A Neglected Version of the Anecdote about Pythagoras's Hammer Experiment

:ToOrbE—$CMaeA DeLrITAReES nn ne itpDNAe4rùa:Tsnh3i0ge8t7=RAMEoeuaPTLRHET.SdO!AYNeoMaryEAdtoceRsA,GETDOAe'SRAIKAretdraLe=el sm £ Eats Da: Eh bqhaFutebheaAdtleLyd pri ce ı:Ae*n RAASTEDde J., A neglected version of the an ecdote about Pytha ican mie _ 668 | RARES A NEGLECTED VERSION OF THE ANECDOTE ANA ABOUT PYTHAGORAS'S HAMMER EXPERIMENT. By Jorgen Raasteé, Roskilde. Cr There is a widely spread story about Pythagoras, which describes how by chance he found a means of demonstrating the simple numeric proportions that govern the musical intervals of the fourth, the fifth, and the octave. Our oldest source for this story is Nicomachus from Gerasa (2nd cent. A.D.), and since his time the story has been told again and again, by Greeks and Romans alike. It was one of the few 'facts' about Pythagoras which were remembered in the Middle Ages, cf. the lovely woodcuts in Franchino Gafori' Theorica Musicae from 1492 which illustrate a number of acoustic experiments, including Pythagoras's hanmers * According to Nicomachus, Pythagoras one day passed by a smithy (napd ti xaAnotuneTov) and heard the sound of hammers against iron on an anvil (Énfxouvoe darotripuv alönpov én' &xuovi Parévtwy). In their different sounds he recognized the octave, the fifth, and the fourth. Full of joy he ran into the smithy (td xaAuetov) and tried to find out what produced these intervals. made him realize that the difference in the sounds was produced by the weight of the hammers (napd tov Ev tots patotñpouv Syuov). It was not produced by the strength of the various blows (nap& TÀv tav faréviwv Biav), nor by the shapes of the hammers (tá Oxñuata tHv owupüv), nor by any change of the 1. «vee Èa telle cata mt dies amm te ae i Eloxywyh povowy) est impossible. D'autre Li orme taire de Porphyre, p. + 121,10-14, 1 A aux Harmoni ca de Ptolémée, . res sources, ’ notamment le late] a 17,16-20 permettent d'inférer qu'il y a eu substitutigp de opópa à ne, oot Fee p. 20, * avoir désigné des récipients, synonyme de dloxoı, Ailleurs, opatpat semble “ lémée considérait comme un A Stougsard Jensen/‘Kebenh avn Des A number of experiments made on the spot 2. ("ApuovıOf Nicomachus's musical works, only the short Enchiridion Scriptore s xôv Éyxecpéérov) has been preserved, see C. Jan, Musici Graeci, Lips. 1895, pp. 209-&2. The anecdote (Enchiridion ch.VI, p; uo. 245 sqq. Jan) must also have been included in his lost ECoaywyñ T: 1,10. musica ione institut De in Boethius by told is on; for it dependency of Boethius on Nicomachus's lost treatise has most rece: s a: ly been demonstrated by Calvin Bower in his article on ’Boethiu m: tione institu De of Sources the ing Nichomachus: An Essay Concern sica' (Vivarium 16, 1978, pp. 1-45). See illustration, opposite.

position of iron In the following, four strings of (A tot ÉAauvouévou otôñpou wetadSeorg). Nicomachus tells how Pythagoras now took identical length and structure, down with weights proportional in the smithy fourth, - and was a fifth, thus weighed able and an octave, to produce intervals stringed With this series of blows on pots gular harps instrument Pythagoras was able other acoustic (Aexíbwv xpobotc), etc. - of flutes, monochords, same simple numeric relations. and the Hammers cannot Any physicist can tell us why possibly be story was believed until am not mistaken. experiments - by means of all of which experiments the Now it the true. is quoted who is for said a Pythagoras The physical 1634, is generally explained as a scholion to Plato, story about to have produced Phaedon 108D, fourth, if I legend the is menwhere Aristoxenos early Pythagorean the truth information on a num- One such experiment the trianhim to observe quite recently - ber of Pythagorean experiments. in a led story about based on ancient - and probably sound - tioned the instrument which Pythagoras constructed for purposes to check a the intervals. Hippasos, fifth, and the octave by means of four brazen 6louor. pasos used had all the same diameter, but differed in thick- 1847 A.J.H. Vincent published five extracts from the The disks which Hipness. In musical treatise in the Paris manuscript Ancien fonds grec 360, the so-called Hagiopolites.” Se One of these texts ("Frag- Notices et extraits des manuscrits de la bibliothèque du roi et autres bibliothèques, XVI,2, Paris 1847, pp. 259-281. For the Hagiopolites, see Christian Hannick, Byzantinische Musik (in Herbert Hunger, Die hochsprachliche profane Literatur der Byzantiner, Bd. 2, München 1978 = Byzantinisches Handbuch im Rahmen des Handbuchs der Altertumswissenschaft, 5. Teil, 2. Band), po. 200-01 and my forthcoming article in Texte und Untersuchungen zur Geschichte der Altchristlichen Literatur Bd. 125 ("The Manuscript Tradition of the Hagiopolites: A Preliminary Investigation on Ancien Fonds Grec 360 and Its Sources"), _ NEUDA) a and to demonstrate the simple then goes on with a detailed description of of demonstration. of of agoras anecdote ent v", Vincent pp. 266-73) contains the Pyth other known sour in a form which differs considerably from all | -235r): , m oe them to the weights of the hammers numeric relations that existed between these basic Nicomachus }NWain+3S.cMaEle,r. ces. h{y ipRdedaelAdeslda La | 6 nap *tottov oùv be utv Adyoe ápxalos tav SUpasev' tia noALTınö gino SpnAAovuevog nvdSaydpac napd TLVL NEARwv wie nadelduevog’ nal Brapdepwv fixwv €F abtob &nov to abtob tabta prac GAng o6onc, ths yalnevontvng” nal aútob vin ai nal &vög onevovuge TOD xadmevovros” nal 106 ÉSE mid an Ev Snep fhaavvovto ta yadnevdueva’ mondv sTOrl nal &notekouuévuv fixwv 5Lapopáv S9ev ylvetat tac opalpas 5h NOAAa onontioag nal épeuviioac’ téAog nodeweve Baputéévéounwev’ 4c nal otaduwcac, nat ebpov thy eodaı TÓ ra pav” tiv 6È, xovgottpay, Eyvw Evrebdev npol a TÜV HWVWV fixwv Sıdwopov’ xual Avalöywc TV TE xovedtnt ñuata fivouv THv oparpüiv Ayrınadeiv Baovtnte, nal TÁ énnxels, &idwopa yeyevñoôar. “EF abrob ual aútde napopund è nateoxevacev and yoodav tecodowy nal uóvov 6pyavov, iad uenAnne uovoruriv* elta &veBiBaoev abrá ele éntamoo t | nados è nudayopınöc guidiaog, Ev TLVL NOVAPATLEL” TMEPL npdc tiva yuvatxa nuSayopetav EUTLSÉUOVOS YPÁP ee thc dipuovinfig vıAoooptag, odTw pdonwv * sie -- TA Là?EIpvSNÈOErbSPiedaACDET:i,LlMsGiTPtaEcAoSrI°TiE=+pLtr.nSiDEM,o.OiINejTdsRtcNanTitLliARea+oEubP,!UitDIh5NeyJTLag7oRU:Y5nceaAshE It runs as follows (Hagiop. fol. 234v yn.. ourraßfic 5.' bEela ueliwv tac ovAAaBds Enet the different sun“ In this version we again fina the smithy, , and the nes coming from it, the metal (} vin À xakuevouévn)one hammer (T But in the Hagiopolites version there is only lware ni meta onedoc tó xalnebov), whereas several pieces of s, as usua "o make being forged (tà xaAnevöueva). Pythagoras the differe many experiments and finds out in the end that eren t weight O of sound (higher or lower) was due to the diff s so close to the owbpaı of a ow, the word owalpaı itcomesugg ests, at first, that ta¢ Ov. the standard version that Ò 6) is preservec > B gg oly pe en in and +) 88,1 (p.1 7d 1,21 Ecl. pl , ter Wa to i ding Accor 4). . 252-5 idion (Jan. Pete a gt e ridg Camb in belle Pythagoreanism, pp . Fhi lao s fr. i (= Philo i laos quotation t Philo ently" quotes from Nicom note ll, the Hagiopolites "evidwas to Burkert. is less evident to me than it achus

i=È gaTLAIO Iveè:s iPluel myeEÀt L sca opalpag is nothing but a scribal error for tác owúpacs hammers), However, the (the Hagiopolites says clearly enough that there was only one hammer - or, at least, one type or size of hammer (tod abtob uai Evdg oxnetoug TOD xaAMEÜOVTOC) plural (tàg ovaipac) for that simple reason makes this first explanation impossible. make is sense, xalueuöueva. pat ed in LSJ The only interpretation which does cvaipar to be synonymous with tà to take at The smiths were obviously working on some ovaiof different opalpa mean in size or different weight. this ("ball, context? globe, In look in connection with the Harmonics of scription of introduced pill"). the meanings instead for Ptolemy (2nd eventual cent. the numeric apparatus ing ovatpüv obpıyyeg, Ftolemy continues with to 13th-14th cent.) 198, 14th cent.). (by ovüpar and BPriton found and as Now, a in the the manuscript G is (Vatican gr. (Vatican gr. the best reprein should not be too readily disregarded. lectio into the difficilior which a traditional hammers Besides, owaLpüv is scribe might easily (opupüv). The reading Die Harmonielehre des Klaudios Ptolemaios, herausgegeben von Ingemar Göteberg 1930 (= Göteborgs högskolas ärsskrift XXXVI.1930:1), 16-17. "möglicherweise das Originalexemplar der Redaktion des Gregoras oder eine Abschrift davon" (Düring, Die Harmonielehre... p.LXI1). Cf. Düring's stemma codicum, m, pp. ibid. p. LXIX, and his description of XLVIII-LIV). NL t - in nov elnev Eni to abtob onuarvouévou). The scholias opal par all probability Gregoras himself - thus suggests that be noand 5toxmo. are synonyms. In this connection it is to Proleny ticed that in the text which we have just quoted from nal for round objects of some kind. In his critof V properly used for a body which is oblong and rounded, men ES as he has just used cvatpüv and 6loxwv about the same td el (totéov St. xatà napdiyxpnoLv cvarpiuòv oüna Agyer nal nai star unxec uÉv, TETOPELLEVOV 5é, dc nal dvatepov Ova púv disks, but would be appropriate to connect a pair of synon or disks of differin G ible. so that their surfaces are as round (owarprxds) as poss ace About the use of the adjective ovarptuôg (for the surf imof a cylindrical body), the scholiast says: "Spieritcal is d but with fi; this conjunction can hardly go with hammers anayms sounds produced 514 text yp-reading des pars of the monochord (the uayééec) . These teMaga nal duotar, scribed by Ptolemy as being navrays9ev Coat (p. 17,16-18 During), the two words are coupled not with inwithin the m-family’; so the testimony of these two sources da di is the 14th century®, and V seems to have a special position 6. Fe strings with sentative for the edition which Nicephoras Gregoras made ppah RAhaeiTnaQdDTATe this passage Diiring mentions a variant read- (for owupiv), 192, ig mw caATAuA A the deinstrument) ent weights), and from empty or filled vessels.” 5. hyENTS„auxa,t. traces of relations between musical TGV ävrooBapüv opunäv fi Sionwv correct Hs experiments. A.D.), (the demonstration Having described avAot, an obvious LE AE “ar by a short description of some imperfect methods attached weichts, ical listfrom G, of ve'P is further supported by a scholion (probably of the cf. Diiring p. C) which concerns in ial description are de- 17 Let us postpone the Pythagorean acoustical the Monochord of demonstrating tervals. Exactly what does Hardly any of sphere, question for a moment and opateat - and the € hofeasetp 43 SE . bi tuCrAe)TA ae Aessfata AfviayApoart ENTE MirES LE PAR RSTr. 4 =. rc 3re Ptolomy's Harmonics were commented by Porphyry in the century.” In his commentaries (pp. 120 and 121, Düring)notwe of find the decisive proof that Ptolemy spoke of owatpaı, menopüpar. In the preceding (p. 119,12sqq.) Porphyry has weights tioned experiments with flutes, strings with attached 13and empty or filled vessels. He then continues (5, 120, r acıda 15): hoabtwg 58 nal Edv TLE Stououc xalnoüg noLticas Sırnl The con- Satépou tov Étepov, ouupovobor xpovéueva 6rd nacav. text leaves no doubt that Porphyry's brazen disks correspond pas: to the “spheres or disks" of Ptolemy. At the end of this sage Porphyry gives a paraphrase of Ptolemy's reasoning about the insufficiency of such acoustical experiments, I quote Ptolemy's text and Porphyry's paraphrase: . è Die Harmonielehre, p. 17,27 sqg. . Ptolemaios, e arsForensics Kommentar zur Harmonielehre (=desGöteb orgs hôgskolas During, ben von Ingemar Düring. Göteborg 1932 skrift XXXVIII. 1932:2).

PTOLEMY (p.17,16-20): PORPHYRY (p.121,10-14): ra napanArfora st ovuBalver tà napanirhcıa 6 ouuBalver nani mal TÜV Matàä obyxpougiy Yıvo- HÉVUV Www, Éni tv Yıvoudvav vévuv KATÀ OUYKDOUOLY dnotovug ÉnivooDo: tobc Sid Tüv dvLoofapiiv opupüv fi Sichwv cpatpiv vaotüv mai ff &yyetwv fowv xual dLOLWV HEvOV TOÙS And túv touBAlwv nevOv TE mal nenAnpuuévuv, TE épywSoug Evtog ndvu TOO tnpeiv Ev &nact is guite dealt with qually clear that “spheres” nal clear that must be synonyms, Porphyry's makes only if The owatpa as a synonym he pasos fi &lonor (Porphyry), Ptolemy MSS), traditions a "hammers". not guite the commented But it understand used same physical ("solid spheres the two words is that he did number to this etext. by Ptolemy But holicu disks”) refer not seems object. or upon to know differthe word of different objects associated experiments: xaluot &touor (Hipto Schol. Plat. Phaedon 108D), according (Ptolemy), ovaïpat tote to 6{0xoc. acoustical of Megapont, opatpar took inference We have now met Pythagorean to adjectives ent objects. with did the two expressions relating use of sense seen, text which Porphyry not with Porphyry As we have already toùrore tó Ev tate Ciare ual Oxñuacoiv abtav &6idmopov. the and AauBavéviwv ÜGwp, Ev Anacı Toig oxfuaorv ánapallaE (av. It 6ionwy Holiwv Suoxepots Svtog ndvu TOD Inpetv toûtorc thy Ev tale late kai Hal fi in awatoar vactol fi (Hagiopolites), owüpaı (Nicomachus; Satotiipes (Nicomachus). Stoxor motAor most We may combine these into a coherent though hypothetical description of how Hippasos's brazen disks were gradually transformed into Pythagoras's hammers: At first, the pat, but by ments was the scholiast told (or his about Hippasos and into ra teaeaLAtLastda understanding of the context in Ptolemy. Evidently, this reconstruction implies a considerable age for the version of the Hagiopolites . ? 0 It furthermore implies that the word owaipa once denoted some kind of object which could be manufactured in a smithy, but that later on this an meaning of the word became obsolete. Now, if we combine idea of Burkert's with a couple of places in Athenaeus's Deipnosophists and Plato's Republic, I think that a case can be made for taking these objects to be some kind of vessels or bowls. Let us begin with Burkert. oras discovered the basic law of acoustics in a smithy is a rationalization - physically false - of the tradition that Pythagoras knew the secret of magical music which was discovered by the mythical blacksmiths. "12 This explanation leaves no room for Hippasos's four brazen disks. To Burkert, they belong to the same tradition as the story about Lasos's exper iments with vessels filled in varying degrees - going back, ultimately, to a tradition "that Lasus and Hippasus, in studywere referred to as an anecdote about Pythagoras; the its author "doit être antériAccording to Vincent, op.cit. p.267, mais peut-être même à Aristote; eur, non seulement à Aristoxéne, écrivain sur la muet il se trouverait être ainsi le plus ancien sique, dont les oeuvres ne sont pas entièrement perdues." (My Italthat were once called ovaLsource) In his treatment of the Pythagoras anecdote*”, he concludes that “the claim that Pythaghis ex- At some later time the description of these expericonverted ¿edema 10. story was periments with some brazen objects 6toxor. aeprid , metalware objects were still called owaipar. Still later aı the anecdote was rewritten, from a source in which owatip had become ovDpar by mistake. Once the hammers had been introduced, there was a possibility to use another word for them, parotñpes. This successive transformation of the story had rs produced the famous anecdote about Pythagoras and the Hamme already in the 2nd century A.D. (the date of Nicomachus from Gerasa), though Ptolemy in the same century still had access to a owatpa-version and knew that the word in this context was a synonym to 6foxor. In the following century, Porphyry no longer knew its exact meaning, but had to rely on his ics). 11. Burkert, op.cit. (cf. note 4), pp-375 sqq. ibid. p. 377. The 'mythical blacksmiths’ are the Idaean Dactyls.

phenomena of resonance, that a subsequent learned the numerical restatement in somewhat cruder laws, form speaks of here, is In Book XI of the Deipnosophists with different kinds of cups, - - arranged Athenaeus quotes Asclepiades in ñxeta. which deals profusely in alphabetical order from Myrlea (lst cent. for the following remarks on Nestor's famous cup: cients, The the one which Greek theatres was produced by empty vessels, B.C.) first to ordain for men the things pertaining to civilized life, being convinced is spherical in shape food to (rdv xéouov elvar opatpoeréf})... thought the things pertaining to their own like the element which encompasses the earth, according the shape it seemed to have (tH nepıexovrı which contains liquid of the universe (nai TPOPAV MHUXAOTEPÈC food, this; are set in for like out, its ornamental is studs a star-spangled cup imitation imitates of the celestial the universe described of Book X of the Republic "curious pasin the Myth of Er near the end (616C sqq.). For although this deno mention at all of any owalpaı, (opóvevio.) of the RA"NDRSeiil#4 aAsiei: sphere with is seen at its worst"} - is Plato's Spindle of Necessity, the whorls i Neswherein philology distorted by allegory and astrology scription contains n, so tha meaning later on fell out of use and was forgotte for the nobody any longer understood the metaphor 'sphere' : Isn't firmament. For this is where our investigation endsafter its it more likely that the celestial sphere was named their name being similar to a bowl than that these bowls got the cup, in other words: Older - and much more interesting than this model of £yatoa as a name for| some vessel is this Apparently, nn no means stranger than the English 'bowl'. formation of these. particularly good stars on the firmament - the eight spindle clearly refer to the planetary and celestial spheres .!® The way in which these whorls are fit- ERRESfpusroh ,sEatlipsp iKa Arr eH=tBriendseg iad 3°PÑs TELi St: ALA 13. ibid., p. 377-76. 14. Athenaeus, Deipnosophistae XI, 489c-d, translated by C.B.Gulick (Loeb), 15. Gulick, vol.V, p. 16. The Harmony of the Spheres is alluded to in 617B. CA ae 170, note a. nre “os. Thus, tion, Plato repeatedly refers to objectsordi nary whor > the whorl is said to be shaped like an on betwee vows oxfiua olanep À tod tvod6e), and in the comparis article ° ” oyévsuior and nádos Plato's use of the defihenite and his reel that he is thinking of some objects which AAAÑAOUG know from their daily life: uaôdnep of ndsoı ol eis ments carried out by Lasos and Hippasos we once od sala tra: vessels, and Hippasos's 8{010ı are to be understo by (DE (tiv fixed stars. sage, ROA St Lödav (the golden Aor e. their rims forming a continuous flat surfac mind were compared to drin al expert” Burkert's explanation of the nature of the acoustic again encoun tó notñprov otv TÓ Sexduevov TAV dypäv example 633) too, Emoincav xatà ulunua tod udcuov) . "14 as Asclepiades points Nestor's Hence, they made circular tor's cup, of uatà thy Hence they made a table round todneCav nuxdAoer Sf}, MATEOKEVÁCAVTO .... ) its the universe da] to make Tob oxhuatog &pouoLoüv). A that |. ni s (xábo eg bowl sp ared toaset or n est of comp ted into one another is had in ) which Er and Asclepiades nm” sphere(sking vessels, not to sels: with "The anwho were the it was only right sEFenrAiEDl$IenO! dai au.tFas tranformed their action to one of striking the vessels. "13 resonance which Burkert and sd195 4 from their likeness to the firmament?