4 9/7 4/3 (Archytas) and 1/1 5/4 13/10 4/3 (Avicenna) approximated in 22- and 3r-tone equal temperament. These examples have been translated into just intonation and are shown in 7-11. An optional 7:8:9:11 chord has been added to Wilson’s original progression for the inverted Archytas’s enharmonic. Although one may limit the harmonization to a single tetrachord, it is more likely that one will want to harmonize all seven tones of the scale. Several solutions to this rather difficult problem using both harmonic and subharmonic chords with varied unit-proportions and different common tones are given in 7-12. In these examples, either the 4/3 or 3/2 is held 135 HARMONIZATION OF TETRACHORDAL SCALES pui | i. [o

constant throughout the progression. A passing chord containing intervals of 13 and 15 is used in number 2 to make the progression smoother. These intervals are conditioned in part by the unit-proportion of the set and in part by the intervals of the tetrachord. The major caveat is to limit the number of chords and extra tones when preservation of the melody of the tetrachord is important. Except for octave transposition of some of the chordal tones and ocassional passing chords there has not been much study of harmonic elaboration (Wilson, personal communication). This is true of the endogenous and tritriadic approaches as well. The standard techniques, however, would appear to be applicable here as in traditional practice, but only more experimentation will tell. Although the majority of this chapter has been presented from the viewpoint of just intonation, these scales and their various harmonizations are equally valid in systems of equal temperament which furnish adequate approximations to the important melodic and harmonic intervals. 9-11, Wilson's expansion technique. The set of ratios are the chordal tones relative to 1/1. (1) is the just intonation version of Wilson'sfirst expansion harmonization with the later addition ofan optional 789 11 chord at the beginning. The original was guantized to 22-tone equal temperament, (2) is the just intonation version of Wilson's second expansion harmonization, The original was quantized to 31tone equal temperament, In both cases, the added tones are in lighter type. The optional chord is in parentheses, I, INVERTED ARCHYTAS ENHARMONIC, HARMONIC CHORDS ON 3/2, UP = +I +1 +2 t/t 5/4 9/7 4/3 3/2 15/8 27/14 2/1 (7 8 9 11) (7/6 4/3 3/2 11/6) 7 21/16 8 3/2 6 9/8 10 15/8 5 6 7 9 15/14 9/7 3/2 27/14 4 5 6 8 1/1 5/4 3/2 2/1 2. INVERTED AVICENNA'S ENHARMONIC, HARMONIC CHORDS ON 3/2, UP = +3 +3 +6 1/1 5/4 13/10 4/3 15/8 39/20 2/1 18 21 24 30 9/8 21/16 3/2 15/8 14 21/20 12 w/t 17 51/40 15 5/4 136 3/2 CHAPTER 7 20 3/2 18 3/2 26

7-12. Trial expansion harmonizations. The successive differences or unitproportions are positive in harmonic chords, negative in subharmonic. The non-scalar added tones are in lighter type. Passing notes are in parentheses. 4. INVERTED DIDYMOS’S CHROMATIC, HARMONIC CHORDS ON 3/2, UP = +2 +3 +5 1/1 I, DIDYMOS’S CHROMATIC, SUBHARMONIC CHORDS ON 4/3, 4/3 30 25 22 20 10/9 4/3 50/33 5/3 25 16/15 3/2 8/5 5/3 I5 1/7 4/3 9/5 15/8 2/1 20 22 25 30 6/5 33/25 3/2 9/5 2/r 17 51/40 10 1/1 Iz 6/5 17 80/51 12 5/3 10 2/1 5/4 4/3 3/2 7/4 15 16 18 21 5/4 43 3/2 7/4 (12) (13) (6/5) (13/10) 15 (18) 3/2 (9/5) 12 3/2 15 15/8 9 Io 9/8 5/4 6 7 1/1 7/6 1/1 28/27 16/15 II 9 8 7 12/11 4/3 3/2 12/7 8 4/3 7 32/21 16 18 20 6 4/3 3/2 5/3 2/1 14 14/9 16 16/9 12 8/5 14 28/15 9 4/3 Il 44/27 13 52/27 8 4/3 10 5/3 Iz 2/1 6. INVERTED ARCHYTAS’S ENHARMONIC, SUBHARMONIG CHORDS ON 3/2, UP= 2-22 1/1 5/4 9/7 4/3 3/2 18 4/3 16 3/2 20 6/5 2/1 16 15/14 6 16/9 12 4/3 10 4/3 6 1/1 UP= +2 II 3/2 14/9 8/5 14 7 28/27 3. ARCHYTAS’S ENHARMONIC, SUBHARMONIC CHORDS ON 4/3, 4/3 3/2 14/9 8/5 8 16/15 12 2/1 1/1 28/27 16/15 4/3 10 10/9 x5/8 z/ı 9 3/2 10 16/15 20 2/1 UP = +2 +2 +2 UP=+I +2 +3 7/6 25 15/8 15 3/2 15 16/9 2. HARMONIC CHORDS, 3/2 COMMON, PASSING NOTES INSERTED, 1/1 20 3/2 5: ARCHYTAS’S ENHARMONIG, 4/3 COMMON, HARMONIC CHORDS, 20 43 20 3/2 15 9/8 UP =—5 —3 —2 ı/ı 16/15 10/9 6/5 5/4 4/3 14 5/4 12 3/2 15/8 27/14 2/1 14 12/7 10 15/8 14 12 10 8 27/16 27/22 3/2 27/14 9 7 6 5 13 II 9 7 28/27 4/3 14/9 28/15 9/8 9/7 3/2 9/5 8 6 x/x 4/3 5 8/5 4 2/x 137 12 1/1 10 6/5 8 3/2 HARMONIZATION OF TETRACHORDAL SCALES

Schlesinger’s harmoniai, Wilson’s diaphonic cycles, and other similar constructs ‘THE HARMONIAI WERE proposed by the English musicologist Kathleen Schlesinger as a reconstruction and rediscovery of the original forms of the modal scales of classical Greek music. Schlesinger spent many years developing her theories by experimenting with facsimiles of ancient auloi found in archaeological sites in Egypt, Pompeii, and elsewhere. Later, she extended her studies to include flutes of ancient and modern folk cultures. As a result of her researches, she questioned the accepted interpretation of Greek musical notation. The results of these studies were previewed in a paper on Aristoxenus and Greek musical intervals (Schlesinger 1933) and were presented at length in her major work, The Greek Aulos (1939). Her writings are a major challenge to the traditional tetrachord-based doctrines of the Aristoxenian and Ptolemaic theorists. While there are compelling reasons to doubt that her scales were ever a part of Greek musical practice, they form a musical system of great ingenuity and potential utility in their own right. This first part of this chapter is devoted to an exposition and analysis of her work, Various extensions and additions are proposed and near the end related materials, including Wilson’s diaphonic cycles, are discussed. The Schlesinger harmoniai Schlesinger’s harmoniai are 7-tone sections of the subharmonic series between members an octave apart. In theory, they are generated by aliquot divisions of the vibrating air columns of wind instruments. The same intervals, however, are obtained by the linear division of half strings. As string lengths are conceptually simpler than air columns, this discussion SCHLESINGER’S HARMONIAI

8-1. The diatonic Perfect Immutable System in the will refer to the former for clarity. The numbers or modal determinants Dorian tonos according ta Schlesinger. Each diatonic assigned to each of the notes are to be understood as the denominators of harmonia may be taken as an octave species ofthis ratios. The sequence 22 20 18 16 is a shorthand for the notes 22/22 22/20 22/18 22/16 or 1/1 11/10 11/9 11/8 above the tonic note 22. systems, (As elsewhere, at variance from Schlesinger, 32 HYPATE HYPATON 28 PARHYPATE HYPATON 26 LICHANOS HYPATON 24 HYPATE MESON 22 PARHYPATE MESON 20 LICHANOS MESON 18 & MESE 16 13 Bm TY TRITE DIEZEUGMENON eo 14 12 II TRITE HYPERBOLAION Io PARANETE HYPERBOLAION 9 NETE HYPERBOLAION 8 mo PARANETE DIEZEUGMENON NETE DIEZEUGMENON Mm 15 PARAMESE bd TRITE SYNEMMENON 2 M.D. PROSLAMBANOMENOS Jade à NOTE ® bypate meson is equated with E rather than F.) Trite synemmenon is required for the bypo-modes, in which it replacesparamese. The diatonic synemmenon tetrachord consists ofthe numbers 16 15 13 and 12. The octave rather than the tetrachord is the fundamental module of these scales. Although the scales can be analyzed into tetrachords and disjunctive tones, the tetrachords are of different sizes which, in general, do not equal 4/3. Furthermore, each interval of the scale is different; the series of duplicated conjunct and disjunct tetrachords of the traditional theorists (chapter 6) is replaced by modal heptachords which repeat only at the octave. The familiar names for the octave species are retained, but each modal octave is, in effect, another segment of the subharmonic series, bounded by a different modal determinant and its octave. 8-1 shows the form the Perfect Immutable System in the diatonic genus takes in her theory. The modal determinants have many of the functions of tonics. As such, they serve to identify and define the harmoniai. Schlesinger also considers that mese itself has tonic functions, a point which is controversial even in the standard theory (Winnington-Ingram 1936). The relations the other octave species have to the central Dorian octave is shown in 8-2. The seven harmoniai may also be constructed on a common tone, proslambanomenos, by assigning their modal determinants to hypate meson. In this case, there are six additional keys or tonoi which are named after the homonymous harmoniai. The Dorian and the other modal octaves are then found at corresponding transpositional levels in each tonos, Con- PS 32 8-2. The diatonic barmiontai as octave species ofthe Perfect Immutable System in the Dorian tonos. Other tonoi are defined by assigning their modal determinants to bypate meson and proceeding through the subharmonic series. The Dorian, however, is the basisfor Schlesinger 's theory. MIXOLYDIAN LYDIAN PHRYGLAN DORIAN HYPOLYDIAN HYPOPHRYGIAN HYPODORIAN HH PH LH HM PM LM M TS PM TD PD ND TH PN NH 28 26 24 22 20 18 16 Ig 14 13 12 IT 10 9 8 28 26 26 140 CHAPTER 8 24 22 24 22 24 22 22 20 20 20 20 20 18 18 18 18 18 18 16 14 16 14 16 I4 16 14 16 (15) 14 16 15 16 IS 13 13 13 13 13 13 12 I2 12 12 12 II II II II

comitantly, there is a seven-fold differentiation of the tuning of the other notes of the Perfect Immutable System. These tonoi are shown in 8-3. Anomalies and inconsistencies The clarity and consistency of Schlesinger’s system, however, is only apparent. Once one goes beyond the seven diatonic harmoniai, anomalies of various types soon appear. Schlesinger explicitly denies harmonia status to the octave species running from proslambanomenos to mese, calling it the bastard Hypadorian or Mixophrygian. She rejects it because it resembles the Hypodorian an octave lower but differs in having 8/7 rather than 16/15 as its first interval. Yet this scale had a name (Hypermixolydian) in the standard theory and was rejected by Ptolemy precisely because it was merely the Hypodorian 8-3. Schlesinger’s diatonic harmoniai as tonoi. Elsewhere she gives differentforms, most notably variants ofthe Lydian, with 27 instread of 26, and Dorian, with 21 instead of 22 (Schlesinger 1939, 1-35, 142). A trite synemmenon could be defined in each tonos, but Schlesinger chose not to do so. Schlesinger conceived ofthe Hypolydian harmonia in twoforms with 15 alternating with 14 (ibid., 26-27). Her theory demands that the Dorian trite synemmenon (15) be employed in all the hypo-modes, but she allows the alternation in the Hypolydian barmonia. transposed by an octave. Each of the diatonic harmoniai also had chromatic and enharmonic forms derived by subdividing the the first interval of each tetrachord and deleting the former mesopyknon. This process is identified with katapyknosis and is analogous to the derivation of the genera in the standard theory (see chapters 2 and 4). These forms are listed in 8-4 for the central octave of the Perfect Immutable System in each homonymous tonos. It is also here that some of the most serious problems with her theory occur. Although all of the diatonic harmoniai occur as octave species of the Dorian, and of each other, the chromatic and enharmonic forms of the other harmoniai are not modes of the corresponding forms of the Dorian harmonia. Rather, they are derived by katapyknosis of the homonymous tonos. The symmetry is broken and the modes are no longer identical in PS HH PH LH HM PM LM M PM A B c D E 6 a be MIXOLYDIAN 44 LYDIAN 40 PHRYGIAN 36 DORIAN 32 HYPOLYDIAN 28 HYPOPHRYGIAN 26 HYPODORIAN 24 40 36 32 28 26 24 22 36 32 32 28 28 26 26 24 24 22 22 20 20 18 141 F 28 26 24 26 24 22 24 22 20 22 20 18 20 18 16 18 16 15 16 15 13 22 20 20 18 18 16 16 14 15 13 13 12 12 II SCHLESINGER'S HARMONIAI TD PD ND TH de f PH NH ga 18 16 14 13 12 II 16 14 13 12 II Io 14 13 12 IX IO 9 13 12 11 10 9 8 12 I1 10 9 8 7 I1 10 9 8 7 13/2

different tonoi. Even the modal determinants of the harmoniai may be changed in different tonoi. Other inconsistencies and anomalies may be noted. The chromatic and enharmonic forms are incompletely separated since the enharmonic and chromatic forms of some harmoniai share tetrachords. Even these presumed canonical forms do not agree with the varieties she derives elsewhere in The Greek Aulos from her interpretation of the Greek notation. Because of certain irregularities in the notation, she claims that the modal determinant of the Lydian harmonia must have been altered at some period from 26 (13) to 27 and that of the Dorian from 22 to 21. These changes of modal determinants would not only have disrupted the tonal relations of the original harmoniai, but would also have affected the tonality of the rest of the system in all three genera. Since the Dorian harmonia was the center of the system, this would not have beena trivial change. The question of modal determinant 15 Another problem is the status of 15 as a modal determinant. Schlesinger strongly denies the existence of a harmonia whose modal determinant is 15. Yet one of her facsimile instruments plays it easily. She also states that hypate hypaton could be tuned to 30 in the Hypodorian harmonia where it generates a perfectly good harmonia of modal determinant 15 with the octave at trite synemmenon (8-2). The inclusion of modal determinant 15 is, on the whole, quite problematical. It enters originally as the Dorian trite synemmenon (Bj), the only 8-4. Schlesinger’s chromatic and enharmonic harmoniai (Schlesinger 1939, 214). It is clear that these scales are not simply modes ofthe Dorian chromatic and enharmonic genera, but are derivedfrom the homonymous tonoî. The chromatic and enbarmonic forms are derived by two successive doublings ofthe modal determinantfollowed by note selection to obtain the desired melodic contours. The upper tetrachords ofthe chromatic and enharmonicforms ofthe Dorian and Hypolydian harmoniai are identical. In the Hypolydian harmonia 30 (15) may replace 28 (14). The Hypophrygian and Hypodorian barmoniai bave a single enbarmonic-chromaticform. accidental in the Greater Perfect System. Although Schlesinger mentions what she calls the conjunct Dorian harmonia where 15 substitutes for 14, and elsewhere allows 15 to freely alternate with 14, she uses trite syn- HARMONIA MIXOLYDIAN LYDIAN PHRYGIAN DORIAN HYPOLYDIAN HYPOPHRYGIAN HYPODORIAN 142 CHROMATIC 28 27 26 22 2019 18 14 26 25 24 201817 16 13 242322 18 1615 14 12 44 42 40 32 28 17 26 22 40 38 36 28 26 25 24 20 3635 34 26 24 23 22 18 32 3130 24 22 2120 16 CHAPTER 8 ENHARMONIC 56 55 544440 39 38 28 525150403635 3426 48 47 46 36 32 31 30 24 44 43 42 32 28 27 2622 40 39 38 28 26 25 2420 36 35 34.26 24 23 22 18 32 313024 22212016

emmenon mainly to construct the diatonic hypo-modes. This is very much at variance with the usage of this note by the standard theorists whose Hypodorian, Hypophrygian, and Hypolydian modes employ only the natural notes of Greater Perfect System. For these theorists, trite synemmenon and the rest of the synemmenon tetrachord are part of the Lesser Perfect System and are used to primarily illustrate the melodic effect of modulations to the key a perfect fourth lower. Bacchios also employs it to illustrate certain rare intervals such as the ekbole, spondeiasmos, and eklysis (chapters 6 and 7). The combination of the Greater and Lesser Perfect Systems to form the Perfect Immutable System is basically a pedagogical device, not a reflection of musical practice. Furthermore, the Lesser Perfect System terminates with the synemmenon tetrachord, but to complete Schlesinger’s hypo-harmoniai the note sequence would have to switch back into the notes of the Greater Perfect System. Although chromaticism and modulation occur both in theory and in the surviving fragments (Winnington-Ingram 1936), this use of synemmenon would seem to be most unusual. Historical evidence Much of Schlesinger’s case for the harmoniai is based on fragmentary quotations from classical Greek writers. This evidence is dubious support at best. Theorists such as Aristoxenos complain about the unstable pitch and indeterminate tuning of the aulos (Schlesinger 1939). Aristoxenos claims that the intervals of music are determined by the performance skill of the player on both stringed and blown instruments and not by the instruments themselves. This polemic may be interpreted either as referring to the inherent pitch instability of the instrument or to the difficulty of bending the pitches so as to approximate a scale system for which it is not physically suited, ie. the standard tetrachordal theory. Whatever the correct interpretation, the passage does suggest that Schlesinger’s harmoniai played little or no role in Greek musical practice in the fourth century BCE. The problem lies with our ignorance of the Greek music and its mode of performance. It is quite possible for an instrument to be musically prominent and at the same time difficult to play in acceptable tune. Schlesinger may well have been right about the natural scales of auloi and still be entirely wrong about their employment in Greek music of any period. SCHLESINGER’S HARMONIAI

The harmoniai in world music Schlesinger also tries to bolster her argument by appealing to ethnomusicology. Her case for the employment of the harmoniai in nonEuropean folk and art music gives the impression of overpleading, especially in her analysis of Indonesian tunings. It is true, however, that wind instruments from many cultures often have roughly equidistant, equal sized finger holes. For example, the scales of many Andean flutes do appear to resemble sequences of tones from the various harmoniai, although the scales may not be identical throughout the gamut (Ervin Wilson, personal communication). The scales on these instruments are usually pentatonic, rather than heptatonic, Often one or more tones will diverge from the heptatonic pattern, particularly with respect to the vent, which is tuned to bring out the pentatonic structure, Nevertheless, some of the harmoniai sound very similar to the scales heard on recordings of Bolivian and Peruvian music. Hence, these data may serve as at least a partial vindication of her ideas. Empirical studies on instruments In The Greek Aulos, Schlesinger made use of a large body of data obtained by constructing and playing facsimiles of ancient auloi. She also studied fipple flutes and other folk wind instruments. These studies deserve critical attention. The chief difficulty one has in evaluating this work is its lack of replication by other investigators. However, there are two published experimental studies which are relevant to her hypotheses. The first is that of Letter, who made the assumption that two of the holes on the surviving auloi were 4/3 or 2/1 apart (Letter 1969). From measurements on these instruments, he determined the probable reed lengths. His measurements and calculations yielded a number of known tetrachords, including 12/11 - 11/10 - 10/9, 9/8 - 88/81 - 12/11, 9/8. 16/15 10/9, 14/13 + 8/7 + 13/12, and some pentachordal sequences, but little convincing evidence for the subharmonic series or the harmoniai. More recently, Amos built modal flutes with holes spaced at increments of one-eighth the distance from the fipple to the open end and the studied the resulting intervals (Amos 1981). This procedure, however, is not really in accord with Schlesinger’s work. She employed rather complex formulae involving corrections for the diameter and certain other physical parameters to determine the spacing of the holes of modal flutes. CHAPTER 8

The pitches of Amos’ flutes were measured by audibly comparing the flute tone to a calibrated digital oscillator and minimizing beats. Amos’s results show that the resulting intervals are subject to wide variation from flute to flute and depend upon humidity, wind pressure, fingering, and other parameters. While not strictly comparable to Schlesinger’s results, the results of these investigators suggest that one should be cautious in extrapolating the tuning of musical systems from the holes of wind instruments. Schlesinger herself made the same caveat and stated that the aulos alone gave birth to the harmoniai. She claimed that the acoustical properties of the aulos are simpler than those of the flute, and therefore, one can accurately deduce the musical system from the spacing of the finger holes of auloi. People who have made and played aulos-like instruments are less certain. Lou Harrison found the traditional Korean oboe, the piri (and the homemade miguk piri), to be difficult to play in tune and noted its tendency to overblow at the twelfth (personal communication). Jim French, who has spent a number of years researching the aulos from both an archaeological and an experimental perspective, has discovered that the type of reed and its processing are far more crucial than Schlesinger implies. His results with double auloi indicate that the selection of a particular reed can change the fundamental by a 4/3 (personal communication), Duplicated tetrachords are thus quite natural on this kind of instrument. He has also found that sequences of consecutive intervals from harmoniai such as that on 16 (Hypodorian) are relatively easy to play on these instruments and may be embodied in historical examples and artistic depictions. Composition with the harmoniai The question of whether or not Schlesinger’s harmoniai are relevant to Greek or world music may be of less importance to the experimental musician than their possible use in composition. Her most fruitful contribution ultimately may be her suggestion that the harmonia be considered a “new language of music” (Schlesinger 1939). Schlesinger tuned her piano to the Dorian harmonia in which C (at 256 Hertz) equals the modal determinant 22. Thus she used only an 11-pitch gamut. For some unstated reason, she did not give a tuning for the note Bj, which would have had the modal determinant 25, though she did include SCHLESINGER’S HARMONIAI

such prime numbers as 17 and 19 and composites of comparable size such as 22 and 24. One would think that the Phrygian harmonia on 24 would make more efficient use of the keyboard, unless there are problems with the altered tension of the piano strings. This, of course, would not be a limitation with electronic instruments. Schlesinger was fortunately able to enlist the composer Elsie Hamilton from South Australia in these efforts. Hamilton composed a number of works in the Dorian diatonic tuning between 1916 and 1929. In 1935, Hamilton trained a chamber orchestra in Stuttgart to perform in the harmoniai. Although several orchestral and dramatic works were composed and performed during this period, it has been impossible to find further information about the composer or discover whether the scores are still extant. From the excerpts in The Greek Aulos, it would appear that Hamilton employed a conservative melodic idiom with straightforward rhythms (86). Schlesinger comments that such a simplification was necessary for both “executant and listener.” The quotations from the score of Agave, brief as they are, seem quite convincing musically in a realization on a retunable synthesizer. Hamilton’s harmonic system is of considerable interest. Although familiar chords are scarce in this system, virtually any interval larger than a melodic second is at least a quasi-consonance. Rather than attempt a translation of tertian harmonic concepts to this tuning, Hamilton instead chose to use the tetrachordal frameworks of the modes as the basic consonances (8-5 and 8-6a). In the Dorian mode, this chord would be 22 16 14 11 (1/1 11/8 11/7 2/1), with 15 (22/15) as an alternative tone. A melodic line may be supported by a succession of such chords taken from all seven of the modes. Hamilton augmented this somewhat sparse 8-5. Harmonization ofSchlesinger's barmoniai. Tetrachordalframework chords. Chordsfront the “conjunct” barmoniai in which 15 replaces 14 are also shown where applicable, MIXOLYDIAN LYDIAN PHRYGIAN DISJUNCT 28:22:20:14 26:20: 18:13 24:18:16:12 CONJUNCT 28:22:16:14 26:20: 14:13, 26:20:15:13 24:18:13:12 DORIAN 22:16:14:1I, 22:16:I5:II 22:16:12:11 HYPOLYDIAN HYPOPHRYGIAN HYPODORIAN 20:15:13:10, 20:14:13:10 1B:13:12:9 16:12:11:8 20:15:11:10, 20:14:1 1:10 18:13:10:9 CHAPTER 8

8-6. Excerpts from Agave by Elsie Hamilton, with ratio numbers. if 13 pie À 15 2 ye eds teu ei nt è 10 A + ma EtI Limame a (a) Tetrachordalframework chords (“Sunrise”), sw 2713 HYPOPHRYGIAN A Li LI Fe 1 13 el TE LYDIAN 13 12 EZAT LE + ! DORIAN fa 7 PHRYGIAN d ei. vi HYPOPHRYGIAN N, PTT MKG TT Pi hd at ui Le À. pen | L 7 Pi _ kl 17 Er LE I 13 10 TA we PHRYGIAN 9 TE En AF Free: HYPODORIAN HYPOPHRYGIAN 9 1 9 10 : 13 é la be EE + + — + — 13 L 1 tf tr t 9 | poe LE MIXOLYDIAN "| pote MIXOLYDIAN „410 RZ = - ie ay.Ir ST =I | È | HYPOLYDUN HYPOLYDIAN oi 71 du ii L Kl tu . HYPOLYDIAN © DORIAN PR IL 4. pe | Martellato (4) Modal tranposition. 8 as 15 12 ud 8 PHRYGIAN 11 — — 5 LE ff pei 8 werden Li 4 DORIAN TT d TS RE 9 T | LT I LUI | (© Combinedframework chords (“Sunrise”). I HYPODO RIAN [TE id 8 + te] T yak er + 15 = fo | L hi ri mre A f — TT | T 4 Cr LYDIAN 7 je + d 81 = LI ve HYPOLYDIAN + ja > nà n ÿ va 9 kw A. + HYPODORIAN SS D) à j + (® Mixed chorus and tetrachords ofresolution (“Funeral March”), + bens ÀFE fi = u 10 9 13 10 8 13 18 15 43 12 ti 8 Thoughtfilly HYPOPHRYGIAN D) 1 15 12 9 12 13 13 15 Con brio PHRYGIAN vi © it 2 147 8 8 8 dy DeÈ I 12 ff 1 8 8 SCHLESINGER’S HARMONIAI fa f re ro i 10 9 „Ah TOUT 8 7

vocabulary with chords formed by the union and intersection of chords from two related harmoniai (8-6b, 8-6c, and 8-7). In the latter case, the chords are resolved to their common dyad. She also discovered that parallel transposition results in changes of modality which are musically exploitable (8-6d), although the given examples are stated to have been approximated to the piano intonation. One would characterize her harmonic techniques as essentially polytonal and polymodal, rather than “diatonic” or “chromatic.” 8-7. Chordal relations between related harmoniat (Schlesinger 1939, 543-44). It is a pity that more examples of Hamilton’s use of the harmoniai are not extant. From this limited sample, it appears that Schlesinger’s system succeeds as a “new language of music.” Schlesinger’s harmoniai have inspired other composers, including D ML HL II 8 Il 7 Io zr 14 10 13 14 20 7 I0 8 II 10 13 «I 14 13 9 14 20 L HP D ML 13 9 9 12 10 13 13 18 11 on» + TETRACHORDAL CHORDS 10 II 14 T2 MIXED CHORDS 9 12 10 13 6 8 13 18 7 10 10 13 13 9 9 12 16/15. Forster has constructed several instruments embodying the ratios of 13 in a Partch tonality diamond context. He has also composed a considerable body of music for these instruments (Forster 1979). II 10 12 16 Extensions to Schlesinger’s system Although Schlesinger’s system suffers from internal inconsistencies and II INTERVALS OF RESOLUTION Il 4 Harry Partch and Cris Forster. Partch devoted a large part of his chapter on other systems of just intonation to her work, citing it as a justification to proceed on to ratios of 13 (Partch [1949] 1974). He correctly identified her harmoniai with his Utonalities, with the addition of the Secondary Ratio, II 14 omissions, her scales form a fascinating system in their own right, independent of their questionable historical status. The most obvious of the corrections or enhancements is to rationalize her enharmonic and chromatic forms so that all three forms of each harmonia are distinct. The next step is the definition of local tritai synemmenon in each of the tonoi so that correct hypo-modes and conjunct harmoniai may be constructed. Finally, new harmoniai based on modal determinants not used by Schlesinger are proposed. These new modal determinants range from 15 to 33. Rationalization of the harmoniai The first and most obvious extension to Schlesinger’s system is to furnish distinct chromatic and enharmonic forms for her diatonic harmoniai. This may be done by katapyknosis of the diatonic with the multipliers 2 and 4. To obtain the corrected chromatic versions, the first interval of each tetrachord of the diatonic harmoniai is linearly divided into two parts. The two new intervals are retained while simultaneously deleting the topmost CHAPTER 8

note of each tetrachord to create the characteristic interval of the genus. By this process, the old diatonic first intervals become the pykna of the new chromatic forms. The enharmonic is created analogously by katapyknosis with four. The first two new intervals are retained, leading to pykna which consist of the chromatic first intervals. This procedure is equivalent to performing katapyknosis with two on the chromatic genera resulting from the operations above. Wilson has suggested performing katapyknosis with 3 to produce trichromatic forms (personal communication). Ptolemy used the same technique to generate his shades. This operation produces two forms, a 1 +1 form in which the two lowest successive intervals are retained and a 1 + 2 form in which the lowest and the sum of the two highest are used. The pykna of the 1 + 1 and 1 + 2 forms are thus different and the 1 + 1 form tends to melodically approximate the enharmonic. A third form, the 2 + 1, potentially exists, but would violate Greek melodic canons (chapter 3). In an analogous manner, katapyknosis by 5 and 6 are possible if the interval to be divided is large enough. These divisors generate what may be called pentachromatic, pentenbarmonic, bexackromatic, and bexenbarmonic genera. The forms of the rationalized harmoniai including the two trichromatic as well as the pentachromatic genera, created from a 2 + 3 division of the pyknon, are shown in 8-8. If one generates all the forms of a harmonia which do not violate accepted melodic canons by katapyknosis with the numbers 1 through 6, nineteen genera result. The Hypermixolydian or “bastard Hypodorian” provides a good example of this process because the first diatonic interval is the comparatively large septimal tone 8/7 (231 cents). The nineteen katapyknotic genera of her “bastard Hypodorian” are shown in 8-9. Local tritai synemmenon Although all of the diatonic harmoniai can be represented as octave species of the Dorian harmonia (plus trite synemmenon) by choosing different notes as modal determinants, in the homonymous tonoi the central octave is occupied by the notes of the corresponding harmoniai. Since all of the tonoi are structurally as well as logically equivalent, the argument which demanded that 15 replace 14 in the hypo-modes of the Dorian requires that a local trite synemmenon be defined in each tonos. Otherwise, the SCHLESINGER’S HARMONIAI

8-8, Rationalized barmoniai. These barmoniai should be compared to Schlesinger’s own as significant differences exist between these and some ofhers in the chromatic and enharmonic genera. Three new genera are also provided; these are hased on katapyknosis by 3 and 5 instead of 2 and 4. To avoidfractions, some numbers bave been doubled. In principle, 14 may be substitutedfor 15 in the bypo-modes. 14 alternates with 15 in the Hypolydian. To preserve melodic contour, the chromatic and enharmonicforms ofthe Hypodorian are derivedfrom the “bastard” harmonia, Theforms ofthe lower tetrachords ofSchlesinger’s preferred barmonia would be 32 31 30 24,48 4746 36,48 4745 36, and 80 78 75 60.. Mixolydian DIATONIC 1413 12 1110987 CHROMATIC 28 27 2622 2019 18 14 TRICHROMATIC I 42414033 30 29 28 21 TRICHROMATIC 2 4241 39 33 302927 21 ENHARMONIC 56 55 5444 40 39 38 28 PENTACHROMATIC 70 68 65 55 5048 45 35 Lydian DIATONIC 1312111098713 CHROMATIC 26 25 24201817 16 13 TRICHROMATIC I 39 38 37 30 27 26 25 39 TRICHROMATIC 2 39 38 36 30 27 26 24 39 ENHARMONIC 52 515040 36 35 34 26 PENTACHROMATIC 65 63 60 50 45 43 4065 Phrygian DIATONIC 121110987136 CHROMATIC 242322 181615 14 12 TRICHROMATIC I PENTACHROMATIC 36353427 24 23 22 18 50 48 45 35 65 63 30 25 TRICHROMATIC 2 Hypophrygian 36 35 33 27 24 23 21 18 ENHARMONIC 48 47 46 36 32 31 30 24 PENTACHROMATIC 60 58 55 45 40 38 35 30 DIATONIC 18161513 1211 109 CHROMATIC 18171613 1223 119 TRICHROMATIC I Dorian 54 52 50 39 36 35 34 27 DIATONIC TRICHROMATIC 2 111098713611 54 52 48 39 36 35 33 27 CHROMATIC ENHARMONIC 222120161427131II 36 35 34 2624 47 23 18 TRICHROMATIC I PENTACHROMATIC 33 32 31 2421 41 40 33 90 86 80 65 60 58 55 45 TRICHROMATIC 2 33 32 30 24 21 20 39 33 ENHARMONIC 44 43 42 3228552722 PENTACHROMATIC 55 53 50 40 35 3465 55 Hypolydian DIATONIC 10987136115 CHROMATIC 2019 18 1413251210 TRICHROMATIC I 30 29 28 21 39 38 37 15 TRICHROMATIC 2 30 29 27 21 39 38 36 15 ENHARMONIC 40 39 38 28 26 512520 150 CHAPTER 8 Hypodorian DIATONIC 161513 12 111098 CHROMATIC 32 30 28 24 22 21 20 16 TRICHROMATIC I 48 46 44 36 33 32 31 24 TRICHROMATIC 2 48 46 42 36 33 32 3024 ENHARMONIC 64 62 60 48 44 43 42 32 PENTACHROMATIC 80 76 70 60 55 53 50 40

three hypo-modes in each tonos would be merely cyclic permutations of the original sequence and would therefore lack modal distinction. These tritai synemmenon are also needed to to form what Schlesinger would probably term conjunct harmoniai. The new tritai synemmenon may be supplied by analogy through katapyknosis of the disjunctive tone by 2. These additions, of course, increase the number of possible scale forms, as the new notes may alternate with the lesser of their neighbors as 15 alternates with 14 in the Dorian prototype. This alternation generates fairly wide intervals in the range of augmented seconds and gives the harmoniai containing them a chromatic or harmonic minor flavor not present in the corresponding modes of the Dorian harmonia. NO. 8-9. The nineteen genera of Schlesinger’s “bastard DI Hypodorian” harmonia. Beyond 6x the intervals are usually too small to be useful melodically. The numbers after the genus abbreviations distinguish the various species. The multiplier refers to the multiplication ofthe modal determinants in katapyknosis. The species are defined by the unit-proportions of theirpykna. The 4x, 5x, and 6x divisions define genera with both enharmonic and chromatic melodic properties, CI TI T2 EI E2 E3 DIVISION MULTIPLIER SPECIES DIATONIC 16 1413 12 111098 IX I CHROMATIC 1615 14121121 108 2X +l TRICHROMATIG 24 23 22 18 33 32 3112 3X I+I 24 23 21 18 33 32 30 12 3x 142 ENHARMONIC/CHROMATIC 32 31 30 24 22 43 2116 4X HIHI 32 31 29 24 22 43 41 16 4x 142 3231282422 43 2016 4X 143 PENTACHROMATIC/PENTRNHARMONIC PI 40 39 38 30 55 27 53 20 SX I+I P2 49 39 37 30 55 27 2620 Sx 142 P3 403936305527 51 20 SX 133 P4 49 39 35 30 55 27 25 20 SX 144 PS 40 38 36 30 55 53 51 20 SX 242 PG 403835 3055535020 5% 243 HE H2 HEXACHROMATIC/HEXENHARMONIC 48 47 46 36 33 65 32 24 6x 48 47 45 36 33 65 63 24 6x Ir 1+2 H3 48 47 44 36 33 65 62 24 6x 143 H4 H5 484743363365 6124 4847423633653024 6x 6x 1+4 145 H6 48 46 43 36 33 64 61 24 6x SCHLESINGER’S HARMONIAI

8-10. Conjunct rationalized harmoniat. These barmoniai are formed in analogy to the conjunct Dorian ofSchlesinger. The Hypodorian forms are based on the “bastard” harmonia. The lower tetrachords of Schlesinger’s preferred form are 32 10 30 24, 48 47 46 36, 48 47 45 36, and 80 78 75 60. TRICHROMATIC I PENTACHROMATIC 36 35 3427 26 25 39 18 50 48 45 75 65 65 55 50 TRICHROMATIC 2 60 58 55 45 40 38 65 30 Hypophrygian DIATONIC 181615 1325 IT 109 CHROMATIG 1817 161325 12 109 TRICHROMATIC I Dorian 54 52 50 39 38 37 30 27 DIATONIC 1110981513611 CHROMATIC TRICHROMATIC 2 Mixolydian DIATONIG 1413 12 1121987 CHROMATIC 282726222120 1614 TRICHROMATIC I 48 47 46 36 35 34 26 24 42 414033 32 31 24 21 PENTACHROMATIC TRICHROMATIC 2 42 41 39 33 32 30 24 21 ENHARMONIC 565554444342 32 28 PENTACHROMATIC 36 35 33 54 26 24 39 18 ENHARMONIC 70 68 65 55 53 50 40 35 2221 201615 Iq12 II Lydian TRICHROMATIC I DIATONIC 13 121110198713 CHROMATIC 262524201918 14 13 TRICHROMATIC I 33 32 31 24 23 22 18 33 TRICHROMATIC 2 33 32 3024 23 21 18 33 54 52 48 39 38 36 3027 ENHARMONIC 36 35 34.26 51 25 20 18 PENTACHROMATIC go 86 80 65 63 60 50 45 Hypodorian 39 3837302928 21 39 44 43 42 32 31 3024 22 PENTACHROMATIC TRICHROMATIC 2 55 53 59 40 35 33 3055 39 3836302927 21 39 DIATONIC 1615 13 12231098 CHROMATIC 32 30 28 24 23 22 1816 TRICHROMATIC I Hypolydian 48 46 44 36 35 3427 24 DIATONIC 2018 1615 13 12 11 10 CHROMATIC TRICHROMATIC 2 ENHARMONIC 52 51504039 38 28 26 PENTACHROMATIC ENHARMONIC 48 46 42 36 35 33 27 24 Phrygian TRICHROMATIC I ENHARMONIC 64 62 60 48 47 46 36 32 PENTACHROMATIC DIATONIC 2422201817 14 13 6 CHROMATIC 2423221817 1613 12 60 58 56 45 43 41 33 30 80 76 70 60 58 55 45 40 65 63 60 55 5048 45 65 2019 18 15 14 13 II IO TRICHROMATIC 2 60 58 5445 43 39 33 30 40 39 38 30 29 28 22 20 CHAPTER 8

New conjunct forms 8-11. Synopsis ofthe rationalized tonoî. The tonoi are transpositions ofthe Dorian modal sequence so that the modal determinant ofeach harmoniafalls on bypate meson. À local trite synemmenon has been defined in each ofthese harmoniai. In the Hypolydian, 15 alternates with 14. When mesefalls on 14, trite synemmenon is 27 (27/22). The Hypodorian also hasa “bastard” form which runs From proslambanomenos to mese in the Dorian tonos. Thefirst tetrachord is 16 14 13 12. The new tritai synemmenon combine with the remaining tones to yield conjunct forms for each of the harmoniai. In order to preserve generaspecific melodic contours, a variation on the usual principle of construction was employed in the derivation of these scales. The procedure may be thought of as a type of inverse katapyknosis utilizing the note alternative to the local trite synemmenon in some cases. These conjunct harmoniai are listed in 8-10 in their diatonic, various chromatic, and enharmonic forms. The tuning of the principal structural notes of the rationalized tonoi is summarized in 8-11. New modal determinants As mentioned previously, one of the most noticeable inconsistencies in NAME MIXOLYDIAN LYDIAN P HH HMM TS 44 40 28 22 21 40 30 20 20 19 P ND 20 14 18 13 Schlesinger’s system is the lack of a harmonia whose modal determinant is 15. Similarly in the new conjunct harmoniai, modal determinants of 17, 19, 21, 23, and 25 are implied by the local tritai synemmenon of the ration- DORIAN HYPOLYDIAN 32 28 22 16 15 28 26 20 15/214 14 11 13 10 . alized tonoi. Schlesinger herself stipulates the existence of harmoniai on 21 and 27 as later modifications of the Dorian and Lydian harmoniai. She PHRYGIAN 36 32 24 18 17 16 12 . , , . . HYPOPHRYGIAN 26 24 18 13 25/2 12 9 claimed that these harmoniai were created by shifting their modal deter- HYPODORIAN 23/2 11 8 minants one degree lower. Additional harmoniai on modal determinants 29 and 31 may be added without exceeding the bounds of the Perfect Immutable System. To these 24 22 16 12 may be added a harmonia on 33, which, though it exceeds the boundaries of the Dorian tonos, is included in the ranges of the tonoi of 8-12 and 813. The normal or disjunct forms of these new harmoniai are shown in 8- 12 and the conjunct, which use their local tritai synemmenon, in 8-13. A summary of these new harmoniai is given in 8-14. 8-12 (next page). New harmaniai. These harmoniai were created tofill in the gaps in Schlesinger’s system, although some, such as tonoi-15, -21, and-27, are implied in her text, Three new genera are also provided; these are based on katapyknosis by 3 and 5 instead of2 and 4. In principle, 14 may be substitutedfor 15 in these harmonia, savefor tonos-15 where the Mixolydian harmonia would result. Similarly, 21 may replace 22 and 27, 26, except when doing so would change the modal determinant. In the diatonic genus when thefirst interval above the modal determinant isroughly a semitone, chromatic alternation with the next highest degree would be melodically acceptable. SCHLESINGER'S HARMONIAI

Tonos-15 Tonos-21 Tonos-27 Tonos-33 DIATONIG 15 13 1211109815 CHROMATIC IS 1413 II 10199 I5 TRICHROMATIC 1 DIATONIC 2119 18 16 Iq 13 12 21 CHROMATIC DIATONIC 27 24 21 20 18 16 14 27 CHROMATIC 33 30 27 24 22 2018 33 21 2019 16 1427 13 21 54 51 48 40 36 34 32 27 33 312924 22 212033 TRICHROMATIC I 63 61 59 48 42 41 40 63 TRICHROMATIC 2 TRICHROMATIC I 81787560 54525081 TRICHROMATIC 2 8x 78 72 60 54 52 48 81 ENHARMONIC 45 4443 33 30 29 28 45 TRICHROMATIC 2 45 44 42 33 30292745 63 61 57 48 42 41 39 63 DIATONIG CHROMATIC TRICHROMATIC I 99 96 93 72 66 64 62 99 TRICHROMATIC 2 99 96 90 72 66 64 60 99 101 102 40 36 35 34 54 33 32 31 24 22 43 21 33 75 71 65 55 50 48 45 75 ENHARMONIC 424140322855 27 21 PENTACHROMATIG 105 101 95 80 70 68 65 105 PENTACHROMATIC 135 129 120 10090 86 80 135 165 1§9 150 120 IIO 106 100 165 Tonos-17 Tonos-23 Tonos-29 DIATONIC 171513 12 II 10917 CHROMATIC 17 1615 12 11211017 TRICHROMATIC I DIATONIC 232120181614 13 23 CHROMATIC 232221181615 14 23 TRICHROMATIC I 5149473633 32 31 SI 69 67 65 54 48 46 44 69 TRICHROMATIC 2 TRICHROMATIC 2 51 49 45 36 33 32 30 51 69 67 63 54 48 46 42 69 ENHARMONIC ENHARMONIC DIATONIC 29 26 24 22 2018 16 29 CHROMATIC 29 28 2722 2019 18 29 TRICHROMATIC I 87 85 83 66 60 58 56 87 TRICHROMATIC 2 87 85 81 66 60 58 54 87 ENHARMONIC 34 33 32 2422 43 21 17 46 45 44 36 32 313013 58 57 56 44 40 39 38 29 85 81 75 6055 53 5085 PENTACHROMATIC 115 111 105 90 80 76 70 115 PENTACHRO MATIC 145 I4I 135 IIO 10096 90 145 Tonos-19 Tonos-15 Tonos-3r DIATONIC 19 18161413 I2 II 19 CHROMATIC 1918 17 141325 12 19 TRICHROMATIC I DIATONIC 25222018 16 14 13 25 CHROMATIC 50 47 22 36 32 30 28 25 TRICHROMATIC I DIATONIG 31 28 26 23 22 2018 31 CHROMATIC 31 29 27 2322212031 TRICHROMATIC I 57 55 53 42 39 38 37 57 75 72 69 54 48 46 44 75 93 89 85 69 66 64 62 93 TRICHROMATIC 2 TRICHROMATIC 2 57 55 51 42 39 38 3657 75 72 66 54 48 46 42 75 TRICHROMATIC 2 93 89 81 69 66 64 60 93 ENHARMONIC ZI 30 29 23 22 43 21 31 PENTACHROMATIC 155 147 135 IIS IIO 106 100 155 ENHARMONIC 30 29 28 22 20 39 1915 PENTACHROMATIG PENTACHROMATIC ENHARMONIC ENHARMONIC 38 3736 28265125 19 50 97 47 36 32 31 3025 PENTACHROMATIC PENTACHROMATIC 125 119 110 90 80 76 70 125 95 91 85 70 65 63 6095 ENHARMONIC PENTACHROMATIC Tonos-2 1: Schlesinger claimed that the Dorian 22 was lowered in the PIS to 21 and that of the Lydian from 27 to 26; tonos-21 is thus the Dorian ofthe PIS. Tonos-25: It has proven difficult to obtain harmoniai whose melodic forms are characteristic ofthe genera. This tonos demands chromatic alternatives (17 for 16, 48 for 47, 23 for 22, 97 for 98, ete.). Tonos-27: This was conjectured by Schlesinger to be the Syntonolydian. Note 21 may alternate with 22. It may be described as the Lydian ofthe PIS. Alternative forms are 27 24 22 20 18 16 14 27, 272625 201817 16 27,and $4 53 52403635 3427. Tonos-29: In the diatonic, 26 may alternate with 27. Tonos-31: These harmoniai admit several variants where 24 and 23, 29 and 30, 28 and 27 are alternatives. In tonos-33, the diatonic bas a variant 33 29 27 24, the chromatic 33 63 30 24, the first trichromatic 99 95 91 72, the second trichromatic 99 95 8772, and the pentachromatic 165 157 145 120 CHAPTER 8

Tonos-15 Tonos-21 Tonos-27 Tonos-33 DIATONIC 151312112118 1615 CHROMATIC 15 14 13 II 21 2016 15 TRICHROMATIC 1 DIATONIC 2119 18 16 15 13 12 21 CHROMATIC DIATONIC 27 2421 2019 161427 CHROMATIC 33 3027 2423 20 18 33 21 2019 16 15 14 12 21 54 51 48 40 38 36 28 27 33 31 29 24.23 22 18 33 TRICHROMATIC I TRICHROMATIC I 81 78 75 60 58 564281 TRICHROMATIC 2 81 78 72 60 58 54 42 81 ENHARMONIC 45 44 43 33 32312445 TRICHROMATIC 2 63 61 59 48 46 44 36 63 DIATONIC CHROMATIC TRICHROMATIC I 99 96 93 72 70 68 5499 54 105 51 40 39 38 28 27 33 32 31 24 47 46 18 33 75 71 65 55 53 504075 TRICHROMATIC 2 63 61 57 48 46 42 36 63 ENHARMONIC 42 41 40 32 31302421 PENTACHROMATIC 105 101 95 80 76 70 60 105 PENTACHROMATIC 135 129 120100 96 90 70 135 165 159 150 120 116 11090 165 Tonos-17 Tonos-23 Tonos-29 DIATONIC 171513 12 23 10917 CHROMATIC 17 1615 12 2311917 TRICHROMATIC I DIATONIC 23 21 20 18 17 14 13 23 CHROMATIC 23 22 21 18 17 16 13 23 TRICHROMATIC I 51494736 35 34 27 51 69 67 65 54 52 50 39 69 45 44 42 33 32 302445 ENHARMONIC 30 29 28 22 43 21 1615 PENTACHROMATIC TRICHROMATIC 2 TRICHROMATIC 2 514945 36 35 33 27 51 69 67 63 54 52 48 39 69 ENHARMONIC ENHARMONIC DIATONIC 29 26 24 22 21 18 16 29 CHROMATIC 29 28 27 22 21 20 16 29 TRICHROMATIC I 87 85 83 66 64 62 48 87 TRICHROMATIC 2 87 85 81 66 64 60 48 87 ENHARMONIC 34 33322447 23 18 17 46 45 44 36 35 34 26 23 58 57 56 4443 42 32 29 PENTACHROMATIC 85 81 75 60 58 55 go 85 PENTACHROMATIC 115 III 105 90 86 Bo 65 115 PENTACHROMATIC 145 141 135 110 106 100 80 145 Tonos-19 Tonos-25 Tonos-31 DIATONIC 19 181614271211 19 CHROMATIC DIATONIC 2522 201817 14 13 25 CHROMATIC 3128262322 20 18 31 191817142713 II 19 50 47 44 36 34 32 26 25 31 29 27 23 22 21 18 31 TRICHROMATIC I TRICHROMATIC I TRICHROMATIC I 57 55 53 42 41 40 33 57 TRICHROMATIC 2 75 72 69 54 52 50 3975 93 89 85 6967 65 54 93 TRICHROMATIC 2 TRICHROMATIC 2 57 55 51 42 41 39 33 57 ENHARMONIC 75 72 66 54 52 483975 93 89 81 69 67 63 54 93 ENHARMONIC ENHARMONIC 38373628 55 5422 19 50 97 47 3635 343625 31 30 29 2345 44 36 31 PENTACHROMATIC 125 119 110 go 86 80 65 125 155 147 135 IIS III 105 90 ISS PENTACHROMATIC 95 91 85 70 68 65 55 95 DIATONIC CHROMATIC PENTACHROMATIC 8-13. New conjunct barmoniai. In this context, conjunct means employing the local tonosSpecific trite synemmenon. 155 SCHLESINGER’S HARMONIAI TRICHROMATIC 2 99 96 90 72 70 66 5499 ENHARMONIC PENTACHROMATIC

8-14. Synopsis ofthe new tonoi. The tonoi are transpositions ofthe Dorian modal sequence so that the determinant ofeach harmoniafalls an bypate meson. A local trite smemmenonfor each ofthe barmoniai has been defined. Certain odd or prime number modal determinants have been expressed asfractions, fe. 21/2, to indicate the higher octave since the modal determinants represent aliquot parts ofvibrating air columns or strings. Modal determinants 14 (28) and 15 (Go) are alternates, Tanos-3 1: in the conjunct form, mese is 23, trite synemmenon is 22. TONOS-I$ TONOS-17 TONOS-19 TONOS-21 TONOS-23 TONOS-25 TONOS-27 TONOS-29 TONOS-31 TONOS-33 P 22 24 28 32 36 36 40 44 48 48 HH 20 22 26 28 32 32 36 40 44 44 HM IS 17 19 21 23 25 27 29 31 33 M II 12 I4 16 18 18 20 22 24 24 TS P 21/2 10 23/2 11 27/2 13 15 14 17 16 17 16 19 18 21 20 22 22 23 22 Harmonizing the new harmoniai - The new harmoniai may be harmonized by methods analogous to those Elsie Hamilton employed with Schlesinger’s diatonic harmoniai. The tetrachordal framework chords of both the disjunct and conjunct forms of the new harmoniai are shown in 8-15. The framework chords from the new conjunct forms are particularly interesting harmonically as they provide a means of incorporating the new harmoniai with the older system. Because many of the modal determinants of the new harmonia are prime numbers, their tetrachordal framework chords do not share many notes with the ones from the older scales. CerND 15/2 17/2 19/2 21/2 23/2 25/2 27/2 29/2 31/2 33/2 tain chords, however, from the new conjunct harmoniai do share notes with the framework chords of the older forms and thus allow one to modulate by common tone progressions. These chords may also be used in progressions similar to those in 8-6c and 8-7. Moreover, these chords may be used to harmonize the mesopykna of the chromatic harmoniai and the oxypykna of the enharmonic which seemingly lay outside of Hamilton’s harmonic concerns. Harmoniai with more than seven tones Although it is quite feasible to define harmoniai with modal determinants between 33 and 44 (the limit of the Mixolydian tonos), it becomes increasingly difficult to decide the canonical forms such harmoniai might take because of the rapidly increasing number of chromatic or alternative tones available in the octave. Rather than omit the extra tones in these and the harmoniai with smaller modal determinants, one may define harmoniai with more than seven tones and utilize the resulting melodic and harmonic resources. 8-15. Harmonization ofthe new barmoniai. Tetrachordalframework chords. HARMONIA-I5 HARMONIA-I7 HARMONIA-I9 HARMONIA-21 HARMONIA-23 HARMONIA-25 HARMONIA-27 HARMONIA-29 HARMONIA-3I HARMONIA-33 156 DISJUNCT 15:11:10:15/2 17:12:11:17/2 CONJUNCT 15:11:8:15/2 17:12:9:17/2 19:14:13:19/2 19:14:11:19/2 21:16:14:21/2 23:18:16:23/2 25:18:16:25/2 27:20:18:27/2 29:22:20:29/2 31:24:22:31/2, 31:23:22:31/2 33:24:22:33/2 21:16:12:21/2 23:18:13:23/2 25:18:13:25/2 27:20:14:27/2 CHAPTER 8 29:22:16:29/2 31:23:18:31/2, 31:24:18:31/2 33:24:18:33/2

8-16. Harmonicforms ofthe Phrygian harmonia. For each ofthe diatonic harmoniai, the harmonic forms are obtained by taking the 2/1 complement of each ratio or interval. FIRST VERSION OF THE INVERTED PHRYGIAN DIATONIC 12 13 14 16 18 20 22 24 CHROMATIC 12 14 16 16 18 22 23 24 ENHARMONIC 24 30 31 32 36 46 47 48 SECOND VERSION OF THE INVERTED PHRYGIAN CHROMATIC 2425 26 32 36 38 40 48 ENHARMONIC 48 49 50 64 72 74 76 96 8-17. Harmonicforms ofthe conjunct Phrygian harmonia. For each ofthe conjunct diatonic harmoniai, the harmonicform is obtained by taking the 2/1 complement ofeach ratio or interval. FIRST VERSION OF THE INVERTED CONJUNCT PHRYGIAN HARMONIAI DIATONIC 1213 14 17 18 20 22 24 Another source of new harmoniai has been suggested by Wilson. One might insert pykna above notes other than the first and fourth degrees of the basic diatonic modal sequence. Interesting variations may also be discovered by inserting more than two pykna, or any number at any location. The final result of this procedure is to generate “close-packed” scales with many more than seven notes. Harmonic forms of the harmoniai Schlesinger’s original harmoniai and all of the new scales generated in analogy with hers are 1- or 2-octave sections of the subharmonic series. These musical structures may be converted to sections of the harmonic series by replacing each of their tones with their 2/1 complements or octave inversions. The resulting harmonic forms may be used in exactly the same way as the originals, save that the modalities of the chords (major or minor) and the melodic contours of the scales are reversed, i.e., the intervals become smaller rather than larger as one ascends from the lowest tone. In general, chords from the harmonic series are more consonant than those from the subharmonic. However, the tones of the harmonic scales are more likely to be heard as arpeggiated chords than are the scalar tones of the subharmonic forms. There is only one form of each of the inverted diatonic harmoniai, but the chromatic, enharmonic and other katapyknotic forms (8-9) have two versions. The first forms are the octave complements of the corresponding subharmonic originals and these forms have their pykna at the upper end of each tetrachord. The second versions are produced by dividing the initial intervals of the two tetrachords of the inverted diatonic forms as in the generation of the chromatic and other kata- CHROMATIC pyknotic forms of 8-9. An example which illustrates these operations 1213 1617 18 22 23 24 is shown in 8-16. The Phrygian harmonia, of modal determinant 12, ENHARMONIC 24 26 34 35 36 46 47 48 SECOND VERSION OF THE INVERTED CONJUNCT PHRYGIAN HARMONIAI CHROMATIC 24 26 27 28 36 38 40 48 ENHARMONIC 48 52 53 54 72 74 76 96 is inverted and then divided to yield the diatonic, chromatic and enharmonic forms. Both versions of the chromatic and enharmonic harmoniai are listed, and the other katapyknotic forms may be obtained by analogy. Conversely, the second of the new harmonic forms may be inverted to derive new subharmonic harmoniai whose divided pykna lie at the top of their tetrachords. These too are listed in 8-16. Conjunct harmoniai may also be inverted to generate harmonic SCHLESINGER’S HARMONIAI

8-18, Wilson's diaphonic cycles. These diaphonic cycles (diacycles) may be constructed on sets of strings tuned alternately a 3/2 and 4/3 apart since the largest divided interval is the 3/2. The order ofthe segments, nodes, and conjunctions may be permuted according to thefollowing scheme: a/b + c/d = ald. cb = 21/1 andc/d. alo =c/b a/d = 2/1. Alternative conjunctions are indicated by primed nodes, i.e. c’, d'. Some diacycles such as number 21 have two independent sets ofnodes and conjunctions. The second is symbolized byefg h. a € 13. Bar 44 serres hrc 33 32 a, c' c d' db bd (3/2 + 4/3; 16/11 - 11/8) (3/2 - 4/3) 2. 12 II 10 9 4a,¢ Id. Gloss Bananen ZÓ en 34 a c d b 8 db (3/2 «4/3; 17/12 « 24/17) (3/2 - 4/3) 3. 18 a 17 16 c 15 14 13 15. Sd, LES GB ann 39 sn. 36 12 bd a 4. 21 20 ac IQ 18 17 16 1$ d 14 b 24 23 ac 22 21 20 19 18 17 16 d 27 26 25 4 22 21 20 c 19 9. c d b 18. 63 nnn 60 nn GO. AS kennen 42 18 a bd c c d' b, d (3/2 + 4/3; 10/7 + 7/5) (3/2 - 4/3) 8. bd (3/2 + 4/3; 10/7 - 7/5) 24 23 a 7. d' 17. GOL. or Alia 40 3/2: 4/3) 6, € 16. 57 56... CERTES 4200000 39 38 ac c d' d b (3/2 - 4/3; 19/14 - 28/19; 19/13 + 26/19) (3/2 + 4/3; 10/7 - 7/5) 5. ce (3/2 » 4/3; 13/9+ 18/13) (3/2 - 4/3) (3/2 + 4/3; 10/7 + 7/5) 19. 66. 64 ann GOL 48... 45 44 a c' c d' db (3/2 > 4/3; 22/15 + 15/11; 16/11 - 11/8) 33 32... Donner 22 ac d b (3/2 «4/3; 16/11: 11/8) 20. 69 Bere 64... Lernen GB. 46 ac c d' d b (3/2 > 4/3; 23/16+ 32/23; 23/17: 34/23) 30cm 28... 21 20 a c db 36. 32 ss 270 24 ac ce’ d 21. Thu JO sus GB. 64... 51 50 49 48 4 6E c € dh f bd (3/2 » 4/3; 10/7 «7/5; 24/17 + 17/12) b (3/2 + 4/3) IO. 39m 36... 27 26 a c db (3/2 - 4/3; 13/9- 18/13) 22. 7h 68... SI a c d II. 42...» GO sum. Jorn, 28 a c d b 23. 78... 76 aen ST resine 52 (3/2 + 4/3; 25/17 + 34/25) a € d (3/2 - 4/3; 26/19+ 19/13) (3/2 + 4/33 10/7. 7/5) 12, 45 Bloemen BO nnen EE 30 a c c d' 50 b 24. Br 4 bd (3/2 + 4/3; 22/16: 15/11) 158 CHAPTER 8 b Bo Tunes COL 56 55 Ge £ d

8-19. Diacycles on 20/13. These diacycles can be constructed on strings 13/10 and 20/13 apart. forms as shown in 8-17. In this case, the disjunctive tone is at the bottom with the two tetrachords linked by conjunction above. These operations may be applied to all of the harmoniai described ‘above. Similarly, the other musical structures presented in the a remainder of this chapter may also be inverted. Ge g d f bh 60 un 56 sar 52 ovnensonrenssonnensenenvenenaveenseenn 42..40 39 ae go € Jhd b (20/13 + 13/10; 3/2 - 4/3; 10/7 - 7/5) BO … 78.76 nnnaansensnosenenensenveneren 60... Sun. 52 a Gt g (20/13 « 13/10; 3/2 - 4/3; 26/19 + 19/13) Other directions: Wilson’s diaphonic cycles Ervin Wilson has developed a set of scales, the diaphonic cycles, which combine the repeated modular structure of tetrachordal scales with the linear division of Schlesinger’s harmoniai (Wilson, personal communication). The diaphonic cycles, or less formally diacycles, may be understood most easily by examining the construction of the two simplest members in 8-18, TOO 99... DO ese Donne 72.470 2. 66 65 a eg € hd f b (20/13 13/10; 10/7 + 7/5:3/2-4/3;16/11- 11/8) 8-20. Triaphonic and tetraphonic cycles on 4/3 and 5/4. (1) may be constructed on three strings tuned to 1/1, 4/3, and 3/2. (2) requires strings tuned to 1/1, 4/3, and 3/2. (3) may be realized on four strings In diacycle 1, the interval 3/2, which is bounded by the nodes a and b, is divided linearly to generate the subharmonic sequence 9 8 7 6 or 1/1 9/8 9/7 3/2. Subtended by this 3/2 is the linearly divided 4/3 bounded by the nodes c and d. This segment forms the sequence 8 7 6 or 1/1 8/7 4/3. Five-tone scales may be produced by joining these two melodic segments with a common tone to yield 1/1 9/8 9/7 3/2 12/7 2/1 (a-b on 1/1, then c-d on 3/2) and 1/1 8/7 4/3 3/2 12/7 2/1 (c-d on 1/1, then a-b on 4/3): 987(6) and 870 (8) 76 (0) 876 tuned to 1/1, 6/5, 147/100 and 42/25. The tones in parentheses are common to the two segments. Diaphonic cycle 2 generates two heptatonic scales which are modes of 20 4, 19 ¢ 18 e 17 d 16 b, 15 f (4/3 - 5/4 + 6/5) DT rccrrscrerarionicren Daenen EN 21 b,f d oe 28 ac (4/3 + 7/6: 9/7) 50 49 Greece ine 42.40 GE fb bd 4 € (5/4 - 6/5 - 7/6 - 8/7) Ptolemy’s equable diatonic genus: 1/1 12/11 6/5 4/3 16/11 8/5 16/9 2/1 and r/ı 12/11 6/5 4/3 3/2 18/11 9/5 2/1. The two forms are respectively termed the conjunctive and disjunctive or tetrachordal form. As the linear division becomes finer, scales with increasing numbers of tones are generated. At number 4, a new phenomenon emerges: the existence of another set of segments whose conjunction produces complete scales. The nodes 4,4 and ¢,b define a pair of diaphonic cycles whose segments are 10/7 and 7/5. These diaphonic cycles can be implemented on instruments such as guitars by tuning the intervals between the strings to a succession of 3/2’s and 4/3’s. The fingerboards must be refretted so that the frets occur at equal aliquot parts of the string length. Wilson constructed several such guitars in the early 1960s. SCHLESINGER’S HARMONIAI

8-21. Divisions ofthefifth, (1) is described as an “aulos-scale (Phrygian, reconstructed by KS)” in Schlesinger 1933. (2) isanother “aulos-scale (Hypodorian),” identified with another unnamed scale of Aristoxenos (Meibomius 1652, 72). (3) isan “aulosscale (Mixalydian),” identified with another unnamed scale ofAristoxenos. (4) is identified with yet another scale ofAristoxenos. (5) spans an augmentedfifth and appears also in ber interpretation of the spondeion. (6) is the “singular major” ofSafiyud-Din (D'Erlanger 1938, 281). The Islamic genera arefrom Rouanet 1922. (8), Isfahan, spans only the 4/3. (9) is labeled “Zirafkend Bouzourk.” Rouanet's last genus is identical to Saftyu-d-Din’s scale ofthe same name. Wilson has also developed a set of simpler scales on the same principles under the general name of “Helix Song.” They consist of notes selected from the harmonic series on the tones 1/1 and 4/3. These have been used as the basis of a composition by David Rosenthal (Rosenthal 1979). Triacycles and tetracycles For the sake of completeness, some new diacycles have been constructed on the interval pair 20/13 and 13/10. These are listed in 8-19. As 20/13 is slightly larger than 3/2, some new diacycles on 3/2 are generated incidentally too. Larger intervals and their octave complements might be used, but the increased inequality in the sizes of the two segments would probably be melodically unsatisfactory. This asymmetry may be hidden by defining three or four segments instead of merely two. A few experimental threeand four-part structures, which may be called triacycles and tetracycles, are SCHLESINGER’S DIVISIONS 1. 24/23: 23/22 . 11/9 + 9/8 2. 16/15 15/14 : 7/6 - 9/8 3. 28/27 9/8 - 8/7 - 9/8 4. 21/20: 10/9 + 9/8 + 8/7 5. I1/10- 10/9 : 9/8. 8/7 ISLAMIC GENERA 6. 14/13: 8/7 - 13/12 - 14/13 - 117/112 7. 13/12 - 14/13 - 13/12 - 287/272 8. 13/12-14/13: 16/14: 16/15 9. 14/13-13/12: 36/35 : 9/8. 10/9 shown in 8-20. Linear division of the fifth Asa final note, it must be mentioned that both Schlesinger (1933) and the Islamic theorists also recognized scales derived by linear division of the fifth instead of the fourth or octave (8-21). Not surprisingly, Schlesinger’s are presented as support for the authenticity of her harmoniai. It is likely that the Islamic forms had origins that are independent of the Greek theoretical system. The genus from Safiyu-d-Din (D’Erlanger 1938) may be rationalized as being derived from the permuted tetrachord, 14/13 + 8/7 : 13/12, by dividing the disjunctive tone, 9/8, of the octave scale into two unequal parts, 14/13 and 117/112. Characteristically, all 24 permutations of the intervals were tabulated. Rouanet’s scales deviate even more from Greek models, though the tetrachordal relationship may still be seen (Rouanet 1922). CHAPTER 8

The Catalog of tetrachords THIs CATALOG ATTEMPTS a complete and definitive compilation of all the tetrachords described in the literature and those that can be generated by the straightforward application of the arithmetic and geometric concepts described in the previous chapters. While the first of these goals can be achieved in principle, the second illustrates Aristoxenos’s tenet that the divisions of the tetrachord are potentially infinite in number. It seems unlikely, however, that any great number of musically useful or theoretically interesting tetrachords has been omitted. Figures 9-1 through 9-6 show that the two-dimensional tetrachordal space is nearly filled by the tetrachords in the Catalog. The saturation of perceptual space is especially likely when one considers the finite resolving power of the ear, the limits on the accuracy and stability of analog and acoustic instruments, the quantizing errors of digital electronics, and our readiness to accept sufficiently close approximations to ideal tunings. Nevertheless, processes such as searches through large microchromatic scales (chapter 7) and propriety calculations (chapter 5) will occasionally turn up new genera, so perhaps one should not be too complacent. ‘The great majority of these new tetrachords, however, will resemble those already in the Catalog or be interchangeable with them for most melodic and harmonic purposes. Organization of the Catalog The tetrachords in the Main Catalog are listed by the size of their largest interval, which, in lieu of an historically validated term, has been called the THE CATALOG OF TETRACHORDS

LARGEST INTERVAL characteristic interval (CI). The term apyknon would have been used except that it has been traditionally employed to denote the sum of the two lower 400 intervals of the diatonic genera. In diatonic tetrachords, this sum is greater than one half of the fourth. Those tetrachords with CIs larger than 425 cents are classed as hyperenharmonic (after Wilson) and listed first. Next come the enharmonic with their incomposite CIs approximating major thirds. Chromatic and diatonic genera follow, the latter beginning when the CI falls below 250 300 200 100 cents. r le) 200 For each CI, the genera derived from the 1:1, 1:2, and 2:1 divisions of the pyknon or apyknon are listed first and followed by the other species of SMALLEST INTERVAL 9-1. Tetrachords injust intonation: smallest vs. largest intervals. Units in cents, The oblique lines are the upper and lower limits ofthe largest interval for each value ofthe smallest. This graph is limited to the tetrachords in the main, reduplicated, and miscellaneous lists. tetrachord with this CI, References to the earliest literature source and a brief discussion of the genus are given below each group. In addition to the genera from the literature, the majority of the Main Catalog comprises tetrachords generated by the processes outlined in chapters 4 and 5. Both the 1:2 and 2:1 divisions are provided because both must be examined to select “strong,” mostly superparticular forms in the Ptolemaic manner (chapter 2). If strict superparticularity is less important than convenience on the monochord or linear order, the 1:2 division is preferable, but recourse to the 2:1 may be necessary to discover the simplest form. For example, the threefold division of the 16/15 pyknon yields the notes 48 47 46 45. Ptolemy chose to recombine the first two intervals and reorder the third to obtain his enharmonic, 46/45 24/23 : 5/4. In general, only the simplest or mostly superparticular divisions are tabulated in this section; occasionally a theoretically interesting tetrachord SECOND INTERVAL 300 | 200 4 100 without any near relatives will be found in the Miscellaneous list. Such isolated tetrachords are relatively uncommon. There are cases, however, in fost which all of the other divisions of a tetrachord’s pyknon or apyknon have very complex ratios, and so closely resemble other tetrachords already o 200 FIRST INTERVAL 9-2. Tetrachords injust intonation:first vs. second intervals, The oblique lines are the upper and lower limits of the second intervalfor each value ofthefirst. This graph is limited to the tetrachords in the main, reduplicated, and miscellaneous lists. tabulated that it did not seem fruitful to list them in a group under the CI in the Main Catalog. “Miscellaneous” is a very elastic category. It consists of a collection of genera of diverse origin that I did not think interesting enough to list in the Main Catalog. The order of intervals within each tetrachord is the canonical small, medium, and large in the case of the historical genera and their analogs. The new theoretical genera are generally listed in the order resulting from CHAPTER 9

their generating process. It should be remembered, however, that all six permutations of the non-reduplicated genera and all three of the 7 er o reduplicated are equally valid for musical experimentation. With the exception of the Pythagorean 256/243 :9/8-9/8 and Al-Farabi’s 10/9 - 10/9. 27/25,the genera with reduplicated intervals are given in the list ait niet FF PRE 22 200 | x © = a 100 1 om: > o T T T (e) L 200 PARHYPATE 9-3. Tetrachords injust intonation: parbypatai vs. lichanoî. The oblique lines are the upper and lower limits of lichanosfor each value ofparhypate. This graph is limited to the tetrachords in the main, reduplicated, and miscellaneous lists. of Reduplicated tetrachords. Those tetrachords defined in either in “parts” of the tempered fourth or which consist solely of tempered intervals are to be found in the Tempered list. Needless to say, these tetrachords are a diverse lot, covering Aristoxenos’s divisions, Greek Orthodox liturgical genera (in two systems — one of 28 parts to the fourth, the other of 30), and those derived from theoretical considerations. As some of the latter contain rational intervals as well, a separate list of Semi-tempered tetrachords is included. No attempt has been made to catalog the very numerous tetrachords and tetrachord-like structures found in the non-zero modulo 12 equal temperaments of 4-17. An index of sources for those tetrachords of historical provenance is provided. In order to show the uniformity with which the set of alt possible tetrachords in just intonation has been sampled in the Catalogs of this chapter, the genera from the Main, Reduplicated, and Miscelianeous lists have been plotted in Le] oO Kad LARGEST INTERVAL Uniformity of sampling 200 100 Le) 200 SMALLEST INTERVAL 9-4. Just and tempered tetrachords: smallest vs. largest intervals. The oblique lines are the upper and lower limits ofthe largest intervalfor each value of the smallest. This graph contains all the tetrachords in the Catalog. 9-1, 9-2 and 9-3. In 9-1, the smallest intervals are plotted against the largest intervals or CIs. As one may see, the area delineated by the two oblique lines is more or less uniformly filled. However, diagonal zones corresponding to genera with roughly equal and 1:2 divisions are evident. The tables are deliberately deficient in genera with commatic and sub-commatic intervals, as these are of little use melodically. The few examples in the tables are taken mostly from Hofmann’s list of superparticular divisions (Vogel 1975) or generated by theoretical operations such as the means of chapter 4. 9-2 isa plot of the first versus the second intervals of the same tetrachords. Although the graph has a different shape, the same conclusions may be drawn. 9-3 is a third representation of the same data. In this case, cumulative rather than sequential intervals have been plotted, This mode reflects the Greek classification of tetrachords into primary genera (enharmonic, THE CATALOG OF TETRACHORDS

chromatic and diatonic) and shades or nuances (chroai) of these genera. The primary distinction is based on the size of the uppermost interval, usually the CI except in Archytas’s and Ptolemy’s diatonics (28/27 - 8/7 - 9/8 and Le) SECOND INTERVAL 5 le) 16/15 : 9/8 - 10/9). The exact nuance or shade is then defined by the size of the first interval. The position of parhypate is equivalent to the size of the first interval and the position of lichanos is an inverse measure of the CI. This graph also reveals the relative uniformity of coverage and the excess of genera with 1:1 and 1:2 divisions. | The tetrachords in the Tempered and Semi-tempered lists were added to the set graphed in 9-1-3, and the entire collection replotted in 9-4-6. Le] The largest empty spaces in the plots are thus filled. In a few cases, the gaps could be filled only by creating new genera specifically for this task. These have been marked in the Tempered tetrachord list. 200 FIRST INTERVAL 9-5. Just and tempered tetrachords:first vs. second intervals. The oblique lines are the upper and lower limits of the second intervalfor each value ofthe first. This graph contains all the tetrachords in the The Main Catalog HYPERENHARMONIC TETRACHORDS Catalog. 500 Hi. CHARACTERISTIC INTERVAL 13/10 | goo | “A O 4n 300 = fai 454 CENTS 80/79 : 79/78 - 13/10 224224454 60/49 - 118/117 : 13/10 29 +15 +454 120/119- 119/117 - 13/10 14 +29 + 454 100/99 : 66/65 : 13/10 17+26+454 WILSON The 13/10 would appear to be the upper limit for a genus-defining CI simply because the pyknotic intervals become too small to be melodically useful, however perceptible they might remain. In general, tetrachords with intervals less than 20 cents or with overly complex ratios will be relegated to the Miscellaneous listing at the end of the Catalog proper, unless there 200 is some compelling reason, such as historical or literary reference, illustration à 100 of theory, or the like, to include them. The pyknon of this hyperenharmonic Mr o genus is the 40/39 (44 cents), which is very close to the Pythagorean double 1 200 comma of 374/238, Number 4 is from the unpublished notes of Ervin Wilson. PARHYPATE AN La See also Miscellaneous. H2. CHARACTERISTIC INTERVAL 35/27 449 CENTS 72/71 + 71/70 + 35/27 24 + 25 + 449 108/107 - 107/105 + 35/27 16 +33 +449 on 9-6. Just and tempered tetrachords: parbypatai vs. lichanoi. The oblique lines are the upper and lower limits of lichanosfor each value ofthe parhypate. This graph contains all the tetrachords in the Catalog. 54/53 « 106/105 : 35/27 64/63 - 81/80 - 35/27 164 CHAPTER 9

This genus divides the 36/35 (49 cents), an interval found in Archytas’s enharmonic and Avicenna’s chromatic. Number 8 is found in Vogel's tuning for the Perfect Immutable System (Vogel 1963, 1967) and Erickson’s (1965) analysis of Archytas’s system (see chapter 6). IO II 12 H3. CHARACTERISTIC INTERVAL 22/17 446 CENTS 68/67 : 67/66 - 22/17 26 + 26 + 446 51/50: 100/99 - 22/17 35 + 17 + 446 102/101 + 101/99 - 22/17 17 +35 + 446 85/84« 56/55 + 22/17 20 + 31 + 446 WILSON The pyknon of this hyperenharmonic genus is 34/33 (52 cents), a quartertone. The intervening genera with pykna between 39/38 and 35/34 have not so far yielded melodically interesting, harmonically useful, nor mathematically elegant divisions, but see Miscellaneous for examples. This genus is replete with intervals of 17. H4. CHARACTERISTIC INTERVAL 128/99 445 CENTS 66/65 » 65/64 : 128/99 26+27 +445 99/98. 49/48 - 128/99 18 + 36 + 445 13 14 15 99/97 : 97/96 + 128/99 35 + 18 + 445 The pyknon of this genus ís 33/32 (53 cents), the octave-reduced thirty-third harmonic and an approximate quarter-tone. 16 Hs. CHARACTERISTIC INTERVAL 31/24 443 CENTS 64/63 - 63/62 : 31/24 27 + 28 + 443 17 18 96/95 + 95/93 : 31/24 48/47 « 94/93 + 31/24 18 + 37 + 443 36 + 19 + 443 This hyperenharmonic genus divides the 32/31 (55 cents), an interval used in Didymos’s enharmonic. 19 20 21 Hé. CHARACTERISTIC INTERVAL 40/31 441 CENTS 62/61 : 61/60 : 40/31 28 +29 + 441 93/92 : 46/45 + 40/31 19 + 38 + 441 93/91 - 91/90 - 40/31 38 + 19 + 441 The pyknon of this genus is 31/30 (57 cents), an interval which occurs in Didymos’s enharmonic. H7. CHARACTERISTIC INTERVAL 58/45 22 60/59 « 59/58 - 58/45 90/89 : 89/87 : 58/45 23 24 45/44 - 88/87 « 58/45 165 29 + 30+ 439 19 + 39 + 439 39 + 20 + 439 THE CATALOG OF TETRACHORDS 439 CENTS

120/119- 119/116 14 + 44 + 439 The pyknon of this hyperenharmonic genus is 30/29 (59 cents). H8. CHARACTERISTIC INTERVAL 9/7 435 CENTS 26 56/55 : 55/54- 9/7 31 + 32+435 27 42/41 + 82/81 - 9/7 84/83 - 83/81 - 9/7 42 +21+435 21 +42 +435 64/63-49/48 - 9/7 70/69 46/45 + 9/7 40/39 + 91/90 : 9/7 27 + 36 + 435 25 + 38 +435 44+ 19 + 435 112/111 - 37/36 9/7 81/80 - 2240/2187 - 9/7 16+47+435 22 +41 +435 9/7: 119/117 . 52/51 435 + 29 + 34 28 29 30 31 32 33 34 WILSON The pyknon of this prototypical hyperenharmonic genus (Wilson, unpublished) is Archytas’s diesis, 28/27 (63 cents). Melodically, this genus bears the same relation to Aristoxenos’s soft chromatic as Aristoxenos’s enharmonic does to his syntonic (intense) chromatic, Number 26 is Wilson’s original “hyperenharmonic” tetrachord. Divisions 29 and 31 are interesting in that their first intervals make, respectively, an 8/7 and a 15/13 with the subtonics hyperhypate (diatonic lichanos meson) and mese, and proslambanomenos and diatonic paranete diezeugmenon as well. Tetrachord number 32 is a good approximation to a hypothetical 1 + 3 + 26 parts, 17 + 50 + 433 cents—see also number 25 above. Number 33 occurs in Vogel’s (1963, 1967) PIS tuning. Number 34 is a summation tetrachord from chapter 4. Ho. CHARACTERISTIC INTERVAL 104/81 35 36 37 54/53 © 53/52 : 104/81 32 + 33 +433 81/79 - 79/78. 104/81 81/80 - 40/39 : 104/81 43 + 22 +433 22 +44 +433 433 CENTS The pyknon of this genus is 27/26 (65 cents). This division is melodically similar to the 9/7 genus, though not harmonically. Number 37, when rearranged, generates a 15/13 with the subtonic. Hıo. CHARACTERISTIC INTERVAL 50/39 39 40 52/51 » 51/50 - 50/39 39/38 - 76/75 : 50/39 78/77 - 77/75 + 50/39 430 CENTS 34+35+430 45 +23 + 430 22 +46 +430 The pyknon is 26/25 (68 cents) and is inspired by Kathleen Schlesinger’s (1939, 214) enharmonic Lydian harmonia. CHAPTER 9

Hir. CHARACTERISTIG INTERVAL 32/25 427 CENTS 35 + 36 + 427 50/49 : 49/48 + 32/25 75/73 + 73/72 32/25 75/74-37/36 + 32/25 46+ 24+ 427 23 + 47 + 427 This genus divides the 25/24 minor semitone (71 cents). The 32/25 is the 3/2’s complement of 75/64, the 5-limit augmented second (5/4 - 5/4 - 5/4. 3/2, reduced to one octave). ENHARMONIC TETRACHORDS Er. CHARACTERISTIC INTERVAL 23/18 424 CENTS 48/47 : 47/46 « 23/18 36 +37 +424 SCHLESINGER 36/35 : 70/69 « 23/18 49 +25 +424 WILSON 72/71. 71/69 : 23/18 24 + 50+ 424 30/29- 116/115 - 23/18 59 +15 + 424 WILSON 60/59- 118/115: 23/18 29 + 45 + 424 This genus divides the 24/23 (74 cents) and lies on the boundary between the enharmonic and hyperenharmonic genera. It is analogous to the 9/7 genus but divides the hemiolic chromatic rather than the soft or intense diesis, Numbers 45 and 47 are from Wilson. Number 44 (Schlesinger 1939, 214) is the lower tetrachord of her enharmonic Phrygian harmonia. Ez. CHARACTERISTIC INTERVAL 88/69 49 50 51 46/45 - 45/44 - 88/69 38 + 39 + 421 69/67 - 67/66 - 88/69 69/68 - 34/33 : 88/69 51+ 26+ 421 25 +52 + 421 421 CENTS The pyknon of this enharmonie genus is 23/22 (77 cents). 52 53 54 E3. CHARACTERISTIC INTERVAL 50/41 421 GENTS 320/313 - 313/306 - 51/40 38 +39 + 421 480/473 - 473/459 + 51/40 25 +52 +421 240/233 - 466/459 - 51/40 51+26+421 The pyknon is 160/153 (77 cents). The 51/40 is the 3/2’s complement of 20/17. 55 56 57 59 E4. CHARACTERISTIC INTERVAL 14/11 418 GENTS 44/43 : 43/42 + 14/11 40 +41 +418 33/32 + 64/63 : 14/11 53 +27 +418 66/65 + 65/63 - 14/11 26 + 54 + 418 88/87 : 29/28 - 14/11 20 + 61 + 418 36/35 : 55/54: 14/11 49 + 32 + 418 THE CATALOG OF TETRACHORDS

50/49 - 77/75 14/11 14/11 - 143/140 - 40/39 418 + 37 + 44 This is a new genus whose pyknon is 22/21 (81 cents). The 14/11 is a supramajor third found in the harmonic series between the fourteenth and 35 +46 + 418 eleventh partials. It occurs in the Partch diamond and other extended systems of just intonation. 62 64 Es. CHARACTERISTIC INTERVAL 80/63 414 CENTS 42/41 : 41/40: 80/63 42 + 42 +414 63/61 + 61/60 - 80/63 56 +28 +414 63/62 - 31/30 - 80/63 27+57+414 The pyknon of this enharmonic genus is 21/20 (84 cents), a common interval in septimal just intonation, 65 66 67 68 69 70 E6. CHARACTERISTIC INTERVAL 33/26 413 CENTS 208/203 - 203/198 + 33/26 42 + 43 +413 312/307 » 307/297 - 33/26 28 +57 +413 312/302 + 302/297 - 33/26 56+29 +413 52/51 34/33 + 33/26 34 +52 +413 26/25 : 100/99 : 33/26 68+ 18 +413 78/77 : 28/27 + 33/26 22 +63 +413 The characteristic interval of this genus is the 3/2’s complement of 13/11 and derives from the 22:26:33 triad. The pyknon is 104/99 (85 cents). E7. CHARACTERISTIC INTERVAL 19/15 71 72 40/39 : 39/38 - 19/15 73 74 60/59 - 59/57 + 19/15 29 + 60 + 409 28/27- 135/133 - 19/15 63 + 26 +409 30/29 + 58/57 - 19/15 44 +45 + 409 409 CENTS ERATOSTHENES 59 + 30 + 409 The pyknon, 20/19 (89 cents), of this historically important genus is very close to the Pythagorean limma, 256/243. Number 71 is a good approximation to Aristoxenos’s enharmonic of 3 + 3 +24 “parts,” and, in fact, is both Eratosthenes’s enharmonic tuning and Ptolemy’s misinterpretation of Aristoxenos’s geometric scheme (Wallis 1682, 170). The next two entries are 2:1 and 1:2 divisions of the pyknon in analogy with the usual Ptolemaic and later Islamic practices. Number 73 is a hypothetical Ptolemaic interpretation of a (pseudo-)Aristoxenian 2 + 4 + 24 parts. An echo of this genus may appear as the sub-4o division found on the fingerboard of the Tanbur of Baghdad, a stringed instrument (Helmholtz [1877] 1954, 517). CHAPTER 9

The last species is an analog of Archytas’s enharmonic and the first makes a 15/13 with the subtonic. E8. CHARACTERISTIC INTERVAL 81/64 408 CENTS 512/499 - 499/486 - 81/64 45 + 46 + 408 BOETHIUS 79 80 384/371 - 742/729 « 81/64 768/755 : 755/729 - 81/64 40/39 - 416/405 - 81/64 128/125 : 250/243 « 81/64 64/63 : 28/27 : 81/64 60 + 31 + 408 30 + 61 + 408 44 + 46 + 408 4I + 49 + 408 27 +63 + 408 EULER WILSON 81 324 /238 . 246/329. 81/64 47 + 43 + 408 36/35 « 2240/2187 - 81/64 49 + 41 + 408 75 76 77 78 82 In these tunings the limma, 256/243 (90 cents), has been divided. Number 75 is the enharmonic of Boethius and is obtained by a simple linear division of the pyknon. It represents Aristoxenos’s enharmonic quite well, but see the preceding 19/15 genera for a solution more convenient on the monochord. In practice, the two (numbers 71 and 75) could not be distinguished by ear. Numbers 76 and 77 are triple divisions of the pyknon, for which Wilson’s division is a convenient and harmonious approximation. Number 78 is an approximation to number 75, as is Euler’s “old enharmonic” (Euler [1739] 1960, 170). Wilson’s tuning (number 80) should also be compared to the Serre division of the 16/15 (5/4 genus). When number 80 is rearranged, the 28/27 will make a 7/6 with the subtonics hyperhypate or mese. In this form, it is a possible model for a tuning transitional between Aristoxenos’s and Archytas’s enharmonics. The purely Pythagorean division (number 81) is obtained by tuning five fifths down for the limma and twenty-four up for the double comma. Number 82 is found in Vogel's tuning (1963, 1967) and resembles Euler’s (number 79). Eg. CHARACTERISTIC INTERVAL 24/19 404 CENTS 38/37 + 37/36 - 24/19 57/55-55/54" 24/19 46 + 47 + 404 62 + 32 +404 57/56 : 28/27. 24/19 76/75 - 25/24 - 24/19 31 +63 + 404 23+ 71 +404 40/39 « 117/95 : 24/19 44 + 50 +404 WILSON The pyknon is 19/18 (94 cents). The interval of 24/19 derives from the 16:19:24 minor triad, which Shirlaw attributes to Ousley (Shirlaw 1917, 434) and which generates the corresponding tritriadic scale. It is the 3/2 complement of 19/16. THE CATALOG OF TETRACHORDS

Ero. CHARACTERISTIC INTERVAL 34/27 399 CENTS 36/35 - 35/34- 34/27 49 + 50 + 399 27/26: 52/51 + 34/27 65 + 34 + 399 go 54/53 > 53/51 + 34/27 32 + 67 + 399 91 24/23 - 69/68 - 34/27 74 + 25 + 399 This genus divides the 18/17 semitone of 99 cents, used by Vincenzo Galilei in his lute fretting (Barbour 1953; Lindley 1984). These genera are virtually equally-tempered and number 88 is an excellent approximation to Aristoxenos’s enharmonic, It is also the first trichromatic of Schlesinger’s Phrygian harmonia. 92 93 94 95 96 97 Err. CHARACTERISTIC INTERVAL 113/90 240/233 + 233/226 + 113/90 51+ 53 +394 180/173 + 346/339 : 113/90 360/353 : 353/339 - 113/90 394 CENTS 69 + 35+ 394 34 + 70 + 394 30/29 - 116/113 - 113/90 59 +45 + 394 40/39- 117/113 - 113/90 44 + 60 + 394 60/59 : 118/113 » 113/90 29 + 75 + 394 These complex divisions derive from an attempt to interpret in Ptolemaic terms a hypothetical Aristoxenian genus of 7 + 23 parts. The inspiration came from Winnington-Ingram’s 1932 article on Aristoxenos in which he discusses Archytas’s 28/27 - 36/35 : 5/4 enharmonic genus and its absence from Aristoxenos’s genera, despite the somewhat grudging acceptance of Archytas’s other divisions. In Aristoxenian terms, Archytas’s enharmonic would be 4 + 3 + 23 parts, and the first division is 3.5 + 3.5 + 23. Number 95 is the 4 + 3 division and 93 and 94 are 2:1 and 1:2 divisions of the complex pyknon of ratio 120/113 (104 cents), Numbers 96 and 97 are simplifications, while number 96 generates an ekbole of 5 dieses (15/13) with the subtonics hyperhypate and mese. E12. CHARACTERISTIC INTERVAL 64/51 99 100 IOI 102 34/33 - 33/32 - 64/51 52 + 53 + 393 51/50 - 25/24 - 64/51 34 + 71+ 393 49/48 : 51/49 - 64/51 36 + 69 + 393 68/65. 65/64 : 64/51 68/67 : 67/64 : 64/51 78 +27 + 393 26 + 79 + 393 393 CENTS The pyknon of this enharmonic genus is 17/16 (ros cents), the seventeenth harmonic and a basic interval in septendecimal just intonation. CHAPTER 9

E13. CHARACTERISTIC INTERVAL 5/4 386 CENTS 32/31 + 31/30 + 5/4 55 + 57 + 386 46/45 + 24/23 - 5/4 38 + 74 + 386 105 48/47 47/45 + 5/4 36 + 75 + 386 106 28/27 - 36/35 - 5/4 56/55-22/21 + 5/4 40/39 - 26/25 + 5/4 25/24« 128/125 - 5/4 21/20 - 64/63 : 5/4 256/243 - 81/80 - 5/4 63 +49 + 386 31 + 81 + 386 44 + 68 + 386 71 +41 + 386 84 + 27 + 386 90 + 22 + 386 112 76/75 - 20/19 : 5/4 23 + 89 + 386 113 96/95 + 19/18 : 5/4 136/135 - 18/17 - 5/4 256/255 - 17/16 - 5/4 68/65 «5/4 + 52/51 18 + 94 + 386 13 +99 + 386 7+105 + 386 78 + 386 +34 103 107 108 109 Io III 114 115 116 DIDYMOS PTOLEMY ARCHYTAS PTOLEMY? AVIGENNA SALINAS PACHYMERES FOX-STRANGWAYS? WILSON HOFMANN HOFMANN These tunings are the most consonant of the shades of the enharmonic genera. Although Plato alludes to the enharmonic, the oldest tuning we actually have is that of Archytas (390 see). This tuning, number 106, clearly formed part of a larger musical system which included the subtonic and the tetrachord synemmenon as well as both the diatonic and chromatic genera (Winnington-Ingram 1932; Erickson 1965). Didymos’s tuning is the 1:1 division of the 16/15 (112 cents) pyknon and dates from a time when the enharmonic had fallen out of use. Number 104 is undoubtedly Ptolemy’s own, but the surviving manuscripts contain an extra page which lists number 107 instead. Wallis believed it to be a later addition, probably correctly. Numbers 104 and 105 are the 1:2 and 2:1 divisions, given as usual for illustrative and/or pedagogical purposes. The Avicenna tuning (D'Erlanger 1935, 154) has the 5/4 first in the original, following the usual practice of the Islamic theorists. In this form, it makes a 15/13 with the subtonic. Number 109 is Euler’s enharmonic (Euler [1739] 1960, 178); Hawkins, however, attributes it to Salinas (Hawkins [1776] 1963, 27). Daniélou gives it in an approximation with 46/45 replacing the correct 128/125 (Daniélou 1943, 175). The Pachymeres enharmonic is attributed by Perrett to Tartini (Perrett 1926, 26), but Bryennios and Serre also list it. Number 111 is given as Rag Todi by Fox-Strangways (1916, 121) and as Gunakali by Daniélou (1959, 134-135). The divisions with extraordinarily small intervals, numbers 114 and 115, were found by Hofmann in his THE CATALOG OF TETRACHORDS

computation of the 26 possible superparticular divisions of the 4/3 (Vogel E14. CHARACTERISTIC INTERVAL 8192/6561 117 4374/4235 + 4235/4096 8192/6561 57 + 57 + 384 118 6561/6283 - 6283/6144 : B192/6561 6561/6422 - 3211/3072 - 8192/6561 75 +39 + 384 37 + 77 + 384 119 120 384 CENTS 374/238 . 227/317 . 8192/6561 47 + 68 + 384 The interval 8192/6561 is Helmholtz’s skhismic major third, which is generated by tuning eight fifths down and five octaves up (Helmholtz [1877] 1954, 432). The pyknon is the apotome, 2187/2048 (x14 cents). It has been linearly divided in the first three tetrachords above, but a purely Pythagorean division is given as number 120. E15. CHARACTERISTIC INTERVAL 56/45 121 30/29 - 29/28 : 56/45 59 + 60 + 379 122 126 45/43 - 43/42 - 56/45 45/44 : 22/21 + 56/45 25/24 - 36/35 : 56/45 80/77-33/32-56/45 , 60/59 - 59/56 - 56/45 79 + 41 + 379 39 + 53 + 379 71 +49 + 379 66+ 53 + 379 29 + 90 + 379 127 40/39 + 117/112 : 56/45 44 + 76 + 379 128 26/25 - 375/364 + 56/45 68 + 52 + 379 123 124 125 379 CENTS PTOLEMY The pyknon is 15/14 (119 cents). Number 121 is Ptolemy’s interpretation of Aristoxenos’s soft chromatic, 4 + 4 + 22 parts. Number 125 is a Ptolemaic interpretation of a hypothetical 4.5 + 3.5 + 22 parts, an approximation to Archytas’s enharmonic (Winnington-Ingram 1932). Number 124 is a simplification of the former tuning, and numbers 122 and 123 are the familiar threefold divisions. Number 128 is a summation tetrachord. 129 130 131 E16. CHARACTERISTIC INTERVAL 41/33 376 CENTS 88/85 - 85/82 - 41/33 60 + 62 + 376 42/41 - 22/21 : 41/33 42 +81 + 376 44/43 : 43/41 + 41/43 39 + 82 + 376 This genus is an attempt to approximate a theoretical genus, 62.5 + 62.5 + 375 cents, which would lie on the border between the chromatic and enharmonic genera. Number 129 is quite close, and numbers 130 and 131 are 1:2 and 2:1 divisions of the complex 44/41 (122 cents) pyknon. CHAPTER 9

CHROMATIC TETRACHORDS 134 135 136 137 138 Ci. CHARACTERISTIC INTERVAL 36/29 374 CENTS 29/28. 28/27 - 36/29 61 + 63 +374 87/85 «85/81. 36/29 40 + 83 +374 81 + 42 + 374 87/83 - 83/81 - 36/29 This genus is also an approximation to 62.5 + 62.5 + 375 cents. The 36/29 is from the 24:29:36 triad and tritriadic scale. The pyknon is 29/27 (124 cents). C2. CHARACTERISTIC INTERVAL 26/21 370 CENTS 28/27 - 27/26. 26/21 63 +65 + 370 SCHLESINGER 21/20 : 40/39 : 26/21 85 + 44+ 370 42/41 : 41/39- 26/21 42 + 87 + 370 24/23 + 161/156: 26/21 74 +55 + 370 This genus divides the pyknon, 14/13 (128 cents) and approximates Aristoxenos’s soft chromatic. Number 135 is from Schlesinger (1933) and is a first tetrachord of a modified Mixolydian harmonia. C3. CHARACTERISTIC INTERVAL 21/17 139 140 141 142 143 144 145 146 366 CENTS 136/131 + 131/126 « 21/17 65 + 67 + 366 102/97 - 194/189 » 21/17 87 + 45 + 366 43 + 89 + 366 204/199+ 199/189 : 21/17 27 + 105 + 366 64/63 : 17/16 + 21/17 34/33 - 22/21 - 21/17 52 +81 + 366 44+ 88 + 366 40/39- 221/210- 21/17 24/23 + 391/378 + 21/17 74 + 59 + 366 63 + 69 + 366 28/27 - 51/49 - 21/17 The pyknon is 68/63 (132 cents). Number 139 is a very close approximation of Aristoxenos’s soft chromatic, 4 + 4 + 22 “parts,” as is number 146 also. Numbers 144 and 146 make intervals of 15/13 and 7/6, respectively, with their subtonics. 147 C4. CHARACTERISTIC INTERVAL 100/81 27/26 « 26/25 - 100/81 65 + 68 + 365 148 81/77» 77/75 + 100/81 87 + 46 + 365 149 150 ISI 152 81/79 : 79/75 - 100/81 45 + BB + 365 81/80+ 16/15 + 100/81 22+ 112 + 365 51/50: 18/17: 100/81 36/35 - 21/20 - 100/81 34+ 99 + 365 49 + 85 + 365 THE CATALOG OF TETRACHORDS 365 CENTS

40/39 - 1053/1000 - 100/81 44 + 89 + 365 135/128 - 128/125 - 100/81 92 +41 +365 DANIÉLOU 24/23 - 207/200. 100/81 74 + 60 + 365 The pyknon is the great limma or large chromatic semitone, 27/25 (133 cents). Daniélou listed his tetrachord in approximate form with 46/45 instead of the correct 128/125. (Daniélou 1943, 175). Number 147 isa close approximation to Aristoxenos’s soft chromatic, but the rest of the divisions are rather complex. Cs. CHARACTERISTIC INTERVAL 37/30 80/77 : 77/74 - 37/30 66 + 69 + 363 363 CENTS 157 158 20/19 - 38/37: 37/30 89 + 46 + 363 40/39 - 39/37 + 37/30 44+ 91 + 363 159 160 30/29: 116/111: 37/30 60/59+ 118/111 37/30 59 + 76 + 363 29 + 106 + 363 PTOLEMY This complex chromatic genus divides the 40/37 (135 cents). Number 156 is Ptolemy’s linear interpretation of Aristoxenos’s hemiolic chromatic, 4.5 + 4.5 + 21 “parts,” with its characteristic neutral third and 3/4-tone pyknon. This division closely approximates his soft chromatic, indicating that Ptolemy’s interpretation in terms of the aliquot parts of a real string was erroneous and that Aristoxenos really did mean something conceptually similar to equal temperament. However, Ptolemy’s approach and the resulting tetrachords are often interesting in their own right. For example, number 157 could be considered as a Ptolemaic version of Aristoxenos’s 1/2 + 1/4+ 1 3/4 tones, 6 +3 +21 “parts,” a genus rejected as unmelodic because the second interval is smaller than the first (Winnington-Ingram 1932). The remaining genera are experimental. C6. CHARACTERISTIC INTERVAL 16/13 359 CENTS 26/25 + 25/24 - 16/13 68 + 71 + 359 168 169 39/37 : 37/36: 16/13 91 +47 + 359 39/38 « 19/18- 16/13 45 +94+ 359 65/64 - 16/15 - 16/13 52/51-17/16. 16/13 40/39 + 169/160 - 16/13 28/27 - 117/112 + 16/13 169/168: 14/13 - 16/13 22/21 + 91/88 > 16/13 27 +112 +359 34 + 105 + 359 44 + 95 + 359 63 +76 + 359 II +128 +359 Br + 58 + 359 The pyknon of this genus, which lies between the soft and hemiolic CHAPTER 9

chromatics of Aristoxenos, is 13/12 (139 cents). Number 169 is a summation tetrachord from chapter 4. 171 172 173 174 175 C7. CHARACTERISTIG INTERVAL 27/22 355 CENTS 176/169+ 169/162 - 27/22 70 +73 + 358 132/125- 250/243 : 27/22 94 + 49 + 355 264/257+ 257/243 + 27/22 47+ 97 + 355 28/27 - 22/21 - 27/22 63 +81 +355 55/54 - 16/15 : 27/22 32+112+355 40/39 + 143/135 + 27/22 44+ 100+ 355 The Wosta ofZalzal, a neutral third of 355 cents, is exploited in this hemiolic chromatic genus whose pyknon is 88/81 (143 cents), an interval found in certain Islamic scales (D’Erlanger 1935). C8. CHARACTERISTIC INTERVAL 11/9 347 CENTS 24/23 +23/22 + 11/9 74 + 77 + 347 18/17 + 34/33 - 11/9 99 +52 + 347 36/35 ‘35/33 - 11/9 49 + 102 + 347 45/44 16/15 + 11/9 56/55-15/14: 11/9 78/77 «14/13: 11/9 20/19 + 57/55 - 11/9 39+ 112 + 347 31+ 119+ 347 22 + 128 + 347 89 + 62 + 347 30/29 + 58/55 - 11/9 59 + 92 + 347 28/27 : 81/77 - 11/9 40/39 «117/110 : 11/9 63 + 88 + 347 44 + 107 + 347 WINNINGTON-INGRAM This genus is the simplest realization of Aristoxenos’s hemiolic chromatic. Winnington-Ingram mentions number 176 in his 1932 article on Aristoxenos but rejects it, despite using 12/11 - 11/9 to construct his spondeion scale in an earlier paper (Winnington-Ingram 1928). In view of the widespread use of 3/4-tone and neutral third intervals in extant Islamic music and the use of 12/11 by Ptolemy in his intense chromatic and equable diatonic genera, I see no problems with accepting Aristoxenos’s genus, 4.5 + 4.5 + 21 “parts,” as recording an actual tuning, traces of which are still to be found in the Near East. Ptolemy, it should be remembered, claimed that the intense chromatic, 22/21 + 12/11 - 7/6, was used in popular lyra and kithara tunings (Wallis 1682, 84, 178, 208) and that his equable diatonic sounded rather foreign and rustic. Schlesinger identifies it with the first tetrachord of her chromatic Phrygian harmonia (Schlesinger 1933; Schlesinger 1939, 214). The pyknon of this chromatic genus is 12/11 (151 cents). Number 176 may THE CATALOG OF TETRACHORDS

be written as 5 + 5 + 20 Ptolemaic “parts” (120 115 110 90), rather than the 4.5 +4.5 + 21 of Aristoxenian theory. A number of other divisions are shown, including the usual 1:2 and 2:1, as well as the neo-Archytan 28/27 and 40/39 types. 186 Cg. CHARACTERISTIC INTERVAL 39/32 342 CENTS 256/245 : 245/234> 39/32 76 + 80 + 342 187 384/373 © 373/351 > 39/32 188 192/181 - 362/351 - 39/32 102 + 53 + 342 64/63 - 14/13 + 39/32 27 + 128 + 342 This genus employs the 3/2’s complement of 16/13, the tridecimal neutral third, found in the 26:32:39 triad. The unusually complex pyknon is 128/117 (156 cents). 190 191 192 193 50 + 105 + 342 Cro. CHARACTERISTIC INTERVAL 28/23 341 CENTS 23/22 + 22/21 + 28/23 76 + 81 + 341 69/65 : 65/63 + 28/23 103 + 54 + 341 69/67 : 67/63 - 28/23 51+ 107 + 341 46/45 : 15/14 + 28/23 38 + 119 + 341 WILSON This neutral third genus is from Wilson, The pyknon is 23/21 (157 cents). 194 195 196 197 198 199 200 201 202 Cir. CHARACTERISTIC INTERVAL 17/14 336 CENTS 112/107 + 107/102 : 17/14 79 + 83 + 336 168/158. 158/153 - 17/14 106 + 56 + 336 168/163 : 163/153 - 17/14 52 + 110+ 336 52/51 - 14/13 - 17/14 34+ 128 + 336 28/27 + 18/17 + 17/14 63 + 99 + 336 35/34 : 16/15 « 17/14 50+ 112 + 336 40/39 » 91/85 - 17/14 44+ 118 + 336 17/14 + 56/55 - 55/51 336 + 31 + 131 17/14 + 56/53: 53/51 336 +95 + 67 This chromatic genus uses Ellis’s supraminor third, 17/14 (Helmholtz [1877] 1954, 455), which occurs in his septendecimal interpretation of the diminished seventh chord, 10:12:14:17. The pyknon is 56/51 (162 cents). C12. CHARACTERISTIC INTERVAL 40/33 333 CENTS 22/21 21/20: 40/33 81 +85 + 333 203 204 205 33/32: 31/30 + 40/33 108+ 57+ 333 33/32 + 16/15 - 40/33 53 + II2 + 333 206 55/54+ 27/25 - 40/33 32 + 133 + 333 CHAPTER 9

66/65 ‘13/12 - 40/33 26+ 139 + 333 18/17 + 187/180 . 40/33 99 + 66 + 333 The pyknon of this genus is 11/10 (165 cents), an interval which appears in Ptolemy’s equable diatonic and elsewhere. Number 208 is a summation tetrachord from chapter 4. 209 210 211 C13. CHARACTERISTIC INTERVAL 29/24 328 GENTS 64/61 : 61/58 + 29/24 83 + 87 + 328 16/15 : 30/29 + 29/24 112 +59 +328 SCHLESINGER 32/31 < 31/29 « 29/24 55+115 +328 SCHLESINGER The interval 29/24 is found in some of Schlesinger’s harmoniai when she tries to correlate her theory of linearly divided octaves with Greek notation (Schlesinger 1939, 527-8). The results agree neither with the commonly accepted interpretation of the notation, nor with the canonical forms of the harmoniai given elsewhere in her book. The 29/24 is also part of the 24:29:36 triad and its 3/2’s complement generates the 36/29 genus. The pyknon is 32/29 (170 cents). 212 213 214 215 216 217 218 C14. CHARACTERISTIG INTERVAL 6/5 316 CENTS 20/19 + 19/18 - 6/5 89 + 94 + 316 ERATOSTHENES 28/27 15/14» 6/5 63 + 119 + 316 PTOLEMY 30/29 : 29/27 - 6/5 59+123 +316 16/15 + 25/24 + 6/5 112 + 71 + 316 DIDYMOS 40/39+ 13/12 + 6/5 44 +139 + 316 BARBOUR 55/54- 12/11 - 6/5 32 +151+316 BARBOUR 65/63 : 14/13 - 6/5 54 +128 + 316 22/21 - 35/33 - 6/5 81 + 102 + 316 21/20 + 200/189 » 6/5 85 + 97 + 316 PERRETT 256/243 « 6/5. 135/128 90 + 316 + 92 XENAKIS 60/59 : 59/54 : 6/5 29 +153 + 316 219 220 221 222 223 224 225 226 227 52/51 - 85/78 - 6/5 100/99 - 11/10: 6/5 34 + 149 + 316 17 + 165 + 316 52+316+ 131 316+ 155 +27 316+41+14I 228 80/77 + 77/72 + 6/5 66 + 116 + 316 24/23 - 115/108 » 6/5 74 + 109 + 316 88/81 : 45/44 « 6/5 143 + 39 + 316 46/45 + 6/5 - 25/23 38+ 316+ 144 229 34/33 6/5 - 55/51 230 231 6/5 + 35/32 - 64/63 6/5 + 2240/2187 : 243/224 177 THE CATALOG OF TETRACHORDS WILSON HOFMANN

This genus is the most consonant of the chromatic genera. Number 212 is the chromatic of Eratosthenes and is identical to Ptolemy’s interpretation of Aristoxenos’s intense chromatic genus. It is likely, however, that Aristoxenos’s genus corresponds to one of the 32/27 genera. Number 213 is Ptolemy’s soft chromatic and is the 2:1 division reordered, Number 214 is the 1:2 division and a Ptolemaic interpretation of a 4 + 8 + 18 “parts.” Didymos’s tuning is probably the most consonant, although it violates the usual melodic canon of Greek theory that the smallest interval must be at the bottom of the tetrachord. In reverse order, this tuning is produced by the seventh of Proclus’s ten means (Heath 1921). Archytas’s enharmonic and diatonic tunings also violate this rule; the rule may either be later or an ideal theoretical principle. Numbers 216 and 217 are from Barbour (1951, 23). Perrett’s tetrachord, like one of the 25/21 genera, is found to occur unexpectedly in his new scale (Perrett 1926, 79). The Xenakis tetrachord (number 221) is from the article, “Towards a Metamusic,” which has appeared in different translations in different places (Xenakis 1971). It also appears in Archytas’s system according to Erickson (1965). The Hofmann genus is from Vogel (1975). Numbers 230 and 231 are found in Vogel's tuning (1963, 1967) and chapter 6. The pyknon is the minor tone 10/9 (182 cents). C15. CHARACTERISTIC INTERVAL 25/21 232 56/53: 53/50 - 25/21 97 +99 + 302 233 234 235 14/13 + 26/25 : 25/21 28/27. 27/25 25/21 21/20 + 16/15-25/21 40/39 + 273/250 è 25/21 128 + 68 +302 63+ 133 +302 84+ 112 +302 44 + 152 + 302 236 302 CENTS PERRETT This genus whose pyknon is 28/25 (196 cents) is inspired by number 235, a tetrachord from Perrett (1926, 80). Number 232 is virtually equally tempered and number 234 is an excellent approximation to Aristoxenos’s 1/3+2/3+1 1/2 tones, 4+8 +18 “parts.” 237 238 239 240 241 242 C16. CHARACTERISTIC INTERVAL 19/16 298 CENTS 128/121. 121/114- 19/16 97 + 103 +298 96/89 - 178/171 : 19/16 131 + 69 + 298 192/185+ 185/171 - 19/16 64 + 136 + 298 20/19 + 19/16- 16/15 89 +298 + 112 KORNERUP 256/243 + 81/76 - 19/16 90 + 110 +298 BOETHIUS 96/95 : 10/9: 19/16 18 + 182 +298 WILSON CHAPTER 9

64/63 « 21/19« 19/16 40/39 - 104/95 : 19/16 27 +173 +298 44+157 + 298 The characteristic ratio for this genus derives from the 16:19:24 minor triad (see the 24/19 genus). The pyknon is the complex interval 64/57 (201 cents). Number 241 is from Boethius (1838, 6). The Kornerup tetrachord (1934, 10) also corresponds toa Ptolemaic interpretation of one of Athanasopoulos's (1950) Byzantine tunings, 6+ 18 + 6 “parts.” As 19/16 20/19. 16/15, itis one of the “mean” tetrachords. 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 C17. CHARACTERISTIC INTERVAL 32/27 294 CENTS ARISTIDES QUINT. 18/17 - 17/16 : 32/27 99 + 105 + 294 27/25 + 25/24 - 32/27 133 + 71 + 294 27/26 + 13/12 + 32/27 65 + 139 + 294 BARBOUR? 28/27 - 243/224+ 32/27 63 + 141 + 294 ARCHYTAS 256/243 + 2187/2048 - 32/27 90 +114 + 294 GAUDENTIUS 81/80 - 10/9 - 32/27 22 + 182 + 294 BARBOUR? 33/32 - 12/11 + 32/27 53 +151 +294 BARBOUR? 45/44 + 11/10 + 32/27 39 + 165 + 294 BARBOUR? 21/20 - 15/14+ 32/27 84+ 119 + 294 PERRETT 135/128 - 16/15 + 32/27 92 +112 + 294 36/35 « 35/32 « 32/27 49+ 155 + 294 WILSON 49/48 - 54/49 : 32/27 36 + 168 + 294 WILSON PS.-PHILOLAUS? 95 + 109 + 294 243/230- 230/216 - 32/27 103 +IOI+ 294 243/229: 229/216 - 32/27 20/19 - 171/160 : 32/27 23/22 + 99/92 : 32/27 24/23 + 69/64: 32/27 40/39 + 351/320 - 32/27 14/13 - 117/112 + 32/27 89 + II5 + 294 77 + 127 + 294 74+ 130+ 294 44+ 160+ 294 128 + 76 + 294 These chrornatic genera are derived from the traditional “Pythagorean” tuning (perfect fourths, fifths, and octaves), which is actually of Sumero-Babylonian origin (Duchesne-Guillemin 1963, 1969; Kilmer 1960), by changing the pitch of the second string, the parhypate or trite. Number 245, the 1:1 division of the 9/8 pyknon (204 cents), is from from the late classical writer, Aristides Quintilianus (Meibomius 1652, 123). Tunings numbers 246 and 254 are of obscure origin. They were constructed after reading a passage in Hawkins ([1776] 1963, 37) which quotes Wallis as crediting Mersenne with the discovery of the 27/25 and 135/128 semitones THE CATALOG OF TETRACHORDS

and their 9/8 complements. However, the discussion is about diatonic genera, not chromatic, and it is unclear to me whether Mersenne really did construct these two chromatic tetrachords. Archytas’s chromatic, number 248, has been identified with Aristoxenos’s 1/3 + 2/3 + 1 1/2 tones by Winnington-Ingram (1932) and number 247 is a good approximation to his 1/2 + 1/2 + 1 1/2 tones. Number 249 is the unaltered Pythagorean version from Gaudentius. The Barbour tetrachords derive from his discussion of different superparticular divisions of the 9/8 (Barbour 1951, 154-156). Although tetrachords are mentioned, it is not clear that he ever actually constructed these divisions. Perrett discovered number 253, like number 235 above, in his scale after it was constructed. Both Chaignet (1874, 231) and McClain (1978, 160) quote (Ps.)-Philolaus as dividing the tone into 27 parts, 13 of which go to the minor semitone, and 14 to the major. Number 257 is the result of this division and number 258 has the parts taken in reverse order. It would seem that number 245 and number 258 are essentially equivalent to Aristoxenos’s theoretical intense chromatic and that numbers 254, 257, 259, and probably 253 as well, are equivalent to Gaudentius’s Pythagorean tuning. The presence of secondary ratios of 5 and 7 in number 253 and number 254 suggests that the equivalences would be melodic rather than harmonic. The last tuning is a summation tetrachord from chapter 4. 264 265 266 267 268 269 C18. CHARACTERISTIC INTERVAL 45/38 293 CENTS 304/287 - 287/270 : 45/38 100 + 106 + 293 456/439 - 439/405 : 45/38 66 + 140 + 293 228/211 : 422/405 : 45/38 134 +71 +293 19/18 - 16/15 - 45/38 94+ 112 +293 76/75 - 10/9 : 45/38 23 + 182 + 293 38/35 - 28/27 + 45/38 142 + 63 + 293 This genus uses the 45/38, the 3/2’s complement of 19/15. The pyknon is 152/135 (205 cents). Number 264 is a reasonable approximation to the intense chromatic and number 269 is similar to Archytas’s chromatic, if rearranged with the 28/27 first. 270 271 272 273 274 Cro. CHARACTERISTIC INTERVAL 13/11 289 CENTS 88/83 - 83/78 - 13/11 IOI + 108 + 289 66/61: x22/117: 13/11 136 + 72 + 289 132/127+ 127/117 > 13/11 67 + 142 + 289 14/13 + 22/21 + 13/11 128 + 81 + 289 40/39 + 11/10- 13/11 44 + 165 + 289 CHAPTER 9

66/65 + 10/9+ 13/11 26 + 182 + 289 27/26 : 88/81. 13/11 65 + 143 + 289 28/27: 99/91 : 13/11 63 +146 + 289 This experimental genus divides a pyknon of 44/39 (209 cents), an interval also appearing in William Lyman Young’s diatonic lyre tuning (Young 1961). The 13/11 is a minor third which appears in 13-limit tunings and with its 3/2’s complement, 33/26, generates the 22:26:33 tritriadic scale. 278 279 280 281 282 C20. CHARACTERISTIC INTERVAL 33/28 284 CENTS 224/211 - 211/198 - 33/28 104 + 110 + 284 68 + 145 + 284 336/323 : 323/297 + 33/28 168/155 - 310/297 : 33/28 139+ 74+ 284 31 + 182 + 284 56/55 : 10/9 : 33/28 16/15-35/32: 33/28 112 + 102 + 284 52 + 284 + 162 34/33 + 33/28 - 56/51 The characteristic interval of this genus is the 3/2’s complement of 14/11, 33/28. The pyknon is 112/99 (214 cents). C21. CHARACTERISTIC INTERVAL 20/17 284 285 286 287 288 290 291 292 293 294 295 296 17/16: 16/15 - 20/17 51/47 + 47/45 + 20/17 51/49 - 49/45 - 20/17 34/33 « 11/10: 20/17 51/50- 10/9 - 20/17 40/39 + 221/200 : 20/17 28/27: 153/140 + 20/17 21/20 + 20/17 : 68/63 68/65 - 13/12 - 20/17 34/31 + 31/30- 20/17 68/61 : 61/60: 20/17 68/67- 67/57 - 19/17 68/67 + 67/60 - 20/17 281 CENTS 105 + 112 + 281 142 +75 + 281 69 + 147 + 281 52 + 165 + 281 34 + 182 + 281 44+ 173 + 281 63 +154 + 281 85 + 281 + 132 78 +139 + 281 160+57+281 188 + 29 + 281 26 + 280 + 193 26+ 191 + 281 The pyknon is 17/15 (217 cents). Intervals of 17 are becoming increasingly common in justly-intoned music. This would appear to be a metaphysical phenomenon of considerable philosophical interest (Polansky, personal communication). 297 298 C22. CHARACTERISTIC INTERVAL 27/23 278 CENTS 184/173 - 173/162 + 27/23 107 +114+278 276/265 + 265/243 : 27/23 70 + 150 + 278 THE CATALOG OF TETRACHORDS

302 303 304 395 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 138/127 + 254/243 : 27/2 144 + 77 + 278 28/27. 23/21 - 27/23 63 + 157 + 278 23/22 - 88/81 - 27/23 77+143 + 278 46/45 : 10/9 : 27/23 38 + 182 + 278 This genus exploits the 3/2’s complement of 23/18, which is derived from the 18:23:27 triad. The pyknon is 92/81 (220 cents). C23. CHARACTERISTIC INTERVAL 75/64 275 CENTS 512/481 : 481/450- 75/64 108 + 115 +275 768/737 © 737/675 « 75/64 384/353 + 706/675 - 75/64 71 + 152 + 275 146 + 78 + 275 16/15 : 75/64: 16/15 II2 +275 + II2 HELMHOLTZ The pyknon is 256/225 (223 cents). The 75/64 is the 5-limit augmented second, which appears, for example, in the harmonic minor scale. Helmholtz’s tetrachord is from (Helmholtz [1877] 1954, 263). C24. CHARACTERISTIC INTERVAL 7/6 267 CENTS 16/15 - 15/14 - 7/6 112 + 119 + 267 22/21 + 12/11» 7/6 81+151+267 24/23 - 23/21 + 7/6 74+157+ 267 20/19 : 38/35 - 7/6 89 + 142 + 267 10/9 + 36/35 : 7/6 182 + 49 + 267 64/63 - 9/8 . 7/6 27 + 204 + 267 92/01 : 26/23 - 7/6 19 + 212 + 267 256/243 : 243/224 - 7/6 90 + 141 + 267 40/39 © 39/35 - 7/6 44+ 187 + 267 18/17. 7/6 - 68/63 50/49 - 7/6 - 28/25 14/13 + 7/6 - 52/49 46/45 « 180/161 : 7/6 28/27 - 54/49 : 7/6 120/113+ 113/105 : 7/6 60/59 + 118/105 : 7/6 30/29 - 116/105 - 7/6 88/81 - 81/77 - 7/6 120/119. 17/15 + 7/6 27/25 + 7/6 + 200/189 26/25 - 7/6. 100/91 99 + 267 +132 35 + 267+ 196 128 + 267 + 103 38 + 193 + 267 63 + 168 + 267 104+ 127 + 267 29 + 202 + 267 59 + 172 + 267 143 + 88 + 267 14 + 217 + 267 133 + 267 + 98 68 + 267+ 163 182 CHAPTER 9 AL-FARABI PTOLEMY PTOLEMY AVICENNA BARBOUR HIPKINS

7/6 : 1024/945- 135/128 The pyknon of this intense chromatic is the septimal tone, 8/7 (231 cents). Number 307 is given by Al-Farabi (D’Erlanger 1930, 104) and by Sachs (1943, 282) in rearranged form as the lower tetrachord of the modern Islamic mode, Higaz. The Turkish mode, Zirgule, has also been reported to contain this tetrachord, also with the 7/6 medially (Palmer 1967?). Vincent attributes this division to the Byzantine theorist, Pachymeres (Vincent 1847). This tuning is also produced by the harmonic mean operation. Ptolemy’s first division (number 308) is his intense chromatic (Wallis 1682, 172), and his second (number 310) is his interpretation of Aristoxenos’s soft diatonic, 6 +9+15 “parts”. In this instance, Ptolemy is not too far from the canonical 100 +150 +250 cents, though Hipkins’s semi-Pythagorean solution (number 314) is more realistic (Vogel 1963). His tuning is also present in Erickson’s (1965) interpretation of Archytas’s system. The Avicenna tetrachord, number 311, (D’Erlanger 1935, 152) sounds, surprisingly, rather diatonic. Barbour’s (1951, 23-24) tuning (number 312) is particularly attractive when arranged as 9/8 : 64/63 - 7/6. It also generates the 16:21:24 tritriadic and its conjugate. Vogel (1975, 207) lists it also. Number 328 is found in Vogel's tuning (chapter 6 and Vogel 1963, 1967). The remaining divisions are new tetrachords intended as variations on the soft diatonic-intense chromatic genus or as approximations of various Byzantine tetrachords as described by several authors (Xenakis 1971; Savas 1965; Athanasopoulos 1950). 329 330 331 332 C25. CHARACTERISTIC INTERVAL 136/117 261 GENTS 78/73 : 73/68 - 136/117 115 + 123 + 261 117/112 + 56/51 - 136/117 76 + 162 + 261 117/107 107/102 : 136/117 155 +83 + 261 52/51. 9/8 - 136/117 34+ 204 + 261 The pyknon of this complex genus is 39/34 (238 cents). Number 332 generates the 26:34:39 tritriadic. C26. CHARACTERISTIC INTERVAL 36/31 333 334 335 259 CENTS 115 + 124+ 259 31/29 + 29/27: 36/31 + 163 + 259 76 93/89 - 89/81 - 36/31 156 + 83 + 259 93/85 - 85/81 + 36/31 The pyknon is 31/27 (239 cents). The 36/31 is the 3/2’s complement of 31/24, which defines a hyperenharmonic genus. THE CATALOG OF TETRACHORDS

339 C27. CHARACTERISTIC INTERVAL 80/69 256 CENTS 46/43 - 43/40: 80/69 117 + 125 + 256 23/21: 21/20.80/69 157 + 85 + 256 23/22. 11/10. 80/69 77 + 165 + 256 38 + 204+ 256 46/45: 9/8 + 80/69 The genus derives from number 339 which generates the 20:23:30 and 46:60:69 tritriadics. The pyknon is 23/20 (242 cents). This and the next few genera are realizations of Aristoxenos’s soft diatonic. 340 341 342 343 C28. CHARACTERISTIC INTERVAL 22/19 254 CENTS 118 + 126 + 254 76/71: 71/66: 22/19 57/52: 104/99- 22/19 159 + 85 + 254 78 + 167 + 254 114/109 - 109/99 - 22/19 SCHLESINGER 19/18: 12/11 - 22/19 94+ 151 + 254 344. 34/33: 19/17: 22/19 52 + 192 + 254 345 40/39 : 247/220 - 22/19 44+ 200 + 254 This genus is a good approximation to the soft diatonic. Number 343 is from a folk scale (Schlesinger 1939, 297). Tetrachord numbers 344 and 345 are close to 3 + 12 + 15 “parts”, a neo-Aristoxenian genus which mixes enharmonic and diatonic intervals. The pyknon is 38/33 (244 cents). 346 C29. CHARACTERISTIC INTERVAL 52/45 250 CENTS 15/14: 14/13 : 52/45 119 + 128 + 250 347 348 349 45/41-41/39 - 52/45 45/43 43/39 - 52/45 24/23 115/104- 52/45 161 + 87 + 250 78 + 169 + 250 74+ 174+ 250 350 40/39 - 9/8 52/45 44+ 204+ 250 351 18/17-85/78. 52/45 99 + 149+ 250 352 45/44:44/39: 52/45 39 + 209 + 250 353 354 65/63 28/25- 52/45 55/52 - 12/11-52/45 54 + 196+ 250 97+ 151 +250 355 60/59: 59/45 : 52/45 29 +219 + 250 356 357 358 20/19-52/45: 57/52 89 + 250 + 149 27/26: 10/9: 52/45 66+ 182 + 250 11/10- 150/143 + 52/45 165 + 83 + 250 This genus lies on the dividing line between the chromatic and diatonic genera. The pyknon of 15/13 (248 cents) is virtually identical to the CI which defines the genus. The first three subgenera are the 1:1, 2:1, and 1:2 divisions respectively. Number 350 generates the 10:13:15 tritriadic scale. CHAPTER 9

DIATONIC TETRACHORDS Di. CHARACTERISTIC INTERVAL 15/13 248 GENTS 104/97: 97/90- 15/13 124 + 126 + 248 360 78/71. 142/135: 15/13 163 + 86+ 248 361 156/149 - 149/135 - 15/13 79 + 171 + 248 362 16/15 + 15/13 + 13/12 112 +248 + 139 SCHLESINGER 363 26/25 - 10/9 + 15/13 68 + 182 + 248 364 256/243 : 351/320 «15/13 90 + 160 + 248 365 «= - 20/19 + 247/225+ 15/13 89 + 161 + 248 366 = 11/10. 15/13 - 104/99 165 + 248 + 85 367 12/11 -15/13 - 143/135 151+248 + 99 368 46/45-26/23 - 15/13 38 +212 + 248 369 40/39 - 169/150 - 15/13 44 + 206 + 248 370 28/27-39/35 - 15/13 63 + 187 + 248 371 91/90: 8/7: 15/13 19 + 231 + 248 This genus is the first indubitably diatonic genus. A pyknon, perse, no longer exists because the 52/45 (250 cents) is larger than one-half the perfect fourth, 4/3 (498 cents). The large composite interval in this and succeeding genera is termed the “apyknon” or non-condensation (Bryennios). Number 362 is the first tetrachord of Schlesinger’s diatonic Hypodorian harmonia. Many members of this genus are reasonable approximations to Aristoxenos’s soft diatonic genus, 100 + 150 + 250 cents. Others with the 15/13 medially are similar to some Byzantine tunings. Some resemble the theoretical genus 50 +200 +250 cents. 372 373 374 D2. CHARACTERISTIC INTERVAL 38/23 244 CENTS 44/41: 41/38 - 38/33 123 + 131 +244 11/10 - 20/19 - 38/33 165 + 89 + 244 22/21-21/19 38/33 81 + 173 + 244 This genus divides the 22/19 (254 cents). 375 376 377 378 379 380 D3. CHARACTERISTIC INTERVAL 23/20 242 GENTS 160/149: 149/138 - 23/20 123 +133 + 242 166 + 90 + 242 120/109. 218/207 : 23/20 81+175+242 240/229- 229/207 - 23/20 231+25+242 8/7. 70/69 : 23/20 44 + 212 + 242 40/39 - 26/23 - 23/20 74 + 242 + 182 SCHLESINGER 24/23 + 23/20- 10/9 THE CATALOG OF TETRACHORDS

384 385 28/27- 180/161 : 23/20 63 + 193 + 242 This genus is derived from the 20:23:30 triad. The apyknon is 80/69 (256 cents), Number 380 is from Schlesinger (1932) and is described as a harmonia of “artificial formula, Phrygian”. Numbers 379 and 381 make intervals of 15/13 and 7/6 respectively with their subtonics. These intervals should be contrasted with the incomposite 23/20 in the tetrachord. Dq. CHARACTERISTIC INTERVAL 31/27 239 CENTS 125+ 134 + 239 72/67 - 67/62 : 31/27 108/103 : 103/93 : 31/27 82+177+ 239 168 + 91 + 239 54/49 + 98/93 - 31/27 32/31 - 9/8 - 31/27 55+204+ 239 The apykmon of this genus is 36/27 (259 cents). Number 385 generates the 24:31:36 tritriadic. 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 Ds. CHARACTERISTIC INTERVAL 39/34 238 CENTS 125 +135 +238 272/253 + 253/234+ 39/34 83 + 178+ 238 408/389 - 389/351 : 39/34 169 + 91 + 238 204/185 - 370/351 + 39/34 44+ 2384217 40/39 - 39/34 + 17/15 The apyknon is 136/117 (261 cents). The 39/34 interval is the 3/2’s complement of 17/13 and derives from the 26:34:39 triad. D6. CHARACTERISTIC INTERVAL 8/7 231 CENTS 128+ 139+ 231 AVICENNA 14/13 - 13/12 - 8/7 19/18 . 21/19 + 8/7 94 +173 + 231 SAFIYU-D-DIN 21/20: 10/9 : 8/7 84+ 182 + 231 PTOLEMY 28/27 - 8/7 + 9/8 63 + 231 + 204 ARCHYTAS 49/48 - 8/7 + 8/7 36+ 231+ 231 AL-FARABI 35/33 - 11/10 - 8/7 102 + 165 + 231 AVICENNA II6 +151 +231 AVICENNA 77/72 + 12/11 «8/7 16/15: 35/32 + 8/7 II2+155 + 231 VOGEL 5O+217+ 231 35/34 + 17/15 + 8/7 25/24 + 8/7 + 28/25 71+231+196 119+ 231 + 147 15/14 + 8/7 : 49/45 40/39 - 91/80 - 8/7 44 +223 + 231 46/45 - 105/92 - 8/7 38 +229+ 231 18/17 - 119/108 - 8/7 99 +168 + 231 17/16 - 8/7: 56/51 105 +231 + 162 52+215 +231 34/33 : 77/68 « 8/7 CHAPTER 9

256/243 : 567/512 : 8/7 90 +177 + 231 This genus divides the 7/6 (267 cents). The Avicenna and Al-Farabi references are from D’Erlanger. Number 390 is also given by Pachymeres (D’Erlanger 1935, 148 referring to Vincent 1847). When arranged as 13/12 14/13 - 8/7, itis generated by taking two successive arithmetic means. Number 394 is especially interesting as there have been reports that it was used on organs in the Middle Ages (Adler 1968; Sachs 1949), but more recent work suggests that this opinion was due to a combination of transmission errors (by copyists) and an incorrect assessment of end correction (Barbour 1950; Munxelhaus 1976). With the 49/48 medially, it is generated by the twelfth of the Greek means (Heath 1921). The scale is obviously constructed in analogy with the Pythagorean 256/243 : 9/8 - 9/8. Similar claims pro and con have been made for number 393 as well. This scale, however, appears to have been the principal tuning of the diatonic in practice from the time of Archytas (390 8GE) through that of Ptolemy (ca. 160 ce). Even Aristoxenos grudgingly mentions it (Winnington-Ingram 1932). Number 397 is from Vogel (1963) and approximates the soft diatonic. It is also found in Erickson’s (1965) version of Archytas’s system. Entry 399 corresponds to 3/8 + 1 1/8 + I tones of Aristoxenos. The Safiyu-d-Din tuning is one of his “strong” forms (2:1 division) and has 21/19 replacing the 10/9 of Ptolemy. Tetrachords 403, 404, and 405 exploit ratios of 17 and are dedicated to Larry Polansky. 407 4o8 409 410 D7. CHARACTERISTIC INTERVAL 256/225 223 CENTS 150/139+ 139/128 : 256/225 132+ 143 +223 225/214- 107/96 + 256/225 87 + 188 + 223 225/203 + 203/192 - 256/225 78+96+223 25/24 : 9/8 . 256/225 71 +204 + 223 The apyknon is the augmented second, 75/64 (275 cents). Number 410 is the generator of the 64:75:96 tritriadic and a good approximation to Aristoxenos’s 3/8 +1 1/8 + 1 tone when reordered so that the 9/8 is uppermost. 411 412 413 414 415 D8. CHARACTERISTIC INTERVAL 25/22 221 CENTS 176/163 : 163/150 - 25/22 133 + 144+ 221 132/119 + 238/225 + 25/22 179 + 97 +221 264/251: 251/225 «25/22 87+189+ 221 16/15 + 11/10 + 25/22 112 + 165 + 221 88/81 : 27/25 + 25/22 143 +133 +221 THE CATALOG OF TETRACHORDS

22/21: 25/22 - 28/25 81 + 221 + 196 28/27 - 198/175 + 25/22 63 + 214+ 221 26/25 - 44/39 « 25/22 68 + 209+ 221 This is an experimental genus whose apyknon is 88/7 5 (277 cents). Number 416 isa fair approximation of Aristoxenos’s 3/8 +1 1/8 + 1 tones, and number 411 is close to a hypothetical 11/16+11/16+1 1/8 tones. 419 420 421 422 423 424 425 426 427 Dg. CHARACTERISTIC INTERVAL 92/81 220 GENTS 133 + 144 + 220 27/25 + 25/23 - 92/81 88 + 190+ 220 81/77 + 77/69 : 92/81 180+98+ 220 81/73 : 73/69 - 92/81 74 + 204+ 220 24/23 : 9/8 + 92/81 66 + 212 + 220 27/26- 26/23 - 92/81 This genus divides the 27/23 (278 cents) and is derived from the 18:23:27 triad. Number 422 is the tritriadie generator, and is an approximation to Aristoxenos’s 3/8+11/8+1 tones (4.5+13.5+12 “parts”) when reordered. Dio. CHARACTERISTIC INTERVAL 76/67 218 CENTS 67/62 - 62/57 - 76/67 134+ 146 + 218 201/181 - 181/171 : 76/67 181 + 98 + 218 201/191 - 191/171 : 76/67 88 + 191 + 218 256/243 - 76/67 : 5427/4864 90+ 218+ 190 EULER This complex genus is expanded from number 427, which is called “old chromatic” in Euler’s text (Euler [1739] 1960, 177). The tuning is clearly diatonic, however, and must be in error. It may have been intended to represent Boethius’s 19/16 (76/64) chromatic. The apyknon is 67/57 (280 cents). 428 429 430 431 432 433 434 435 436 437 Dir. CHARACTERISTIC INTERVAL 17/15 217 CENTS 135 + 146 + 217 40/37 « 37/34: 17/15 10/9 18/17: 17/15 182+99+ 217 KORNERUP 20/19+ 19/17+ 17/15 89 + 192 + 217 PTOLEMY 15/14 + 56/61: 17/15 119+ 162 +217 80/77 : 77/68 « 17/15 66 + 215 + 217 12/11 - 55/51 - 17/15 I§l +131 +217 120/109: 109/102 : 17/15 166+ 115 +217 104+ 177 +217 120/113 + 113/102 + 17/15 24/23 - 115/102 : 17/15 74 +208 +217 160/153 : 9/8 - 17/15 77+ 204+ 217 This genus divides the 20/17 (281 cents). Number 429 is Kornerup’s (1934, CHAPTER 9

to) Lydian. Genus number 430 is Ptolemy’s interpretation of Aristoxenos’s intense diatonic, 6 + 12 + 12 “parts” (Wallis 1682, 172). Kornerup refers to it as Dorian. Number 432 is a hypothetical Ptolemaic interpretation of 4.5 + 13.5 + 12 “parts”, a mixed chromatic and diatonic genus not in Ptolemy. Number 437 generates the 34:40:51 triad and tritriadic. The remaining divisions are experimental neo-Aristoxenian genera with a constant upper interval of 12 “parts.” 438 439 440 441 442 Diz. CHARACTERISTIC INTERVAL 112/99 214 CENTS 136 + 148 + 214 66/61 - 61/56 - 112/99 90 + 195 + 214 99/94: 47/42 + 112/99 184 + 100 + 214 99/89: 89/84- 112/99 182 + 102 + 214 10/9 : 297/280- 112/99 81 + 204+ 214 22/21 + 9/8 - 112/99 This very complex genus divides the 33/28 (284 cents). Number 442 generates the 22:28:33 tritriadic and its conjugate. 443 444 445 D13. CHARACTERISTIC INTERVAL 44/39 209 CENTS 151 + 139 + 209 12/11: 13/12 + 44/39 187 + 102 + 209 39/35 « 35/33 44/39 QI + 198 + 209 39/37 - 37/33 : 44/39 209 + 204 +85 44/39 - 9/8 - 104/99 YOUNG The first division is William Lyman Young’s “exquisite 3/4-tone Hellenic lyre” (Young 1961, 5). The apyknon is 13/11 (289 cents). Number 446 generates the 22:26:33 tritriadic scale. 447 448 449 450 D14. CHARACTERISTIC INTERVAL 152/135 90/83 : 83/76- 152/135 140 + 153 + 205 135/128 : 64/57: 152/135 92 + 201 + 205 135/121 - 121/114< 152/135 190 + 103 + 205 20/19 + 9/8 . 152/135 89 + 204 + 205 205 CENTS This genus derives from the 30:38:45 triad and divides its upper interval, 45/38 (293 cents). Number 450 generates the 30:38:45 tritriadic and its conjugate. Dis. CHARACTERISTIC INTERVAL 9/8 451 452 453 454 64/59 - 59/54 9/8 48/43 - 86/81 - 9/8 96/91 - 91/81 - 9/8 256/243 : 9/8 - 9/8 189 204 CENTS 141 + 153 + 204 190 + 104 + 204 93 + 202 + 204 90 + 204 + 204 THE CATALOG OF TETRACHORDS SAFIYU-D-DIN SAFIYU-D-DIN PYTHAGORAS?

16/15 - 9/8 - 10/9 2187/2048 - 65536/59049 -9/8 9/8 « 12/11 - 88/81 13/12 - 9/8 - 128/117 14/13 + 9/8 : 208/189 9/8 : 11/10 + 320/297 9/8 - 15/14 : 448/405 112 + 204 + 182 114+ 180 + 204 204 + 151 + 143 139 + 204 + 156 128 + 204 + 166 204 + 165 + 129 204 +119 + 175 462 9/8 - 17/16. 512/459 204 + 105 + 189 463 9/8 - 18/17 : 272/243 204 + 99 + 195 9/8 - 19/18 : 64/57 204 + 94+ 201 56/51 © 9/8 - 68/63 162 + 204 + 132 9/8. 200/189 : 28/25 204 + 98 + 196 184/171 : 9/8 . 76/69 127 + 204 + 167 32/29 + 9/8 . 29/27 170 + 204 + 124 121/108 «9/8. 128/121 197 + 204 + 97 PARTCH 9/8. 4096/3645 : 135/128 204 + 202 + 92 9/8. 7168/6561 : 243/224 204 +153 + I4I 35/32 - 1024/945 : 9/8 204 + 139 + 204 The apyknon of this genus is 32/27 (294 cents). Numbers 451 and 452 are Safiyu-d-Din’s weak and strong forms of the division, respectively. The 455 456 457 458 459 460 464 465 466 467 468 469 470 471 472 PTOLEMY, DIDYMOS ANONYMOUS AVICENNA AVICENNA AVICENNA AL-FARABI attribution of the tetrachord number 454 to Pythagoras is questionable, though traditional—the diatonic scale in “Pythagorean” intonation antedates him bya millennium or so in the Near East (Duchesne-Guillemin 1963, 1969). The earliest reference to this scale in a European language is in Plato’s Timaeus. Number 455 is attributed to both Ptolemy and Didymos because their historically important definitions differed in the order of the intervals, Ptolemy’s is the order shown; Didymos placed the 9/8 at the top. Ptolemy’s order generates the major mode in just intonation. Its retrograde, 10/9 - 9/8 - 16/15, yields the natural minor and new scale of Redfield (1928). Number 456 is a “Pythagorean” form extracted from the anonymous treatise in D’Erlanger (1939). In reverse order, it appears in the Turkish scales of Palmer (1967?). Numbers 457-460 are also from D’Erlanger. Numbers 457 and 458 generate the 18:22:27 and 26:32:39 tritriadics and their conjugates. These and the tetrachord from Al-Farabi, number 459, resemble modern Islamic tunings (Sachs 1943, 283). Numbers 464 and 465 generate the 16:19:24 and the 14:17:21 tritriadics. In theory, any tetrachord containing a 9/8 generates a tritriadic and its conjugate, but in practice the majority CHAPTER 9

are not very consonant. Examples are numbers 467 and 468 which generate the 38:46:57 and 24:29:36 tritriadics with mediants of 23/19 and 29/24. Number 469 is an adventitious tetrachord from Partch (1974, 165). Numbers 470-472 are from chapter 4. The last two resemble some of the Islamic tunings of the Middle Ages. The remaining tunings are proposed approximations to Islamic or syntonic diatonic tetrachords. 473 D16. CHARACTERISTIG INTERVAL 160/143 194 CENTS 165 +139 + 194 11/10: 13/12 - 160/143 AL-FARABI This tetrachord is from Al-Farabi (D’Erlanger 1930, 112). It did not seem worthwhile to explore this genus further because the ratios would be complex and often larger than 160/143 itself. 474 475 476 D17. CHARACTERISTIC INTERVAL 10/9 182 CENTS 12/11 - 11/10 + 10/9 151 +165 + 182 10/9- 10/9 - 27/25 182 + 182 + 133 10/9 « 13/12 » 72/65 182 + 139 + 177 PTOLEMY AL-FARABI AVICENNA The apyknon is 6/5 and the majority of potential divisions have intervals larger than the 10/9. Number 474 is Ptolemy’s homalon or equable diatonic, a scale which has puzzled theorists, but which seems closely related to extant tunings in the Near East. Ptolemy described it as sounding rather foreign and rustic. Could he have heard it or something similar and written it down in the simplest ratios available? It certainly sounds fine, perhaps a bit like 7-tone equal temperament with perfect fourths and fifths. The Avicenna and Al-Farabi references are from D’Erlanger (1935), and Ptolemy (Wallis 1682). Reduplicated tetrachords These genera are arranged by the reduplicated interval in descending order 477 478 479 480 481 482 483 484 485 486 ofsize. 11/10: 11/10 400/363 165 +165+168 12/11 + 12/11 + 121/108 ISI +151 +197 13/12 + 13/12 + 192/169 139+139+221 14/13 - 14/13 - 169/147 128+128 +241 15/14: 15/14: 784/675 119+119+259 2187/2048 «167772 16/14348907 + 2187/2048 114+271+114 17/16: 17/16. 1024/867 105 +105 +288 AVICENNA AVICENNA AVICENNA AVICENNA PALMER RI 2 R3 R4 RS RÓ R7 18/17 - 18/17 + 289/243 99 +99 + 300 R8 256/243 256/243 -19688/16384 22/21. 147/121 «22/21 90+90+318 81+337+81 RQ THE CATALOG OF TETRACHORDS

25/24: 25/24 -768/625 488 28/27: 28/27.243/196 63 +63 +372 RIT RI2 489 34/33 + 34/33 - 363/289 52 +52 +395 RI3 490 491 492 36/35-36/25-1225/972 49 +49 +401 RI4 40/39: 40/39 : 507/400 44+44+410 RIS 46/45 - 46/45 675/529 38+38+422 RIG While a number of other small intervals could be used to construct analogous genera, the ones given here seem the most important and most interesting. Number 477 is an approximation in just intonation to the equally tempered division of the 4/3. See number 722 for the semi-tempered version. The Avicenna genera are from vol. 2, pages 122-123 and page 252 of D’Erlanger. The Palmer genus is from his booklet on Turkish music (1967?). This genus is very close to Helmholtz’s chromatic 16/15 - 75/64 - 16/15. The 18/17 genus is also nearly equally tempered and is inspired by Vincenzo Galilei’s lute fretting (Barbour 1951, 57). Number 486 is nearly equal to 1/1 1/3 4/n 4/3, a theoretical genus using intervals of 11 to approximate intervals of x. Numbers 487 and 488 come from Winnington-Ingram’s (1932) suggestion that Aristoxenos’s soft and hemiolic chromatics were somewhat factitious genera resulting from the duplication of small, but known, intervals. The remaining tetrachords are in the spirit of Avicenna and Al-Farabi. Miscellaneous tetrachords The tetrachords in this section are those that were discovered in the course of various theoretical studies but which were not judged to be of sufficient interest to enter in the Main Catalog. Many of these genera have unusual Cls which were not thought worthy of further study. The fourth and fifth columns give the ratio of the pyknon or apyknon and its value in cents. 493 494 176/175 - 175/174. + 29/22 25/19 ‘931/925 - 148/147 10+ 10 + 478 475 +11 +12 88/87 76/75 20 23 MI M2 This tetrachord is generated by the second of the summation procedures 495 496 of chapter 5. 128/127 + 127/126 » 21/16 21/16 + 656/651 124/123 14+ 14+ 471 471+13+14 64/63 64/63 27 27 M3 M4 52/51 52/51 34 34 M5 M6 Another summation tetrachord from chapter 4. 497 498 104/103 + 103/102 - 17/13 17+17+464 17/13 :429/425 + 100/99 464+ 16+ 17 Another summation tetrachord from chapter 4. CHAPTER 9

98/97 + 97/96. 64/49 SOI 502 92/91 91/90: 30/23 18 + 18 + 462 19+19+460 49/48 46/45 90/89 - 89/88 - 176/135 88/87-87/86. 43/33 19+20+459 20+20+458 45/44 44/43 593 86/85 - 85/84- 56/43 84/83 - 83/ - 82 82/63 20+20+457 43/42 21+21+456 41 42/41 MII 504 505 42 MI2 21+22+455 41/40 43 M1 82/81-81/80: 160/123 36 38 39 40 M7 M8 M9 MIO These genera contain intervals which are probably too small for use in most music. However, Harry Partch and Julián Carrillo, among others, have used intervals in this range. 506 13/10 - 250/247 : 76/74 454+21+23 40/39 44 MIQ Another summation tetrachord from chapter 4. 507 508 509 510 SII 78/77 77/76 152/117 76/75 76/75 74/57 74/73 : 73/72 48/31 22+23+453 23+23+452 24+24+451 39/38 38/37 37/36 45 46 47 MIS MIG MI7 70/69 : 69/68 - 136/105 254254448 35/34 so mM18 22/17 + 357/352 + 64/63 446+24+27 34/33 52 MIg 29/28 61 M20 Another summation tetrachord from chapter 4. 512 58/57 57/56 + 112/87 513 514 87/80 : 43/42 + 112/87 20+41+437 29/28 61 M2I 87/85-85/84. 112/87 40+20+437 29/28 61 M22 The preceding are a set of hyperenharmonic genera which divide the dieses 30+31+437 between 40/39 and 28/27. Similar but simpler genera will be found in the Main Catalog. Small intervals in this range are clearly perceptible, but have been rejected by most theoreticians, ancient and modern. 515 68/53 + 53/52: 52/51 431+33+34 53/51 67 M23 516 136/133 + 133/130 » 65/51 34+34+420 68/65 78 M24 517 68/67 : 67/65 : 65/51 34/33 : 66/65 + 65/51 68/67 : 67/54 + 18/17 26+52+420 52+26+420 26+373+99 68/65 68/65 72/76 78 78 125 M25 M26 M27 25/24: 32/31 + 31/25 68/55 : 55/54« 18/17 71+55+372 367 +32+99 100/93 55/51 126 131 M28 M29 68/67 : 67/63 : 21/17 68/65 : 65/63 + 21/17 26 + 107 +366 78+54+366 68/63 68/63 132 132 M30 M31 36/35: 256/243 + 315/256 49+90+359 1024/9045 139 M32 + 315/256 64/63 - 16/15 27+112+359 1024/9455 139 M33 518 519 520 521 522 523 524 525 Numbers 524 and 525 are from Vogel's PIS tuning of chapter 6. THE CATALOG OF TETRACHORDS

64/63 : 2187/2048 896/729 36/35 «135/128 - 896/729 27+114+357 49+92+357 243/224 243/224 141 I4I M34 M35 This tuning is a close approximation to one produced by the eighth mean 528 529 530 531 532 (Heath 1921) of chapter 4. It also occurs in Erickson’s analysis of Archytas’s system and in Vogel’s tuning (chapter 6 and Vogel 1963, 197). 28/27-2187/1792-256/243 63 + 345 +90 7168/6561 153 M36 This tetrachord appears in Erickson’s commentary on Archytas’s system with trite synemmenon (112/81, B,-) added. 16/15-2240/2187-2187/1792 112+41+345 7168/6561 153 M37 28/27-128/105:135/128 63 +343 +92 35/32 141 M38 Numbers 528-530 are from Vogel’s PIS tuning of chapter 6. 17/16-32/31.62/51 105+55+338 34/31 160 M39 20/19: 57/47-47/45 89 +334 +75 188/171 164 M4o Number 532 is a possible Byzantine chromatic. 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 360/349-349/327-109/90 54+113+332 120/109 166 M4I 24/23.115/109- 109/90 74 +94 +332 120/109 166 M42 Number 534 is a hypothetical Ptolemaic interpretation of 5 +6+ 19 “parts”, after Macran (1902). 240/229 -229/218- 109/90 81+85+332 120/109 166 M43 19/18 - 24/23 + 23/19 94+74+330 76/69 167 M44 15/14 - 36/35 98/81 119+49+330 54/49 168 M45 Number 537 occurs in Other Music’s gamelan tuning (Henry S. Rosenthal, personal communication). 28/27. 16/15 - 135/112 63 +112+323 448/405 175 M46 24/23 115/96 - 16/15 74+313 +112 128/115 185 M47 A Ptolemaic interpretation of Xenakis’s 5+19+6 “parts” (1971). 256/243 - 243/230 - 115/96 90 +95 +313 128/115 185 Maß 68/67 - 67/56 - 56/51 26+310+162 224/201 88 M49 68/57- 19/18 - 18/17 305+94+99 19/17 193 M50 15/14: 266/255 - 68/57 119+73+305 10/17 193 MSI 256/243: 243/229: 229/192 90+103+305 256/192 193 M52 32/31-13/12: 31/26 240/227. 227/214+ 107/90 55+139 +304 96+102+300 104/93 120/107 194 199 M53 M54 360/347: 347/321-107/90 64+135+300 120/107 199 M55 This genus is related to (Ps.)-Philolaus’s division as 6.5 + 6.5 + 17 “parts”. See also chapter 4. 7168/6561: 36/35-1215/1024 153 +49+296 4096/3645 202 194 CHAPTER 9

552 553 554 16/15-1215/1024-256/243 112+296+90 4096/3635 202 M57 28/27 - 1014/9045 1215/1024 63+139+296 4096/3635 202 M58 Numbers 548-5 50 are from Vogel’s PIS tuning of chapter 6. 120/113 - 113/106 - 53/45 104+111+283 60/53 215 M59 180/173 + 173/159 + 53/45 69+146+283 60/53 215 M6O 90/83 - 166/159 - 53/45 140+75+283 60/53 215 M6I 24/23: 115/106 - 53/45 74+141+283 60/53 215 M62 Number 554 is a hypothetical Ptolemaic interpretation of 5 + 9 + 16 “parts.” The others, numbers 551, 552, and 553 are 1:1, 1:2 and 2:1 divisions of the 555 556 557 pyknon. 34/29: 58/57: 19/17 10/9 + 117/100 - 40/39 120/113 - 113/97 + 97/90 275+30+193 58/51 182+272+44 400/351 104+ 264+130 388/339 223 226 234 M63 MÓ4 M65 This genus is a Ptolemaic interpretation of Xenakis’s 7+ 16+7 “parts.” 558 13/12: 55/52: 64/55 139+97 +262 55/48 236 M66 This genus is generated by the second ratio mean of chapter 4. 559 560 561 562 563 564 68/65. 65/56. 56/51 78+258+162 224/195 240 M67 12/11: 297/256 - 256/243 151+257+90 1024/8901 241 M68 28/27 : 81/70 + 10/9 63+253+182 280/243 245 M69 This tetrachord is also found in Erickson’s article on Archytas’s system with trite synemmenon (112/81, B,-) added. It also occurs in Vogel’s PIS tuning of chapter 6. 81/70 2240/2187 + 9/8 253+41+204 280/243 245 M70 81/70: 256/243 + 35/32 253+90+155 280/243 245 M71 135/128. 7168/6561 «81/70 92+153+253 280/243 245 M72 These three tetrachords are from Vogel’s PIS tuning of chapter 6. 565 566 60/59: 59/51-17/15 40/37 - 37/32: 16/15 29+252+217 135+251+112 68/59 128/111 246 247 M73 M74 This is a Ptolemaic interpretation of Athanasopoulos’s 9 + 15 + 6 “parts.” 569 16/15 : 280/243 - 243/224 36/35 - 9/8 - 280/243 8/7 - 81/80 : 280/243 570 571 46/45 + 132/115 - 25/22 16/15-12/11: 55/48 567 568 112+245+14I 49+204+245 231+22+245 81/70 81/70 81/70 253 253 253 M75 M76 M77 These three tetrachords are from Vogel’s PIS tuning of chapter 6. 38+239+221 112+151+236 115/99 64/55 259 262 M78 M79 This is an approximation to the soft diatonic of Aristoxenas, 1/2 + 3/4 + 1 1/4 tones, 6 + 9 +15 “parts.” THE CATALOG OF TETRACHORDS

575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 59° 591 592 593 10/9 - 63/55 : 22/21 182+235+81 220/189 263 M8o This is another tetrachord from Partch ([1949] 1974, 165), presented as an approximation to a tetrachord of the “Ptolemaic sequence,” or major mode in s-limit just intonation. 30/29 - 116/103 - 103/90 59+206+234 120/103 264 MBI 360/343 : 343/309 - 103/90 84+181+234 120/103 264 M82 40/39 - 143/125 - 25/22 44+233+221 600/429 265 M83 68/65 » 65/57 - 19/17 78+227+193 76/65 271 M84 256/243 - 729/640 - 10/9 90+225+182 2560/2187 273 M85 30/29 58/51 - 17/15 59+223+217 34/29 275 M86 23/21 - 14/13 - 26/23 158+128+212 46/39 286 M87 23/22-44/39: 26/23 77+209+212 46/39 286 M88 14/13 260/231. 11/10 128+205 +165 77/65 293 M89 4096/3645 - 35/32 243/224 202+155+I4I 1215/1024 296 MgO From Vogel’s PIS tuning of chapter 6. 38/35-35/32: 64/57 142 +155+201 19/16 298 Mor 19/17: 17/16-64/57 193 +I05 +201 19/16 298 M92 11/10 : 95/88 . 64/57 165 +135 +201 19/16 298 M93 The apyknon of genera numbers 583-585 is 19/16. The 1:2 division is listed as DIS (9/8), number 464. 240/221. 221/202 . 101/90 143+156+200 120/101 298 MO4 15/14: 112/101 101/90 II9+179+200 120/101 298 M95 120/113 - 113/101 + 101/90 104 +194+200 120/101 298 M96 533/483 - 575/533 + 28/25 171+131+196 25/21 302 M97 A mean tetrachord of the first kind from chapter 4. 19/17-85/76. 16/15 193 +194 +112 304/255 304 M98 19/17 + 1156/1083 - 19/17 193+113+193 68/57 305 M99 Two tetrachords from Thomas Smith (personal communication, 1989). 68/63-21/19: 19/17 132+173+193 68/57 305 MICO 10/9 + 108/97 - 97/90 182+186+130 97/90 368 MıoI Tetrachords in equal temperament The tetrachords listed in this section of the Catalog are the genera of Aristoxenos and other writers in this tradition (chapter 3). Included also are those genera which appear as vertices in the computations of Rothenberg’s propriety function and other descriptors, and various neo-Aristoxenian genera. These are all divisions of the tempered fourth (500 cents). CHAPTER 9

The “parts” of the fourth used to describe the scales of Aristoxenos are, in fact, the invention of Cleonides, a later Greek writer, as Aristoxenos spoke only of fractional tones. The invention has proved both useful and durable, for not only the later classical writers, but also the Islamic theorists and the modern Greek Orthodox church employ the system, though the former have often doubled the number to avoid fractional parts in the hemiolic chromatic and a few other genera. Until recently, the Greek church has used a system of 28 parts to the fourth (Tiby 1938), yielding a theoretical octave of 68 (28 + 12 + 28) tones rather than the 72 (30+ 12 + 30= 72) or 144 (60 + 24 +60 = 144 in the hemiolic chromatic and rejected genera) of the Aristoxenians. The 68-tone equal temperament has a fourth of only 494 cents. Note that a number of the Orthodox liturgical tetrachords are meant to be permuted in the formation of the different modes (echoi). This operation may be applied to the historical and neo-Aristoxenian ones as well. ARISTOXENIAN STYLE TETRACHORDS 594 595 597 2+2+26 2.5 +2.5 +25 2+3+25 3+3+24 2+4+24 2+5+23 7/3 + 14/3 +23 4+3+23 3-5 + 3-5 +23 2+6+22 4+4+22 8/3 + 16/3 + 22 3+5+22 4.5 + 3,5 + 22 247421 3+6+21 4.5+4.5 +21 4+5 +21 6+3+11 6+20+4 10/3+20/3+20 197 33 +33 +433 42+42+417 33 +50+417 50 + 50 +400 33 + 67 +400 33 +83 + 383 39 + 78 + 383 67 + 50+ 383 58 + 58 + 383 33 + 100 + 367 66 + 66 + 367 44 + 89 + 367 50 + 83 + 367 75+58+367 33 +117 +350 50 + 100 + 350 75 +75 +350 67 + 83 +350 100 + 50 + 350 100 + 333 +67 564111 +333 THE CATALOG OF TETRACHORDS CHAPTER 4 CHAPTER 4 CHAPTER 4 ARISTOXENOS CHAPTER4 CHAPTER 4 CHAPTER 4 CHAPTER 3 CHAPTER 4 CHAPTER 4 ARISTOXENOS CHAPTER 4 CHAPTER 4 ARISTOXENOS CHAPTER 4 CHAPTER 4 ARISTOXENOS CHAPTER 4 ARISTOXENOS SAVAS CHAPTER 4 TI T2 T3 T4 T5 T6 T7 T8 T9 TIO TII TI2 T13 TI4 TIS TI6 T17 T18

618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 5+5+20 83 +83 + 34 5:5+5:5 +19 11/3 + 22/3 + 19 5+19+6 5+6+19 2+10+18 3+9+18 4+8+18 92+92+317 61 +122 +317 83 +317 + IGO 83 + 100 + 317 33 + 167 + 300 50 + 150 + 300 67 +133 +300 45+7.,5+18 75 +125 +300 6+6+18 5+7+18 6+18+6 13/3 +26/3 +17 100+ 100 + 300 83 +117 + 300 100 + 300 + 100 72 +144 + 283 6.5+6.5 +17 108+ 108 + 283 2+16+12 14/3 + 28/3 + 16 5+9+16 8+16+6 7+16+7 2+13 +15 3+12+15 4+II+1S S+IO+IS 6+9+15 7+8+15 33 +267 +200 7547-5415 9+15+6 2+14+ 14 4+14+12 5+11+14 16/3 + 32/3 +14 8+8 +14 45+ 13.5 +12 S+12+13 4+13+13 78 +156 +267 83 +150 + 267 133 + 267 + 100 II7 +267 + 117 33 +217 +250 50 + 200 + 250 67 +183 +250 83 +167 +250 100+ 150 +250 117 +133 +250 125+125 +250 150 +250 + 100 33 + 233 + 233 67 +233 + 200 83 +183 + 233 89 +178 +233 133+ 133 +233 75 +225 +200 83 +200 +217 67 +217 +217 17/3 +34/3 +13 94 +189 +217 8.5+8.5 +13 142+ 142 +217 198 CHAPTER 9 CHAPTER 4 CHAPTER 4 CHAPTER 4 XENAKIS MACRAN CHAPTER 4 CHAPTER 4 ARISTOXENOS CHAPTER 4 ARISTOXENOS CHAPTER 4 ATHANASOPOULOS CHAPTER 4 CHAPTER 4 CHAPTER 4 CHAPTER 4 WINNINGTON-INGRAM SAVAS XENAKIS; CHAP. 4 CHAPTER 4 CHAPTER 4 CHAPTER 4 CHAPTER 4 ARISTOXENOS CHAPTER 4 CHAPTER 4 ATHANASOPOULOS CHAPTER 4 ARISTOXENOS WINNINGTON-INGRAM CHAPTER 4 CHAPTER 4 ARISTOXENOS CHAPTER 4 CHAPTER 4 CHAPTER 4 CHAPTER 4 T22 T23 T24 T25 T26 T27 T28 T29 T30 T31 T32 133 134 135 136 137 138 139 T40 T4I T42 T43 T44 T45 T46 T47 T48 T49 T50 TSI T52 T53 T54 T55

100+ 200+ 200 ARISTOXENOS T59 654 Savas, Xenakis and Athanasopoulos all give permutations of this tetrachord in their lists of Orthodox church forms. 12+11+7 200 + 183+117 XENAKIS T60 Xenakis (1971) permits several permutations of this approximation to Ptolemy’s intense diatonic. 10+8412 167 + 133 + 200 SAVAS TOI 655 The form 8 + 12 + 10 is Savas’s “Barys diatonic” (Savas 1965). 12+9+9 200 + 150 + 150 AL-FARABI; CH. 4 653 T62 656 8+11+11 133 + 183 + 183 CHAPTER 4 This tuning is close to 27/25 - 10/9 : 10/9. 657 9.5+0.5 +11 158 +158 + 183 CHAPTER 4 T64 10+10+10 166 + 167 + 167 AL-FARABI T65 Tiby’s Greek Orthodox tetrachords of 28 parts to the fourth of 494 cents. 12+13 +3 212 +220+53 TIBY T66 I2+5+ II 212 +88 +194 TIBY 167 12+9+7 212+159+124 TIBY T68 9+12+7 159+212+ 124 TIBY T69 See Tiby (1938) for numbers 659-662. 658 659 660 661 662 763 TEMPERED TETRACHORDS IN CENTS 663 664 665 666 22.7 +22.7+454.5 37.5 +37.5 +425 62.5 +62.5 +375 CHAPTER 5 CHAPTER 5 CHAPTER 5 T70 T7I T72 Tetrachord numbers 663- 665 are categorical limits in the classification scheme of 5-9. 95+115 +290 T73 This tetrachord was designed to fill a small gap in tetrachordal space. See 9-4 9-5, and 9-6. 667 668 669 670 671 672 673 674 89 +289 +122 87.5 + 287.5 +125 83.3 + 283.3 + 133.3 CHAPTER 5 CHAPTER 5 CHAPTER 5 T74 T75 T76 75+275+150 CHAPTER 5 T77 100 +275 +125 CHAPTER 5 T78 55+170+275 This tetrachord was designed to fill a small gap in tetrachordal space. T79 66.7 + 266.7 + 166.7 233.3+ 16.7 + 250 T80 T81 199 THE CATALOG OF TETRACHORDS CHAPTER 5 CHAPTER 5

CHAPTER 5 CHAPTER 5 CHAPTER 5 CHAPTER 5 T82 T83 T84 T85 678 225+25 +250 66.7 + 183.3 + 250 75+175+250 125+125+250 679 105 + 145 + 250 680 110 + 140 + 250 Tetrachord numbers 679 and 680 fill possible gaps in tetrachordal space. 681 682 87.5 + 237.5 +175 + 166.7 + 100 233.3 212.5+ 602,5 +225 675 676 677 683 684 685 686 687 688 689 690 T86 T87 CHAPTER 5 CHAPTER 5 CHAPTER 5 CHAPTER 5 CHAPTER 5 CHAPTER 5 T88 T8g T90 TOI T92 T93 CHAPTER 5 T94 + 212.5 100+ 187.5 CHAPTER 5 T95 212.5+ 137.5 +150 200 +125 +175 145 + 165 + 190 CHAPTER 5 CHAPTER 5 196 T97 T98 225+75+200 225 +175 + 100 87.5+ 187.5 +225 212.5+ 162.5 +125 This tetrachord was designed to filla small gap in tetrachordal space. Semi-tempered tetrachords The tetrachords in this section contain both just and tempered intervals. Two 692 of these genera are literal interpretations of late Classical tuning theory. A number are based on the assumption that Aristoxenos intended to divide the perfect fourth (4/3), a rather doubtful hypothesis. The remainder are mean tetrachords from chapter 4 with medial 9/8. Formally, these latter tetrachords are generators of tritriadic scales, In all cases they span a pure 4/3. 16/(9V3) - 16/(9V3) - 81/64 45 +45 +408 SI Number 692 is Barbera’s (1978) literal interpretation of Nicomachos’s enharmonic as 1/2 semitone + 1/2 semitone + ditone, where the 1/2 semitone 693 is the square root of 256/243, also written as 16 - V3 /27. 126376: 1.05321 : 1.00260 405+88+4 s2 This mean tetrachord ofthe second kind is generated by mean 9. 694 (310 « (4/310. (4/38/10 50+ 50+ 398 s3 This tetrachord is a literal interpretation of Aristoxenos’senharmonic under Barbera’s (1978) assumption that Aristoxenos’s meant the perfect fourth 4/3. In Cleonides’s cipher, it is 3 + 3 + 24 parts. CHAPTER 9

(4/3)/15 . (4/3)?/15 . (4/3) 66 + 66 + 365 s4 This tetrachord is a semi-tempered interpretation of Aristoxenos’s soft chromatic. In Cleonides’s cipher, it is 4 + 4 + 22 parts. 696 (4/3)329 . (4/3)7/60. (4/3) 115 75 +58 + 365 ss This tetrachord is a semi-tempered interpretation of a genus rejected by Aristoxenos, It somewhat resembles Archytas’s enharmonic. In Cleonides’s cipher, it is 4.5 + 3.5 + 22 parts. 697 (4/30 (4/3732. (4/3710 75 +75 +349 sé This tetrachord isa semi-tempered interpretation of Aristoxenos’s hemiolic chromatic. In Cleonides’s cipher, it is 4.5 + 4.5 + 21 parts. 698 699 700 (4/3) » (4/3) 10. (4/3)710 100 + 50 + 349 87 This tetrachord is a semi-tempered interpretation of a genus rejected by Aristoxenos, In Cleonides’s cipher, it is 6 + 3 + 21 parts. 1.21677 : 1.03862 : 1.05505 340 + 66 + 93 s8 This mean tetrachord of the first kind is generated by mean 9. (4/3) + (4/3)! + (4/3) 100+ 100+ 299 89 This tetrachord is a semi-tempered interpretation of Aristoxenos’s intense chromatic. In Cleonides’s cipher, it is 6 + 6 + 18 parts. Jor (4/329 + (4/3) (4/3) 66+ 133 +299 sıo This tetrachord is a semi-tempered interpretation of a genus rejected by Aristoxenos. It closely resembles Archytas’s chromatic In Cleonides’s cipher, Joz itis 4 +8 + 18 parts. 3V2/4 - 3V2/4 32/27 102 + 102 + 294 SII This tetrachord is implied by writers such as Thrasyllus who did not give 793 numbers for the chromatic, but stated only that it contained a 32/27 anda 1:1 pyknon (Barbera 1978). The semitones are the square root of 9/8. 1.18046 - 1.06685 - 1.05873 287 +112 +99 SI2 This mean tetrachord of the second kind is generated by mean 5. 704 1.05956 - 1.06763 : 1.17876 100+ 113 + 285 s13 This mean tetrachord of the first kind is generated by mean 13. 705 1,17867 + 1.06763 : 1.05956 285+ 113+ 100 SI4 This mean tetrachord of the second kind is generated by mean 14. 706 707 1.17851 + 1.06771 - 1.05963 284+ 113 +100 This mean tetrachord of the second kind is generated by mean 17. 1.17851 : 1.06771 : 1.05963 282+ 1144 101 This mean tetrachord of the second kind is generated by mean 6. 201 THE CATALOG OF TETRACHORDS

(4/3)!9 + (4/3)?10. (4/3)? 100+ 149 + 250 s17 This tetrachord is a semi-tempered interpretation of Aristoxenos’s soft diatonic. In Cleonides’s cipher, itis 6 + 9 + 15 parts. 709 1.07457: 1.07457 + 1.154701 125+125 +249 s18 This mean tetrachord of the first kind is generated by mean 2. The corresponding tetrachord ofthe second kind has the same intervals in reverse order. 710 Wa? 66+ 232 + 199 sig This tetrachord is a semi-tempered interpretation of Aristoxenos’s diatonic with soft chromatic diesis. In Cleonides’s cipher, itis 4 + 14 + 12 parts. 1.13847 - 1.1250 1.0410 225 + 204 + 70 $20 This mean tetrachord of the third kind is produced by mean 5. 712 (4/3)°79 - (4/3)920. (4/3)? 75 +224 +199 $21 This tetrachord is a semi-tempered interpretation of Aristoxenos’s diatonic 73 714 715 716 717 with hemiolic chromatic diesis. In Cleonides’s cipher, it is 4.5 +13.5 + 12 parts. 1.13371 + 1.1250 + 1.04540 217+ 204+ 77 $22 This mean tetrachord of the third kind is produced by mean 14. In reverse order, itis generated by mean 13. 1.13315 + 1.1250: 1.04595 216+ 204 +78 $23 This mean tetrachord of the third kind is produced by the root mean square mean 17. 1.09185 - 1.07803 - 1.13278 152 +130+216 824 This mean tetrachord of the first kind is produced by mean 6. 1.09291 + 1.078328 - 1.13137 164+ 131 +214 $25 This mean tetrachord of the first kind is produced by mean 17. 1.09301 + 1.07837 + 1.13122 I54+ 131 +213 $26 This mean tetrachord of the first kind is produced by mean 14. In reverse order is the tetrachord of the second kind generated by mean 13. 718 1.09429 - 1.07874 - 1.12950 156+ 131 +211 527 This mean tetrachord of the first kind is produced by mean 5. 719 720 1.12950 + 1.1250 - 1.04930 2II + 204 + 83 s28 This mean tetrachord of the third kind is produced by mean 6. 1.08866 : 1.1250: 1.08866 147+ 204+ 147 $29 This mean tetrachord of the third kind is produced by the second or geometric mean. CHAPTER 9

(4/35. (4/3)?/5 + (4/3)?/5 100+ 199+ 199 $30 This tetrachord is a semi-tempered interpretation of Aristoxenos’s intense diatonic. In Cleonides’s cipher, it is 6 + 12 + 12 parts. 722 4/3)3 +(4/3)!2 + (4/3) 166 + 166 + 166 $31 Number 722 is the equally tempered division of the 4/3 into three parts. It is the semi-tempered form of Ptolemy’s equable diatonic and of the Islamic neo-Aristoxenian approximation Io + 10 + 10. 723 (4/3)? + (4/3)39 - (4/3310 200+ 149+ 149 532 Number 723 is the semi-tempered version of the Islamic neo-Aristoxenian genus 12 +9 + 9 parts, Source index The sources of the tetrachords listed below are the discoverers, when known, or the earliest reference known at the time of writing. Further scholarship may change some of these attributions. Because the Islamic writers invariably incorporated Ptolemy’s tables into their compilations, they are credited with only their own tetrachords. The same criterion was applied to other historical works. Permutations are not attributed separately except in notable cases such as that of Didymus’s and Ptolemy’s mutual use of forms of 16/15 - 9/8 - 10/9. Doubtful attributions are marked with a question mark. For more information, including literature citations, one should refer to the entries in the Main Catalog. Uncredited tetrachords are those of the author. AL-FARABI 307, 394, 460, 473, 475: 655, 658 ANONYMOUS TREATISE: 456 (FROM D'ERLANGER) ARCHYTAS: 106, 248, 393 ARISTIDES QUINTILIANUS: 245 ARISTOXENOS: 597, 604, 607, 610, 612, 622, 624, 638, 643, 647, 652 ATHANASOPOULOS: 626, 641 AVICENNA: 108, 311, 390, 395, 396, 457, 458, 459, 476, 478, 479, 480, 481 BARBERA: 692, 604 BARBOUR: 216, 217, 247?, 250}, 251?, 252?, 312 BOETHIUS: 75, 24I DANIÉLOU: 154 DIDYMOS: 103, 215,455 ERATOSTHENES: 71,212 THE CATALOG OF TETRACHORDS