The Continuing Concept of the Platonic-Pythagorean System and Its Application to the Analysis of Fifteenth-Century Music

The Continuing Concept of the Platonic-Pythagorean System and Its Application to the Analysis of Fifteenth-Century Music Author(s): Margaret Vardell Sandresky Source: Music Theory Spectrum, Vol. 1, (Spring, 1979), pp. 107-120 Published by: University of California Press on behalf of the Society for Music Theory Stable URL: http:/www.jstor.org/stable/745782 GS 26 DA x VARDELS, Accessed: 16/07/2008 12:11 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.orp/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www .jstor.org/action/showPublisher?pubtisherCode=ucal. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. http://www.jstor.org

Society for Music Theory The Continuing Concept of the Platonic-Pythagorean System and Its Application to the Analysis of Fifteenth-Century Music Author(s): Margaret Vardell Sandresky Reviewed work(s): Source: Music Theory Spectrum, Vol. 1 (Spring, 1979), pp. 107-120 Published by: University of California Press on behalf of the Society for Music Theory Stable URL: http://www.jstor.org/stable/745782 . Accessed: 06/02/2012 06:40 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. University of California Press and Society for Music Theory are collaborating with JSTOR to digitize, preserve and extend access to Music Theory Spectrum. http://www.jstor.org

of the Continuing Concept its System Platonic-Pythagorean of FifteenthApplication to the Analysis Music Century The by MargaretVardellSandresky In view of the corpus of researchnow available concerning the culturaland intellectuallife of the fifteenth century, it has become possible to investigate in some detail the theory that analytical procedures which are rooted in fifteenth-century thoughtcan greatlyclarify the compositionalpracticesand aesthetic views of quattrocentocomposers. The specific relationships which suggest such an approachcan best be understoodby discussing briefly some examples derived from primary sources. In the well-known woodcut from De Harmonia musicorum instrumentorum1 by the early Renaissance music theorist and humanistFranchinusGafurius, Gafuriusis shown lecturingto his pupils on the "harmoniousdiscord" resultingfrom the two unequal consonances drawn from two dissimilar proportions, 3 : 4 and 4: 6, or perfect fourth and perfect fifth, which to1See ClaudioSartori,"FranchinusGaffurius,"Die Musikin Geschichteund Gegenwart, IV, col. 1238. This woodcut also appearsin an earlier Gafurius treatise,Angelicum opus Musicae, 1508. getherform the octave. The sonnetbelow the pictureinformsus that in order to comprehendthe celestial harmonies,one must know Mercury, Apollo, Orpheus, Aristotle, Plato, and Pythagoras. By the celestial harmonies,the poet refers to Plato's doctrineof the creationof the world as describedin his dialogue Timaeus. In brief, Plato wrotethata MasterWorkmancreateda harmonious universe by placing the planets in their orbits aroundthe earthaccordingto the Pythagoreannumberratiosof the musical intervals. The woodcut furtherdemonstratesthe Renaissance idea of space by showing different uses of the ratios 3 : 4: 6. Their relationto sound is symbolizedby threeorganpipes on the left, to space by threeparallellines on the right,to the universeby the astronomer'sdividersbelow, andto time by the hourglass at the rightof the lectern. Not only is the place of music theory in the quadriviumof the liberal arts here defined, but also the idea is implied that musical pitch, rhythm, and duration all find a commonbond withinthe same ratios, a conceptbasic to Renaissance musical thought. So that the intervalof 4 : 6 as a perfect

fifth is equated with a sesquialter rhythm as well as with a whole section of music in tempus imperfectum in relation to another section in tempus perfectum.2 Because fifteenth-century Platonic-Pythagoreanism was the reinterpretation of a concept which had a long and continuing influence from the time of classical Greece, the channels through which this concept developed need to be sketched. A comparison of the early fourteenth-century concept of the universe which Dante describes in his Divine Comedy and the fifteenth-century idea of the universe as illustrated by a woodcut from the Practica musicae3 by Gafurius illustrates the relationship between these two periods. Gafurius retains the same plan of the spheres of heaven each with its particular planet and muse, and with the same concept of a measurement of their distances that Dante describes in the Paradiso. On the other hand, Dante places Christian saints and the Virgin Mary in the circles nearest to God in the highest realm, while Gafurius in typical humanist fashion places Apollo and the three Graces in the highest heaven of the universe, a concept entirely foreign to Dante. However, the relationship between God and Apollo which is implied in the Gafurius woodcut is in fact a reference to the analogy between early Christian and Greek symbols. This analogy is illustrated by a mosaic from a fourth-century preConstantinian necropolis under St. Peter's Basilica in Rome where Christ is in fact depicted as Apollo, the sun-god in his chariot.4 An interesting medieval French illumination shows God in his role as Plato's Master Workman measuring the universe with his astronomer's dividers, and illustrates the theological impli2This idea is discussed by M. van Crevel in his edition of Jacob Obrecht's Missa Maria Zart, Volume 7 of Obrecht'scomplete works (Amsterdam,1964) particularlyon pages LXXI and LXXVIII. 3There are two translationsof the Practica musicae, one by Clement A. Miller (Dallas:AmericanInstituteof Musicology, 1968) and the otherby Irwin Young (Madison: University of Wisconsin Press, 1969). 4See Jean Lassus, The Early Christianand ByzantineWorld (London:Paul Hamlyn, 1969), p. 3. cations which made possible the Church's acceptance of the liberal arts as a part of Christian thought.5 The twelfth-century West Portal of Chartres Cathedral, showing the cycle of the liberal arts in which Boethius symbolizes music, also demonstrates the Church's recognition of the liberal arts as a means to the knowledge of God. Such iconographic programs in the visual arts of this period are important in illustrating the continuing influence of the Platonic-Pythagorean concept of a musical universe.6 A basic function of the new quattrocento humanistic spirit consisted of a transformation of Platonic-Pythagoreanism from a purely speculative concept into an active practical system. The harmonious ratios of the musical intervals became the basis for a new idea of beauty. They symbolized for Renaissance man the artistic truth of nature as represented by natural law, and they provided artists a means for solving the problem of proportion and perspective. With regard to music, such an empirical process is described by Gafurius in his Practica musicae when he writes: "The function of numbers . . . deals with sounds according to their progression in time. . . . First, in the disposition of sounds by consonant interval, which belongs to the domain of the theorist; and second, in the temporal quality of these sounds according to the numerical value of the notes, a matter ascribed to the active or practical discipline."7 In other words, the music theorist measures the intervals, but the creative artist transforms these sounds into active experience by arranging them within rhythmic and formal space according to the laws of beauty and proportion. The same point of view is also central to the concept of beauty 5See ArthurC. Clarke, "Space and the Spirit of Man," Horizon, January, 1959, p. 26. 6Fora detail of the West Portalof Chartres,see TwelfthCenturyEuropeand the Foundationsof ModernSociety, ed. Clagett,Post, andReynolds (Madison: University of Wisconsin Press, 1966), p. 11. 7FranchinusGafurius,Practica musicae, trans.anded. IrwinYoung, p. 166.

The Platonic-Pythagorean System borrowall ourrulesfor harmonicrelationsfromthe musiciansto whom this kind of numbersis extremely well known." l According to Wittkower'sanalysis, Alberti's design for the facade of Santa Maria Novella in Florence "can be exactly circumscribedby a square.A squareof half the size of the large square defines the relationshipbetween the two storeys. The main storey can be divided into two such squares, while one encloses the upperstorey. In otherwords, the whole building is relatedto its mainpartsin the proportionsof one to two, which is in musical terms an octave."12 Piero della Francesca's painting The Flagellation of Christ dates from approximatelythe same period. KennethClarkhas shown that the basic unit of measureis the black bar above the beardedfigure in the foregroundand that mathematicalratios relate this unit to the ceiling panels, pavement squares, columns, figures, and to the circle in which Christ is bound.13 Marilyn Aronberg Lavin in her monographon this painting came to the conclusion that the abstractdesign of the picture, quite apartfrom the visual representationof figures, "must, in its own time, have formed part of its meaning."14 Such a point of view when translatedinto musical termsleads to the hypothesisthatthe design of a musicalcomposition, quite apart from its surface patternsof pitch and rhythm, possibly formedan integralpartof its meaning. Indeed, such an assumption seems more closely related to musical form than visual design, since it is based on the concept of measuringmusical space. Example 1 illustrates the development of this PlatonicPythagorean geometric-musical space from the ratios of an octave, fifth, andfourthto those basedon the squareandcube of two and three. From these the three mean proportionalsharmonic, arithmetic,and geometric-are derived. If the two series are multipliedby six, then the mean proportionalscan be calculated in terms of whole numbersratherthan fractions.15 From the mean proportionals,the other musical intervals are measuredas shown in Example2. This is a chartof Pythagorean measurementswhich are not stated in terms of vibrationsper second but are in terms of numberrelationships.The multiples of the series based on the squareare all octave pitches and are placed vertically below the letter names of the notes, while the 8See The Architectureof Leon Battista Alberti in Ten Books, trans. James Leoni (London, 1726) reprint(London:J. Rykwert, 1955). 9The firstedition was publishedin 1949. My quotes aretakenfromthe fourth edition (New York: W. W. Norton, 1971). 'Ibid., p. 9. "Ibid., p. 110. '2Ibid., p. 46. 13KennethClark, Piero della Francesca, 2nd edition, London and New York: Phaidon, 1969), p. 35. 14MarilynAronbergLavin, Piero della Francesca: The Flagellation (New York: Viking Press, 1972), p.13. 15Example 1 is derived from Wittkower's summary of this material in ArchitecturalPrinciples, pp. 111, 112; but see also Crevel, p. LXXIV for historicalbackgroundon the use of the numbersix as a multiplier. in the visual arts. An excellent primarysourcefor this viewpoint is provided in the work of Leon BattistaAlberti, the fifteenthcenturyhumanistand architect.Alberti, the first to write about architecturesince the Roman Vitruvius, broughtout his work De re aedificatoria8in 1452. One of Alberti's most perceptive interpretersin moder times has been the scholar Rudolf Wittkower;and from his book ArchitecturalPrinciples in the Age of Humanism, it is possible to reconstructthis relationship between architecturaland musical space which was central to the fifteenth-centuryidea of beauty.9 Wittkowersays thatfor Alberti"music is geometrytranslated into sound, and that in music the very same harmonies are audible which inform the geometry of the building."1' He quotesAlbertias follows: "The numbersby meansof which the agreementof sounds affects our ears with delight, are the same which please our eyes and our minds. . . . We shall therefore

Ex. 1. The mathematicalstructureof Plato's universe in Pythagoreanmeasurement(* geometric mean) 4th 5th 8ve 4 3 2 6 1 2 4 6 12 24 3 9 18 54 Harm. Mean 8 16 32 Arith. Mean 9 18 :36 I M2nd 8* X6 48 27 X6 162 6 6 ': Series 6 12 24 P4th I1p P5th I Serles 12 24 48 P4th P5th P8ve Series 6 18 54 Harm. Mean 9 27 81 Arith. Mean 12 36 108 Series

The Platonic-Pythagorean System Ex. 2 Numbers in parenthesesare the distances between pitches E D B C 6 A G F E 8 12 /16 24 48 48 (6) 54-' 96 96 (12) 108 192 192 (24) 216 (27) 243 < 384e" (48) 432d" (54) 486C (26) 512b' (64) (6 57' 576a' 768e' (96) 864d (108) 972c (52) 1024b (128) 1536e (192) 1728d (216) 1944C (104) 2048B (256) (32) (139) (117) 2187b 288 (36) 324 (72 648729 6729f 1152a (144) 1296g (162) 2304A (288) 2592G (324) 384 (39) 768e' 1458f (78) 1536e

multiples of the series based on the cube are all octave plus fifth pitches and are indicated by arrows.16 The remaining examples present a summary of the PlatonicPythagorean number patterns found in a group of fifteenthcentury cantus firmus Masses by Guillaume Dufay. In order to show the development of the form, these mature Masses of Dufay are introduced by one of the earliest extant cantus firmus Masses, Alma redemptoris mater by Leonel Power. The patterns shown in these examples were derived from a tabulation of the number of semibreves contained in different structural elements of the works.17 The various elements consist of musical materials with cantus firmus underlay compared to those with no cantus firmus, one mensuration compared to another mensuration, or a comparison of the lengths of cantus firmus phrases. The examples imply that the compositional procedure must have been first to outline the total form and its substructure in terms of Pythagorean ratios before actual composition of pitches and rhythms began. The graphs show that the composer must have considered the various elements listed above as separate entities and that an integral part of this pre-compositional process must have consisted in arranging these entities in a pattern of interwoven designs by means of the proportional number ratios. For instance, Example 3 shows that Leonel Power's cantus-firmus-based material extends for forty-eight semibreves in all tempus perfectum sections of not only the Gloria but also of the entire Mass. Furthermore, the non-cantus firmus sections are always the duration of twelve or twenty-four semibreves, and the ratio of these elements is 48 : 12: 48: 24 or 4 : 1 : 2: 1; in Pythagorean terms, the octave and the double octave. These are the proportions Alberti used in his design for the facade of Santa Maria Novella. The structural ideas shown in Example 4, Missa Se laface ay 16Example2 is an extension of the work of Jacquesde Liege as presentedin Crevel, pp. LXXXII and LXXXVIII. 17Intempusimperfectumdiminutum,I equatedthe brevewith a semibreveof tempus perfectum. pale, are very close to Alma redemptoris mater. While Missa Se la face ay pale is a mensuration Mass, the cantus firmus is still regularly broken up by non-cantus firmus material and the entire pattern is either doubly or triply augmented by means of mensural canons. There is, however, some flexibility introduced into the form by means of an occasional difference in the lengths of non-cantus firmus material, and the cadences are sometimes extended. The large design is nevertheless quite clear. 18 In Missa Caput, 19since the cantus firmus was taken from a melisma, it was necessary for the composer to decide on the note values and lengths of phrases. The cantus firmus appears twice in each movement, once in tempus perfectum and once in tempus imperfectum, and it never varies except for two phrases which are omitted in the Agnus dei. While the cantus firmus is broken up in a regular pattern by insertions of non-cantus firmus material, the lengths of this material vary widely from 112 semibreves in the Credo to as few as three. For these reasons, the plan of the work may be more clearly introduced by Example 5, a graph of the cantus firmus alone.20 Here a study of the example reveals an interwoven pattern of Pythagorean ratios (reading both vertically and horizontally) between phrases, groups of phrases, and mensurations. Example 6 illustrates the design for the entire Mass. While the non-cantus firmus material seems to lack structure, it also has its own pattern. The total length of all non-cantus firmus or threevoice writing in all tempus perfectum sections of the Mass is 378 semibreves. These may be divided into three equal parts each containing 126 semibreves. The first part extends from the beginning of the Mass through the first twenty-four semibreves 18Forfurtherstudyof architecturalparallelsin Dufay's work see CharlesW. Warren, "Brunelleschi's Dome and Dufay's Motet," Musical Quarterly, 59 (1973), 92-104. 19Dufay's authorshipof Missa caput has recently been questioned. 20The cantus firmus phrase structure used here agrees with Manfred Bukofzer's analysis in his Studies in Medieval and Renaissance Music (New York: W. W. Norton, 1950), p. 260.

The Platonic-Pythagorean System Ex. 3. Leonel Power, Missa Alma redemptorismater 4: 0 Gloria Credo Sanctus Agnus Totals NCF CF NCF CF NCF CF 108 27 48 48 48 48 12 12 12 12 48 48 48 48 24 24 24 24 48 48 48 48 192 48 192 96 192 CF 0 : ? each mvt. 144: 144 NCF 0 : 4 each mvt. 36 : 32 (9.8) TotalCF 0 : ( 576: 576 Total NCF 0 : ? 144: 128 NCF 36 96 CF NCF CF NCF CF 40 40 40 40 16 16 16 16 76 76 76 76 16 16 16 16 28 28 28 28 160 64 304 64 112 Ratios of irregularNCF: Credo 108 : Sanctus 27 = 4:1 Credo 108 : Credo 36 = 3:1 Credo 36 : Sanctus 27 = 4:3 Agnus 96 = geometric progression from regularCF:NCF = 12:24:48:96 perfectum. The total non-cantus firmus material in all tempus imperfectum sections21 is 448 semibreves. These are divided into two large equal sections of 224 semibreves. The first section extends from the beginning of this mensuration at the Christe through the non-cantus firmus tempus imperfectum sections of the Gloria and through the first fifty-six (a quarter of 224) semibreves of the Credo tempus imperfectum prelude and a clausula vera on C. The second section begins immediately after this cadence with the text "Crucifixus etiam pro nobis" in the superius and includes all non-cantus firmus material in tempus imperfectum from this point to the end of the Mass.22 21Signs for both tempus imperfectumas well as diminutumare found in the sources, and both work out to Pythagoreanratios. I have chosen the former, because the ratio seems more logical and more balanced:1458 : 1296 over 162 where the divider is equal to the difference, as opposedto the diminutumwhich would make the ratio 1456 : 648. For a discussion of this mensurationsign, see AlejandroPlanchart'sedition of the work in Missae caput (New Haven: Yale University, 1964), pp. 159, 160. 22One furtherresult of this analysis is of interest. Because the Kyrie was copied into the choirbooks of Cambraiin 1463, a date which is late for the style of the work, it has been suggested by Bukofzer and others that this movement was written some years later than the other movementsof the Mass. In such a case, the composerwould then have been compelled to returnto his earlierstyle and to imitatethis in orderfor the Mass to be a unified work. I believe that the of the Gloria prelude and a clausula vera on G. The next section of 126 semibreves begins immediately after this cadence and extends through the first eighteen semibreves of the Sanctus prelude ending at the penultimate semibreve to a clausula vera on D. The last section consists of all the non-cantus firmus material from this point to the end of the Agnus dei tempus

Ex. 4. Dufay, Missa Se la face ay pale Kyrie Gloria Credo Sanctus Osan. I Agnus Total 1 2 3 4 5 6 7 8 9 Canon Duplo Triplo Duplo Ut jacet Triplo Duplo Ut jacet Duplo NCF CF A 108 162 108 54 162 108 54 5 54 Cad. 2+6 Cad. 2 CF B 72 108 72 36 108 72 36 Cad. 2+9 2+6 4 2+9 2+6 4 2+9 Total 279 108 972 2+6 33 NCF 78 27 18 9 27 18 9 117 36 36 54 132 561 72 72 648 2+4 2+9 74 Duplo 54 36 18 54 36 18 36 18 270 3+6 2+6 NCF column 2 = 270 Cad. column 4 = 33 NCF column 5 = 561 Total = 864 = pitch d' Cad. columns 6 & 8 = 108 NCF Total = 972 = pitch c 6 2+8* 8* 4 4 34 725 725 510 353 2592 CF A = 972 = pitch c CF B = 648 = pitch g' Ratio A: B = 3: 2 (* fermatain O equals 9; in C, 8) Finally, it should be noted in Example 6 that the ratio of the total number of semibreves between the two mensurations of the cantus firmus is 6: 5 while the ratio of the total number of semibreves between the two mensurations of the entire Mass is 9: 8. Since these are the ratios of the intervals between the first five pitches of the cantus firmus, we are led to the conclusion analysis presented here proves that the Mass was definitely conceived and writtenin its entirety during one time span. that the composer may have derived the large plan for his work from the opening cantus firmus pitches. Missa Ecce ancilla domini-Beata es Maria shown in Example 7 uses two cantus firmi from different sources. Ecce is set in tempus perfectum and Beata in tempus imperfectum diminutum, except for the Kyrie, in which Beata is also set in tempus perfectum. Note in the graph that the lengths of cantus firmus phrases as well as those of non-cantus firmus insertions begin to be treated with more freedom and that the pattern is less

The Platonic-Pythagorean System Ex. 5. Guillaume Dufay (?), Missa Caput. Organizationof cantus firmus in each movement 0 C Al Bl A2 21 18 15 B2 15 A3 21 54 36 14 14 A1 B1 8 14 A2 B2 14 A3 28 32 20 20 [16 '180 Cl 108 C1 36 C2 36 dl d2 E Total 1 15 18 21 216 54 formal. An examination of the music shows that the cantus firmus is not invariably stated in long note values but contains points of embellishment. Every movement ends on a complete authentic cadence in C major with an open fifth tonic. These triads are made up of the tones G, B, D and C, which are also found as Pythagorean numbers in the large sections of the mass. In Missa L'homme arme, the many exceptions to a pattern necessitate a graph of great detail, and I have therefore excluded it from my summary. A study of the music shows that the cantus firmus itself is frequently embellished or extended by melismas which make the tenor assume the quality and shape of the other voices. Missa L'homme arme has been a very difficult work to analyze, and until only recently I believed that my procedures would not apply to this Mass. After all, would a high embellished, free, and irregular cantus firmus be channeled into a neat pattern of number ratios? I have found two designs, one summarized in Example 8a which follows the analytical procedures of the previous Masses. The other, shown in Example 8b, treats the musical materials as a continuum, but separates the three *90< W-< C2 D1 D2 E canons and the section with three simultaneous time signs from the main body of the Mass. In Example 8b the musical events which delineate structural divisions are as follows. In the Kyrie there are 210 semibreves. These are exclusive of the canon consisting of twenty-three semibreves shown in the right hand column. In the Gloria tempus perfectum23 there are seventy-eight semibreves to the close of cantus firmus phrase A1, 144 to the close of phrase C1, and forty-six to the cadence; in the tempus imperfectum section there are 170 semibreves to the beginning of the Amen in the contratenor and seventeen to the close of the movement. In the Credo tempus perfectum section there are 216 semibreves to the close of cantus firmus phrase B3 and the beginning of a duet between the superius and contratenor to the text "Deum verum de deo vero." Then there are thirty-three semibreves to the 23Accordingto Leo Treitler'sanalysis of the cantusfirmus. See "Dufay the Progressive,"Dufay QuincentenaryConference, ed. Allan W. Atlas (Brooklyn College, 1976), p. 115.

Ex. 6. Guillaume Dufay (?), Missa Caput NCF A1 B1 A2 B2 A3 NCF C1 NCF C2 D1 D2 E 12 3 6 12 15 36 36 36 36 36 27 6 9 9 9 36 36 36 36 36 15 15 15 15 15 18 18 18 18 18 21 21 21 21 21 0 0 0 0 8 28 28 28 28 28 40 0 32 12 0 32 32 32 32 0 20 20 20 20 0 20 20 20 20 20 16 16 16 16 16 0 Kyrie Gloria Credo Sanctus Agnus 63 48 57 51 48 21 21 21 21 21 18 18 18 18 18 15 15 15 15 15 15 15 15 15 15 21 21 21 21 21 C Kyrie Gloria Credo Sanctus Agnus 80 76 112 48 40 14 14 14 14 14 14 14 14 14 14 8 8 8 8 8 14 14 14 14 14 CF 0 : C each mvt. = 216 : 180 (6:5) CF plus NCF total for entire Mass equals 1458 : 1296 (9:8) Opening pitches of CF: B6:5D9:8C D B 14 14

The Platonic-Pythagorean System Ex. 7 Guillaume Dufay, Missa Ecce ancilla domini-Beataes Maria Beata Ecce NCF Cad NCF 0 60 48 0 Kyrie Gloria 0 O NCF A 12 30 48 60 Credo 0 0 90 42 48 48 0 0 Sanctus Agnus 0 O 48 18 48 42 6 0 Totals 0 270 264 6 42 A 18 16 NCF B 0 18 8 10 NCF C 39 15 10 0 NCF D 0 21 16 0 E 54 6 Cad 9 3 : 44 16 12 10 0 10 8 16 8 2 11 I 11 ? 46 64 10 10 0 0 10 10 0 0 10 10 0 0 16 16 6 6 3 3 O 18 0 18 39 15 0 21 54 9 ? 52 20 40 0 40 8 64 26 11 B 36 24 NCF 0 0 11 ? 66 9 ? 48 84 24 24 6+12c 0 0 12 84+6c 6 30 30 18 0 330+6c 168 24+12c Cad 54 NCF ? 262 CF Ecce A plus B equals 432 (pitch d") plus final cadences 54 equals 486 (pitch c') NCF Ecce sections O plus cadences equals 648 (pitch g') CF plus NCF Beata (? equals 512 (pitch b') Beata O (Kyrie): E = 54 A + B = 36 intervalbetween 486 and 432 C + D = 36 72 intervalbetween 648 and 576 NCF + cadence = 48 intervalbetween 432 and 384

Ex. 8a. Guillaume Dufay, Missa L'hommearme 0+?t=233 Kyrie 0 Gloria Credo Sanctus Agnus 268 307 422 299 187 18 3 =362 120 Total 1296=g 972=c 70 Total 2268 is not a Pythagoreannumber contratenor change of meter into tempus imperfectum. This is the beginning of the passage shown in the right hand column consisting of three different mensuration signs at once. Next there are fifty-eight semibreves from here to the close. In the Credo tempus imperfectum there are 256 semibreves to the beginning of the canon "Scindite pausas," shown in the right hand column, consisting of eighty-eight semibreves to the end of the movement. In the Sanctus tempus perfectum section there are 149 semibreves to a highly embellished cadence beginning with the word "Sabaoth" in the superius and consisting of eighteen semibreves. The Pleni consists of 144 semibreves and the Benedictus'of 120. In the Osanna II there are seventy-eight semibreves to the beginning of phrase C and 33 to the close. After the Agnus dei I prelude of twelve semibreves, 108 remain. In the Agnus II there are seventy semibreves. In Agnus III the cancrizan canon shown is the right hand column opens this section after which there are sixty-five semibreves to the close of the Mass. Missa Ave regina caelorum carries the development shown in the previous Masses further and is Dufay's last work in this form. My detailed analysis is still incomplete, but the large structure illustrated in Example 9 is shown as the ratio of the three mensurations to each other, 1296 : 1458 : 162. Note the similarity of these proportional ratios to those of Missa Caput in Example 6. In closing I should like to discuss some of the questions which needed to be answered during the analytical process. One of the first which presented itself was whether or not small rests occurring in the cantus firmus were conceived as an integral part of its design. The procedure which I adopted was to count all rests of a breve or less as part of the cantus firmus, with rests of more than a breve carrying enough weight to be considered as non-cantus firmus material. The problem of cadences is twofold. First, there is the question of whether or not to count the cadences as a definite duration, and, second, if they are counted, what the duration should be. While Apel says the duration of the final long "was not considered an exact value,"24 it is a significant fact that number patterns do emerge when cadences are counted as a 24Willi Apel, The Notation of PolyphonicMusic, 2nd edition (Cambridge, Mass., Mediaeval Academy of America: 1944), p. 104.

The Platonic-Pythagorean System Ex. 8b Guillaume Dufay, Missa L'homme arme Kyrie O+? canon: 23 210 288 Gloria 0 432 78 144 46 648 216 Gloria ? Credo O 972 170 17 216 324 *18,0 33 729 Pleni O Benedictus ( Osanna II 0 58 256 149 18 144 120 78 Agnus I 0 33 12 Agnus II ? Agnus III 0 70 65 Totals GrandTotal 2025 2268 Credo 4 Sanctus 0 canon: 88 243 648 486 288 108 243 *See Charles Hamm, A Chronology of the Worksof GuillaumeDufay (Princeton, N.J.: PrincetonUniv. Press, 1964), pp. 144-45. canon: 105 (plus 9 rests)

Ex. 9 Guillaume Dufay, Missa Ave regina caelorum 0 C "3" Total Kyrie Gloria Credo Sanctus Agnus 344 193 333 301 125 268 366 358 192 274 20 14 40 40 48 632 573 731 533 447 Total 1296=g 1458=f 162* 2916=F *162 is the difference between 1296 and 1458 O:C = 1296: 1458 =8:9 162 definite duration. This does not mean that when one adds up a series of perfect longs and compares it to a series of imperfect longs one is bound to come up with a 3 : 2 ratio, because in many instances Dufay extends the cadence in one or two voices. I counted these extensions as additional semibreves to the final long in order to obtain the results presented here. A good illustration of this is shown in Examples 4 and 7, where the charts show cadential lengths in a separate column. In dealing with the duration of the fermata, I followed Charles Warren's research and his citation of De Grocheo as the authority for holding the note under a fermata in tempus perfectum for the duration of a perfect long consisting of nine semibreves.25 However, no reference is made to the meaning of a fermata written over a note in tempus imperfectum, such as one finds in Missa Ave regina caelorum. In this case, the note was counted as an imperfect maxima consisting of eight semibreves. Finally, in setting up a system for analysis, there is always the problem of defining a set of guidelines to which the theorist feels 25CharlesW. Warren,"PunctusOrganiand CantusCoronatusin the Music of Dufay," Dufay QuincentenaryConference, p. 134-36. he can adhere with honesty. When all the evidence is assembled for building principles of construction, there are undoubtedly problem areas and seeming inconsistencies which have to be worked out or perhaps can never be solved. Nevertheless, I feel it is important to work toward developing procedures which reveal a composer's ideas with greater precision and insight, and I hope and believe that the process outlined in this paper is one further step in that direction.