The consonant eleventh and the expansion of the musical tetractys:a study of ancient pytagoreanism

THE CONSONANT 11TH AND THE EXPANSION OF THE MUSICAL TETRACTYS + ANCIENT AND MEDIEVAL HARMONIC THEORY — A STUDY OF ANCIENT PYTHAGOREANISM SO JOURNAL OF MUSIC THEORY, 1984, 28, 2, p 191-223 sg 7 THE CONSONANT ELEVENTH AND THE EXPANSION OF THE MUSICAL TETRACTYS: A STUDY OF ANCIENT PYTHAG 1350 DAR a8 André Barbera The interval of the eleventh, that is, the octave plus fourth, stands at the crease between rationalism and empiricism in ancient musical treatises. For convenience, I refer to the interval as the eleventh, and to its neighbor as the twelfth, although most ancient and medieval treatises refer to these intervals, respectively, as the diapason plus diatessaron and diapason plus diapente. By extracting the eleventh, | perform a biopsy on ancient Pythagoreanism and thus hope to elucidate the Pythagorean world view as well as to indicate some avenues along which ancient Pythagoreanism may have traveled to the Middle Ages.! By tracing the Pythagorean treatment of the eleventh, 1 show the role of number in early musical theory to be both causal and descriptive. The causal nature of number would seem to be more primitive and therefore older than the descriptive role played by number in Pythagorean harmonics. Modern musical theory tends to use numbers as a general, abstract system or metaphor for sound. Thus, our modern disposition invites us to see a development from a time when numbers determined the consonance or dissonance of an interval to a time when numbers simply described an interval, that is, provided a quantitative metaphor without evaluating the interval according to its consonant or dissonant character. The history of mathematics reflects this development

from numerical cause to numericaldescription. By studying the perception of the eleventhin antiquity, we shall see this developmentonly in a very general and sporadic way. In fact, the causal properties of numbersremainedvital for musical theory well beyond antiquity, and some would arguethat this numericalvitality persiststo the present. Mathematically,the most primitivetreatment of the eleventh is the exclusion of the interval from the category of consonancebecause it cannot be generatedor accommodatedby the numericalquaternary1, 2, 3, 4. Here the numbersdeterminethe consonance or dissonanceof the interval. Nevertheless,some of the ancient theorists who treat the eleventh heard it as consonant. Thus their sensory perception confronted their rational beliefs. In some cases, the apparentparadox of rationalism and empiricismis left unreconciled. Some authors recognized the variabilityand uncertaintyof sensoryperceptionandtherefore opted for causal numbers.Others crafted a new numericalrepresentation that relied on the old (?) causal numbers,but one that used these numbers to justify sensory perception. In other words, some treatises synthesize two causal, numerical quaternaries(1, 2, 3, 4 and 6, 8, 9, 12) in order to characterizethe consonant eleventh. By doing so, they take elements of causal number and transformthem into acoustical metaphor. A discussion of the nature of consonanacefalls largely outside the bounds of this investigation.I am concernedprimarilywith the classification of intervalsas consonant or dissonant.One can study this classification from the rationalist'sand empiricist'spoints of view without addressingthe question: what is consonance? Indeed, several ancient treatises eschew the largerphilosophicalquestion in favor of classification. A few remarksabout the perception of consonance,however,are in order. In his De sensu, Aristotle is concerned with sensory perception in general, and in particularwith his argumentthat a single sense cannot perceive two objects simultaneously(447b 10-13).2 In the course of his discussion,Aristotle querieswhethera consonanceis a relation,mixture, or unity. Since in most cases the notion of consonancerequirestwo different pitches, one is tempted to rule out unity, but Aristotle observes that a consonance is like the mixture of two colors, for example,blue and yellow, to form a third (439b26ff). Just as the eyes can see only one color in the mixture, green in this case, so the ears sense one consonance and not two different pitches. In other words, becausethe ears cannot hear two distinct pitches at the same time, they perceivea consonance only if the two sounds have formed a singlemixture.Whether two distinct but dissonant pitches would form a single dissonance is uncertain. There exists some evidence in the musical treatises of antiquity,

especially the Pythagoreanones, that Aristotle's argumentheld true. The discussion in Sectio canonis of the single blend formed by consonant notes seems to stem from the Aristotelianpassage(see below, p. 194). The Sectio stressesthe unity formed from consonantnotes by claiming that the notes must be related by a single name or one term (see n. 14). Furthermore,many ancient treatises-not Sectio canonisuse the phrase "struck at the same time" to refer to the individual pitches making up the consonance.3 This distinction is especiallyimportant in light of the essentialmelodic natureof ancient Greekmusic. Boethius goes one step further when he distinguishesbetween an interval (intervallum),the distance between two pitches struck alternately, and a consonance (consonantia), the mixture of two pitches struck simultaneously.4 Thus most ancient authors seem to associate the notion of consonancewith a harmonicratherthan melodic event.s Finally, one can use the eleventh not only to set in relief issues of rationalism and empiricism but also to record the transmissionof musicalscience. Treatmentsof the eleventhin antiquity and in the early Middle Ages may serve as markersfor the transmissionof theory, indicating especially what did and did not pass from Hellenisticto Latin culturesthroughthe De musicamanuscriptsof Boethius. Boethius writes in the second book of his De musica: "What,then, if we join a fourth and an octave consonance:will they not make-according to the Pythagoreans-anotherconsonance? Not at all."6 Boethius had previouslydefined consonanceas "a mixture of high and low sound sweetly and uniformly falling on the ears."' But here he tells us that the Pythagoreansreject the eleventh because "it falls into the superpartient variety of inequality, and it preservesneitherthe orderof mulIn other words, the tiplicity nor the simplicity of superparticularity."'' Pythagoreans'rejection apparentlyrests not on the degreeof sweetness or uniformityof mixture, but ratheron a rationalisttheory of relations, that is, Pythagorean proportional theory. As is well known, what Boethiushad to say on the matteris quite important,becausehis treatise is the portal through which many treasuresof ancient Greek musical theory flowed to the West. We should not, however,necessarilyascribe the Pythagoreantheory and opinions of the De musicato Beothiushimself. The De musica is certainly derivedfrom an earlierwork or works and in largepart is probablya translationof Nicomachus'On music.9 In earlierchaptersof the De musica and in its companionwork, the De arithmetica, which is a Latin translationof Nicomachus'Introduction to Arithmetic, Boethiusdevelopedthe proportionaltheory that he invokes here to dismiss the eleventh. I base my presentation of this

theory, below, on the work by Nicomachusand on Theon of Smyrna's Expositio. ' There exist five categories of ratios or relations between unequals that are orderedin the following manner(Diagram#1). (1) multiple: the greaterterm contains the lesserexactly some number of times greaterthan one. (2) superparticular:the greater term contains the lesser once plus one part of the lesser.1 (3) superpartient: the greater term contains the lesser once plus more than one part of the lesser. (4) multiple superparticular:the greater term contains the lesser more than once plus one part of the lesser term. (5) multiple superpartient:the greaterterm containsthe lessermore than once plus more than one part of the lesser. Each cateogry also includes a reciprocalrelation, the submultiplefor example, which amounts to a simple restatementof the originalrelationship in the passivevoice. This qualitativeorderof ratiosrepresentsa departurefrom both unity and equality found in the ratio 1:1. or x:x. By relatingany two terms from the quaternaryknown as the tetrad, that is, 1, 2, 3, 4, we see that the resultantratios all fall into the first two categories, multiple and superparticular(Diagram#2).12 This is precisely how the Pythagoreansdefined the numericalcharacterization of musical consonance. The EuclideantreatiseSectio canonis, which is purportedto date from as early as 300 B.C.,'3 states: And we know regardingnotes that some are consonantand some are dissonant, and that [two] consonant [notes] produce a single blend from the two, and dissonant [notes] do not. And so this being the case, it is reasonablethat consonantnotes, sinceboth producea single blend of sound, when relatednumericallyto one anotherby a single 14 name, are either multiple or superparticular. Here we find the connection between the notion of a blend or mixture and categories of ratios. The famous astronomerPtolemy, in his Harmonics, confirms this to be the Pythagorean position on consonance,is and all this appearsin the De musica(1.4-7) of Boethius and thus was transmittedto the MiddleAges. To understandexactly why the eleventh was rejected, let us review the Pythagoreanassociationof consonance with numericalratios. Our earliestauthorityfor this is Philolaus,a Pythagoreanof the fifth century B.C., who states: "The measure of harmonia is the syllaba and the dioxeian... . The syllaba has a sesquitertianratio, the dioxeian a sesquialterratio, andthe diapasona doubleratio."16 Certainlythe terminology used by Philolaus, syllaba for fourth and dioxeian for fifth, is

All ratiosare reducedto their lowest terms. NAME ALGEBRAIC FORM (1) multiple nx:x,n > 1 (2) superparticular (x + 1): x, x > 1 (3) superpartient (x + m):x, x > m > 1 (4) multiplesuperparticular (nx + 1):x, x > 1 and n > 1 (5) multiplesuperpartient (nx + m):x, x > m > 1 and n > 1 Diagram1. PythagoreanProportionalThe

tetrad= 1,2, 3, 41 2:1=octave multiple ratios 3:1 = octave + fifth ( "twelfth") 4:1 = double octave 3:2 = fifth ratios superparticular 4:3 = fourth Diagram2

striking.These remarksserve not only as the earliestnumericalcharacterization of musical intervalsin the West,but also as the earliestgrammatical or linguisticrepresentationof music.17 Returningto the issue at hand, we can compose an eleventh from Philolaus'remarks.The sum of 4:3 plus 2:1-one "adds"ratiosby treatingthem as fractionsand multiplying-holds the ratio 8:3. Boethius composes an eleventh using 8:6 and 6:3, arrivingalso at 8:3.18 fourth + octave = eleventh 4:3 + 2:1 : : 8:3 Boethius,De musica,2.27 fourth + octave = eleventh 8:6 + 6:3 :: 8:3 Thus the eleventh is a multiple superpartient,the category of ratio recognizedby the Pythagoreansas the furthestremovedfrom unity and equality. In addition to the quality of ratio involved,the Pythagoreans rejected the eleventh for another reason, that is, because the ratio involved the number 8. Orthodox Pythagorean theory recognizes five consonances: fourth, fifth, octave, twelfth, and double octave; and these are representedby the multiple and superparticularatiosderived from the tetrad (Diagram#2). The number8 obviouslydoes not belong to the tetrad. On a much more generallevel, quaternariessuch as the tetradplayed a role in Pythagoreanismthat equalsand perhapssurpassesthat of trinities in Christianity.Severalauthors from aroundthe second and third centuries A.D. inform us that the quaternary1, 2, 3, 4 assumedthe position of an oath in Pythagoreancircles.The Pythagoreansnamedthe number 10 tetractys because it was the sum of the first four numbers. The most common oath associatedwith the tetractyswas: Nay, [I swear] by him who bequeathedto our soul the tetractys, Source containingthe root of everlastingnature.19 Anotheroath, transmittedby Iamblichus,states: Whatis the oracle at Delphi?The tetractys; that is, the harmonyin which the Sirenssing.20 The Sirensmust be those of Plato'smyth of Er (Republic 616Bff). Authors of late antiquity specify that the tetractys referredto in these oaths is the quaternary1, 2, 3, 4. Severalcenturies earlier,Aristotle cited Plato'sDiscourseson Philosophy (lost) as the source for connecting reason (vokc) with unity, science (&taor•l'a)with two, opinion (66'a) with the number of a plane surface, and sensory perception

The authorof the Theologou(aoOrlow) with the number of a solid.21' mena arithmeticae,Nicomachus,lamblichus,or both, who may be quoting from Speusippus(ca. 407-339 B.C.), makes overt the connection between the tetrad and spatial dimensions (from none to three): one with the point, two with the line, three with the triangle,and four with the tetrahedron.22Thus the venerationof the tetrad, so common in late antiquity, stretches back at least to Plato's Academyif not to an earlier time. Besidesthe tetrad, ancientwritersconcernedthemselveswith the important quaternariesof the four elements (fire, air, water, earth) and the four geometric figuresassociatedwith these elements(tetrahedron, octahedron, icosahedron, cube). Theon lists eleven such quaternaries, using the word tetractys to refer to them,23 and to these should be addedthe four mathematicalsciences.Archytas,a Pythagoreanscientist and politician of the fourth century B.C., was the first to specify the four mathematicalsciences.24Severalcenturieslater, lamblichussanctified this organizationof science by quoting from Pythagoras:"And indeed there are four stepsin set orderto ascendto wisdom:(1) arithmetic, (2) music, (3) geometry, (4) spherics."2s Finally, Boethius christened this quaternaryquadruviumat the beginningof his De arithmetica.26 Withinthe realmof music, one can easily find additionalquaternaries such as the tetrachord-the basic system or arrangementof notes in ancient Greekmusic-and the helicon-a four-stringdevice used to test intervals.Accordingto the Pythagoreans,however,the most important musical quaternarybesides the tetrad was the musicaltetractys 6, 8, 9, 12. To appreciatethe importanceof this quaternary,let us delve a bit more into Pythagoreanarithmetic. In additionto rations,which involvetwo terms, the Pythagoreans,in their discussions of sound, employed numericalmeans, which require three terms. Archytas tells us that there are three means peculiar to music (Diagram#3).27 (1) The arithmetic mean between two terms is such that the first term exceeds or falls short of the mean by the same amount that the mean exceeds or falls short of the second term. (2) The geometric mean between two terms is such that the first term is related to the mean in the same way as the mean is related to the second term. (3) The harmonicmean between two termsis such that by whatever part of itself the first term exceeds or falls short of the mean, the mean exceeds or falls short of the second term by the same part of the second. The connection between the consonant ratios of the tetrad and the musical means of Archytasoccurs in the musical tetractys 6, 8, 9, 12,

Supposex > z. (1) y is an arithmeticmean between x and z if: x - y = y - z. Example:x =3, y =2, z = 1 (2) y is a geometricmean between x and z if: Example:x=4, y=2, z= 1 (3) y is a harmonicmean between x and z if: Example:x = 6, y = 4, z = 3 3- 2=2 - 1 x + y = y - z. 4 2 =2 + 1 (x - y) + x = (y - z) + z. (6 - 4) + 6 = (4 - 3) + 3 Diagram3. MusicalMeans

for this quaternary embodies not only three of the five consonant ratios,but also two of the three musicalmeans. 12:6 holds the duple ratio, 2:1. 12:8 and 9:6 hold the sesquialterratio, 3:2. 12:9 and 8:6 hold the sesquitertianratio, 4:3. 9 is the arithmeticmean between 6 and 12, and 8 is the harmonic mean. Finally, the fournumbersembracea geometricproportionor analogy, 12:9::8:6. There exists evidence that this quaternarymay also antedate the Academy. Iamblichusinforms us that Pythagorashimself brought the ultimate proportion(rcXe1wrd',drt vaXoyia),12:9:: 8:6, from Babylonia to Greece, and this proportion, called (mousikE),28contains the same consonant ratios as those mentioned by Philolaus.29The Pythagoreans added together the numbersof 6, 8, 9, 12, as they had 1, 2, 3, 4, deriving special numerological significancefrom the sum. We read in the "Excerptaex Nicomacho": The first tetractys, even the root of these tetrachords,is in a way the source of all the divisionsby genera,perfecting,as I said, the number 36 in conformity with the addition of the stable monad.30 Here the sum of the four numbers6, 8, 9, 12 is 35, but the authoradds the monad, makingthe sum 36, which is the squareof the first perfect number, 6. Rather than adding the monad, Aristides Quintilianus multiplies the sum 35 by the perfect number 6, obtaining 210, which he observesto be the gestation period in days for seven-monthfetuses. Similarly, 35 plus the tetractys 10 equals 45, and Aristidesmultiplies this sum by 6, obtaining 270, the gestation period of nine-month fetuses."' Ancient lore held that seven-and nine-monthfetuses survived birth whereaseight-monthfetuses did not. The tetractys called mousikUappearsin very many musicaltreatises of late antiquity and the Middle Ages, often in the form of a delightful myth that has Pythagorasdiscoverthe numericalratiosby investigating the weights of hammersused in a forge. Recordedfor the first time during the second century A.D., the myth exists in many versions, all of which contain an acoustical fallacy.32 Without actually relating the myth, Ptolemy observed that the substances and methods involvedhammersand suspendedweights-were unreliable.3"That the acoustical experiments of the myth did not work out must have only enhanced the perception of Pythagorasas magical.After all, the experimentsdid work when he performedthem. Boethius recounts the myth and also providesa discussionof why the Pythagoreansrejectedthe eleventhfromthe categoryof consonance.

Boethius underscoreshis role as reporter, not theoretician, here by noting that Nicomachushad much to say about this interval(no such remarks exist in the extant writings of Nicomachus), and Boethius distinguishes the Pythagorean position of Nicomachus from that of Ptolemy (see below).34 Thus the Pythagoreanposition on the matter was passed on to the Middle Ages. All this would not amount to very much for medieval theorists or for us if there were not a sense and a theory that judged the eleventhto be consonant. Such a theory of consonance was developed by Ptolemy in his Harmonics,where he established four categories of intervals:homophonic, symphonic, emmelic, and ekmelic. He states: "We define homophones as the [intervals] which, taken at the same time, producea singleimpressionon the hearing."asIn this category he includes the octave and the double octave. Ptolemy'sposition here is similarto the one taken in the AristotelianProblems(918a78), where the lower note of the octave intervalis said to contain the higher.Ptolemy continues: Next after homophones come symphones such as the fifth and the fourth, and they are put together from themselvesand from homophones. Next after symphones come emmelic [intervals] such as whole tones and all the rest of such kind. Just as homophones are put together from symphones, so are symphones from emmelic [intervals].36 Later Ptolemy tells us: "The emmelic are those of the superparticulars after the 4:3." 7 All other intervalsare ekmelic. Ptolemy's remarks here are confusing, especially when he tells us that symphonesareput togetherfromthemselvesandfromhomophones, and then a sentence later, that they are composedof emmelicintervals. Ptolemy's intentions, however, are clear. Of the four categories of intervals, three concern melody: homophonic, symphonic, and emmelic. Of these three categories,two are consonant: homophonic and symphonic. Ptolemy wants to construct a system in which the smallest melodic intervals combine to form larger intervals. In particular, emmelic intervals such as 10:9, 11:10, and 12:11 combine to form symphonic intervals such as 4:3. Symphones such as 4:3 and 3:2 in turn combine to form homophones such as 2:1 (Diagram#4). At the same time, Ptolemy wants to classify as consonant those intervalsthat are formed by joining the octave to a consonance, a classification that had been established several centuries earlier by Aristoxenus.38 Thus we read that symphones are composed from emmelicintervalsas well as from themselvesand homophones. Ptolemy stated in his definition that some symphonic intervalsare composed of homophones and symphones, and this is the category,

L v-liv 10:9 11:10 LI~V I C 9:8 12:11 Ilv 10:9 12 11:10 Lv---L---I 4:3 3:2 Diagram4. Ptolemy,Harmonics,1.7

symphonic, into which the eleventh and the twelfth-octave plus fifthfall. Ptolemy observes: In generalthe consonance of the octave-since the notes do not distinguish is in function from a single note-, when added to some other [interval], preservesthe form of that intervalunaltered,just as the decad does, that is, when addedto other numbers.39 In other words, if we add the number 10 to some other number,let us say 2, the 2 is preservedin the result, 12.40 From this theoreticalstandpoint,Ptolemycriticizesthe Pythagoreans for their rejection of the eleventhfrom the categoryof consonance.He maintains that they allow their rationalism,based on the tetrad and qualitativeproportional theory, to interfere with sensory perception. Accordingto Ptolemy, the eleventhsoundsconsonantbecausethe fourth is consonant-all ancient theorists agree on this interval-and because the octave, when combined with the fourth, preservesthe sound of the fourth. We have here, then, the elements of what appearsto be a confrontation between rationalism and empiricism. In addition to the orthodox Pythagoreanview regardingthe eleventh, Boethius transmits Ptolemy's argumentthat the eleventhis consonant.Thusthe MiddleAges had both points of view from whichto choose, andthe compositenature of Boethius'De musicareflects the equivocalstatusof the eleventh.41 Although one can clearlyseparatethe two positions on the eleventh, one cannot divide up the ancient authors neatly into Pythagoreanand Ptolemaic camps.For instance,some Pythagoreanauthorssuch as Nicomachus and lamblichus do not mention the eleventh at all. Neitherdo some Latin authorssuch as Censorinusand Macrobius,who areworking from a mixture, if not a synthesis, of Pythagoreanand Aristoxenian musicaltheory. We have seen that Ptolemy and Boethiusare quite specific in ascribing the rejection of the eleventh to the Pythagoreans.Plutarchin his On the E(psilon) at Delphi, written aroundthe turn of the second century and thus before the treatises of either Ptolemy or Boethius, also rejectsthe eleventhexplicitly, althoughhe neithercites the Pythagoreans nor specifies the mathematical principles that underlie his rejection. Plutarchstates (389D-E): For the primary business of harmonics, so far as it can be said in words, is concerned with consonances.Common sense makes clear to anyone who wishes to seek these things on stringsand boringsby means of sensory perception in lieu of reason, that there are five [consonances] and no more; for they all originatefrom numerical ratios. And the ratio of the fourth is sesquitertian,that of the fifth is sesquialter,that of the octave is duple, that of the octave plus fifth is

triple, and that of the double octave is quadruple.And that which the harmonistsintroduce, namingit the octave plus fourth, extending beyond measure,is not worthy of acceptance,becauseby doing so we would be favoringirrationalhearingover-as it were-rational law.42 Plutarch'sposition is Platonic, that is, a priori mathematical.From a purely mathematicalstandpoint,one adoptedby Socrates(Plato) in the Republic, there is no need to discussthe eleventh, and it is noteworthy that Plato does not do so, not even in the Timaeus.Only if the subject matter is real sound, as was the case with the Pythagoreans,would the eleventhbe a suitabletopic for discussion.43 There are several ancient authorsbesides Ptolemy who classify the eleventh as consonant.44Of these, Theon is the most exasperating;for he tells us that consonancesare numericallyrelated accordingto muland that: "The tetractys [1, 2, 3, 4] tiples and superparticulars,45 encompassesall the consonances."46But Theonalso claimsthat by joining the octave to other consonances, one producesnew consonances, and as one of his exampleshe givesthe eleventh.47 We have seen alreadythat Theon presentsseveralquaternariesin his Expositio, one of which is mousiki, 6, 8, 9, 12. After this, he observes that 8:3 representsthe eleventh and 3:1 the twelfth,48 and then he incorporates,partially,the two musicalquaternaries1, 2, 3, 4 and 6, 8, 9, 12 (Diagram#5).49 He does this by doublingthe 12, thereby obtaining 24. He notes that 24:6 representsthe double octave, that is, a quadrupleratio, andthen he proceedsto the triple ratio, 24:8 or 18:6. Theon notes further that the twelfth plus a fourth yields the double octave, and he gives two numerical characterizationsof this composition. He neglects, however, to discuss the eleventh in this passage, and so does Chalcidius,who, in his commentaryon Plato's Timaeus, follows Theon.o5 ChargingTheon with inconsistency here does little to enlighten us about the classificationof consonancesand their numericalrepresentations. With Theon's Expositio, we may have severalhistoricallayersof Pythagoreanharmonics, which are inconsistent only when they are comprehended simultaneously. For instance, the tetractys 1, 2, 3, 4 may very well have delimited the numerical characterizationof consonant intervals at one time. It is possible that from this quaternary came the generalrule stated in Sectio canonis about multiple and superparticularratios. In a different acoustical realm, and probably at a different time, Pythagoreansdirected their attention to the incomposite consonancesof the fourth and the fifth, their compositionto form the octave, and their numerical representationin the musical tetractys 6, 8, 9, 12. From this state of musical theory, one can easily imaginethe

fourth twelfth iI1 6 8 9 12 16 twelfth double octave twelfth + fourth = double octave 24:8 + 8:6 :: 24:6 18:6 + 24:18 :: 24:6 Diagram5. Theon of Smyrna,Expositio, H

acoustical experiments that would cause considerationof composite intervals such as the octave plus fourth, octave plus fifth, and double octave. Two of these intervalswere representedalreadyby the tetractys 1, 2, 3, 4, leaving the status and perception of the octave plus fourth open to dispute and neglect. I turn now to Gaudentius,an ancient author about whom we know virtually nothing except that he lived sometime between the first and sixth centuries A.D. His Introduction to Harmonics is a remarkably sectional treatise, the first nine chaptersbeing devoted to Aristoxenian theory, the next eightto Pythagoreantheory, and the last five to species of consonances and to notation. In the Aristoxenianportion of the treatise, we read that it is through hearing alone that one recognizes consonance and dissonance, and reason will be of no help in making such judgments.51The tone of the treatise changes dramaticallyat chapter ten where the author takes up Pythagoreantheory. In this second section we are treated again to the myth about Pythagorasat the forge, although Gaudentiusinjects an equivocalnote regardingthe validity of these experiments."But being dissatisfiedwith this attempt alone [hammersand suspendedweights], he [Pythagoras] tested the method in another way."s2 This other way was, of course, by measuring lengthsof stringand was, therefore,acousticallysound. The first Pythagoreanchapter is the most remarkableof all, because here the author not only shifts abruptly from Aristoxenianto Pythagorean theory, but also expandsthe musicaltetractys 6, 8, 9, 12 in order to accommodatethe five consonancesof the tetractys 1, 2, 3, 4 as well as the eleventh.Weread: Ratios of consonancesare found in numbersand are tested in every way, of which [consonances]: the fourth is sesquitertian,which is 24:18; the fifth is sesquialter,which is 24:16; the octave is double, which is 24:12; the octave together with the fourth is multiple superbipartient,which is 24:9. And again the octave plus fifth is triple, which is 24:8; the double octave is quadruple, which is 24:6.53 24:18 = fourth 24:16 = fifth 24:12 = octave 24:9 = eleventh 24:8 = twelfth 24:6 = double octave Thus Gaudentiusincorporates,or synthesizes the tetractys 1, 2, 3, 4 with the one called mousike and transmittedby the myth. In so doing, he createsa muscialheptactysthat representsnumericallythe consonant eleventh,thereby abrogatingthe old Pythagoreanrule regardingmultiple and superparticularratios. Such a synthesisstraddlesthe fence between rationalism and empiricism, and pricks the modern sense of logical

theory. By admitting the eleventh into the category of consonance, Gaudentiusapparentlyacknowledgesthe empiricalfact of the eleventh's consonance. By creating the numericalheptactys that accommodates the eleventh, he providesa numericalmetaphorfor apparentacoustical truth. In so doing, he relies on those same numericalquaternariesthat had excluded the eleventh by their causal nature. Thus the characterization of the consonant eleventh as 24:9 indicates the demotion of numericalratios from cause to metaphorin Pythagoreanharmonics. One might wonder now if the synthesis of Gaudentiuswas passed on to the Middle Ages. If it was, Boethius did not do it, nor did Martianus Capella. Boethius relies on Pythagoreanproportionaltheory, not quaternaries,when he classifies the eleventh as dissonant, and at no place in the extensive De musica does he effect a synthesis of the two quaternaries1, 2, 3, 4 and 6, 8, 9, 12. The De nuptiis philologiae et mercurii of Martianus,which draws its discussion of music in part from the De musica of AristidesQuintilianus,servesas another source for the transmissionof Greek musical theory to the MiddleAges. In the De nuptiis, Martianustakes up the matter of the eleventh and recognizes it as consonant, but like Boethius he does not treat the intervalin terms of musicalquaternaries.54 Besides Gaudentius,I am awareof only one authorin late antiquity, Cassiodorus,who uses the number 24 in an attempt to characterizethe eleventh. Cassiodorusis also the only authorin all of antiquity to mention Gaudentiusby name, citing him as an authority on music.ss And the eleventh? Cassiodorusjudges it to be consonant and lists all six consonanceswith their characteristicratios.56 The fourth is 4:3. The fifth is 3:2. The octave is 2:1. The eleventhis 24:8. The twelfth is 3:1. The double octave is 4:1. Of course, the eleventh is 24:9, not 24:8. If we can trust the manuscript tradition of the Institutiones, Cassiodorushas apparentlymade this intervaltoo consonant by renderingit as a triple ratio. 7 It is noteworthy that Cassiodorusrepresents the other five consonances with ratios drawn from the tetrad, that is, with the smallestintegerscapable of expressingsuch ratios. In the case of the eleventh, however, he has chosen 24:9, or 24:8, ratherthan 8:3, or even 16:6. Ratherthan choosing the smallest integers capable of representingthe eleventh, Cassiodorus has chosen a ratio that involvesthe number24. As we have seen already, Cassiodorusdid not get this representationof the eleventh from Boethius' De musica, at least not in the form that we now know the treatise.Thereare,however,numerousconnectionsbetween Cassiodorus and Boethius, although all of these may have been superficial.

Both Cassiodorusand Boethius were members of the Roman government under Theodoric, the Ostrogoth king, and from Cassiodorus' letters we learn that Boethius' services were highly valued for a time by Theodoric.58Cassiodorustells us also that Boethiustranslatedworks on: arithmeticby Nicomachus,the De arithmetica;geometryby Euclid, presumablylost; astronomyby Ptolemy, presumablylost; and music by Pythagoras,presumablythe De musica.s9Thusthere exists the possibility that Cassiodorusderived some of his information about musical science from writingsand translationsof Boethiusthat no longerexist.60 Of all the extant ancient sources,however,only Gaudentius,in his synthesis of the two musicalquaternaries,uses the number24 to represent the eleventh. The severalcenturiesthat we call medievalcomprisesignificantmusical creativity as well as musical adoption and adaptation.FromMusica disciplina and Musica enchiriadisto the treatises of Jacquesde Liege and Marchetto da Padova,the musicologisttries to piece together the musical history of these centuries, although frequently lacking all the pieces to do so. Commensuratewith the wide span of time and profound musical developments of the MiddleAges, the musical treatises of this period are variedboth in scope and in purpose.61My intent here is not to tackle the entire rangeof medievalmusicaltheory, nor even to trace the treatment of the eleventh throughout this period, although such a trophy hunt might be the kind of study that ultimately would facilitate the writing of a history of medieval music theory-not an imminent achievement.Ratherthan these ambitiousprojects,I confine my attention here to the appearanceand treatmentof the eleventhin a few musical treatises or manuscripttraditionsof the Middle Ages. The treatment of the eleventh in these works proves to be of little philosophical interest. A study of the eleventh in these treatises,however, may shed some light on the transmissionof musical science from late antiquity to the MiddleAges. My investigation,primarilytheoreticalor conceptual,cannotreplacethe moretangibleworkof manuscriptstudies, but rathermay serveas a guide to subsequentwork with primarysources copied duringthose centuries. The matter of the eleventh appearsin many medievalmanuscripts. A perusalof MartinGerbert'sScriptoresproducesno fewer than fifteen "treatises"that discussthe matter.62 Thesediscussionsvarywidely from mere mention of the intervalby Aurelianand Reginoto extensivetreatment in Musica enchiriadis.Some authorsnever mention the eleventh, and the extent to which the eleventhis discussedin a treatisemay be a function of the author'sintention for writingthe treatise.Thus Guido,

who is concerned primarilywith teaching the singingof chant, makes no mention of the matter, althoughhe does transmitthe Pythagorean myth of the hammersat the end of the Micrologus.63 The mixture in Guido's writings of Pythagoreanharmonic science with the practical, cantus tradition connected to the church marks a middle period in medieval musical theory. The synthesis of music as skill (peritia) with music as knowledge (scientia) had its roots at least in the reappearance on the continent of Boethius'De arithmeticaandDe musicaduringthe ninth century.64I show below that the two works by Boethiusplus the De nuptiis of Martianusas we know them do not account entirely for the transmissionof scientia to the MiddleAges. The Epistola de harmonica assigned to Regino treats the eleventh in cursory fashion, rejects it from the category of consonance, and repeats the standardPythagoreanargumentreceivedthroughBoethius.65 Hucbald,on the contrary,identifiessix consonancesin his De harmonica institutione. In other words, he classifiesthe eleventhas consonant,although he providesno specialjustificationfor his classification.Hucbald adopts the notion of simpleand compoundconsonancesfrom Boethius' De musica (5.12), which in turnderivedthe classificationof consonance from Ptolemy's Harmonics (1.7). But whereas Ptolemy and Boethius divide the six consonancesinto three groups(octave and double octave, fourth and fifth, eleventh and twelfth), Hucbald divides the six into two groups. The simple consonancesare the fourth, fifth, and octave. The compound consonances are the eleventh, twelfth, and double octave.66 By the time of Hucbald,that is, the early tenth century,the De musica of Boethius not only had reappearedon the continent but also had permeatedthe musical theory of the time.67 The treatment accorded the eleventh in three treatises preceding the De harmonica institutione, that is, Musica disciplina,Musica and Scolica enchiriadis, provides us with a more perplexingpicture of the eleventh's status, a picture that appearsto depend less than wholly on the De musica of Boethius. With the Musica disciplinaof Aurelian,mid-ninthcentury, we have an early confluence of musicaltheoriesandtraditions.As is well known, the Musica disciplina is devoted primarilyto the understandingand classificationof the cantus tradition.In these endeavors,Aurelianrelies on modal classification from Byzantium, and indeed his work is the first theoretical treatmentof such kind in the West.68 Aurelianalso presents the scientific theory of music that he had learned from Boethius and apparentlyfrom Cassiodorus.Musicadisciplinais the first medieval treatise to quote substantialportions of Boethius'De musica. Wehave earlierreferencesto the De musica, however, by Remigiusof Auxerre, and the De arithmeticaof Boethius was discussedby medievalwriters at a time no later than the beginningof the ninth century.69

In the sixth chapter of Musica disciplina,Aurelianappearsto vacillate between five and six consonances. He states: "Music, moreover, consists of all six consonances,fifteen sounds, and eight tones."70He follows this with a list that includes the eleventh. Who is Aurelian's source here? A few sentenceslater, Aurelianrefersto Boethius,lists the consonanceswith their characteristicratios,but omits the eleventh.Unless Aurelianhad been relyingon the fifth book of Boethius'De musica -and he certainlywas not-, there is no reason to expect a numerical treatment of the eleventh as a consonanceat this point. Upon concluding his forayinto Pythagoreanarithmetic,however,Aurelianagainrefers to the six consonances.The six consonances,I think, come ultimately from Cassiodorus,the five from Boethius. I have shown alreadythat Cassiodorusdid not deriveall of his musicalscience from the De musica of Boethius, but rather relied in part on the lost Latin translationof Gaudentius' Introduction to Harmonics. Therefore, when Aurelian classifiesthe eleventh as consonant,he is relyingultimately on Gaudentius and not on the fifth book of Boethius'sDe musica, which in turn was a loose translationof the first book of Ptolemy'sHarmonics. I turn now to the perplexingMusica enchiriadisand its equallyperplexing companion work, the Scolica enchiriadis. With the Musica enchiriadis we find a self-conscious mixture of the cantus tradition with the harmonic science of antiquity." We find also the famous early discussion of organum, and one interval at which the treatise postulates organumis the octave plus fourth. The treatise even refers to the intervalas the eleventh (undecimum).72As for the statusof the eleventh, the author of our treatise determinesit to be consonant and cites Ptolemy as the ultimate authority on the matter. In fact, Musica enchiriadisrefers specifically to Ptolemy's fifth book on music, quotes from the fifth book of Boethius'De musica (ch. 9), and concludesthe quotation with a citation of Boethius: "HaecquidemBoethius."73This portion of Boethius'treatiseis explicitly based on Ptolemy'sHarmonics (1.7.15-16). Although there exist a few soft spots in the quotation from Boethius,74we can safely conclude that Boethius is indeed the source for the argument presented in Musica enchiriadis. Thus the Musica enchiriadis claims that the eleventh sounds the same as the fourth because the octave preservesany consonance with which it is joined, just as does the decad when joined to another number.7s Furthermore,one of the diagramsthat accompaniesthe eleventh chapter of Musica enchiriadispresents the two-octave system and a poem that refers to intervalsof the system. Regardingthe eleventh we read: "Takecare so that Ptolemy's dictum prevailshere."76 The Scolica enchiriadisis divided into three parts,the first of which is devoted to the cantus tradition as representedin the companion Musicaenchiriadis.The secondsectionintroducessomeancientharmonic

science and the third, even more. The second partbeginswith a division of the six consonancesinto two groups, simple and compound.77This classificationis the same as that found later in Hucbald'sDe harmonica institutione, which in turn was an adaptationor distortionof the classification found in Ptolemy'sHarmonicsand transmittedby Boethius.As this section of the Scolica proceeds, we find relianceon the De arithmetica of Beothius78and on the Institutiones of Cassiodorus.79The Scolica then expands the musical tetractys in the same way that Gaudentius did, and discussesall the possiblenumericalcharacterizationsof the six consonances(Diagram#6),80 referringto the intervalin question as the eleventh (undecima). One must wonderwhetherthese numerical characterizationsare originalwith the Scolica, whether they represent an elaborationand correction of those characterizationsfound in Cassidorus' Institutiones (see above, p. 207), or whether they are based on yet other sources. In the thirdsection of the Scolica, the authortells us: "Theprinciples are 1, 2, 3, 4; for the numericalquaternaryrelates completely all the consonances."81 Again the tetractys is expandedto 24, and even to 48, but no mention is made of the eleventh. We have seen this apparent contradictionor vacillationbetween five and six consonancesbefore in Musica disciplina. But the Middle Ages inherited the problem from antiquity, although they did not do so from the works of Boethiusthat we know. Although the De musica of Boethius presents both the Pythagorean position (five consonances) and the Ptolemaic (six consonances), the treatise clearly distinguishesbetween the two positions. The same cannot be said for Theon's Expositio (see above, p. 204) nor for Musicadisciplinaor Scolica enchiriadis. That the number of and numerical representationof consonances were confusing to the medieval theorists, as they are to us who try to read the writingsof these theorists, is exemplifed by Gerbert'sAnonymous I. The author of the manuscriptexpand the tetractys to 24 but replaces 16 with 14. Thus the characterizationof the eleventh-if we can trust Gerbert'sedition-becomes 14:6, which the author labels a "double sesquitertian"ratio (duplex sesquitertia [sic]) rather than a duple plus sesquitertian.82 The consonant nature of the eleventh and the interval'snumerical characterizationattracted some attention throughout the later Middle Ages. For most of the later authorswe can discerntheir sources-most often, Boethius-and thus the eleventh no longer servesas a guide to the transmissionof musical science.83Byzantine theorists also devoted some attention to the eleventh and its numericalcharacterizationin their revivalof ancient mathematicalthought, although they relied, of course, on the ancient Greeksourcesratherthan on Boethius.a4 In contrastto early medievaltheoristsin the West,some later writers

24:9 eleventh fourth 16:12 24:18 ) 9:6 18:6 12:8 24:8 twelfth 18:12 i fifth 24:16 24:6 12:6 octave ) double octave Diagram6. Scolica enchiriadis,S.2.202ff

mounted original philosophical argumentsagainst the Ptolemaic position. WalterOdingtonattacks,in particular,Ptolemy's analogybetween the diapasonand the decad.85Relyingon Boethius'contributionto the analogy (see n. 40), Odington observesthat the 2 is indeed preserved when added with 10 to form 12, but he counters that intervals are properly representedby ratios, not by numbers.Odingtongives as an example the ratio 10:5. He notes that this is duple and thus consonant. By adding 2 to the 10 in this ratio, one produces 12:5, a superpartient ratio, which is, of course, dissonant. Jacques de Liege takes a slightly different tack than Walter."6He arguesthat the value of an intervalmust be judged accordingto its extremes, and not accordingto its constituentparts.Thejustificationfor this eminently Pythagorean position stems from Jacques's refusal to allot special status to the octave. Jacquesnotes that two stringssounding the octave do not sound like one string,thus denyinga basic part of Ptolemy's argument.If there is not a special category of consonance, such as Ptolemy's homophones, then there is no point in dealingwith special combinations of consonant intervals, such as the octave plus fourth. Noting that the octave, representedby 2:1, is indeed consonant, Jacques observes that such an interval can be composed from the intervals8:7 and 7:4, neither of which is consonant.Just as one does not judge the octave according to the parts 8:7 and 7:4, one should not judge the eleventhaccordingto the octave plus fourth. Ratherthan this, the eleventh should be judged by its extremes, that is, 8:3, or 16:6, or 24:9. Each of these ratios, of course, is a multiple superbipartientand thereforerepresentsa dissonantinterval. The evidence that I have presentedhere covers a wide span of time, from the Presocraticsto the late MiddleAges, and is largelyconceptual or theoretical ratherthan corporeal.The embodimentof these theories in manuscriptscovers a much smallerspan of time, beginningwith the ninth century and extending to the late Middle Ages. Although the classificationof intervalsis a ratherairy topic, I believe that the scope of my study is sufficiently narrowso as to shed light on the perception of consonance and also, perhaps, to provide some clues regardingthe transmissionof musicalthought. Wehaveseen that some ancient authors make a synthesisof the two musicalquaternariesthat lie at the heart of Pythagorean theory. Theon expands the musical tetractys and thus represents the quadruple ratio 24:6 and the triple 24:8. Gaudentius goes one step further and representsthe eleventh, 24:9. It seems more than likely, furthermore,that Cassiodorusknew of this expansion. The major sources of Greekmusicaltheory for the medievalwriters,

Boethius' De musica and Martianus'De nuptiis, treat the eleventh,but not in the context of the musical tetractys 6, 8, 9, 12.87 And yet we find medievaltreatises, especially the Scolica enchiriadis,that treat the eleventh in the same fashion as did Gaudentius.Now the representation of the eleventh as 24:9 is not a difficult arithmeticformulation,and any authorcould havehappenedupon it. But why use 24:9 to represent the eleventh ratherthan 8:3 or 16:6? The numericalheptactys that we find in ancient and medievaltreatises, 6, 8, 9, 12, 16, 18, 24, could be used to representa two-octave system, but it is ill suited to the Greater Perfect or Immutablesystem. If one were to attempt such a characterization according to either frequency or string length, the movable notes Paranete diezeugmenonand Lichanos hypaton would be represented in the diatonic genus at the expense of the standingnotes Paramese and Hypatehypaton. Frequency 24 18 16 12 9 8 6 Nete hyperbolaeon Nete diezeugmenon DiatonicParanetediezeugmenon Mese Hypatemeson Diatonic Lichanoshypaton Proslambanomenos Stringlength 6 8 9 12 16 18 24 The numerical sequence 4, 6, 8, 9, 12, 16 would characterizethe frequencies of the Greater Perfect System better than the heptactys, omitting only one standingnote, Hypate hypaton, while includingno movablenotes, and such a numericalsequencecan be found in medieval and Renaissancetreatises. Frequency 16 12 9 8 6 4 Nete hyperbolaeon Nete diezeugmenon Paramese Mese Hypate meson Proslambanomenos The original sequence from 6 to 24 would be equally ill suited to represent the medieval revision of the GreaterPerfect System, which centered attention on the tetrachord of the modal finals (d, e, f, g). One must conclude, therefore, that the choice of 24:9 to representthe eleventh ratherthan 8:3 or 16:6 was made under the strong influence of the musicaltetractys.

All this is pretty scanty stuff from which to arguethat the authorof a medievaltreatise,for example,Musicadisciplinaor Scolica enchiriadis, was drawing from Gaudentiusor even from Cassiodorus,although the evidence that I have presentedindicates that this connection is worth pursuingfurther.Have portionsof Boethius'De musicabeen lost in the transmission?The incompleteversionof book 5 would seem to indicate so. Moreover,the letters of Cassiodorusascribeto Boethiustranslations of works on the ancient mathematicalsciences, the quadrivium.We have only two of these four works. These missingor incompleteworks as well as the Latintranslationby Mutianusof Gaudentius'Introduction to Harmonicsmay have served directly, or indirectly throughCassiodorus, as important sources for the transmissionof ancient musical learning.Furthermore,Byzantinesingersand sourcesmayhaveprovided the West with a link to the musical knowledge of antiquity, although the extant Byzantine sources are late (see n. 84). In any event, the evidence that I have presented argues for the long-livedeffect of the musicaltetractys,discoveredby Pythagorasin Babylonia. Regardingthe perception of consonance, is the eleventhconsonant? The question drawsone to the center of the philosophicaldebate over rationalismversusempiricism.It is no surprisethat some of the practical medieval treatises on music, Guido's Micrologusfor instance, do not take up the matter at all. Unless the practitionerpostulates singingin parallelelevenths,as does Musicaenchiriadis,there is no need to discuss the interval. But if the disposition of a musical treatise is why rather than how, then the eleventhis an importantmatter. Breakingdown Ptolemy's argumentinto its parts, the first is that the octave is consonant. Historically in the West, this claim is irrefutable and has been for millennia. Rationalists and empiricistsagree here, as do most modern students of acoustics. The second part of the argumentclaims that the fourth is consonant, and again from a historical point of view, this is true throughout antiquity and well into the late Middle Ages. The third part of Ptolemy's argumentis not that a consonance plus a consonance yields a consonance, although Walter Odington, in his rebuttal to Ptolemy, demonstratesthat a fifth plus a fifth produces a dissonance.88Ptolemy knew this and demonstratedit also in his Harmonics.89 Ptolemy claims, rather,that the octave, along with the double octave, is more consonantthan the other consonances. Finally, the octave is so consonant that it almost soundslike a unison, thus when added to another consonance the octave preservesthe consonant characterof the other consonance. I believe that the third part of Ptolemy's argument-that the octave is more consonant than the fourth or fifth, for example-is also borne out historically in the West, although it may be going too far to claim unanimity on the matter. That the octave is so specialthat it preserves

the characterof a consonance to which it is added is another matter entirely. Ptolemy claims that he hears this to be the case. In effect, so do Gaudentiusand the author of Musica enchiriadis.Wehave the basis here for an empiricalfact, that is, a fact contingent upon sensory perception. Of course, sensory perception varies from person to person. From this argument,then, the status of the consonant eleventh could never attain a position higher than or different from one contingent upon unanimity of sensory perception. As we have seen, the eleventh does not even get that far. In what amounts to a synopsis of Boethius' remarkson the Pythagoreannotion of consonance,Jehandes Murswrites: The Pythagoreansrejected the octave plus fourth [from the category of] sounding a good harmony because it fell on the hearing neither sweetly nor pleasantly. Inquiringwhy, they discovered [the interval] itself to be outside of the multiple and superparticular varieties."9 In other words, the Pythagoreansrejectthe eleventh from the category of consonance because it is neither sweet nor pleasingto the ears.Why? Becauseit is neithermultiple nor superparticular.This is the connection between numerical and acoustical truth first made by Sectio canonis (see above, p. 194). WhenPtolemy makes his empiricalclaim, this entailsthe qualitative notion of a sweet and pleasingblend or mixture of sounds. These metaphors of mixture and sweetness were used to describe consonance throughout the early history of Westernmusic. But we have seen that ancient and medievalwriters differ on whether the eleventh is a sweet mixture. This is so, in part, because such a descriptionis both metaphoricaland qualitative.The numericalratiosused by the Pythagoreans, however, are neither metaphoricalnor qualitative.Clearlythe quantitative aspect of the Pythagoreanargumenthas the advantageof being immutable. But there is more than this. The orthodox Pythagoreansclaim that numbersdo not representreality,but ratherthey are reality."9Thus the Pythagoreansavoid metaphor in their discussion of consonance. This is the point made by Jehan des Murs when he states that the eleventh is not pleasingto the earsbecauseit is neither a multiple nor a superparticularatio. The issue of consonance and dissonance was for the Pythagoreans not a matter of devisinga theory that was harmoniouswith their hearing, but ratherone of hearingthe numericaltruth that they discovered to be inherentto nature.

NOTES NOTE: I read a short version of this article at the Annual Meeting of the American Musicological Society (Boston, November, 1981). 1. Barbara Miinxelhaus surveys some of the information presented here in her Pythagoras musicus. Zur Rezeption der pythagoreischen Musiktheorie als quadrivialer Wissenschaft im lateinischen Mittelalter (Bonn-Bad Godesberg: Verlag fuir systematische Musikwissenschaft, 1976), 88-94. The scope of my study is considerably more narrow than hers and, I think, more philosophical and exegetical. 2. See Carl Dahlhaus, "Ein vergessenes Problem der antiken Konsonanztheorie," Festschrift fur Walter Wiora zum 30. Dezember 1966, eds. Ludwig Finscher and Christoph-Hellmut Mahling (Kassel-Basel: Birenreiter, 1967), 164-169. 3. The phrase "d4a Kpo6w," meaning to strike or play at the same time, is usually rendered with KpoV~w in a participial form. See Aristides Quintilianus, De musica libri tres, ed. Reginald P. Winnington-Ingram (Leipzig: Teubner, 1963), p. 10.2 and 10.4; Gaudentius, Introduction to Harmonics, in Musici scriptores graeci, ed. Karl von Jan (1895; reprint, Hildesheim: Olms, 1961), p. 337.8-9 (hereafter JanS); Nicomachus, Manual of Harmony, JanS 262.2-3; Porphyry, Porphyrios Kommentar zur Harmonielehre desPtolemaios, ed. Ingemar Diiring (1932; reprint, New York: Garland Publishing, 1980), p. 114.11; Theon of Smyrna, Expositio rerum mathematicorum ad legendum Platonem utilium, ed. Eduard Hiller (Leipzig: Teubner, 1978), p. 51.3. Boethius translates this phrase into Latin as simul pulso, De institutione arithmetica libri duo. De institutione musica libri quinque, ed. Godofred Friedlein (1867; repr. Frankfurt: Minerva, 1966), pp. 302.3 and 349.7. 4. De musica, F. 195.6-8 and 348.23 - 349.1. 5. But see the discussion of intervals by Aristides Quintilianus where he implies that consonant and dissonant intervals are melodic, De musica 1.7. In all matters regarding Aristides Quintilianus, see Thomas J. Mathiesen's translation of and commentary on the De musica, On Music. In Three Books (New Haven: Yale University Press, 1983). 6. ibid., F. 259.10-13. "Quid igitur, si diatessaron ac diapason consonantias iungamus, ullamne secundum Pythagoricos efficient consonantiam? Minime." 7. ibid., F. 195.6-8. "Consonantia est acuti soni gravisque mixtura suaviter uniformiterque auribus accidens." 8. ibid., F. 259.13-15. "in superpartiens inaequalitatis genus cadit, nec servat vel multiplicitatis ordinem vel superparticularitatis simplicitatem." 9. See Calvin M. Bower, "Boethius and Nicomachus: An Essay Concerning the Sources of De Institutione Musica," Vivarium 16 (1978): 1-45, and Ubaldo Pizzani, "Studi sulle fonti del De Institutione Musica di Boezio," Sacris erudiri 16 (1965): 5-164. 10. Nicomachus, Introductionis arithmeticae libri ii, ed. Richard Hoche (Leipzig: Teubner, 1886), pp. 44.8 - 72 and 119.19 - 144.19. See also my "Republic 530C - 531C: Another Look At Plato And The Pythagoreans," American Journal of Philology 102 (1981): n. 29. 11. Nearly all modern writers on the subject of Pythagorean arithmetic represent

the superparticular ratio as (x + 1): x, assuming the ratio to be in its lowest terms. This representation is incorrect because it admits 2:1, a multiple ratio, into the category of superparticular. Both Nicomachus, Introductionis arithmeticae, H. 49.1-4, and Theon, Expositio, H. 76.21 - 77.2, carefully define superparticular so as to exclude 2:1. 12. The word rerpd4 means, literally, the number four. The word also refers to the first four integers, 1, 2, 3, 4. See: Armand Delatte, Etudes sur la litterature pythagoricienne (1915; reprint, Geneva: Slatkine Reprints, 1974), 253ff. Thus I distinguish, as do some ancient authors, between the tetrad, 1, 2, 3, 4, and a that is, a quaternary. tetractys (rerpaKrr3), mathematical and acoustical principles contained in Sectio canonis 13. Many of the are doubtless very old. Far less certain is the date of composition of the treatise as we know it, that is, introduction and twenty propositions. Sectio canonis is transmitted in three versions: one ascribed to Euclid; one contained in Porphyry's commentary on Ptolemy's Harmonics, D. 99 - 103.25; and one contained in the De musica of Boethius, F. 301.6 - 308.15. A comparison of these three versions reveals an unstable, probably living tradition of Pythagorean acoustical science that may have received an embodiment in the form of the Sectio at a time considerably later than that of Euclid. v ouvwapwvov~ roiv 14. JanS 149.17-24. "rw6oK'o ev 6 Ka r .Ov O6yyYwv0z l EV K wloav Ka ToTV '6vraq, rovk 6 rvI &lr~ov KpaY UoLjyvoup &ataOvovo, ojr. oTo'rwvoirw xd6vrwvYelK6Crokv oavA0crrotowvraq,r70o Sbe 6taWvovo rTq7Ow~wC,elvat Q~ZO6yrov, &ret6? lav rr7v dLOgvrolovtovratlKpdatLV voov v& vi buvdoart 7rpde dhhrhov p dpt0pCv, 'rrot 7roXhha7rarCAV hXeyop•vwv oiovC 'dvrac ' ~ srtop ovz." Immediately preceding this passage, we read: "Of these, the multiple and the superparticular are ordered one to another by a bi 6vdsart 66 oi pesv 7rohXXa7rXdCtot single name." (rotrwv Kal "a single name" or JanS 149.14-16.) The phraserrt•s6dpto heyovoratrrp6 d&hhXXovq, "one term" bv6~aart)is difficult to interpret here. Thomas Mathiesen "one term" means consonant, citing Porphyry's commentary maintains that(uvi on Ptolemy's Harmonics (D. 98.3-6) as support, "An Annotated Translation of Euclid's Division of a Monochord," Journal of Music Theory 19 (1975): n. 12. See my discussion of this phrase in "Placing Sectio canonis in Historical and Philosophical Contexts," The Journal of Hellenic Studies, forthcoming. 15. Ptolemy, Die Harmonielehre des Klaudios Ptolemaios, ed. Ingemar Diiring (1930; reprint, New York: Garland Publishing, 1980), 11. 16. Hermann Diels, Die Fragmente der Vorsokratiker, 10th ed., rev. Walther ~z rpeOde o07t Kranz, 3 vols. (Berlin: Weidmann, 1960), 44 B 6. "&ppoviac jWXLOV, Kai . . . &6 uvhhat& n7rpTrov,TO ro 6r b6t'btetdv avXXhhapa' 6t'6tetdv" 6t rraodv6UburXhdov." 17. The analogy between the rational combination of letters to form syllables, syllables to form words, and the rational combination of notes to form intervals, intervals to form melodies, occurs frequently in ancient treatises. See my "The Persistence of Pythagorean Mathematics in Ancient Musical Thought" (Ph.D. diss., Univ. of North Carolina at Chapel Hill, 1980), ch. 5, nn. 20 and 37, as well as JanS 254.4ff and 417.12-18. 18. Boethius, De musica 2.27. 19. Aitius, Placita 1.3.8, in Hermann Diels, Doxographigraeci, 4th ed. (1879; repr. Berlin: De Gruyter, 1965), 282.

"ob j& 7rbv AerLpqL4/vX napa6dvra rerpaKrv', vav." 7rayv devaov douewo'pitwsd r'oXorliber, ed. Cf. Iamblichus, De vita pythagorica Ludwig Deubner (Leipzig: Teubner, 1937), p. 85.4-5, and Sextus Empiricus, Adversus mathematicos 4.2, in Sexti Empirici opera, ed. Hermann Mutschmann (vols. 1 and 2) and and Jilrgen Mau (vol. 3) (Leipzig: Teubner, 1912 (vols. 1 and 2), 1954 (vol. 3)) 3: 133.16-17. Theon, Expositio, H. 94.6-7, is identical to Sextus Empiricus above. 20. Iamblichus, De vita pythagorica, D. 47.15-16. 'r71art rd v AeXhoSq,avelov; TerpahrTv. e drep ariv h&ppovia,v Va!tLepiveq." I have adopted Walter Burkert's translation of this oath; see: Walter Burkert, Lore and Science in Ancient Pythagoreanism, trans. Edwin L. Minar, Jr. (Cambridge, Mass.: Harvard University Press, 1972), 187. See, also, Paul Kucharski, Etude sur la doctrine pythagoricienne de la titrade (Paris: Les Belles Lettres, 1952), 75-77. 21. Aristotle, De anima 404b18-27. Cf. Aetius, Placita 1.3.8, Diels, Doxographi graeci, 280-283. 22. Iamblichus, Theologoumena arithmeticae, ed. Victorius de Falco (Leipzig: Teubner, 1922), 82-85. 23. Theon, Expositio, H. 94ff. 24. Diels, Die Fragmente der Vorsokratiker 47 B 1. 25. Iamblichus, Theologoumena arithmeticae, de F. 21.8-10. "r7'rrapeC z v Kai raltaoiac nrrpdOpat, sIovUoud yewserpla oi0atp(td, d P ~• 6 eray&pt0rt7Ord& 26. F. 7.21ff. 27. Diels, Die Fragmente der Vorsokratiker 47 B 2. 28. Iamblichus, In Nicomachi arithmeticam introductionem, ed. Hermanegildus Pistelli (Leipzig: Teubner, 1894), pp. 118.19-24 and 122.26-27. 29. Modern scholars have tended to dismiss many of lamblichus' remarks as fanciful, but this skeptical assessment results in part from our inability to unearth his written sources, if in fact they ever existed. 0 rpTcr77 erpaKTrC,7 Kat rTwvrerpaX6p6 30. JanS 282.10-14. "' 6 o6v Y ToTWV ro % Kara Ydv7l6tatpoemwv, d&rore'piTa, rpd7rrov r7T rraaocv •art 1rapeKrt rC<Ov y W T• Xofaaoc dLTv rv Xhe pt0d6v 7Tr &rrXavoi olKelov rrpoO'aKrJ jToved6oq." 31. Aristides Quintilianus, De musica 3.18. 32. Of the many renditions of this myth, Nicomachus presents the earliest in his Manual of Harmony, JanS 245.19 - 248.26. See Barbera, "The Persistence of Pythagorean Mathematics," 313-315. 33. Ptolemy, Harmonics, D. 17.19-20. See also Flora R. Levin, "7rkryr' and rdaot in the Harmonika of Klaudios Ptolemaios," Hermes 108 (1980): 205-229. 34. Boethius, De musica 2.27. 35. Ptolemy, Harmonics, D. 15.10-12. "bpt?e'dowav 66%tApv6,Ad4wvoot pjv ot Kara rv canvSjavaw &v6qdvrhlX nw EtvLo.folvreC raC Tq&oaTq." 36. ibid., D. 15.12-17. "o'j?Awvot 66 oL ~YYvrdrTarwrv bDjo0crWwv, oti 6LStd Kal ol • abrrwvKai bso0 vwwvUvvrLVOT 7rcvre Kal ot 6d TreoadlpWV evot, rr<v Twv UavjApivwwv, S ot CCot Tovtalot KLa< AeXjehez t77vrTLTr70 a Tv rotoLV)rYoi Xomot'. 6& Kat ~ovvrT7Oeral rrwc oi wlv blNd~Awvot ro7iC avjU/WvotC, oli 86 taVJPwvot Tro70i 6LLe X'dot."

37. ibid., D. 16.24-25. "?ppe• 76v wrlr7ptrov rov •7rqjloplwv." be6' ol perdro 38. Aristoxenus, Aristoxeni elementa harmonica, ed. Rosetta da Rios (Rome: Typis Publicae Officinae Polygraphicae, 1954), pp. 25.18 - 26.1 and 56.10-19. 66t& rraaoCAv 39. Ptolemy, Harmonics, D. 13.3-7. "KaOdhoov yp aouvCwvia, rTOv &•vd, 6rav 7rpoaaz aopotvrvwv Kar&77V 6bdva#pv rowtodrwvyalrrjyV0Odyw77ov 4 6EK6d ro ?Ket'vnCelboC7rlpel, KaOdeTrep &rrapd7iperrrov hh&XXwv, ?00t rtvt r(V & bqr' abr v dpL•Oodq." eXec,O6'peeierZv,rrpodTrov 40. The specific example of 10 plus 2 is Boethius' contribution to Ptolemy's reremark, De musica, F. 360.11-14. 41. Cf. also the intervallic tests contained in Ptolemy's Harmonics 1.8 and Boethius' De musica 4.18. As Bower has pointed out, the chapters are related, but whereas Ptolemy tests all of the consonant intervals (4:3, 3:2, 2:1, 8:3, 3:1, 4:1), Boethius tests only four (4:3, 3:2, 2:1, 3:1), omitting both the eleventh and the double octave, "Boethius and Nicomachus," 37-38. Bower proposes that we have an "edition" of Ptolemy preserved in the De musica, undertaken by Nicomachus, and that the omission of 8:3 in Boethius' tests stems from the Pythagoreans' rejection of this interval. This may be so, although we must question why the double octave, 4:1, is also absent from Boethius' tests. 42. Plutarch, Moralia, 7 vols. (Leipzig: Teubner, 1925-1967), vol. 3 ed. W. R. Paton, M. Pohlenz, and W. Sieveking (1929), pp. 12.22 - 13.7. "r6 yap hrXelorov r7tLouvOwviaC boariv,ai'rat 6' elrreuv 'pyov lo) &piAovu•W4 ,rept &Xopale Kal 7povrrlacat 4eXEyXel r6vd ij7- 7vreVE Kal6',ro4 6b h6do o6 7rrhelovz, ,raura OrlpdaZ&6ywe 7r alaOo4aetpovhX6Aevov. rrdoaty&p ~v hd6otl ryI7vfveaw raodpwv &pt0pwv haapp3vovot - Kai Xyoc o7 rirrqv/ 6%tdare ir7ppoL , 7r4 6e 6tL wirevre~lhXotcC, 6trhXdlto66~rCz 6td raouv, 7r-q6etd rraoLy Kal6t& r 'vre otv rptrrhdacoi , KaLrerparrXdacoo 7-r 6t 6dta7raa<Ov. Iv 6g ra6raatz&rreaod'yovUw Kal 6ut reaadpwv bvojsdovrez 6orwpjpov palvovuav, 6btd &PJOVZnKOI %reaoC; d rrap 7r6vhX&ovco~rrep vdp6 oKli ldv 4r7T 6~Xeao at rc &KO 7edr &X7yq Xaptopervovu." 43. Barbera, "The Persistence of Pythagorean Mathematics," 1-5 and 61-83. 44. For example, Cleonides, JanS 194.9-12, and Baccheius, JanS 294.13-16. 45. Theon, Expositio, H. 50.16-19. 46. ibid., H.58.13. "7rrdoa64 rd aouvpOwv'iac 4 Terpanrivc." ,repxetex 47. ibid., H. 52.1-9. 48. ibid., H. 62.1 - 63.2. 49. ibid., H. 63.2-24. 50. Chalcidius, in Platonis Timaeum, W. 113.1 - 114.19. 51. JanS 330.15-20. 52.JanS 341.12-13. "'Ahh' ob6be 7 rrelpq rodrOwv paoaviret dpKneoOelsi•,vJ pOo6ov." 7rpOdrov &Xhov7rhvE 53. JanS 339.21 - 340.3. "A6yoc 6b e'lww r<o; ovulavconvV. r77bpna6voc leo av dp P KaL6osaaO'uvre &KP3oP. rrdvr7a rpd6rovrir4 /%v 6std reaodapwv Tl'7UpTro(, v e'XE 7rdKs6 rpdoZr t7 -rTZ 6e6%t taIrl~ e6hyOWedo, 8V ~Xet6 KS6rrp6 7rdvtC be 6& rraC 0v 6 waoivre 7r b eXet b Ks6 rpod r6vy t r6e Stdrra%;ov 6trhduloo, 0. Ka( ypa Kal sdt reaadpwv 6t7rXaactert6tLMocpo,6v 'Xet b Ks rrpbC76v r7adhx rw b C p6v 6tdarraowV Kald6a v7rere T7pL7rXdchlo,6 V'Xec Ks rrpb 76yV 7"riC brK7.-, b 6 6~StL&rraov Ks6 VU." 7dv V'Xec 6 7rpO 7ETrpa7rXh•dlo, , 54. Martianus Capella, De nuptiis philologiae et mercurii, ed. Adolf Dick (1925; reprint, Stuttgart: Teubner, 1978), pp. 507.14 - 509.5.

55. Cassiodorus, Cassiodori Senatoris institutiones, ed. R. A. B. Mynors (Oxford: Clarendon Press, 1937), pp. 142.13-17 and 149.16-19. 56. ibid., M. 144.21 - 145.19. 57. ibid., M. 145.13-15. 58. Cassiodorus, Varie, ed. Theodore Mommsen (Berlin: Weidmann, 1961), 1.10 and 45, 2.40. As is well known, Theodoric's devaluation of Boethius' services around 524 culminated in Boethius' execution. 59. Cassiodorus, Varie, 1.45. See Calvin M. Bower, "The Role of Boethius' De Institutione Musica in the Speculative Tradition of Western Musical Thought," Boethius and the Liberal Arts. A Collection of Essays, ed. Michael Masi (Berne: Peter Lang, 1981), 160-162. 60. The role played by Cassiodorus in the transmission of ancient knowledge to the Middle Ages has generated considerable interest and speculation on the part of historians. The famed library at the monastary Vivarium apparently did not survive Cassiodorus' death around 583, but the contents of this library may have been dispersed throughout early medieval Europe. In this context, see: Pierre Courcelle, Late Latin Writersand Their Greek Sources, trans. Harry E. Wedeck (Cambridge, Mass.: Harvard University Press, 1969), 334; James O'Donnell, Cassiorodus (Berkeley: University of California Press, 1979); and Giinter Ludwig, Cassiodor. Uber den Ursprung der dbendlandischen Schule (Frankfurt am Main: Akademische Verlagsgesellschaft, 1967). 61. See Lawrence A. Gushee, "Questions of Genre in Medieval Treatises on Music," Gattungen der Musik in Einzeldarstellungen. Gedenkschrift Leo Schrade, ed. Wulf Arlt, Ernst Lichtenhahn, and Hans Oesch (Bern: Francke, 1973), 365-433. 62. Martin Gerbert, Scriptores ecclesiastici de musica sacra potissimum, 3 vols. (1784; reprint, Milan: Bollettino Bibliografico Musicale, 1931) (hereafter GS). 63. Guido, Guidonis Aretini Micrologus, ed. Joseph Smits van Waesberghe, Corpus scriptorum de musica (hereafter CSM), 4 ([Rome]: American Institute of Musicology, 1955), 228-233. 64. See Bower, "The Role of Boethius' De Institutione Musica," 157-174. 65. Region of Priim, Epistola de harmonica, GS 1.238. 66. De harmonica institutione, GS 1.107. 67. Bower, "The Role of Boethius' De Institutione Musica," 172. 68. See especially "De tonis octo," ch. 8, sentences 1-21, and subsequent chapters, Aureliani Reomensis Musica disciplina, ed. Lawrence Gushee, CSM XXI (1975), 78-79ff. The "De tonis octo" most likely antedates the rest of Musica disciplina. 69. Bower, "The Role of Boethius' De Institutione Musica," 167-170. 70. Ed. Gushee, 71. "Constat autem omnis musica simphoniis sex, sonitibus quindecim, tenoribus octo." 71. Bower, "The Role of Boethius' De Institutione Musica," 171-172. See: Musica et scolica enchiriadis, una cum aliquibus tractatulis adiunctis, ed. Hans Schmid (Munich: Verlag der Bayerischen Akademie der Wissenschaften, 1981), ch. 19, especially lines 33-40, and the review of Schmid's edition by Nancy Phillips, Journal of the American Musicological Society 36 (1983): 128-142. 72. Musica enchiriadis, ed. Schmid, 11.40. 73. ibid., 16.39. 74. Cf. Boethius, De musica, F. 359.1 and 359.19-20 with Musica enchiriadis, ed. Schmid, 16.17-18 and 32-33 respectively.

75. Ed. Schmid, 16. For the analogy between the octave and the decad, see 11. 76. Ed. Schmid, 30. "attende ut dictum Ptolomei regnet in ore." 77. ibid., 2:1-3. 78. For example, cf. Scolica enchiriadis, ed. Schmid, 2.165-168 with Boethius, De arithmetica, F. 8.8-11. 79. For example, cf. Scolica enchiriadis, ed. Schmid, 2'160-161 with Cassiodorus, Institutiones, M. 130.19-20. 80. Ed. Schmid, 2:202ff. 81. ibid., 3:225-226. "Principales sunt I II III IV. Quaternarius enim numerus omnes symphonias perfecte absolvit." 82. Musica, GS 1.334. 83. Berno of Reichenau presents the quaternary 1, 2, 3, 4 and notes the numerological value of its sum, 10, drawing analogies with the ten-string psaltry used in praise of God and with the four evangelists, GS 2.66. When it comes to the eleventh, however, he sides with Ptolemy and cites Boethius' instance of the analogy to the decad, GS 2.64-65. Wilhelm of Hirsau, like Berno, deems the eleventh to be consonant, Willehelmi Hirsaugensis Musica, ed. Denis Harbinson, CSM 23 (1975), 39-40. Marchetto, in the Lucidarium, reviews the entire matter and discusses quaternaries, referring to the four evangelists, GS 3.84. (In this regard, see also Aurelian's remarks about four and the name of God, Musica disciplina, ed. Gushee, 79.) Marchetto ultimately sides with Ptolemy and in what amounts to a radical transformation of theory in order to accommodate empirical evidence, states: "of the five kinds of inequality in music, only three are necessary, namely the multiple, . .. the superparticular,.. . and the multiple superbipartient" (italics mine), GS 3.88. (de quinque generibus inaequalitatis in musica, omnino tria sunt necessaria, scilicet multiplex,.... & multiplex superbipartiens .) superparticulare, ... .... treatise on the quadrivium traditionally ascribed to 84. See: the eleventh-century Michael Psellus, Anonymi logica et quadrivium, cum scholiis antiquis, ed. J. L. Heiberg, in ser. Det Kgl. Danske Videnskabernes Selskab historisk-filologiske Middelelser 15, 1 (Copenhagen: 1929); Georgius Pachymeres, Quadrivium de Georges Pachymere, ed. Paul Tannery, text ed. E. St6phanou, Studi e testi no. 94 (Vatican: Biblioteca Apostolica Vaticana, 1940), 124-126; and Manuel Bryennius, The Harmonics of Manuel Bryennius, ed. and trans. G. H. Jonker (Groningen: Wolters-Noordhoff Publishing, 1970), pp. 98.27ff and 144.21ff. See also: Thomas J. Mathiesen, "Aristides Quintilianus and the Harmonics of Manuel Bryennius: A Study in Byzantine Music Theory," Journal of Music Theory 27 (1983): 31-47. 85. Walteri Odington Summa de speculatione musicae, ed. Frederick F. Hammond, CSM, 14 (1970), 72-73. 86. Jacobi Leodiensis Speculum musicae, ed. Roger Bragard, 7 vols., CSM 3 (19551973), 2:248-252 and espeically 252-258. 87. Isidore of Seville, who relied on Cassiodorus for much of his information about music, does not take up the matter of the eleventh at all. See: "de numeris musicis," Isidori Hispalensis episcopi Etymologiarum sive originum libri xx, ed. Wallace M. Lindsay (Oxford: Clarendon Press, 1911), 3.23; and Bower, "The Role of Boethius' De Institutione Musica," 162-163. 88. Summa de speculatione musicae, 72-73.

90. Musica speculativa secundum Boetiam, GS 3.271-273. "Pythagorici noluerunt diapason & diatessaron sonare bonam harmoniam, eo quod nec dulciter nec suaviter veniret ad auditum. Causam inquirentes invenerunt, ipsam esse extra genus multiplicis & superparticularis." 91. Barbera, "Interpreting an Arithmetical Error is Boethius's De Institutione Musica (iii.14-16), Archives internationales d'histoire des sciences 31 (1981): 40-41. O