and in the harmonika of Klaudios Ptolemaios

in the harmonika of Klaudios Ptolemaios In: Hermes, | . Y 108 — 1980 - p. 205 - 229 DADA Er Lern. RR. NAPS D / Music Levin F. R., nAnyh and téoç in the Harmonika of Plotemaios {pols contre Nicomaque] ; cf, N° 3593, gue Levin F. R., many und dou in the Harmonika of Klaudios Plolemuius : Hermes CV III 1980 205-229. | Already in the 2nd cent. A.D. Plolemy, on the basis of experiments that he conducted, was in the presence of, Mf not indeed possessed, the knowledge which precipllated M. Mersenne’s discovery in the 17th cent. Ptolemy was led to this knowledge by bis dissatisfaction wilh uccounts of various Pythagorean experiments; Lhe discussion in which this knowledge Is evinced forms part of a polemic directed against Nicomachus of Gerasa.

Source: Hermes, Vol. 108, No. 2 (1980), pp. 205-229 Published by: Franz Steiner Verlag Stable URL: http://www.jstor.org/stable/4476160 Accessed: 18/01/2010 05:47 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/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?publisherCode=fsv. 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 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. Franz Steiner Verlag is collaborating with JSTOR to digitize, preserve and extend access to Hermes. http://www.jstor.org

kumulierten>>dati<< und ?notizie di ogni genereozu kommen24.Aber mussen wir denn uberhauptannehmen,daBetwa Epidem. I1 in der uns vorliegenden Form von einem >>Verfasser<( stammt? Konnen nicht, von wem und wann auch immer, die beiden >>physiognomischen(< Abschnittein diese >>Schrift(( ohne Rucksichtauf deren sonstigenInhalt hineingestopftworden sein? Ein Zeugnisfur echte )>Weiterarbeit.( ist jedenfalls, fur unser Thema, hier nicht gegeben. Ganz andersin Morb. IV: Hierist das archaischeUngerade-Prinzipfest in die aligemeinenVorstellungendes Vfs. integriert(s. o.). Wennwir nun GRENSEMANNSSpatdatierungdieser Schrift (s. o.) akzeptieren,dann hatten wir hier, von unseremThema aus gesehen, einen eigentumlichenFall: Ein Arzt, der eigentlichdie knidisch-koischeSchultradition,und damitauch sowohldas modifizierteUngerade-Gerade-Prinzip in Epidem. I wie die ganz andereKrisenarithmetikdes Prognostikon gekannt haben sollte, hatte dennoch das archaisch-exklusiveUngerade-Prinzipvorgezogen. Mir personlich schiene dergleichen,bei diesemVf., nicht vOlligausgeschlossen. Kiel FRIDOLFKUDLIEN 24 Vgl. Di BENEDETTOa. 0. 257. Zilry' AND act6cntIN THE HARMONIKA KLAUDIOS PTOLEMAIOS OF The discovery that music is ruled by number, that its constituents - pitch and interval - can be accurately expressed by numerical ratios, was one of the most significant intellectual events of antiquity. It founded the science of acoustical physics; moreover, it gave rise to a period of speculation out of which issued important developments in mathematics, philosophy, cosmology and astronomy 1. According to general opinion no discovery of comparable magnitude was made until the seventeenth century, when MARIN MERSENNE, the French mathematician, philosopher and scientist, first explained in his 'Harmonie Universelle' the relations obtaining between tension and the frequency of vibration of a stretched string, relations subsequently codified in 1 See, for instance, A. DELATTE, Etudes sur la litterature pythagoricienne, Paris 1915, 258- 259; Sir TH. HEATH, Aristarchus of Samos, The Ancient Copernicus, Oxford 1913, 46-47; F. R. LEVIN, Synesis in Aristoxenian Theory, TAPA 103, 1972, 217- 220.

'MERSENNE'S Laws' 2, If this opinion is correct, then a period of some 2000 years intervened between the discovery of the musical ratios and that of the relation between tension and frequency of vibration. This paper is written in the belief, however, that already in the second century A. D. Ptolemy, on the basis of experiments that he conducted, was in the presence of, if not indeed possessed of, that knowledge which precipitated MERSENNE's discovery3. It will be argued that Ptolemy was led to this knowledge by his dissatisfaction with accounts of various Pythagorean experiments and, further, that the discussion in which this knowledge is evinced forms part of a polemic directed against Nicomachus of Gerasa4. I. The Musical Ratios According to numerous ancient authorities, the first one to have revealed the numerical ratios determining the consonant intervals of the musical scale was Pythagoras of Samos. Because of the deep and far-reaching implications which the discovery had for such fundamental branches of knowledge as mathematics, cosmology and astronomy - implications that extended far beyond its immediate utility in converting the sensory distinctions of pitch and interval into objective numerical form - it was treated by the ancients as a divine revelation 5. The musical formulae vouchsafed to Pythagoras impressed the Pythagoreans as a key of universal application, leading them to postulate number as the element of all things and to promote the study of number per 2 M. MERSENNE, Harmonie Universelle, The Books on Instruments, Paris 1635, trans. R. E. CHAPMAN, The Hague 1957. See note 52 below. The discovery of these relations was made indeSee A. WOOD,The Physics of Music, pendently by GALILEO,the contemporary of MERSENNE. London 1975, 90. London 1944, rev. J. M. BOWSHER, 3 On the basis of astronomical observations made by him between 125 A.D. and 141 or 151, Ptolemy is assumed to have lived during the first three quarters of the second century A.D. Little is known of him otherwise. See Sir TH. HEATH, A History of Greek Mathematics, Oxford 1921, vol. 2, 273. 4 Nicomachus' floruit is judged to be c. 100 A.D. See L. TARAN, Asclepius of Tralles: Commentary to Nicomachus' Introduction to Arithmetic, Transactions of the American Philosophical Society n.s. 59. 4, 1969, 7 - 8; F. R. LEVIN,The Harmonics of Nicomachus and the Pythagorean Tradition, American Classical Studies 1, 1975, 8- 10; W. C. McDERMOrr,Plotina Augusta and Nicomachus of Gerasa, Historia 26, 1977, 193. That Nicomachus was an older contemporary of Ptolemy is assumed in part from his apparent ignorance of Ptolemy's work on Harmonics. See K. VON JAN, Musici Scriptores Graeci, Leipzig 1895, 21 1; M. L. D'OoGE, F. E. ROBBINS, L. C. KARPINSKI, Nicomachus of Gerasa, University of Michigan Studies, Humanistic Series 16, New York 1926, 71. The earliest reference to the Harmonika of Ptolemy appears in Nicomachus Exc. 4 (JAN 275, 7 - 9), but may be an interpolation by the excerptor. See i. DURINC;,Die Harmonielehre des Klaudios Ptolemaios, Goteborg 1930, lxxiii. 5 Nicomachus Harmonikon Enchiridion 6 (JAN 246, 6-7); Boethius De inst. mus. 1, 10 (FRIEDLEIN 197, 3 -- 4).

se6. In its own turn, music was henceforwardto be treatedwith the rigorof mathematics,its components- pitchand interval- beingnow susceptibleof study throughthe mediationof number. That the discoveryof the numericalratioswas appreciatedby the ancients for whatit was - a monumentof humangenius - is evident.But that Pythagoraswas in fact its author,as the ancienttraditionmaintains,has beencalled into questionon two fundamentalgrounds:first, thereexist no contemporary reportsof Pythagoras'sdiscovery,the earliestdetailedaccountscoming from the secondcenturyA. D. and later;second,the experimentsdescribedin these late accounts are mathematicallyso implausiblethat their association with Pythagoras can scarcely be credited; somewhere in the transmissionthe apocryphalmust have made its appearance7. The earliest referenceto Pythagoras'sdiscoverycomes via a circuitous route from Xenocrates,who is cited by a certainHerakleidesapud Porphyry In Ptol. Harm. (DORING 30, 1 - 8). Apart from the fact that the identityof this Herakleidesis itself a disputed issue8, it is hard also to ascertainat a criticalpoint in the passagewho in fact is being cited by Porphyry9.Despite these difficulties, however,the passageis valuable for its implicationthat it was on the basis of experimentswhich he conductedthat Pythagorasarrived at his discovery. At the same time, since the passage appearsin a context whichconcernsspeedof motion (-uaXCla (popa)as a determinantof pitch(and not quantity), the referenceto Pythagorasappearsgratuitousand unmotivated. Herakleideswritesabout this [sc. TaXtia (popadin his Introductionto Musicas follows: >>Pythagoras, as Xenocratessays, discoveredthat the intervalsin music have no originapartfrom number.For thereis intrinsicto thema comparativerelationof one quantityto another(noooi ntp6 nopoov). Ac6 Aristotle Met. 985b 31 - 986a 2. Cf. Aristoxenus (fr. 23 WEHRLI): HutOay6pct4... ntvta tai npyLatra dinFtxa(i tv roi4 dptO1goi1. See H. CHERNISS, Aristotle's Criticism of Presocratic Philosophy, Baltimore 1935, 386. 7 See W. K. C. GUTHRIE, History of Greek Philosophy, vol. 1, The Earlier Presocratics and the Pythagoreans, Cambridge 1971, 223 - 224; W. BURKERT, Lore and Science in Ancient Pythagoreanism, trans. E. L. MINAR, Jr., Harvard: Cainbridge 1972, 375 - 376. 8 The identification of this Herakleides with the celebrated Herakleides Pontikos has been assumed by numerous scholars, the most recent of whom are E. A. LIPPMAN, Musical Thought in Ancient Greece, New York and London 1964, 142 and G. H. JONKER, The Harmonics of Manuel Bryennius, Groningen 1970, 381, note on 94. However, F. WEHRLI, Die Schule des Aristoteles, vol. 2, Aristoxenus, Basel 1945, 102, comm. on fr. 122, and 113, argues that this identification is without any real basis in fact. The various arguments for and against the identification are reviewed by BURKERT (n. 7 above), 380-381, who hinmselfsubscribes to WEHRLI'S position. 9 DORING30, 6 - 7: Xai aVEXO6Vrni'PjV -YV4OGV TvC, (PriVqir ?(1] in which the speaker can be either Xenocrates or Herakleides. On the difficulty of interpretation, see GUTHRIE(n. 7 above),

cordingly, he proceededto investigateunder what circumstancesthe intervals become consonancesand dissonances, studying as well the whole questionof attunementand discordancy<.And returningto the originof sound, he said: >>There mustbe movementof some sort, if the interval meant to be heard is a consonance such as results from an equalityof proportion.(< The referenceto Pythagorasin the above passageis thus interposedbetween two partsof a discussionwhose subjectis motion, not quantity;and the pertinenceof quantity(no0o6) to the contextis not clear.However,the reference to quantityand the subsequentdiscussionsuggestthat therewas availablein Xenocrates'saccount a descriptionof experimentsconducted by Pythagoras1. This description,to judge from the discussion, presumablydealt not with speed of motion but with quantityas the determinantof pitch. Furthermore, if as Xenocratesstates, the relationthat Pythagorasdiscoveredwas one holdingbetweenquantityand pitch, it is reasonableto supposethat Pythagoras'sexperimentswereconducteduponthe Monochord-Canon,an instrument on whose single stringthe >>comparative relationof one quantityto anothero< could be accuratelydemonstrated". Assumingthen that Pythagorashad first surmisedthat stringlengthwas the quantitativefactorthat determinedthe pitchof the sound, a logicalprocedurewouldhavebeen to sectionoff variouslengthsof the stringand to correlate themwiththe differentpitchesthusproduced12. Thiswouldhaverevealed the relationsof quantityto quantity,as Xenocratessaid, or one stringlength to another.Correlatingtheselengthswith the variouspitchesproducedwould have demonstratedthat the longer the string, the lower the pitch; and, conversely,the shorterthe string,the higherthe pitch. Translatingthesequantitative relations into numbers would then have shown that an Octave was describedby two stringlengthsin the doubleratio, 2: 1; a Fifth in the hemiolic 1O DURING 30, 4: COXOntEITOTOiVUV.See BURKERT ( n. 7 above), 376, who believes it to be unlikely that Xenocrates failed to mention how Pythagoras arrived at his discovery or that he omitted reference to the instrument on which Pythagoras conducted his experiments, namely, the the Pythagorean name for Monochord. See Nicomachus Harmonikon Enchiridion 4 (JAN xaCtvcbv, 243, 14 - 15). 11 Significantly, Pythagoras on his death-bed is said by Aristides Quintilianus De mus. 3, 2 (WINNINGTON-INGRAM 97, 3 - 7) to have advocated experimentation on the Monochord (~tovoXop8i~Et,v)to his disciples in order to study the relation between pitch and quantity (tvp6-tror T^v tv pouotxfi ... Al' dpt0opv). 17, 27ff.) for detertnining the relative string lengths necessary for the production of the consonant intervals. And in fact Gaudentius Harmonike Isagoge 11 (JAN 341, 12ff.) explains that Pythagoras conducted experiments of this sort on the Canon because he was not satisfied with the single procedure thus far adopted by him involving the use of weighted strings. 12 This is the method recommended by Ptolemy Harm. 1, 8 (DURING

ratio, 3: 2; a Fourth in the epitritic ratio, 4:3. It is possible that in Xenocrates's original attribution of the discovery to Pythagoras he included a description of the experiment that Pythagoras conducted in arriving at his results perhaps one similar to that outlined above. Unfortunately, no such description appears in the passage cited. If it existed in the Xenocrates source, it was presumably omitted by Herakleides (whose words Porphyry reports). Such an omission would account for the sense of incoherence in Porphyry's citation and would suggest a lacuna of indeterminate length in line 6 of DURING'S text. If a description of the experiments with string lengths had been included in Xenocrates' account, the authority of the ancient tradition ascribing to Pythagoras the discovery of the musical ratios would have been rendered secure in its foundations 13. Since such a description is missing from Xenocrates's account, however, we are forced to rely on those that have come down to us. Unfortunately, in these accounts the descriptions of Pythagoras's experiment encourage little credence, contravening, as they do, the laws of physics and mathematics 14. This is especially true of the account recorded by Nicomachus, Harmonikon Enchiridion 6 (JAN 245, 18-248, 26), in which the celebrated tale of Pythagoras and the blacksmith's hammers makes its first appearance 15 Nicomachus begins his account with Pythagoras pondering the problem of how to translate the sensory testimony of the ear into some accurate and visible measure such as is available in the case of the other sensory perceptions 16, According to Nicomachus, Pythagoras, while thus deliberating, chanced to walk by a smithy, where by a fortuitous circumstance he heard the smith's hammers beating out upon the anvil a medley of sounds. These sounds registered upon his unaided ear as the consonances - Octave, Fifth and Fourth - while between the latter two intervals he heard a single dissonance the whole-tone. Amazed at this god-given phenomenon, Pythagoras rushed 13 Of the findings based upon string length dimensions, it is observed by J. BURNET, Early Greek Philosophy, London 1930, 107: >>Onthe other hand, the statement that he [sc. Pythagoras] discovered the 'consonances' by measuring the lengths corresponding to them on the monochord is quite credible and involves no error in acoustics.(< 14 BURNET (n. 13 above), 106- 107 argues that while it would be a waste of time to rationalize these stories, their absurdity demonstrates the existence of a real tradition in that >they are not stories which any Greek mathematician could have invented((. 15 See BURKERT (11. 7 above), 376; LEVIN (n. 4 above), 69- 70; H. CHERNISS, Plutarch's Moralia, xiii, part 1, Loeb Classical Library: Harvard 1976, 306- 307, n. a. 16 The problem is discussed by Plato Philebus 56b 3-c 6, in which music is judged by Plato to be of all the technai the least susceptible of accurate and systematic study precisely because it lacks the proper tools of measurement. For inasmuch as it is dependent upon the guesswork of the ear for proper tunings, it is in Plato's estimation paradigmatic of the imprecise fechnai. The need for instruments to serve the senses is discussed at length by Bryennius Harm. 2, 415 (JONKER 174, 12ff.); cf. Harm. 2, 404 (JONKER 150, 20ff.) on the 6)oyog of the senses.

into the smithy,wherehe determinedby varioustests that it was hammersof different weights that were responsiblefor the sounds he had heard. After ascertainingthe weightsof the hammers,he hurriedhome to duplicateby a controlledexperimenthe resultsobtainedin the smithy. His methodfor setting up the experimentis explained by Nicomachusas follows (JAN 246, 22 - 247, 7): He planteda singlestakediagonallyin the walls in orderthat no difference might arise from this implantation,or, in brief, that no variation mightbe detectedfrom the use of severalstakeswith theirown peculiar properties. From this stake he suspended four strings of the same material,madeof an equalnumberof strands,equalin thicknessand of equal torsion. He attacheda weightto the bottom of each, suspending each by each in succession. And he contrivedthat the lengths of the stringsshould be exactly equal. Then, pluckingtwo stringssimultaneously in alternation,he found the aforementionedconsonances, one pair producingone consonance,anotherpairproducinganotherconsonance. Nicomachusgoes on to explainthat the suspendedweights,which had been arrangedin a graduatedseries,were of the values: 6, 8, 9, 12; that the string suspendedby the weightsof 6 and 12 unitsproducedwhenpluckedan Octave, those suspendedby weights8 and 12 produceda Fifth, those suspendedby 9 and 12 units produceda Fourth. In addition,the pairs6 and 8 werefound to producea Fourth,while6 and 9 producedthe Fifth. And the dissonantwholetone was discoveredto be producedby the pair, 8 and 9. Therebyit was demonstratedthat (JAN 248, 1 - 6): The Octave is a compoundeither of the Fifth and Fourthin conjunction, just as the double ratio (2: 1) is a compoundof the hemiolic(3:2) and the epitritic(4:3), as, for example, 12, 8, 6; or conversely,it is a compoundof the Fourthand Fifth, just as the double ratio is a compoundof the epitriticand the hemiolic,as, for example,12, 9, 6, in such order. After applyingthese findingsin tests upon variousother instruments,such as Monochords(JAN 248, 16), Pythagorascorrelatedthe numericalvaluesthus adducedwith the pitcheshe heard(JAN 248, 18- 23): Hypate (E) Mese (A) Paramese(B) Nete (E') 6

Inasmuchas the experimentdescribedby Nicomachuscould not possibly yield the resultsthat he reports,it is clearthat he did not himself attemptto verify it. Instead, he seems to have transmitteda tale long since absolvedof absurditiesby the weightof ancienttradition17. The initialabsurditytakesthe form of an interchangingof the activeand the passivein the processleadingto pitchvariation.Accordingto Nicomachus(JAN246, 7 - 8), the musicalconsonances heard by Pythagoraswere producedby four hammersof different weights strikingupon a single anvil (tn' dxigovt). Moreover,he assertsthat Pythagoras ascertainedthis fact after having studied four factors - the weights of the hammers,the dimensionsof the hammer-heads,the force of the hammerblows, and the anvil (on which the iron was being forged; JAN 246, 16-20). And from this investigationPythagorasconcludedthat it was the different weights of the hammersthat determinedthe variation in the pitchesproduced.This accountis preposterous,however,sincepitchvariation producedby percussionof this primitivesort dependsnot on the object used in striking the blow, but on the object affected by the blow. Thus, even though the affected object is one of the four factors among which the pitch determinantwas to be locatedby Pythagoras,and even thoughit is this factor which in actuality does determinethe pitch, Nicomachus has Pythagoras rejecting it (JAN 246, 19- 20: oV&tnap& Tiv toO AAauvogtvouot6ijpou pt6a olv) and selectingratherthe irrevelantfactor of the hammerweights. In neglectingthe role playedby the affected object (the anvil) in determining pitch, and in focussingon that of the hammerweights, Nicomachusappears to have been ignorant of a basic fact: four hammers of different weight strikingan anvil at the same point would have producedidenticalpitches pitcheswhich would vary in loudnessonly, dependingupon the force of the hammer blows18. In this respect an anvil, when set into vibration by the hammerblow, would functionno differentlythan a bell, cymbal,tuning-fork or a singlebar of a xylophone,whichsimilarlygive off each a particularpitch irrespectiveof the object with which they are struck'9. In order for four differentpitchesto have been produced,it would have been necessarythat a (single)hammerstrikefour anvilsdifferingin eithertheir size, their shape or theirmaterialcomposition.Differencesin any of thesefactorsor combination of factorswould influencethe pitch producedby each anvil. Indeed,an experimentin whichthe aforementionedprinciplesare brought to bear is said by Aristoxenus (fr. 90 WEHRLI= Schol. Plato Phaedo 108 d) to have beenconductedby Hippasusof Metapontum.Using four bronzediscs of identicaldiameter,the comparativethicknessesof which were in the pro4 above), 73 -74. 18 Cf. GUTHRIE(n. 7 above), 223. 17 Cf. LEVIN (n. 19 See WOOD (n. 2 above), 14*

portions, 4:3, 3:2 and 2:1, Hippasus is reported to have found that these discs produced when struck the consonances, Fourth, Fifth and Octave, respectively, this enabling the musician Glaukos of Rhegium to play a tune upon them. Were the tradition which makes Hippasus a contemporary and dissident disciple of Pythagoras more reliable, it could be argued that Hippasus's experiment was conducted along lines proposed by Pythagoras20. As matters stand, however, the uncertainty regarding Hippasus's date and position among the Pythagoreans does not permit us to refer Aristoxenus's evidence even inferentially to Pythagoras. Nonetheless, it does demonstrate that the ancients understood certain basic principles of sound production which are found utterly perverted in Nicomachus's account21. It remained for Ptolemy to incorporate the primitive causes of sound - percussion (ii), substance (oDtoia)and motion of air (rCovd&pwvxivretr) - into a comprehensive theory, one in which the attributes of sound - pitch and timbre - are referred respectively to the appropriate categories of material quantity and quality22. As Ptolemy demonstrates, the role of percession in the production of sound is far more complex than had thitherto been realized. Furthermore, although this was not its intent, Ptolemy's exposition of the several factors involved in percussion effectively disposes of Nicomachus's misrepresentation of the facts, and in this sense it may be regarded as a critical commentary on Nicomachus's text. Like Archytas before him, Ptolemy Harm. 1, 3 (DURING 6, 14ff.) refers the cause of sound to percussion (ntXjyf) occasioned by the impact of one object against another23. An example of such a (primitive) type of percussion is the blow of a hammer against an anvil24. But Ptolemy recognized that percussion is also manifested - if less obviously - in the operation of string and wind instruments. Ptolemy thus proposed that there are various means by which percussion is induced and that these means, in connection with other factors, account for differences in sound. In his discussion of this passage DURING argues that Ptolemy intends by nkyiiy the characteristic sound produced by diverse instruments, as for exam20 The problems surrounding Hippasus are discussed by J. A. PHILIP, Pythagoras and Early Pythagoreanism, Toronto 1966, 26 - 27. 21 Cf. BURKERT(n. 7 above), 377. 22 Ptolemy's contribution to musical theory is acknowledged by MERSENNE(n. 2 above), Second Book of String Instruments, Prop. IV, 84: ))Ptolemy is the most knowledgeable of all those who have taught us Greek music((. 23 Archytas Vors. 47B 1 (DIELS-KRANZ432, 10-433, 24 See note 44 below.

ple, the )>gong<(Schlag) of a metal plate, the >>twang(< (Ausschlag)of a pluckedstringor the )>blast<< (StoI3)of a wind instrument.For this reasonhe ist die finds it difficultto expressthe senseof nktiyi'by a singleword(>>Ferner Bedeutungvon ikriyy schwermit einem Wort zu fixieren<<)25.But DORING'S interpretationof nkXiyii,apartfrom raisingthe difficultynoted by him, does not do justice to the line of investigationadopted by Ptolemy. For Ptolemy treats tXilyynot as the (characteristic) sound (>>dieReaktion<<, as DOURINGhas it) but as the immediatecause of the sound26.By doing so, he orders the acoustical facts in a manneridenticalto that advancedby modernacousticians. This orderingprovidesthe researcherwith the means of studyingpercussionand its reactionas two separateevents.By distinguishingthen between the activeand passiveagentsinvolvedin percussion,acousticiansreferthe initiator of sound to the former, and to the latter the pitch and timbre of the sound produced.This analysismakesit possibleto comprehendunderpercussion not only the primitivetype of percussion,but also the methodsby which stringsand air-columnsare activated:pluckingof stringsby fingersorplectra and stimulationof air-columnsby the actionof mouth-pieces27.Ptolemyanticipatesthis scientificmethod of dealingwith acousticalphenomenawhen he construesthe active percussiveagent in wind instrumentsto be the mouthToiOnirXtrovTo;(DORING 9, 2- 4), and that in piece: 6(pokpiou,TOUT.OTI stringsto be the act of pluckingthe string:xpou6vTov (DURING 7, 4)28. Each of these mechanismsis understoodby him to be the initiatorwhich sets into motion (apXfivTfg xit 'o?Cog;DURING 6, 22) the entirevibratingsystem,be it air-columnor string, and whichresultsin the productionof the characteristic sound. In Harm. 1, 3 (DURING 6, 14- 22), Ptolemy associateswith these several typesof percussionthreepossiblecausesof pitchvariation:1) forceexertedby the active percussiveagent (Pjv ToO5it7ttOVT0o j3av); 2) distance (,nrv betweenthe active and passiveentities; 3) corporealcompositionof d7to%XTv) the passivereceptor(dt,i oawxtoxa'gouoTdasl.;ToU... nknItopEvou) and of 25 1. DORING, Ptolemaios and Porphyrios ilber die Musik, Gbteborg 1934, 149. 26 Loc. cit.: >)furdie Reaktion im Aulos oder in der Luftrohre wurde man eher 'Stol3' sagenx<. By referring nITyis to the active principle in percussion, Ptolemy evidently followed the analysis n otoovaa. 6to xai a&uvcvTov tv6g 6vto of Aristotle De an. 419b 10- 12: nlXlyfl ydp Aottv yeveoeat g6oqov. ETEpOV -Y&pTO6rTotov xati T6 uTc6gTE;vov. 27 See C. A. TAYLOR, The Physics of Musical Sounds, London 1965, 89; A. H. BENADE, Horns, Strings and Harmony, New York 1960, 39f.; 62; 142 f.; 165f. 28 K. SCHLESINGER,The Greek Aulos, London 1939, 71, believes that the mouth-piece, must refer to the single )>beating-reed by Ptolemy as TOV irXi-ttovTo;, of the Aulosu(. This interpretation of TOV nlTXrrovro4, >the beating thing<<, as she translates it, is then claimed to constitute the only literary reference to the existence of a single 6(pokXiou, described mouthpiece reed Aulos. Since SCHLESINGER'Sproposal rests upon a forced interpretation, lemy can hardly be treated as the evidence SCHLESINGERalleges. the passage in Pto-

the active agent (TOi 6t' ot5 ' iTXI )29. In characteristically difficult language, Ptolemy approachesthe problemof pitch variationas follows30: A differenceobtainsbetweensounds,as betweenall otherthings,which is qualitativeand quantitative.It is impossibleproperlyto demonstrate underwhich of these aforesaidcategorieshighnessand lownessshould be subsumeduntil we examinethe causesof such an occurrence.These causes seem to me to be factorswhich are somehowintegralto the differencesarisingfrom percussionof diversetypes. For the sensationsresultingfrom these varioustypes of percussiondiffer with respectto the force of the agentexecutingthe blow, the corporealcompositionof the item struckand of that by whichthe blow is executed,and, further,the distanceof the item struckfrom the point of originof the motion. Of the three factorsadducedby him for theirpossiblebearingupon pitchvariation,Ptolemyrejectsforceon the groundsthat it produceschangesin loudness only, having no effect upon the pitch at all (DURING 6, 27-7, 1): ,iia p?YE8ou;&v ( tiVEl j6vov ccizic xt i oCx 6,TTITo;, q 3Ip6Tnro. Further, the corporealcomposition of the agent of excitation is mentionedby Ptolemy only insofaras it concernsthe qualityof the soundsproducedby windinstruments. For inasmuchas mouth-piecesof wind instrumentsare reckonedby him to be percussiveagents31,he ascribesdifferencesin sound qualityto differences in the mouth-pieces employed - reeds and lips, for example (DURING 7, 1 1): olov Tov yXCoaawv xaci tcov cyTogto cov. Apart from this reference to mouth-pieces,the compositionof the active percussiveagent (the initiator) enters not at all into Ptolemy's analysis. The remainingfactors - compositionof the passiveagentand distance - are therebydeterminedas responsiblefor pitch variation. The factor, distance(dQtoX',&ctao-rao,),is treatedby Ptolemyonly as it pertainsto string and wind instruments;that is, in speakingof the distance betweenthe object struckand the point of origin of the motion, Ptolemyis not consideringthe distanceobtainingin the case of simpleor primitivepercussion, as, for example,that occurringon the impactof a hammerupon an anvil. For distancein a percussionof this type would have no influenceupon 29 The material of the active agent in percussion is treated as a causative factor inasmuch as it must be rigid enough for sound to be produced at all. Cf. Aristotle De an. 419b 6-8; Porphyry In Ptol. Harm. (DURING 39, 4- 5). In discussing this problem we follow Porphyry In Ptol. Harm. (DURING 38, 23 - 24) in treating the active agent in conjunction with the passive receptor, thus as constituting a single causative factor. 30 The infelicities in Ptolemy's style throughout the treatise appear to result more often than not from his grappling with the inadequacies of the language. Cf. R. P. WINNINGTON-INGRAM, Mode in Ancient Greek Music, Cambridge 1936, 67. 31 See p. 213 above.

the pitch produced, but only upon the force exerted32. Rather, Ptolemy intends distancein the case of pluckedstrings, wherethe distanceextendsbetweenthe fixed bridgeand the movablebridgeof a Monochord,for example. In this situation,the lesserthe distancebetweenthesebridges,the higheris the pitch produced(DORING 8, 27-9, 2)33. In respectto wind instruments,the same principleobtainsbetweenthe mouth-pieceand the fingerholes. For the sounds thus producedgrow progressivelyhigherin pitch as the finger holes are succesivelyopened in the directionof the mouth-piece(DURING 9, 1 -4), this operationbeing equivalentto shorteningthe length of the air-column34. On this basis, the factor, distance,and hencemeasurablelength,is determined by Ptolemyto be an attributeof quantity,and this quantityis specificallycorrelatedwith pitch, whosehighnessand lownessis found to be reciprocallyproportionalto the distance(DURING 8, 15-21)35: On this basis, it appearsthat the differencebetweensoundswith respect to highness and lowness is a species of quantity (too65TllTo4 ?i60og sIvcti Tt) and, moreover,dependentupon the inequalityof the distancesbetweenthe item affectedby the percussion(nrowcpt nltXrtrogvou) and the agent executing the percussion (T.oi37XntttovTo;).For quite clearly, the differencein pitch is implicatedin the size of these distances.Highness of pitch is consequenton the lesserdistancesowing to the vigor (of motion) issuingfrom a more limiteddimension;lownessof pitch is consequent on the greaterdistancesowing to the slackening(of motion) issuing from a largerdimension.The resultis that soundsare in a reciprocal proportionto the distances. Ptolemy next considerspercussionin its relationto the factor, corporeal composition(DURING7, 5-8): The difference between sounds which proceeds from the means by which percussionsare executeddependsthen on the primarysubstance of the corpus(Tdg npwt Tagro) o4toaToSouatdast;), that is, on those elementsaccordingto whicheach thingis accountedto be rareor dense, thin or thick, smooth or rough. 32 See Porphyry In Ptol. Harm. (DURING 53, 11 - 14), who explains that distance, as it pertains to simple percussion, is a factor inseparable from force and hence has no bearing upon pitch variation. 33 Cf. Porphyry In Ptol. Harm. (DURING 54, 15-21). 34 Cf. Porphyry In Ptol. Harm. (DURING 54, 21-26; 55, 20-22). See SCHLESINGER (n. 28 above), 43 - 44. 35 At the same time the sounds are inversely proportionate to the speed of motion. Cf. Nicomachus Harmonikon Enchiridion 4 (JAN 244, 6- 11). See DORING (n. 25 above), 151 on 8, 17.

Accordingly, Ptolemy argues that the corporeal material of t6 ntX1t-r6givov ultimately determines not only the timbre of the sound produced, but the pitch as well. Thus the substance of the item struck is referred by him to the categories of quality and quantity alike36. For the material attributes of T6 are, he says, not only responsible for the characteristic sounds XIT?rot-r6pcvov (timbres) produced, as for example (DURING 7, 13): TtaWYoi XaCi O1not Xaci (pwvCIixcii xXayycii xaci rnpicple oac ToMiuTC- but also for their inherent pitch range. He thereupon explains (DURING 7, 29) that striking upon metal produces a higher pitch than striking upon wood. As we have seen, the material attributes which bear upon the pitch and timbre of the sounds produced are recognized by Ptolemy to be density and rarity, thickness and thinness, roughness and smoothness. Of these attributes, roughness and smoothness are referred by him to the category, quality, inasmuch as their effect on the sounds produced is specifically qualitative and imsicii material to pitch change (DURING 7, 17): Oti xcai ciUtiai ntot6Ttg xupito. But since density and rarity, thickness and thinness, admit of quantitative differences, and these differences have a direct bearing on the highness or lowness of the sound, it is to these quantitative factors that pitch must be referred (DURING 7, 22 - 31) 37: For that which is denser contains a greater mass of substance distributed within the same radius; and that which is thicker contains a greater amount of the same material distributed over the same length. The more compact and the thinner the item, the higher is the pitch produced; the more rarified and the thicker the item, the lower is the pitch produced. In other words, the higher is said to be such by virtue of its association with the thinner item, just as the duller is associated with the thicker item. For the thinner items produce a more concentrated percussion in proportion to their ability to be penetrated more quickly; while the denser items produce a more concentrated percussion in proportion to their greater mass. For this reason bronze produces a higher pitch than wood; and a lyre string produces a higher pitch than a thread, for the former items are of a denser material. In the foregoing passage, Ptolemy refers the pitch difference that obtains between two objects to quantitative factors - greater and lesser density and thickness - in the material of which those objects are composed. Thus, the greater the density of the material, the higher is the pitch produced; on the other hand, however, the greater the thickness of the material, the lower is the 36 Cf. Porphyry In Ptol. Harm. (DURING,44, 5 -9). 37 Thus Porphyry In Ptol. Harm. (DORING 44, 9- 10): 6qiXov oov i; p06yyWv Vai f O'6VTn; ciC TriV nOG6Ta &\'UX8n?Tar. 1 PapJkiplrt6)v

pitch produced 38. In this analysis pitch is shown to be a function of factors that inhere in the material properties of an object. Later in the chapter (DURING 7, 30- 8, 8), Ptolemy shows that there is another respect in which quantity affects pitch. As we shall see, this respect concerns not only the nature of the material of which an object is composed, but also the size and shape which that material assumes or is made to assume. Before explaining how changes in the size and shape of an object may yield variation in pitch, Ptolemy first considers the basic problem of inherent pitch, i. e., that pitch produced on percussion of a single object without any alteration in its size or shape. Such an object, Ptolemy argues, produces when struck a single and undeviating pitch, which is itself a consequence of the continuous and unchanging compression and rarefaction generated in the surrounding air by the percussion (DURING 8, 12- 14)39. The inherent pitch of an affected object is described by him accordingly (DORING 6, 24-27): The difference between sounds construed with respect to the composition of the object struck would be negligible (ov66wog) or at least imperceptible (oux acioOrln'ys) owing to the fact that the change in air motion remains constant in its progress to the ear. By attributing the inherent pitch of an object to its material composition and to the unchanging periodic motion produced in the air on its percussion, Ptolemy expresses what modern acousticians term free vibration. Thus, WOOD40 Any source of sound if set into vibration and left to itself vibrates in its own natural frequency, producing a note which gradually dies away as the vibrations decrease, but remains constant in pitch [italics mine]. In order for an individual object to produce on percussion a change in its natural pitch, it is required that some alteration be made in its corporeal composition. As Ptolemy explains it, a deviation in any one of the three factors adduced by him will cause a consequent change in the sound (DOJRING 6, 22- 24)41: 38 MERSENNE (n. 2 above), Third Book of String Instruments, Ninth Rule, 178, demonstrated this effect on pitch using strings of brass, gut, steel, gold and silver. Applying the same amount of tension to strings of different density, he found that the string of less dense material (as, e.g., gold) produced a lower pitch than that of more dense material (as, e.g., steel). To bring such strings into unison he computed the amount of tension that had to be exerted on the string of lesser density. The effect of thickness on pitch is explained by him in his Fifth Rule (177), in which he computes the amount of tension required to bring strings of unequal thickness into unison. 39 On this physical definition of sound, see DORING (n. 25 above), 150- 151 on 8, 12. Cf. Aristotle De an. 419b 21 -27. 40 WOOD (n. 2 above), 23. 41 See p. 214 above.

For clearly,assumingotherthingsto be equal,eachof the aforesaidfactors impartsits own peculiarpropertyto the sensation,wheneverand in whatever respect it may deviate (6'rav au5t6 Sl?vtyxi xa6' 6vttva ouv tp6iov). Thus, he says, a deviationin force producesa changein loudness(DURING7, 1- 5); a deviationin distance, as in stringlength or air-column,producesa changein pitch (DURING8, 18ff.); and a deviationin the densityor thickness in the substance of 'r6 ntX1tt6Rivov also produces a change in pitch (DURING 7, 5 ff.). This latter type of change and its effect on pitch is describedby WOODin connectionwith the single bar of a xylophone42: The barsaremadeof hardwood, and are hit withhammers... If we lessen the stiffness we tend to slow the vibrationsand flattenthe pitch of the note. On the otherhand, if we thin the bar at its ends we reducethe masswhichthe stiffnesshas to move, and so speedup the vibrationsand sharpenthe pitch. This same effect is describedby Ptolemywith surpassingclarity(DURING),7, 30-8, 8): In the case of bronze items of similarand equal density, that which is thinnerproducesthe higherpitch; whilein the case of stringsof similar densityand equallength,the thinnerproducethe higherpitches;hollow vessels producehigherpitchesand, moreover,the denserand the thinner the air channels,the higherare the pitches. Each of these instances of higherpitch occurs not specificallybecauseof the densityand thinness itself, but because of the elasticity (68t lr sC)'rovov),seeing that items of such composition happen to be more elastic43. And the more elastic the item, the more agitatedit becomes on percussionand this produces a greater condensation and a higher pitch. Wherefore, if somethingis otherwisemore elastic, as, for example,if it be stiffer or generallymore(compact),it producesa higherpitch, since, inasmuchas both attributesare able to producea similarresult [i.e., higherpitch], an excesson eitherstandardof proportionis the prevailingfactor. By referringthe originof sound to nkiy in its activeand passiveaspects and proceedingtherefromto an analysis of the propertiesof vibratingsystems, Ptolemy accuratelydeterminedthe principlesunderlyingall vibration, and thereforeall sound - meansof excitation(n6 7tXfiTTov),comprehending striking,blowingand plucking,and motionof the affectedobjectconditioned by its materialproperties.His approachto the study of sound is one that is 42 WOOD (n. 2 above), 148. 43 On the meaning of sTrovov, see DURING (n. 25 above), 150 on 8, 2.

sharedby moderntheorists,who commencetheir analysesfrom an identical point of departure44: Whatkind of bodiesvibrate?Let us startwith solid matterand consider first of all 'three-dimensional'objects, that is, objects in which no one dimension is of a different order of magnitude from the others ... If they are made of hard material - glass or steel - it is possiblethat a note of specificpitch may be emittedif the object is struckwith a hammer; the characteristic'ring' of an anvil is perhapsan example. The notes of such objects are usuallyof relativelyhigh pitch and, since it is not easy to change either the shape or the state of stress, the pitch is fixed. Ptolemy's recognitionof the principalfactors in the productionof pitch by percussion - r6 nXrtll6g?vov and its materialproperties- testifies not only to his own insightinto the natureof sound; it revealsalso the fallacyunderlyingthe accountin whichNicomachusdescribesthe activitiesof Pythagoras in the smithy45.Fromthis initialfallacythereproceedthe furtherabsurdities in Nicomachus'saccount. On one issue only is Nicomachusin agreement with Ptolemy - the negligibleeffect on pitch producedby the factor force46. For he has Pythagorasreject force (HarmonikonEnchiridion6; JAN 246, 11- 18), presumablybecause it affects only the loudness of the sound (cf. HarmonikonEnchiridion4; JAN 243, 5-8). On the other hand, however, that factorwhichscarcelyentersinto Ptolemy'sanalysisof soundgeneration, namely the agent of percussionor r6 niXflrtov47,becomes in Nicomachus's account the source from which Pythagorasderivedthe pertinentnumerical data. Thus the smith's hammersor nt kiXi'ttovta,whose function, as initiators of sound, is purelytrivialin percussion,are advancedby Nicomachusin place of the anvil or T6 intXiltro'.Vov, the actual sourceof the sound. 111. iaoic On the basis of the data obtainedfrom the smith'shammers,Pythagoras, we are told by Nicomachus,devisedthe ingeniousexperimenton stringswhich led him to postulatethat the musicalratiosweredirectlyproportionateto the weightsof the hammersstrikingupon the anvil48.The fact is, however,that 44 TAYLOR (n. 27 above), 8 -9. 45 See pp. 210-212 above. 46 See pp. 214 above. 47 See p. 209f. above. See p. 210 above.

the experiment described by Nicomachus would not produce the results he adduces, since strings stretched by weights in the proportions of 6:8:9:12 would not yield the consonant intervals at all. Inasmuch, therefore, as Nicomachus' account of this experiment with weighted strings was accepted without demur by the numerous ancient authorities who transmitted it - whether verbatim or with minor variations - down to the time of Boethius49, it is a safe inference that none of these writers attempted to reconstruct the experiment; had they done so, they would have been unable to duplicate the findings reported by Nicomachus50. For because the frequency of vibration productive of a string's pitch is directly proportional to the square root of the tension, the tension would have to be quadrupled (not doubled) in order to raise the pitch to the higher octave note. Thus the fact that none of the writers who transmitted the story of Pythagoras's experiment with weighted strings expressed even a modicum of skepticism at the results purportedly derived from it has led scholars to conclude that the ancients were ignorant of the behavior of strings under tension If, as Nicomachus has it, two strings are suspended by weights whose ratios are 2: 1, they would be subjected to tensions in a double ratio. But the interval yielded by their respective pitches would not only be less than an Octave, it would also be less than a Fifth and greater than a Fourth. Thus, if one were to reconstruct the experiment described by Nicomachus, he would be struck by the incompatibility between the intervals produced and the ratio of the weights. It was not until the seventeenth century, however, that the factor, tension, was shown, by MERSENNE, to be the source of this disparity. MERSENNE traced the disparity to the peculiar relation obtaining between tension 49 The account of lamblichus De vita Pyth. 115 - 120 (DEUBNER 66, 12- 69, 5) is taken verbatim from Nicomachus. An abridged version appears in lamblichus In Nic. Arithm. Intro. 171 - 176 (PISTELLI 121, 15- 125, 13). Similar versions are given by Gaudentius Harmonike Isagoge 11 (JAN 340, 4- 341, 25); Censorinus De Die Natali 10 (HULTSCH17, 19- 19, 2); Macrobius Comm. in Somnium Scip. 2, 1, 8 - 14 (WILLIS 96, 16 - 97, 25); Fulgentius Mitologiae 3, 9 (HELM 75); Chalcidius Comm. in Platonis Tim. 45 (WASZINCK93 - 94); Boethius De inst. mus. 1, 10- l 1 (FRIEDLEIN196- 198). Cf. Plutarch De an. proc. in Tim. 1021a; Aristides Quintilianus De mus. 3, 1 (WINNINGTON-INGRAM94, 11-95, 7); Porphyry In Ptol. Harm. (DURING 119, 29-120, 7); isadore Etym. 3, 16, 1 (LINDSAY); Hagiopolites (VINCENT,Notices, 266-268). 50 As numerous scholars have observed of this experiment, the ratio 6:12 does not correctly express the relation between a string's pitch and the amount of tension required to produce the higher octave note. See TH. H. MARTIN, letudes sur le Timee de Platon, Paris 1841, vol. 1, 391; A. E. CH UIGNET,Pythagore et la philosophie pythagoricienne, Paris 1873, vol. 2, 134; JAN (n. 4 above), 141; BURNET(n. 13 above), 106; B. L. VANDERWAERDEN,Die Harmonielehre der Pythagoreer, Hermes 78, 1943, 179ff.; GUTHRIE (n. 7 above), 223-224; BURKERT(n. 7 above), 375 - 376; CHERNISS(n. 5 above), 307, note b. 51 See BURKERT(n. 7 above), 376, note 24.

and frequency,wherethe latteris the squareroot of the former52. This relation requiresthat the Octavebe expressedby the ratio 4:1, the Fifth by 9:4, the Fourthby 16:9. In order,therefore,to raisethe pitch of a stringto an Octave, the tensionexertedupon it must be four times greaterthan that productive of the lowernote; to producea Fifth, the tensionmust be 21/4times greater; to producea Fourth,it mustbe 17/9timesgreater53.Thus, the experiment of Pythagoras,in which the tension is appliedto four stringsin the proportions of 6:8:9:12, would producea seriesof nondescriptintervals. In light of the unanimitywith which the ancients accepted the results derivedfrom Pythagoras'sexperimentwith weightedstrings, Ptolemy's unequivocalrejectionof these same resultsmust be accountedas singular.No less remarkableis the censurewhichhe appliesto those scholarswho accepted the resultsof such experiments(Harm. 1, 8; DURING 16, 33- 17, 2): Representationsof the aforesaid[sc. ratios]using auloi and syrinxesor stringssuspendedby weightsmustbe rejected.For suchdemonstrations cannotattainto completeaccuracy,but ratherintroducegroundsfor indicting the investigators (atXkaX 8tacoXfi; paXXov 'titowtcv 6(popgiai tolg ltlpWitvot4). Ptolemy's reasons for rejectingthe wind instruments,auloi and syrinxes,as proper vehicles for experimentationare the same as those lodged against weightedstrings - both methodsintroduceso manyvariablesinto the experiments that accurateresultscannotbe expected 4. EventhoughPtolemydid take this strongexceptionto the experimentwith weightedstrings as describedby Nicomachus, and even though he was the only ancientauthorityto have done so, his criticismis not consideredby scholars to reflectany advancedknowledgeon his part respectingthe behaviorof strings under tension. Rather, his rejectionof this particularexperimentis judged importantonly as implyingthe priorityof Nicomachus'account55. Settingaside the questionof priority,however,thereis noticeablein the language which Ptolemy uses a vehemencethat is disproportionatefor the mere expressionof dissatisfactionwith the experiment.For Ptolemynot only repudiatesthe findingsadduced;he also directshis attackad homines.That Ptolemy would have expressedhimself in this peremptorymannerwithout good and sufficientreasonsseemsunlikely.It will be arguedhere that lying behind 52 MERSENNE (n. 2 above), First Book of String Instruments, Prop. XVI, 59; Third Book of String Instruments, Prop. VIl, Sixth Rule, 177. 53 See WOOD (n. 2 above), 91. 54 For experiments conducted by the Pythagoreans on wind instruments see Plutarch De an. proc. in Tim. 1021a; Porphyry In Ptol. Harm. (DURING 119, 13 ff.). Cf. Athenaeus 4, 184e. SS Cf. BURKERT(n. 7 above), 376.

the reasonshe in fact offers was a greaterknowledgeof the problemthan has hithertobeen attributedto him. Of the variousextant accountsof Pythagoras'experimentwith weighted strings,only two can be placedin the secondcenturyA. D., the periodof Ptolemy's activities - those of Adrastusthe Peripateticapud Theon of Smyrna (HILLER57, 1- 4; cf. 60, 7 - 8; 66, 22 - 23) and of NicomachusHarmonikon Enchiridion6 (JAN 245, 21 - 248, 26). To these accountsshouldbe addedthe commentsof Theon on the factor tension (HILLER65, 10- 24). Since other versionsnow lost may have been currentat the time of Ptolemy and hence knownto him, it is possiblealso that his polemicwas directedagainstthem. As mattersstand, however,the only availableevidenceagainstwhich Ptolemy's argumentsmay be weighedis that providedby Adrastusand Nicomachus, other accountscoming from later than Ptolemy'sera56. Adrastusomits the storyof Pythagoras'sobservationsin the smithy,referring only to Pythagoras's >>ratherwell-known experimentwith weighted strings<( (HILLER 57, 4): yvcopti6spov xaTd Tiv tgaptrIcv twOV kap6v. Apart from this reference,Adrastusoffers no details about the mannerin which the experimentwas conducted,remarkingonly that tension was introduced, as it was in the case of an experimentin whichtuningpegs wereused (HILLER 57, 2 -3): rig ta&@o; YtvoRtv1l; xat-id TfV oTrpo(piv tCV xoXXap1Vo57. This leaves Nicomachus'detailedaccount as a likely sourceof information for Ptolemy. In fact, the variouspoints of contact betweenNicomachus'saccountand Ptolemy'scriticismsuggestverystronglythat Nicomachus may indeed have provided Ptolemy with the basis on which to frame his argument. In the passagecitedabove(pp.210), Nicomachusdescribedin detailthe experimentof Pythagorasin whichfour weightedstringsweresuspendedfroma single stake. The purposeof this single stake was, he explained,to provide againstthe introductionof variableswhichwouldderivefrom the use of multiple stakes (JAN 246, 23-247, 1): tva RN xdx to6'ro) 8tl(popa tI ;no(peiviltattfl6top i?ovoftatl waodov i8tczovtcwvirapacty. Accordingly,a single stake was implantedin the walls58, from it the four stringswere suspendedeach by each (txa'tllv t(p' txaorllg), and a weight was attachedto the free-hangingend of each string.Evidently,Nicomachuswas underthe impressionthat the behaviorof the stringsthus weightedwould have been in56 See note 49 above. S7 Such an experiment is mentioned also by Nicomachus Harmonikon Enchiridion 6 (JAN 248, 10 - 13). 58 The stake was presumably implanted into the opposite angles formed by the four walls, thus occupying the position of an hypotenuse. Accordingly, M. MEIBOM,Antiquae Musicae Auctores Septem, Amsterdam 1652, Notae ad Nic., 47; Ab uno angulo transversum ad alium. Cf. Aristides Quintilianus De mus. 3, 3 (WINNINGTON-INGRAM99, 1 - 7).

fluencedvariablyif suspendedfrom multiplestakes.Therefore,a singlestake, by rulingout this variability,would guaranteethe validityof the final results. The fact is, however, that the single stake neitherprovidesagainstvariables nor influencesthe behaviorof the stringsat all. It merelyaffords a rigidsupport for the strings.In orderproperlyto conductsuch an experiment,the researchermust focus upon the stringsthemselvesand not upon the stake, for it is exclusivelyfrom the weightedstringsthat the resultsmust be ultimatelyderived. Significantly, Ptolemy commenceshis critique by arguing this very point; moreover,his statementto this effect appearsto addressthe text of Nicomachus(DORING 17, 7 -12): In the case of the experimentin which weightsare attachedto strings, thereis no way at all to maintainthe stringsin a relationto one another in which there are no variables(Ahi 8taocoEogvov cnapaaXxtrwv d&itlkat; navtadiacn xxv xop&bv);when indeed the task is to discover how each individual string behaves by itself (6n0r6? xai ztp6;S acirfv O ~dteiovollViow; XouCav F.6pEXvtpyov). Nor will it be possible any longer for the ratio of the weights to harmonizewith the sounds produced by the strings (ou5xtrt 5uvac6v tonrat toI)g tiv ?(papi6oalt toiCt ytvopttvot; 6t' w5tCov x6(po) Piapcov k6youg even by having denser and thinner strings under the same tensions produce higher sounds. The beginningof Ptolemy's statementabove apparentlyhas referenceto the claimof Nicomachusthatthe use of a singlestakein the experimentresults in the elimination of variables. Ptolemy points out that no such result is achieved. For the fact is that when strings are stretchedperpendicularlyby weights,as Nicomachusdescribed,theirreactionis not a functionof the support from whichthey are suspendedbut, rather,of the weightsby whichthey are stretched.And since the diverseweightsaffect each string in a different way, the experimentis influencedat the veryoutset by the introductionof variables. These variablesaccount for the inconsistencybetweenthe intervals producedby the stringsand the ratio of the weights. In order to isolate the factorresponsiblefor this inconsistency,it is necessary,accordingto Ptolemy, to studythe behaviorof each individualstring.Testingwhetherthe inconsistency could be due to the dimensionsof the strings, Ptolemy first variedthe thicknessand density of the stringsunderthe same tensions and found that the inconsistencypersisted.Ptolemythereforeconcludedthat the pitchesproduced must be referredto some factor other than the dimensions of the strings. In discreditingthe experimentwith weightedstringsby showingthat variablesare indeed introduced,Ptolemy was in effect counteringNicomachus's claim concerningthe eliminationof 7rapackkay' with his own gt ... &utapax?a.xtov, while at the sametime treatingNicomachus'ssubordination

of each individual string (txa6arTv ?(P' txdorlg) to the domination of the supporting stake as wholly inappropriate (txa-rfiv ... ci6pC1v Epyov). These correspondences in language between Ptolemy's argument and the text of Nicomachus seem too pointed to be fortuitous. Ptolemy next appears to address himself to Nicomachus's stipulations concerning the lengths of the strings used by Pythagoras. While Nicomachus asserts that the lengths were identical, it is hard to determine from his account whether they were equal before or after the application of the weights (JAN 247, 1-7): [From the stake] he suspended four strings of the same material, made of an equal number of strands, equal in thickness and of the same torsion. He hung them up each by each in succession, having attached a weight to the bottom end of each. And having contrived that the lengths of the strings were in every respect absolutely equal, he thereupon plucked simultaneously two strings in alternation and found the aforementioned consonances. Nicomachus appears to be saying that the four free-hanging strings, each weighted by a different unit of measure, were somehow equalized in length after the attachment of the weights. It is possible, however, that Nicomachus means that the strings were to be equal in length before the attachment of the weights - the order of the steps in his account being non-significant. In this case, however, the problem arises that four strings of equal length would, when submitted to four different weights, be stretched to four different lengths59. If, on the other hand, Nicomachus indeed meant that the strings turned out to be equal in length after the attachment of the weights, this would require that they were initially unequal in length and, moreover, that their inequalities were in the correct proportions so as to allow for their all being pulled by the different weights to the same length. It is unlikely, however, that Pythagoras had determined the lengths of string required for this result before commencing the experiment. For in order to do this he would have had to know those facts which could only have been derived from the experiment itself. In any case, Ptolemy's critique makes it evident that the experiment was to have commenced with strings of equal length (DURING 17, 12 -16): Even if one were to assume a priori that this were possible and, further, that the lengths of the strings were equal, the heavier the weight, the more will it increase the extension of the string attached to it owing to 59 The elongation would be proportional to the downward force exerted by the weight, according to Hooke's Law, for which see M. Y. COLBY, Sound Waves and Acoustics, New York

the greater tension (Tfi n)tXiovt atost); and the more dense will it make the string (tuxvoXVoFtgta?Xov). The result of this experiment therefore is that the rise in the sounds turns out to be inconsistent with the ratio of the weights (nap t6v A.6yovr6vT capcov). This statement, by assuming that the strings were of equal length before the application of the weights, is highly useful for the interpretation of Nicomachus's text. Of more significance, however, is the basis upon which it refutes the claims made by Nicomachus. For to argue, as Ptolemy does, that strings of equal length cannot remain equal after the attachment of the weights, is to challenge Nicomachus's representation of the experiment at its outset. And to assert, as Ptolemy does, that the pitches produced by such weighted strings do not harmonize with the ratio of the weights is to repudiate the final results of the experiment. The burden of Ptolemy's argument then was to substantiate this proposition, namely, that the ratio of the weights does not accurately express the intervals produced by the strings (DORING 17, 9-11). The inconsistency noted by Ptolemy between the ratios of the weights (6:8:9:12) and the intervals produced by the four strings derives from Nicomachus's assumption that the musical ratios which are embodied in the weights are necessarily parallelled by the tensions on the strings. In reality, of course, these ratios express string length proportions, not tension60. The assumption is evident in the following passage, where Nicomachus says (JAN 247, 7-9): >>He[sc. Pythagoras] found that the string stretched (tctvoFiv11v) by the greatest weight produced, when compared with that stretched by the smallest, an Octave.(( On the basis that the heaviest weight exerted a tension on the string equivalent to the unit of weight (12), thereby producing the highest note, the largest number in the series is assigned to the higher note of the Octave, Nete, the smallest number assigned to the lower note, Hypate (JAN 248, 19 - 23). This is in effect to assume that the rise in pitch is consistent with the ratio of the weights and the ratio of the weights is in direct proportion to the tension on the strings. Therefore the ratios 6:8:9:12 are determined by Nicomachus to represent the tension required to produce the rise in pitch from Hypate through the consonances, Mese and Paramese, culminating in Nete6 . A similarly misleading assumption is made by Theon who, in rationalizing these numerical assignments in which the larger number is allotted to the higher note, evidently intuited that a difference obtains between the ratio of the weights and the force of the tension. But by arguing that this difference was somehow mutually compensatory, he convinces himself that the final rew See p. 208 above. 61 See p. 210 above.

sults of the experiment with weighted strings are consistent with those derived from experiments based upon string length dimensions (HILLER 65, 10- 24): He [sc. Plato] says that it is proper to assign the larger numbers to the lower notes even though this appears not to accord with certain cases involving tension, as, for example, when tension is introduced by the suspension of weights. For when two strings are equal in length and thickness and are alike in other respects as well, the greater weight will produce a higher note because of the greater tension. For since the greater weight produces a greater tension, it adds extrinsically a greater force to the higher note, which note embodies a force peculiar to itself that is less than the weight. Conversely, it is clear that a lower note, by possessing a force peculiar to itself that is greater than the weight, suffices to preserve the proper harmonia and consonance. So that the larger number must be assigned to the greater force. And there is agreement between these and other factors. Theon here apparently means that the two weighted strings produced an Octave (dipiiovica)in a double ratio (cf. HILLER 58, 3 -4), the higher note of which he refers to the heavier weight62. He then proceeds to argue that an increase in the other factors - length and thickness (HILLER 65, 4 - 66, 5) produces a result that is the reciprocal of that induced by tension. Accordingly, he says that the assignment of the larger number to the higher note must be based upon the added force exerted by tension to the intrinsic force of the higher pitch, while the assignment of the larger number to the lower note is based upon the concomitantly greater intrinsic force of the lower pitch over the increase in length and thickness. The illogicality of Theon's argument derives from his according to pitch an intrinsic force (XTiviSiav ioXOv),which he treats as independent of the factors producing it. Theon's tortuous attempt to provide a theoretical basis for numerical assignments derived from tension proceeds on the implicit acceptance of the experiment with weighted strings. And it is evidently against construals of this sort that Ptolemy levelled the charge of &ta4o?i' (DURING 17, 2). For by so expressing himself, Ptolemy was in effect taxing the experimenters with a total disregard for scientific procedures, the primary task of which was to provide against the introduction of uncontrolled variables (DURING 17, 19- 20): tpyw8oug 6v-rog icdvu toi5 trnptiv ... dtnpaU)aiLav. Thus he argued that there was no way in which strings of equal length and density could be made 62 According to Nicomachus Harmonikon Enchiridion 9 (JAN 252, 5 - 6) &ptovia was the Pythagorean term for Octave (td6 8a n,aoCov).Cf. Philolaus Vors. 44B 6 (DIELS-KRANZ Nicomachus Harmonikon Enchiridion 9 (JAN252, 17- 18).

to remainidenticalin these respectsonce the differentweightswereattached to them. For in everycase these dimensionswere increasedas the weightsbecame heavier (DORING 17, 13- 15). Therefore, although the difference in weightsplayeda role in producingthe pitchvariation,that difference,in view of the inconsistencybetweenthe weight ratios and the pitch intervals,could not be the determiningfactor. For the ultimateand realcauseof the inconsistency was that the tension exertedon the stringsby the weightsin these ratios was insufficientto producethe postulatedintervals.Facedwith this inconsistency, which he attributedto the as yet undeterminedpropertiesof tension, Ptolemy advocatedthe use of the Canon63in preferenceto experimentswith weightedstrings(DURING17, 20-24): A stringstretchedover a so-calledcanonwill demonstrateto us moreaccuratelyand with greateraccessibilitythe ratiosof the consonances,not by admittingtension on a randombasis (od'Pv b; 9tuxuekapoiocta Pv tdatv) but first by providingthroughpreliminarytests againstthe anomaly that would resultfrom the preparationof the experiment. The inferencefrom this statementis that Ptolemy conductedthe experiment with weightedstringsand was unableto duplicatethe findingsadvancedby Nicomachus;that in place of the expectedconsonancesassertedby Nicomachus, he found an &vcogaXia. The consonanceswhich Ptolemyexpectedto be producedby the weighted stringsare describedthus by him in Harm. 1, 7 (DURING 15, 26- 16, 6): Of the homophones64,the fairestand the most unifiedis the Octave;so that with this homophonethe double ratio is in harmony(?(papgo',6tv) ... After the homophones,the first consonanceswhichmost nearlydivide the Octavein two are the Fifth and the Fourth;so that the Fifth is assumedby the hemioleand the Fourthby the epitrite. Ptolemy then proceededto carryout the experimentin accordancewith the dictates of Nicomachus65.He suspended four strings of identical length, thicknessand density. (These strings,were they to be held stationaryat both ends, would produceunisonsor iso6ovot)66.To these free-hangingstringshe 63 See note 12 above. Homophones are defined by Ptolemy Harm. 1, 7 (DURING 15, 10- 12) as intervals which when sounded simultaneously create in the ear the impression of one note. Such intervals are Octaves and combinations of Octaves (6S oi 8&t4ictctv xai oit FE oUT)Cv ouvn0Eii6vot). 65 See pp. 222- 223 above. 66 Isotones are defined by Ptolemy Harm. 1, 4 (DORING 10, 1 - 2) as sounds >whichare indistinguishable in pitch< from one another.

attached weights in amounts analogous to the musical ratios. When plucked each string indeed produced a pitch which rose in proportion as the weights increased. But the rise in pitch was, as Ptolemy observed, >>inconsistentwith the ratios of the weights(( (DORING 17, 15 - 16). Thus while there was a rise in pitch (iv toi; xgo'(poi; 5i'npoXiiv) from string to string, it was not sufficient to produce the Octave, Fifth and Fourth prescribed by these ratios. Since the intervals thus produced were anomalous (DURING 17, 24), Ptolemy evidently employed strings of different thickness and material (density) in an attempt to correct this deficiency. As expected, the thinner and the denser the strings, the higher were the pitches produced67. But varying these factors did not eliminate the anomaly. For submitting the strings to the same tensions (i. e., weights) as before, Ptolemy found the same anomalous intervals occurring, only at a higher pitch register. Thus he concluded that no accord was possible between the intervals produced and the ratios of the weights even when the pitch range was heightened by the use of thinner and denser strings (DU3RING 17, 11- 12): Tr( xai iwxvoTpaq xvti XrotpCTFpaictVCd (w0tgiq TaGECIv OwuTpou; (pOyyou;g iowEiv. Having eliminated the factors, thickness and density, as responsible for the deviant results, Ptolemy reached a critical stage in the experiment. For the only recourse left to him was alteration of the tension exerted by the weights. But this would have required him to depart from the prescribed ratios. Since, however, he had already observed that the weights as prescribed increased the lengths and density of the strings (DURING 17, 13 - 15), it is a fair assumption that he saw the solution as implicit in the relationship between the weights and the pitches. Thus the problem became for him how to determine what increase of tension and consequent elongation of the strings had to be applied in order to raise the pitches to the required height. Having arrived at this stage in the experiment, Ptolemy was on the brink of discovering the true proportions between tension and pitch variation. The problem facing Ptolemy was the obverse of that dealt with by MERSENNE. For while Ptolemy commenced the experiment with strings of equal length (thus with potential unisons) which he sought to convert into consonances by using weights in the musical ratios, MERSENNE commenced the experiment with strings whose lengths differed in proportion to the musical ratios (thus with potential consonances) which he sought to convert into unisons. Thus the problem for Ptolemy consisted in determining what weights were necessary to make strings of equal length be at the different tensions required for the production of the consonances. For MERSENNE?the difficulty consists simply in knowing what weight is necessary to make the double string be at a tension equal to that of half the length(<68. In other words, MERSENNE'S 67 See pp. 215 - 218 above. 68 MERSENNE(n. 2 above), First Book of String Instruments, Prop. XVI, 59.

task was to raisethe pitchof the longerstringan Octavein orderto producea unison with the shorter string. He discoveredthat a fourfold weight was necessaryto put the string of double length (productiveof the lower octave note) in unison with the shorterstring(productiveof the higheroctave note). He thus concluded69: When, first of all, the stringsare double in length, the force or weight whichdrawsthe doublestringtightmustbe four timesthe weightwhich drawsthe half lengthtight, whenone wishesto put them in unison; and if the string were quadrupledin length, there would be necessarya weight sixteentimes the size, that is to say, which would weigh sixteen times as much as that which tightensthe string one-fourththe length. Thus the lengthfollows the singleratio of lines and of the roots, and the size of the weightsfollows the double ratio of planesor squares. Despitethe differencesin theirapproaches,the issue for both Ptolemyand MERSENNE was the same: how to raisethe pitch of a stringan Octave.For to convertan Octaveinto a unison, as with MERSENNE,and to converta unison into an Octave,as with Ptolemy, the same solutionobtained:the pitchof one string had to be raised an Octave by a requisiteincrease in the tension70. WhetherPtolemy in fact producedthe desiredresults must remainan open question. The fact that he had isolatedthe problem, however,places him at the very thresholdof discovery71. New York FLORAR. LEVIN 69 Loc. cit. 70 See MERSENNE (n. 2 above), Third Book of String Instruments, Prop. VII, First Rule, 176, which pertains to the experiment as conducted by Ptolemy with strings of equal length and thickness. 71 References are made to the following texts: L. DEUBNER, lamblichi De Vita Pythagorica Liber, Leipzig 1937; H. DIELS and W. KRANZ, Die Fragmente der Vorsokratiker 12, Dublin/Zurich 1966; 1. DtORING, Die Harmonielehre des Klaudios Ptolemaios, G5teborg 1930; I. DORING, Porphyrios Kommentar zur Harmonielehre des Ptolemaios, Goteborg 1932; G. FRIEDIEIN, Anicii Manlii Torquati Severini Boetii De Institutione Musica, Leipzig 1867; R. HELM, Fulgentii Mitologiarum Libri Tres, Leipzig 1898; E. HILLER, Theonis Smyrnaei Expositio rerum mathematicarum ad legendum Platonem utilium, Leipzig 1878; FR. HULTSCH, Censorini De Die Natali Liber, Leipzig 1867; G. H. JONKER, The Harmonics of Manuel Bryennius, Groningen 1970; H. PISTELLI, lamblichi In Nicomachi Arithmeticam Introductionem Liber, Leipzig 1894; A. J. H. VINCENT, Notices et extraits des manuscrits de la bibliotheque du Roi et autres bibliotheques, vol. XVI, part 2, Paris 1847; J. H. WASZINK, Chalcidii Commentarii in Platonis Timaeum, London and Leiden 1962; 3. WILLIS, Macrobii Commentarii in Somnium Scipionis, Leipzig 1963; R. P. WINNINGTON-1NGRAM, Aristidis Quintiliani De Musica, Leipzig 1963; W. M. LINDSAY, Isidori Hispalensis Episcopi Etymologiarum sive Originum Libri XX, Oxford 1911.