The vibrating string of the Pythagoreans

SCIENTIFIC ‘ablished zu AMERICAN December 1967 Volume 217 Number 6 La | ARTICLES 19 INFECTIOUS DRUG RESISTANCE, by Tsutomu Watanabe A newly recognized infectious agent can make bacteria resistant to several drugs. THE EARLIEST APES, by Elwyn L. Simons 28 The oldest known skull of a precursor of apes and man has been found in Egypt. 36 X-RAY STARS. by Riccardo Giacconi The strongest source of X rays found by rockets has now been shown to be a star. 62 ZONE REFINING, by William G. Pfann This simple technique can produce industrial materials of extraordinary purity. 76 INGH-ENERGY SCATTERING, by Vernon D. Barger and David 8. Cline What happens when two particles meet is a main theme of high-energy physics. > 92 THE VIBRATING STRING OF THE PYTHAGOREANS, by E. Eugene Helm The relations among musical tones comprise a cord that binds science to music. 104 NON-CANTORIAN SET THEORY, by Paul J. Cohen and Reuben Hersh A mathematical advance is explained by analogy with non-Euclidean geometry. THE WATER BUFFALO, by W. Ross Cockerill This tractable, hardworking and productive animal is just beginning to be studied. DEPARTMENTS THE VIBRATING STRING LETTERS HSEL.M OF THE PYTHAGOREANS 50 AND 100 YEARS AGO 431 1467 ART AND ARCHITECTURE IN ITALY: 1600— 14 THE AUTHORS 1750. Rudolf Wittkower. Longmans 48 SCIENCE AND THE CITIZEN Canada Ltd., 1959. 127 E. EUGENE HELM (“The Vibrating 134 String of the Pythagoreans”) is associate professor of music at the University of 140 154 158 lowa. A graduate of Southeastern Loui- CONVERSATIONS WITH STRAVINSKY. Igor Stravinsky and Robert Craft. Doubleday & Company, Inc., 1962, Tue HARMONIES OF THE WoRrLD: V. Josiana College, Louisiana State University and North Texas University (where he received a Ph.D. in musicology). Ilelm writes: “My interest in resounding T soaro OF torrosstringé undoubtedly goes back to my undergraduate days, when I was a mishannes Kepler in Great Books of the Western World: Vol. XVI, edited by Robert Maynard Hutchins, Encyclopaedia Britannica, Inc., 1952. HE MATHEMATICAL Basis OF THE ARTS. Joseph Schillinger. Philosophical Libräry, 1948. ser ouvseruer Buided physics major. 1 remember that Music IN THE MEDIEVAL AND RENAISI was the only student in sophomore since UNIVERSITIES. Nan Cooke CarPRODUCTION DEPARIMER COPY DEPARTMEN physics lab who, during our search for. penter. University of Oklahoma Press Joops and nodes, could make the piece j958, , erw; wang tll paper jump off the string of the mono- «pig New CoLLEGE ENCYCLOPEDIA OF chord at the very first pluck. Being à Music. J. A. Westrup and F. L. I. Harmisguided violinist at the time, I knew CIRCULATION MANAGE . ion. W. W. Norton & Company ADVERTISING MANAGE assestanr to rue pusciswe Perfectly well where the loops were. a ao — PUBLISHED MONTHLY BY SCIENTIFIC AMERICAN, INC., 415 MADISON AVENUF, NEW YORK, N.Y. 10017, COPYRIGHT © 1987 BY SCIENTIFIC AMERICAN, INC. ALL SIGHTS RESERVED. SECOND-CLASS POSTAGE PAID AT NEW YORE, M.v., AND Ai ADOUIONAL MAILING OFFICES. AUTHORIZED AS SEC- OND-CLASS MAIL BY IME POST OFFICE DEPARTMENT, OTTAWA, CANADA, AND FOR PAYMENT OF POSTAGE IM CASH. SUBSCRIPTION RATE; $6 PER YEAR.

MONOCHORD, the most effective demonstration of the Pythagorean laws of proportion, consists of a single string stretched on a frame. In this print, from a 12th-century manuscript, the 1lth-century theoretician Guido d'Arezzo is explaining the divisions of the

The Vibrating String of the Pythagoreans The monochord gave rise to far more than Western musical scales. For the Greeks and those who followed them music was number, and the ratios of the scale were manifested in nature and in the arts Ca by E Eugene Helm length. For instance, if the total length the same sets of numbers seemed to keep produces C. two-thirds of the string will cropping up in other matters. A single to mathematics itself, but certainly to produce G. (The interval C-G is called a fifth because five lines and spaces on the evidently governed all the rhythms. of nature: the four seasons, the two tides, gor Stravinsky once observed that musical form is “far closer to mathematics than to literature—not perhaps order, expressible in number and ratio, something like mathematical thinking musical stall are traversed in going from and mathematical relationships.” Straone to the other, counting € and €.) the pendulum-like fluctuations of Invinsky spoke from the viewpoint of a Three-fourths of the string will yield a pitch a fourth higher than the total man allairs-in fact, all the machinery composer who, like many of his colleagues, has consciously made “something like mathematical thinking” an integral element of his music. This element is widely regarded as a hallmark of modern music. What is seldom appreciated, even in the musical world, is that the roots of the relation between music and mathematics extend deep into antiquily. The way in which the two were betrothed and the course the marriage has taken over the centuries make a begniling and instructive tale. length (F, if the total vields C); cightninths, a whole step, or a second, higher (D if the total vields €). and so on. In time the fractious become more complex and the two notes represented by the resulting intervals become more dissonant if they are sounded together, The fraction for the half-step, for example, is 243 256. Greeks—more precisely, by the school of The Pythagoreans used these musical facts lo construct scales. Gradually the relation of the vibrating portion of a string to the entire string came lo be expressed in ratios, such as 1:2 for the Pythagoras in the sixth century 8.c., aloctave,2:3 for the fifth and so forth. though their courtship certainly began, Soon it was noted that the most perhaps even earlier, among the Chaldeans, Egyptians, Babylonians and Chinese. During the next few centuries the simplest ratios. For example,2:3 (the They were plighted, these two, by the Pythagoreans joined music and number irrevocably by means of a vibrating string. Whether the string was actually mounted and measured or was only described in speculative treatises and imagharmonious intervals were those with the fifth) was much more harmonious than 8:9 (the whole step). Both the intelleeof the universe from microcosm to macrocosm. The most suecinel expressions of the numbers and ratios governing the orderly events of nature seemed to be precisely those that applied also to music. It is small wonder, then, that Pythagorean thinkers came to regard music as a kev to all being. A notable example of their thinking is the myth of the “music of the spheres.” The Pythagoreans believed that bodies moving in space produced sounds unheard by ordinary mortals, and that bodies moving rapidly produced sounds of higher pitch than the sounds produced by bodies moving slowly. Greek astronomy of the sixth century n.c. held that the greater the distance of a planet from the earth, the more rapidly the planet moved. Moreover, the distances between planets and the ratios between speeds of tual appeal of simplicity and the sensuous appeal of harmoniousness made the planets (relative to the earth) were both simple relations come to be regarded as that is, expressible in whole-number rasuperior to complex ones. Harmonious tios. Therefore the sounds produced by thonght to be harmonically determined, inatively or pedantically discussed, has vibrated in the conscious and unconsounds, said the Pythagorcans, are prothe planets harmonized with one anduced other. hy ratios expressible as whole scious minds of musicians, mathematinumbers, and the simpler the ratios— cians, philosophers, astronomers, physithe smaller the whole numbers expresscists and architects ever since, ing Chem—the more consonant the sound. “Music of the spheres” also meant the mathematical ordering of the physical universe: the regulation of the world by It followed from these attitudes that laws of musical proportion, In this sense rr its simplest form the basic Pythagorean doctrine relating number to music can be described as follows. Pluck a stretched string of any Jength aud allow it to vibrate; it will sound a certain pitch. Allow only half of it to vibrate, and the pitch will rise an octave. If two-thirds of the string vibrates, the pitch will rise a fifth above the one produced hy the total the octave, the fifth and the fourth were regarded as musically superior to other music was mathematics. Thus it was primarily the Pythagoreans who established intervals, Indeed, since carly medieval music as a mathematical discipline, causlimes the octave, filth and fourth have been called the “perfect consonances.” ing it eventually to be put into the qua-drivium—the standard three-vear course Until the end of the Middle Ages these of study in medieval universities—along intervals were the theoretical basis of with the other mathematical studies of arithmetic. geometry and astronomy, almost all Western polyphony. To the fascination of the Pythagoreans Plata, under the influence of Pythago93

ANSE di | HECHA f 4 - ( FOURTH N A | N WHOLE STEP É N WHOLE STEP | ha _ ro 1 WHOLE STEP WHOLE STEP | = Pa e) NOTE A B E D E FS FINGER POSITION OPEN 1/9 FROM NUT TO BRIDGE 1,0 FROM B TO BRIDGE 1:4 FROM NUT TO BRIDGE 1,2 FROM NUT TO BRIDGE 1/9 FROM E TO BRIDGE STR 8/9 ese > PRE JOY 0/8 - 440 SECOND. = Ati fae gi 13 24 de 37 440 27 - 16/07 9: - DTM: dag = ì > n : i = UOo y = PYTHAGOREAN SCALE | go : . Gu ga. TH OF LEN tnot quite the same as the Cu BRIDGE medern per second. (Frequencies are a modern concept, unknown to the “pqual-tenperedTM sealed is derived by dividing a string, as ill Pithasoreans.t Stopping the string hs placing the finger 1/9 of the trated here with the f «tring af the violin, Sounding the total length was from the nat to the bridge leaves 8 9 of the string open, pro: from nnt to bridge produces an cf, with a frequency of 810 eveles ducing a 47, anıl <a on, The only intervals used to constrnet the Py|

reanism, probably did as much to raise music to this level as the Pythagoreans themselves did. He gives a description of the music of the spheres in Book X of the Republic. His classic view of a world created according to musical proportion appears in the Timaeus. Pythagorean and Platonic doctrines of music as number and proportion and music as part of science aud mathematics were passed on almost intact into the Christian era. It is no coincidence that Ptolemy, the great astronomer of the second century, was also a leading music theorist. Nor is it surprising that the medieval thinkers who followed him could hardly begin to scrutinize the universe quantitatively without first taking a long look at music. The chief medieval transmitter of the ancients’ ideas on musical proportion a G' A "9 FROM F* 1,2 FROM NUT 10 BRIDGE TO BRIDCE was Boethius, the sixth-century Roman philosopher and mathematician. He classified the proportions and helped to establish a 1 I . 89. 16/27 = in the Pulse?” Occasionally the mathematical way of thinking affected music is most remembered, however, for an- 2 x 441) == gan division of music into three types. One that inaugurated a new musical age and that had to be justified in traditional the spheres; the second was musica humathematical fashion. By the end of the mana, the harmony of the human soul 14th century French secular music had and body, and the third was musica inarrived at a degree of rhythmic complexstrumentalis, the actual playing and singity undreamed of by De Vitry and hardly ing of music, The third category, which equaled in the 20th century. (This music, incidentally, has recently been widely admired by a number of conmusic, was in the time of Bocthius regarded as the lowest of the three forms. temporary composers and avant-garde Playing and singing were only tinny musicians.) reflections of the greater harmonies emas Boethius had defined them. Philoso- One would be remiss to discuss the laws of musical consonance as they applied to music alone in the Middle Ages. Musical proportions appear almost everywhere in the thought and art of medieval Europe. To St. Augustine the octave seemed to be rooted in the very being of even the most untutored man and therefore must have been implanted in man's nature by God himself as a means of conveying to human ears the meaning of redemption. To the School phers and mathematicians, as well as of Chartres in the 12th century the garmusic theorists and composers, commonbled fragment of the Timacus then available had almost the stature of Holy Writ. The scholars of Chartres discovered mubodied in the other categories. The true musician was the man who knew music as part of a celestial order; the mere performer was only a servant. As Boethins put it, the performer was “separated from music” because “physical skill obeys like a handmaid while reason rules like a mistress.” Medieval treatises on music were often slavish restatements of the old doctrines lv felt obliged to pay their respects to music as a traditional mathematical disvipline before proceeding to more conlemporary matters. Slandard operating procedure scemed to be to quote respectfully from Roethins (or to steal his ideas that must be derived from the simpler ones. in the most direct way. Philippe de Vitry’s treatise Ars Nova, which dates from about 1320, presented a revolutionary system of metrical organization was musica mundana, the harmony of is now considered the main aspect of thagorean scale are the octave, fifth, fourth and whole step: the other intervals, such as the third (A to C-xharp) have complex ratios tion, the all-pervading ratios and the music of the spheres were not far distant. The great monasteries, and later the cathedral schools and the universities, kept the old mathematical traditions alive and flourishing. For example. one of the writings of Pietro d'Abano, who in 1315 was Distinguished Professor of Medicine, Philosophy and Astrology at the University of Padua. was a treatise entitled “Is Musical Consonance Found new terminology agli. other refinement of the old doctrines: the "ts 312 Greek limes, consisted of a single string mounted on a flat surface and was provided with movable bridges and ası accurate means of marking off serial lengths of the string. Throughout the Middle Ages this speculative side of the musical art ran parallel to music making, Although folk music and the music of such practical artists as troubadours and minnesingers were relatively untouched by speculathem in words: proportio dupla for 1 : 2, proportio sesquialtera for 2 : 3, proportio sesquitertia for 3 : 4 and so on. His name = 243/128 » 440 = the monochord became widely used ag a means of illustrating the ancient musical laws. This instrument, dating back to without acknowledgment) and sical proportions in the dimensions of Solomon's temple and built such proportions into their own great cathedral. Indeed, Otto von Simson has shown to inin his recent book The Gothic Cathedral clude a drawing of Pythagoras striking that not only the cathedral of Chartres sonorous bodies or adjusting strings of but also nearly -all the great Gothic various lengths. cathedrals were conceived as “music in An instrument called

: founder of physical astronomy and the '*,* thinker who placed Copernicus’ churchyet it is in this treatise that he sets forth his famous third law of planetary motion. The law is one result of Kepler's long interval, letting the two outer notes of the interval represent the greatest and et's interval into a dillerent pitch regis- ‚oger whose lifelong view of the universe attempt to relate musical harmony to planetary motion and, more specifically, Lo find a “musically harmonious” relation between the distances of the planets was essentially that of Pythagoras and from the sun. The distances vary as the velocities to the average distances from Plato. His Jargest work, De Harmonice Mundi (Concerning the Harmonies of the planets revolve around the sun, and so the sun. When this approach did not dis- Kepler's first calculations were based on World), published in 1619, is one of the most astonishing events in the history the greatest and Jeast distances. When calculations in distances failed to proclose God's harmony, Kepler substituted the period of revolution for the average of ideas. It is a latter-day explanation duce a concord, Kepler turned his attenof the music of the spheres, which Kepler “~ shaking heliocentric theory on a firm mathematical foundation. On the other hand he was a mystic soaked in medieval speculation and an occasional astrolleast velocities. Then he put each planter, which was determined by the planet's average distance from the sun. Next he tried to relate the average angular angular velocity. Here he found the relation he had been seeking. Kepler's third Jaw is usually stated as the various parts of the universe are tion to the angular velocities of the planets. (Velocities and distances are related, since the closer a planet approaches to the sun, the greater is its angular velocity with respect lo the sun.) arranged in accordance with abstract Kepler associated the varying angular sun and K is a constant. The values of T and D are known for the earth: T is notions of the beautiful and harmonious; velocity of each planet with a musical one year and D is 93 million miles. believed was heard ouly by the Being that animates the sun. The whole basis of the treatise is Kepler's conviction that A | « comi ds pro; mundanur cause äutem monochordum “nr Teen, Hic& Intetvallis compoés «nantiis E hocmododepinximus. “had a mathematical formula: 7/0? = K, where T is a planet's period of revolution, D is its average distance from the Therefore K can be computed, so that one can compute any other planet's average distance from the sun if the period of revolution is known or the period of revolution if the average distance is known. The formula looks antiseptic, but like so much of miuthematies it springs from the aesthetic power of natural order. PR + te fo a IS. Fo È AP ie s ve = 187 / wt Fa Hi a e “en i N (È Ze NS Essentially it is this same aesthetic power that lies at the root of musics relation to number. As “duality” is the word for the atti- 2% BR È tude of thinkers during and immediately after the Renaissance toward the ja relation of music and number, so “reca vt onciliation” is the word for the period % from 1600 to 1750. This period, the age En au: PES of the Baroque, is one in which feeling seems at first to have predominated over ie mathematical tradition. Below the sur- : face of Monteverdi's operas, Corelli's ik is. sonatas and Bach's fugues, however, is a continning and fundamental respect for number as a basis of music. The reconciliation was particularly evident in the work of the Freuch theorist and composer Jean Philippe Rameau, in Se. 2 se who stands as perhaps the foremost theorist in the history of Western music. In the first half of the 18th century he pro- Ohvitn.a cluced a series of treatises based on what he called “natural principles.” These principles are expressions of the age-old laws of proportion. Ramean's explanations of such basic harmonic concepts as the progression of chords, the invertibility of chords, chord-building by thirds and the relation of chords and melody make him the creator of the modern scicuce of harmony. Yet his writings are GOD'S HAND tunes the “monochord of the world” in an illsteation from a book published in 1617, The classical elements and the planets appear, along with the musical ration. 98 based on and reconciled with the previous discoveries of such theorists as Zarlino—Uhinkers who walked the well-worn path leading back to antiquity. It was also in the carly 18th century wwos

pret SIMPLE RATIOS were employed by the Italian architect Alberti in designing the façade of Santa Maria Novella in Florence, as shown by Rudolf Wittkower of Columbia University, The black form a square half the width of the full story. Squares of the same lines indicate how the facade is made up of squares related in the width of the side bays (C) and twice the height of the attic (D). ratio 1 : 2, as in an octave (left), The same ratio rerurs in the <ub- The ratio of the width to the height of the entrance hay IE) is 2 : 3. position is serially determined not only composition. Al face value they seem too simple; when they are applied sublimition) of the fundamental one, with the riods in Western history when most comin pitch but also in rhythm and in density (the number of instruments playing at any given moment). It is not only a model of the uncompromisingly mathematical in music but also a manifesto: one of the lines of its poetry, which was written by Kfenek, states “What looks ahead subordinates itself to number.” most important of these notes lying an octave, a fifth and a fourth apart. The posers did not perceive mathematical order in the structure of music and stren- For the most part, however, mathematics swims seductively just below the phenomenon is also known as the harmonic series and has been called the uously try to find a meeting ground besurface of music. It is a naiad gazing at tweeu the old mathematical Jaws and the composer, seemingly within reach the creation of new music. The attempts but actually unreachable. Its presence is explained by the unshakable principle that music, like all other arts, must be nally or “in principle.” they seem to disappear altogether. To put the matter another way. composers usually realize the mathematical bases of composition only after the composing is done. Nonetheless, there have been few pe- Sauveur's discovery marks a major dihave become more subtle, but the ques- ‘vision in the history of musical thought. ‚Before his time the power of music was tion remains: Are there any valid mathematical approaches to composition? largely explained through mathematical flights of reason and fancy. Since his Among the groups of composers who would answer the question affirmatively _ time most theorists have tried to discovare the serialists, whose “12-tone system” naiad, is it any wonder that Athanasius Kircher, the 17th-century German mather the secret of music in acoustical laws, is based on the hannonie and melodic equivalence of the 12 tones of the harematician and self-styled musical encyclopedist, built a composing machine such as the law of overtones. The name of Pythagoras still figures importantly in most books on acoustics, but to regard the modern science of acoustics as a continuation of Pythagoreanism would be to 3 size enclose the pediment and entablature (B); these are twice the that the French physicist Joseph Sauveur achieved perhaps the most concrete theoretical reconciliation of old and new with his discovery of overtones: the fact that a vibrating string sounds not only a fundamental note but also a subsidiary series of higher notes that are integral multiples (as defined by speed of vibra- “chord of nature.” ’ anit. rights, The sides of the central bay of the upper stary 14) equate two different modes of thought. founded on some kind of order. Considering the seductiveness of the monic scale (the white and black keys whose from one E to the next on a piano). Sedictate pitch, rhythm and tempo to any rialism was invented some 50 years ago composer willing to use it? (Nobady seemed willing.) Is it presumptuous to by Arnold Schoenberg, who saw the numerical combinations would technique as a imilving device for atonal believe that Mozart, Haydn, Handel and music, or music without a kev. In this others may not have had their tongues completely in their cheeks when they wrote music dictated by the throw of Against the historical background that - A I have sketched let us consider fur+. ther the role of mathematics from the ed, the 12 notes are arranged in some specific order that is then used as the > Viewpoint of the composer. The fact is basis for a composition. The series of mathematical preoccupations of . that the string of the Pythagoreans has tones can be represented as at series of contemporary composers Kienek, always vibrated more in minds than in numbers. A recent chamber work indicates how Karlheinz Stockhausen, Milton Babbitt : been able to put the proportions to difar the serialist techniqne has moved. It tolerant of music generated by comi’. q y Tect use during the process of musical is Sestina, by Ernst Kienck. This computer? “ ears, because the composer has seldom 2e technique, which has since been extenddice? Can we begin to understand the as such and Pierre Boulez? Should we be more