ptolemy on sound_ harmonics 1.3 (6.14-9.15 düring)

copy Koninklijke Brill NV Leiden 2009 DOI 101163156852509X339897

Mnemosyne 62 (2009) 548-585 brillnlmnem

Ptolemy on Sound Harmonics 13 (614-915 Duumlring)

Miguel Bobo de la PentildeaConservatorio Profesional de Muacutesica Luis Moya Blanco 261 33203 Gijoacuten Spain

miguelbdeducasturprincastes

Received February 2008 accepted April 2008

AbstractPtolemyrsquos acoustics develops throughout his Harmonics chapter 13 He defi nes sound as πάθος ἀέρος πλησσομένου expressing itmdashas most authors at the timemdashin terms of a stroke (πληγή) and thus linking the study of sound attributes to that of the strokes His tripartite analysis of sound stroke represents an original description of sound production by means of an agent (τὸ πλῆττον) exciting a medium (τὸ διrsquo οὗ ἡ πληγή) which in turn stimulates the air (τὸ πληττόμενον) However clear Ptolemyrsquos explanation is a wide consensus on its interpretation has not been reached since almost each scholar has read Ptolemyrsquos three factors in a diff erent way As a result several problems arise in diff erent authors especially the contradiction between ἡ τοῦ διrsquo οὗ ἡ πληγὴ σφοδρότης seen as responsible for pitch and ἡ τοῦ πλήττοντος βία thought of as cause of loudness but also the understanding of ἡ ἀποχὴ τοῦ πληττομένου πρὸς τὴν ἀρχὴν τῆς κινήσεως as a diff erential cause of sounds Th is paper tries to settle the aforesaid factors and to clear up the diffi culties arising as well as to comment on some fundamental aspects of Ptolemyrsquos acoustics

KeywordsPtolemy harmonics acoustics ancient Greek music

M Bobo de la Pentildea Mnemosyne 62 (2009) 548-585 549

1 Ptolemyrsquos Concept of Sound

Ptolemyrsquos theory of sound as appears in his Harmonics1) is based on an original re-arranging of previous notions so that they concur with practi-cal experience I will thus start by setting them out

11 Background

Th e study of the production transmission and properties of sound is ancient even within Greek tradition Some aspects of such a study were not in my opinion diffi cult to establish For example it is easy to check experimentally that sound can be produced by some kind of impact it is enough to clap the hands or to consider the beat of a drumstick on a drum or that of the plectrum on the string it strikes once this has been stated the same production mechanism can after thoughtful refl ection be recog-nized in other cases Likewise when the transmission of sound is consid-ered the channel between the impact produced and our hearing it is obviously air (sometimes water) so it was easy to realize that somehow such an impact interacts with the air and the latter in turn with the hear-ing Th ese common-sense considerations could not but be familiar to ancient philosophers and in fact we fi nd them expressed by authors such as Anaxagoras (56A106 DK) Empedocles (31A93 DK) Alcmaeon (24A6 DK) Diogenes of Apollonia (19A21 DK) and Democritus (68A135 DK) they all include the idea of an impact though placing its signifi cance in a diff erent stage of the process

Th e concept of sound in terms of a stroke (πληγή)2) can be found early in Pythagorean acousticsmdashfi rstly in Archytas (47B1 DK) though he refers to previous possibly Pythagorean authorsmdashand the description of sound

1) Ptolemyrsquos Harmonics (Ptol Harm) is quoted by page and line number from Duumlring 1930 without any other indication though occasionally Wallis 1699 is used Also Porphy-ryrsquos commentary (Porph in Harm) is quoted by page and line number from Duumlring 19322) Th e noun πληγή (sometimes πλῆξις) lsquostrokersquo and its corresponding verb πλήττω (πλήσσω in the Ionian variant which can be found once [32] in Ptolemy) lsquoto strikersquo are commonly used to refer to the knock or impact other verbs as τύπτω or προσπίπτω appear sporadically

550 M Bobo de la Pentildea Mnemosyne 62 (2009) 548-585

as a lsquostroke of airrsquo3) became usual afterwards among the members of this sect according to the Peripatetic Adrastus (apud Porph in Harm 81) Th is description was doubtless very common from the fi fth and fourth centuries BC onwards and can thus be seen in Plato (Ti 67b2-4 also cf Ps-Plu Plac 901F9-11) Xenocrates (fr 87) Aristotle (de An 419b9-11) and the Peripatetics (Ps-Arist Aud 800a1-3 Pr 901a17) the Stoics (Chrysipp SVF II fr 425) Epicurus (cf Ps-Plu Plac 902C9-11) and Pseudo-Euclid (SectCan 1486-7 Jan) even the grammarians made use of this explanation (cf Ps-Plu de Mus 1131D2-3) It was a really wide-spread notion in antiquity with the outstanding exception of Aristoxenus a defi nition of sound in terms of stroke cannot be found in his works even though he could not be unfamiliar with it4) and he only alludes (Harm 174-6 554-7) to the Pythagorean doctrine of sound as air motion5) in order to criticize it

Th e description of sound as an eff ect of a stroke was then well-established although there was not a consensus on the terms of such a description the stroke was sometimes considered to occur between the bodies producing sound (Arist de An 419b19-22) or between them and the particles thought of as constituents of air (Ps-Arist Aud 800a1-13) sometimes between these particles themselves as a transmission channel

3) Th e question whether sound is lsquoair struckrsquo or lsquostroke of airrsquo depends on the type of defi ni-tion required cf Porph in Harm 1117-29 AlexAphr in Top 45023-4 and Simp in Ph 94261-64) Perhaps Aristoxenus considered such a defi nition as not befi tting harmonic but physical science (cf Beacutelis 1986 76) whereas basically his treatise on the former is the only work of his preserved However as a follower of Aristotle whom he expected to succeed he could not be ignorant of the Aristotelian theory of sound which explains it (cf Arist de An 419b9-11) by means of a stroke5) In Pythagorean acoustics (cf Porph in Harm 95-12 2927-33 4522-4) the stroke determines a motion postulated as a cause of high pitch when it is fast and of low pitch when it is slow Th e fi rst testimony to it Archytas (47B1 DK) adds the strokersquos strength as a justifi cation for high pitch however later Pythagorean theory (cf Adrastus apud Porph in Harm 81-5) shows the strokersquos intensity only as responsible for the loudness of sound thus isolating speed as a cause of pitch Th is concept became virtually unanimous (cf Pl Ti 67b6-c1 Arist de An 420a31-3 Ps-Arist Aud 803a5-6 Aelianus the commentator on Plato apud Porph in Harm 3328-9 AlexAphr in Top 10629-72 Ps-TiLocr 2206-7 etc) with some exceptions such as Th eophrastus (fr 89 115-7)


(Chrysipp SVF II fr 425) or between them and our ear (Pl Ti 67b2-4) In any case it seems that even before Ptolemy there were some facts accepted by most authors namely a) that the origin of sound lies in the collision of two sounding bodies6) b) that such a collision causes an aff ec-tion7) in the air susceptible of transmission through it and c) that its reception occurs when the air aff ected impinges on our auditory organ Th ere are in correspondence with these facts three phases of the auditory process which we could call respectively sound lsquoproductionrsquo lsquotransmis-sionrsquo and lsquoreceptionrsquo

12 Ptolemyrsquos Formulation

Th ree are the defi nitions of sound in Ptolemyrsquos treatise Th e fi rst two appear after that of harmonic science combined in a single one which we could label lsquocompound defi nitionrsquo8)

ψόφος δὲ πάθος ἀέρος πλησσομένου τὸ πρῶτον καὶ γενικώτατον τῶν ἀκουστῶν (32-3)

Sound is understood as πάθος ἀέρος πλησσομένου (lsquoaff ection of air struckrsquo ie lsquowhat the air experiences when struckrsquo) a physicoacoustic defi -nition along the lines indicated above Besides sound is said to be τὸ πρῶτον καὶ γενικώτατον τῶν ἀκουστῶν (lsquothe fi rst and most generic9) of

6) Not every body is considered as sounding ie capable of making sound (cf Arist de An 419b6-9 419b35-20a4)7) lsquoAff ectionrsquo in the meaning of lsquoaction or process of aff ecting or being aff ectedrsquo Such an aff ection is described as a motion among the Pythagoreans (cf n 5) and virtually all the authors who refl ect that concept8) A similar combination occurs earlier (if the source is not corrupt) in Diogenes of Baby-lon (c 2nd century BC) ἔστι δὲ φωνὴ ἀὴρ πεπληγμένος ἢ τὸ ἴδιον αἰσθητὸν ἀκοῆς (apud DL 7552-3) More or less contemporary with Ptolemy is Ps-Plu de Mus 1131D2-3 ὁρίζονται τὴν φωνὴν οἱ ἄριστοι γραμματικοὶ ἀέρα πεπληγμένον αἰσθητὸν ἀκοῇ9) In the upward process of lsquogeneralizationrsquo toward the universalsmdashinverse of the down-ward of lsquospecifi cationrsquo leading to the particularsmdasha genus is identifi ed which cannot be conceived as a species of another superior or lsquomore genericrsquo τὸ γενικώτατον Th is one could be called lsquoextreme genusrsquo and philosophers later than Aristotle such as Porphyry will iden-tify it with the substance (οὐσία) itself in such a sense sound is understood as lsquothe most generic of the audiblesrsquo (about the term lsquoaudiblersquo cf n 10) it is the very substance of an


the audiblesrsquo)10) so as to show a more conceptual typically Peripatetic approach Th is addition establishes without doubt that the term lsquosoundrsquo (ψόφος) is taken in its widest sense11) while the physicoacoustic defi nition describes what this lato sensu sound is from the point of view of its material production

While the physicoacoustic defi nition considered sound mainly from the point of view of its production harmonic science studied instead the attributes of sound12) once it has been produced and especially one of these attributes pitch (practically identifi ed with sound itself ) Now in strings the greater the tension (τάσις) the higher the pitch whereas less tension means lower pitch which does not vary while the tension remains constant then it is not diffi cult to understand that the tension of a string ended up by being identifi ed with the pitch of the sound it produced13)

audible and without which it is not yet so At the opposite end of the scale are τὰ εἰδικώτατα the lsquoextreme speciesrsquo which cannot be divided into more specifi c subspecies and will be the particulars τὰ κατὰ μέρος Cf Porph Intr 42-51610) Th e usual Greek system of nominalizing adjectives by means of the article is also used in English for a few current adjectives from philosophical texts which become lsquoregularrsquo nouns Such is the case for lsquouniversalrsquo and lsquoparticularrsquo and less often for lsquosensiblersquo and lsquointelligiblersquo (which occur especially in American English texts) thus I have chosen to render τὰ αἰσθητά by lsquothe sensiblesrsquo (in contrast with τὰ νοητά lsquothe intelligiblesrsquo) and τὰ ἀκουστάmdasha particular case of the formermdashby lsquothe audiblesrsquo Th us a lsquosensiblersquo is an lsquoentity perceptible through the sensesrsquo and an lsquoaudiblersquo is a lsquosensible proper to the hearingrsquo that is an entity perceptible through the sense of hearing (as sensible) and only through it (as proper to hear-ing) cf Arist de An 418a7-13 and Porph in Harm 1127-811) Th e terminological diff erence between ψόφος φθόγγος and φωνή is clear and constant in Ptolemy ψόφος refers to sound lato sensu as I have just said φθόγγος (lsquosound which occupies a single and identical tonersquo 1019) is the musical sound here translated as lsquonotersquo and φωνή (lsquojust the most beautiful of soundsrsquo 1026) is the lsquovoicersquo a species of φθόγγος (obviously the lsquointervallicrsquo one (ἡ διαστηματικὴ φωνή) used in singing opposed to the lsquocontinuousrsquo one (ἡ συνεχὴς φωνή) used in speaking cf Porph in Harm 934-104) Th e gradation is then ψόφος gt φθόγγος gt φωνή from more generic to more specifi c12) I refer to pitch (by which we mean one sound is higher or lower than another) loudness (by which we mean one sound is more intense than another) duration and timbre (by which we diff erentiate sounds of similar pitch loudness and duration)13) A testimony to it might perhaps be traced as far back as Terpander (c 7th century BC) Indeed he uses τόνος (lsquotonersquo but also lsquotensionrsquo cf ἄτονος lsquoatonicrsquo lsquowithout tensionrsquo) which has the same root t(eo)n- as τάσις (both of them derive from τείνω lsquoto tensersquo cf Chantraine 1999 sv τανυ-) in the compound ἑπτάτονος to refer to the φόρμιγξ in


Th is together with the great prestige of chordophones like the kithara (κιθάρα) or the lyre (λύρα)14) and maybe allied to the fact that early acous-tic experimentation was often carried out on chordophones led to the terminology and concepts characteristic of strings being extrapolated to other instruments and musical fi elds So metaphorically from strings (cf Cleonid 1817-8) a particular pitch was fi rstly identifi ed with the corre-sponding tension of a string and consequently named τάσις15) later the metaphor went further and τάσις was identifi ed with pitch itself (cf Aris-tox Harm 1816-8) as we can see mutatis mutandis in Ptolemy himself (102-5) As a result the concepts of lsquoincrease in tensionrsquo (ἐπίτασις) and lsquodistensionrsquo or lsquolooseningrsquo (ἄνεσις) of strings adopted16) the more abstract senses of lsquopitch risersquo and lsquopitch fallrsquo respectively while each particular pitch started to be called17) φθόγγος

Th us the concept of lsquonotersquo (φθόγγος) had been long-established in terms of τάσις and Ptolemy synthesizing previous harmonic notions uses it18)

φθόγγος ἐστὶ ψόφος ἕνα καὶ τὸν αὐτὸν ἐπέχων τόνον19 (1019)

such a way that his ἑπτάτονος φόρμιγξ (fr 4 Gostoli) can be read as lsquophorminx of seven tensionsrsquo (cf the lsquophorminx of seven knocksrsquo (ἑπτάκτυπος φόρμιγξ) from Pi P 270-1) as well as lsquophorminx of seven notes or pitchesrsquo (also cf the lsquophorminx of seven tonguesrsquo (ἑπτάγλωσσος φόρμιγξ) from Pi N 525) in any case referring to the number of strings Also cf Ion Chius fr 32 West and E HF 68314) Which can be appreciated if we take into account that for example most of the names of the notes in the lsquocomplete systemrsquo (τέλειον σύστημα) are connected with those of the lyre strings (cf West 1992 219) an early example of this extension might perhaps be found in Philolaus (44B6 DK)15) Which started to mean lsquopermanencersquo (μονή) lsquostabilityrsquo (στάσις) lsquostillnessrsquo (ἠρεμία ἐνηρέμησις) and even lsquoidentityrsquo (ταὐτότης) of the voice cf Aristox Harm 172-4 Anon Bellerm 620-1 Cleonid 18019 Aristid Quint 628 WI Gaud 3298 and Nicom Harm 2431-216) Cf Aristox Harm 1514-8 Cleonid 18020-12 Anon Bellerm 122-5 and AristidQuint 629-73 WI Also cf Hagel 2005 63 n 3517) Cf Aristox Harm 2016-9 Cleonid 1799-10 Gaud 3297-8 Bacch 29215-6 and Nicom Harm 2614-7 yet it is occasionally possible to see φθόγγος and τάσις as syn-onyms (Cleonid 1817-9) or φθόγγος with the abstract value of τάσις (Anon Bellerm 1415-6)18) Although the term τάσις is here replaced by its synonym τόνος cf n 1319) lsquoNote is sound which occupies a single and identical tonersquo


Th is defi nition showed the note to be a species of sound initially consid-ered in the treatise from the physicoacoustic point of view mentioned above so a link between both ψόφος and τάσις was needed whence the third defi nition of sound20) given in the treatise

τάσις γάρ τίς ἐστι συνεχὴς τοῦ ἀέρος ὁ ψόφος ἀπὸ τοῦ τοῖς τὰς πληγὰς ποιοῦσιν ἐμπεριλαμβανομένου διήκουσα πρὸς τὸν ἐκτός21) (812-4)

Ptolemyrsquos eclecticism22) in defi ning sound is then patent nevertheless such eclecticism is not gratuitous since his various descriptions reveal diverse interests Th e most conceptual defi nition of Peripatetic origin has an epis-temological character23) whereas the physicoacoustic one connected to a Pythagorean tradition is aimed at studying the attributes of sound How-ever as the signifi cant term in the development of the concepts of har-monic science is pitch expressed as tension the author resorts to the concept of sound in terms of τάσις

2 Factors Involved in Sound Production πληγη τινος πρός τι διά

τινος

Once sound has been defi ned as πάθος ἀέρος πλησσομένου the author analyzes sound attributes in terms of the stroke which produces it and identifi es the strokesrsquo features contributing to diff erentiate the sound they produce

τὴν περὶ τὰς ὀξύτητας καὶ βαρύτητας ἐν ὁποτέρῳ γένει τῶν εἰρημένων θετέον οὐκ ἔστιν ἀποφήνασθαι προχείρως πρὶν ἐπισκέψασθαι τὰ αἴτια τοῦ τοιούτου συμπτώματος ἅ μοι δοκεῖ κοινά πως εἶναι καὶ τῶν ἐν ταῖς

20) Whose connection with the previous one we will later deal with (cf sect 332)21) lsquoIndeed sound is a certain air tension a tension stretching with continuity from the air comprehended between the bodies producing the strokes to the outer airrsquo22) On eclecticism in Ptolemyrsquos philosophical concepts cf Long 198823) By using sound lato sensu in his defi nition of harmonic science Ptolemy widens its fi eld of speculation which now accepts the nature of the harmonizedmdashthat is abiding by the harmonic science rulesmdashmelos (ἡ τοῦ ἡρμοσμένου φύσις) as habitual as well as that of the inharmonious (ἡ τοῦ ἀναρμόστου) with the idea that a science must comprehend what is its proper subject and what surrounds it (cf Porph in Harm 620-9)


ἄλλαις πληγαῖς παραλλαγῶν γίνεται γὰρ τὰ ἐξ αὐτῶν πάθη διαφέροντα παρά τε τὴν τοῦ πλήττοντος βίαν καὶ παρὰ τὰς σωματικὰς συστάσεις τοῦ τε πληττομένου καὶ τοῦ διrsquo οὗ ἡ πληγή καὶ ἔτι παρὰ τὴν ἀποχὴν τοῦ πληττομένου πρὸς τὴν ἀρχὴν τῆς κινήσεως σαφῶς γὰρ τῶν ἄλλων ὑποκειμένων τῶν αὐτῶν ἕκαστον τῶν εἰρημένων ἴδιόν τι ποιεῖ περὶ τὸ πάθος ὅταν αὐτὸ διενέγκῃ καθrsquo ὅντινα οὖν τρόπον24) (615-24)

Sounds are specifi ed again as lsquoaff ections derived from strokesrsquo (τὰ ἐξ αὐτῶν [sc τῶν πληγῶν]25) πάθη) and four diff erential causes of sounds are recog-nized namely 1) ἡ τοῦ πλήττοντος βία lsquothe force of what strikesrsquo 2) ἡ σωματικὴ σύστασις τοῦ πληττομένου lsquothe bodily constitution of what is struckrsquo 3) ἡ σωματικὴ σύστασις τοῦ διrsquo οὗ ἡ πληγή lsquothe bodily constitu-tion of that whereby the stroke happensrsquo and 4) ἡ ἀποχὴ τοῦ πληττομένου πρὸς τὴν ἀρχὴν τῆς κινήσεως lsquothe space from what is struck to the origin of the movementrsquo Th ese four causes are enunciated according to a series of factorsmdashτὸ πλῆττον τὸ πληττόμενον τὸ διrsquo οὗ ἡ πληγή and ἡ ἀρχὴ τῆς κινήσεωςmdashwhich imply a description of the striking process the source of sound Consequently it would be appropriate for a correct understanding of Ptolemyrsquos theory to determine fi rst of all how many and which of them are the factors essential to such a description and then their respec-tive contributions to the sound process

For a start we have to point out that the expression ἡ ἀρχὴ τῆς κινήσεως is coreferential with τὸ πλῆττον26) as it stems from the comparison between

24) lsquoIn which of both the aforesaid genera [sc that of quantity and that of quality] must be placed the diff erence concerning highness and lowness is something which cannot be dis-played off hand before the causes of such a characteristic have been researched causes which I fi nd to be somehow common also to the variations in the other strokes Certainly the aff ections derived from them [sc from the strokes] prove to be diff erent depending on the force of what strikes on the bodily constitutions of what is struck as well as of that whereby the stroke happens and also on the space from what is struck to the origin of the movement Indeed it is clear that if the other underlying factors remain identical each one of the aforesaid ones produces a proper peculiarity concerning the aff ection when it itself is made diff erent in any wayrsquo25) I think αὐτῶν should not be referred to ψόφων as Barker (2000 38 ff ) seems to do (ldquothe pathē of soundsrdquo) sounds are themselves the πάθη resulting from the strokes (remem-ber that sound is lsquoaff ection (πάθος) of air struckrsquo this was just the reason why causes of variation of strokes lato sensu were studied)26) Raff a (2002 271) seems to have reached the same conclusion and for the same cause I


the syntagm παρὰ τὴν ἀποχὴν τοῦ πληττομένου πρὸς τὴν ἀρχὴν τῆς κινήσεως from the previous text and its paraphrase ἐκ τῆς τῶν ἀποχῶν τοῦ τε πληττομένου καὶ τοῦ πλήττοντος ἀνισότητος (817-8) made by Ptolemy himself when developing the study of this fourth diff erential cause of sounds In fact the characterization of the agentmdashhere τὸ πλῆττονmdashas origin of the movement (or as cause of it) is a Peripatetic notion (cf Arist Ph 194b29-32)27) the stroke is understood as movement (κίνησις) or change from potential sound (state of rest) to eff ective or actual sound where the diff erences between sounds become evident (cf Arist de An 420a26-7) We can then identify ἡ ἀρχὴ τῆς κινήσεως with τὸ πλῆττον simplifying the description of the striking process of sound

Secondly we should refl ect on the formal expression of the three remain-ing factors τὸ πλῆττον τὸ πληττόμενον and τὸ διrsquo οὗ ἡ πληγή whose formulation is made in such a way as to make it impossible to identify each of them with any other but itself My statement will be better understood if the aforesaid factors are comparedmdashwith reference to their formal expressionmdashwith others from a diff erent analysis one of fi ve components carried out by Ptolemy in a work on his theory of knowledge

Τὰ μὲν οὖν ἄκρα τοῦ ζητουμένου κριτηρίου καὶ αὐτόθεν εἴη δῆλα τὸ μὲν ὂν ὡς ὑποκείμενον καὶ κρινόμενον ( ) τὸ δrsquo οὗ ἕνεκεν ἡ ἀλήθεια ( ) τὰ δὲ μεταξὺ τῶν ἄκρων ( ) λάβοιμεν ἂν ἀπὸ τῆς ἐνεργείας αὐτῆς ποιούμενοι τὴν ἐπὶ τὰ καθόλου ἔφοδον ( ) γὰρ ( ) οὐκ ἀλόγως ἄν τις ἐφαρμόσειεν

mention here however he had previously (p 257) identifi ed ἡ ἀρχὴ τῆς κινήσεως with τὸ διrsquo οὗ ἡ πληγή the latter clearly distinguished from τὸ πλῆττον Barker (2000 50) in turn assimilates the origin of the movementmdashin the case of a vibrating stringmdasheither with its oscillation centre or with one of its ends which makes it hard for him to interpret the fourth diff erential cause of sounds27) Porphyry expresses τὸ πλῆττον as lsquoresponsiblersquo (αἴτιον) for the stroke ἐκ τρίτων δὲ διάφοροι γίνονται αἱ πληγαὶ καὶ παρὰ τὴν ἀποχὴν τοῦ πληττομένου καὶ τοῦ αἰτίου τῆς πληγῆς ἄλλως γὰρ ἐπὶ πλεῖστον ἀπεχόντων ἀλλήλων διάστημα τοῦ τε πλήττοντος καὶ τοῦ πληττομένου γίνεται ἡ πληγή ἄλλως δrsquo ἐπrsquo ἔλαττον (in Harm 395-9) Th e com-mentator specifi es the voice more (ibid 4027-11) distinguishing between the agent strictly speaking (lsquothe natural blow (πνεῦμα)rsquo) and the origin of the movement (lsquothe impulse (ὁρμή) to blowrsquo) nevertheless only in that case such a distinction is pointed at and it is Porphyry not Ptolemy who does so


τὴν μὲν αἴσθησιν τῷ διrsquo οὗ κρίνεται τὸ κρινόμενον τὸν δὲ νοῦν τῷ κρίνοντι τὸν δὲ λόγον ᾧ κρίνει τὸ κρῖνον28) (Ptol Judic 54-14)

Th e comparison I have proposed clearly shows the formal similarity between τὸ πληττόμενον and τὸ κρινόμενον (τὸ ὄν lsquothe beingrsquo which is the object of examination according to Ptolemy himself ) between τὸ πλῆττον and τὸ κρῖνον (ὁ νοῦς lsquothe understandingrsquo) and between τὸ διrsquo οὗ ἡ πληγή and τὸ διrsquo οὗ κρίνεται τὸ κρινόμενον (ἡ αἴσθησις lsquothe sense the sensationrsquo) Now since the three factors in this second analysis aim at totally diff erent references which we should not confuse with each other it seems obvious that the ones we are concerned withmdashformally identical to the othersmdashshould also be carefully distinguished any confusion29) between them could distort the acoustic theory in the treatise and should be avoided in order to respect the terminological coherence usual in its author

In short when Ptolemy defi nes sound as lsquoaff ection of air struckrsquo he is focusing on its production identifi ed as a stroke where three diff erent irre-ducible factors cooperate τὸ πλῆττον lsquothe striking agentrsquo which triggers the process τὸ διrsquo οὗ ἡ πληγή lsquothe medium transmitting the strokersquo whereby the stroke happens and τὸ πληττόμενον lsquothe thing struckrsquo as fi nal receiver of the stroke

Th is lay-out is not original a tripartite scheme of sound production can already be recognized in Aristotle30) But this philosopher perhaps thought

28) lsquoTh us the ends of the criterion sought would even by themselves be manifest on the one hand the being as subject and matter of the judgment ( ) on the other the thing in the interest of which the judgment is the truth ( ) As for the intermediate terms between the ends in turn ( ) we could obtain them from our own activity when we travel the road toward the universals for ( ) it would not be illogical if one made the sense corre-spond with the medium whereby is judged what is judged the understanding with what judges and the reason with the procedure by which judges what judgesrsquo29) Such confusions have nevertheless been frequent especially as regards τὸ διrsquo οὗ ἡ πληγή and to a lesser extent τὸ πληττόμενον I will refer to them later when studying each factor individually30) Cf de An 419b9-11 (γίνεται δrsquo ὁ κατrsquo ἐνέργειαν ψόφος ἀεί τινος πρός τι καὶ ἔν τινι πληγὴ γάρ ἐστιν ἡ ποιοῦσα) and 419b19-20 (οὐκ ἔστι δὲ ψόφου κύριος ὁ ἀὴρ οὐδὲ τὸ ὕδωρ ἀλλὰ δεῖ στερεῶν πληγὴν γενέσθαι πρὸς ἄλληλα καὶ πρὸς τὸν ἀέρα) the adjective στερεός lsquosolidrsquo refers to the bodies which produce sound in contrast to air or water (ibid 419b18) as receivers and transmitters of the disturbance


about a most elementary stroke when one solid body impacts on anothermdashboth of them on equal termsmdashwithin the air (the third factor involved in the process cf de An 420b14-5) and the latter undergoes somehow the result of such an impact and he expresses it (de An 419b10 cf n 30) as a πληγή τινος πρός τι ἔν τινι a stroke of something (a solid body) against something (another solid body) inside something (the air) Instead Ptol-emy was aware that every factor involved in the process made a diff erent contribution to the attributes of sound since it could be easily proved by practical experience that for example none of the plectrumrsquos features made any contribution to pitch while the stringrsquos are an essential determinant of it31) It is what he advanced in his analysis of the diff erential causes of sounds whence by a clear distinction of every factorrsquos functions he estab-lished a more sophisticated hierarchy between them that is to say a body (the striking agent) triggers the stroke which is transmitted through a second body (the transmitting medium) and ends up at its fi nal receiver (the thing struck) which is none other than the air (cf sect 21) He recog-nized then the same factors as Aristotle in sound production but explained their contribution to the process in an entirely diff erent way which repre-sents in fact an original explanation by him we could express his concept of sound with a paraphrase of Aristotle as an aff ection derived from a πληγή τινος πρός τι διά τινος a stroke of something (the agent) against something (the air) through something (the medium)

Now on the basis of this account the movement (κίνησις) earlier iden-tifi ed as change from potentiality to actuality could be assimilated to the oscillatory motion of which sound consists this motion is then triggered by the striking agent as origin or cause of the process (ἀρχὴ τῆς κινήσεως) and transmitted through the medium up to its fi nal term the air becom-ing actual sound Th us Ptolemy recognizes a peculiar nature in the stroke which causes sound since he admits besides the body hitting and the body hit a medium transmitting the stroke from the former to the latter In fact diverse bodies could be conceived in the striking process of sound

31) It is certainly laid down by practical experience that pitch depends essentially on the characteristics of the instrument by means of which it is produced and very little or noth-ing at all on the agent or the air while loudness for example depends on the force the agent uses


colliding with each other that is striking and being struck32) In an atom-ist description of the process (the only one giving a fair account of sound production transmission and reception) the particles of the medium could be thought of as hitting each other in some cases and always hitting those of the air (in the phase of production) and also the particles of the air in turn hitting each other and hitting the hearing organ (in the phases of transmission and reception respectively) In any case Ptolemy focuses on the phase of sound production exclusively and identifi es a single agent (τὸ πλῆττον) striking but not struck which triggers the process and a single receiver of the impact (τὸ πληττόμενον the air) struck but not striking33) where sound production fi nishes and sound transmission starts and yet the medium (τὸ διrsquo οὗ ἡ πληγή) for its part both is struck (by the agent cf n 39) and strikes (the air)34) Herein lies the peculiarity of these factors shown by practical experience through their diff erent contribution to the attributes of the sound they help to produce in fact the authormdashas an empirical scientistmdashproposes nothing else but an analytical formulation deduced from practice namely that sound happens when an agent (a drumstick a plectrum the human blow) excites a medium (a membrane or a taut string the air column in a pipe the vocal folds) which in turn stimulates the air the fi nal receiver of the disturbance

32) Cf Arist de An 419b19-20 and Ps-Arist Aud 800a1-333) Th e air does not strike as the end of the striking process in the phase of sound produc-tion Nevertheless it is the striking agent in aerophones or in the human voice (cf sect 22) then called πνεῦμα lsquoblowrsquo better than ἀήρ Also separately should be considered the strokes caused by the air in the phases of transmission or reception cf Pl Ti 67b2-4 (sound as stroke by action of air (ἡ ὑπrsquo ἀέρος πληγή)) commented in Porph in Harm 4628-30 (the air transmitting (διαφέρων) the stroke after being itself struck)34) Cf Porph in Harm 5314-20 ἡ δὲ παρὰ τὰς σωματικὰς συστάσεις τοῦ πληττομένου [sc αἰτία πληγῶν διαφορᾶς] διῃρεῖτο εἴς τε τὰς τοῦ ἀέρος διαφοράς (ἦν γὰρ καὶ οὗτος τῶν πληττομένων) καὶ εἰς τὰς τῶν στερεῶν ἢ ἄλλων σωμάτων διαφοράς ὧν πάλιν τοῦ ἀέρος παρεθέντος αἱ σωματικαὶ συστάσεις τοῦ τε πλήττοντος καὶ τοῦ πληττομένουmdashπλήττοντος δ᾿ οὐχ ὡς ἡ βία ἡμῶν πλήττει (αὕτη γὰρ ἐκβέβληται) ἀλλ᾿ ὡς δι᾿ οὗ πλήττομεν σώματοςmdashδιῃροῦντο εἴς τε τὰς πυκνώσεις καὶ μανώσεις (punctuation is mine) also cf Barker 2000 37 ldquoIn many cases of course sound is caused when one solid body impinges on another as when a plectrum strikes a string or a stick strikes a metal disk It is not this impact between solids as such however that is held to cause sound but the resulting impact on the air made in these cases by the string and by the vibrating surface of the diskrdquo (italics are mine)


21 Τὸ πληττόμενον

As I have already pointed out this factor is the air which stems from the very defi nition of sound ψόφος δὲ πάθος ἀέρος πλησσομένου35) Th us it is understood by Barker (2000 37) ldquoIn Ptolemyrsquos treatment as in that of most of his predecessors from Archytas onwards the lsquothing struckrsquo is always the airrdquo36) and a little further on ldquoHere [sc referring to ἡ σύστασις τοῦ πληττομένου] too then and in all cases the relevant lsquothing struckrsquo is the airrdquo Such an interpretation otherwise is the one which prevails when Ptolemy elucidates the constitution of the thing struck as a diff erential cause of sounds

τῶν δὴ τῶν ψόφων ἡ μὲν παρὰ τὴν σύστασιν τοῦ πληττομένου διαφορὰ ἢ οὐδόλως ἂν γίνοιτο ἢ οὐκ αἰσθητή γε διὰ τὸ καὶ τὴν τῶν ἀέρων παραλλαγὴν οὕτως ἔχειν πρὸς τὴν αἴσθησιν37) (624-7)

since diff erences resulting from the constitution of the thing struck are at this point linked to variations in air Th ere is little doubt that Porphyry interprets τὸ πληττόμενον here as the air38) though he does not use the articular participlemdashthat is does not single out a specifi c body struckmdashfor he admits another body struck in the striking process the medium39) how-ever we cannot thence deduce that he identifi esmdashas Raff a (2002 250) asserts40)mdashboth of them but can at the very most say that the air is included

35) Also cf Porph in Harm 811-536) I have already referred to sound specifi ed as stroke of air (or air struck cf n 3) which occurs in Pythagoreans (cf Adrastus apud Porph in Harm 81) Aristotle (de An 419b21-2) and the Stoics (Chrysipp SVF II fr 425) to quote but a few37) lsquoNow between sounds the diff erence from the constitution of the thing struck would either not occur at all or at least be imperceptible since also the variation of airs proves to be such [sc imperceptible] for the sensersquo38) Cf Porph in Harm 3933 403-4 and especially 405-7 (ἀλλ᾿ αἱ μὲν αὐτοῦ [sc τοῦ ἀέρος] κατὰ σύστασιν διαφοραὶ οὐδεμίαν παραλλαγὴν εἰς ὀξύτητα καὶ βαρύτητα ἐμποιοῦσι τοῖς ψόφοις) also cf 4024 538-11 and 5314-20 (n 34)39) It (τὸ διrsquo οὗ ἡ πληγή) can be seen as πληττόμενον in Porph in Harm 4024-6 also cf id ibid 536-11 In id ibid 5517-8 we fi nd even the specifi c musical instrumentmdashthat is the transmitting medium (cf sect 23)mdashas struck40) According to this scholar Porphyry identifi es τὸ πληττόμενον with τὸ διrsquo οὗ ἡ πληγή in in Harm 393-6 since he regards αἱ συστάσεις τοῦ πληττομένου καὶ τοῦ διrsquo οὗ ἡ πληγή


among the πληττόμενα σώματα41) Ptolemy certainly by using the defi nite article singles out a specifi c πληττόμενον σῶμα which Porphyry always just interprets as ὁ ἀήρ in his comments on 624-7 (cf n 38) Neverthe-less Raff a (2002 253) refuses to identify τὸ πληττόμενον with the air42) precisely on the grounds of this passage ldquoOra se le espressioni τὸ πληττόμενον e ὁ ἀήρ indicassero entrambe sempre la stessa cosa cioegrave lrsquoaria la forza argomentativa del nesso causale διὰ τό κτλ si annullerebbe in una inconcludente tautologiardquo In fact there is not such a tautology in the text what in the main clause is brought up is the diff erence between sounds owing to the constitution of the thing struck (τῶν ψόφων ἡ παρὰ τὴν σύστασιν τοῦ πληττομένου διαφορά) while in the causal subordinate clause this diff erence is linked to variation of airs (τὴν τῶν ἀέρων παραλλαγήν) What Ptolemy actually saysmdashas has been pointed out by Barker (2000 38)mdashis that imperceptible variations (those of the air) do not have any eff ectmdashor in any event have an imperceptible onemdashon the corresponding sounds the statement is debatable (cf id ibid and sect 31) but can by no means be described as a tautology

Another surprising interpretation of this factor for diff erent reasons is Levinrsquos Th is scholar puts (1980 216) τὸ πληττόμενον on the same level as τὸ διrsquo οὗ ἡ πληγή and thence the former is assigned some characteristics by her on the grounds of 75-8 which Ptolemy explicitly attributes to

as a single diff erential cause Raff a however does not bear in mind that what the commen-tator regards here as singlemdashand so paraphrasing Ptolemy himselfmdashis the fact that both are cause since they are συστάσεις instead Porphyry eg in ibid 405-7 recognizes that both kinds of συστάσεις have very diff erent contributions to sound attributes hardly can he consider then the factors holding them as identical41) Cf sect 2 on the diff erent ways for a body to strike or be struck in the process Anyway though admitting the instrument among the πληττόμενα σώματα Porphyry recognizes the air as πληττόμενον (τοῦ δrsquo ἀέρος πληττομένου διὰ τὸ περικεχύσθαι τὰ κρούοντα καὶ τὰ κρουόμενα in Harm 413-4)42) Raff a (2002 257) is the only scholar who clearly isolates the three factors distinguished by Ptolemy However though he initially proposes reading the medium (τὸ διrsquo οὗ ἡ πληγή) as ldquoil corpo per mezzo del quale avviene la percussionerdquo he immediately identifi es it with ldquolrsquoorigine del movimentordquo (ἡ ἀρχὴ τῆς κινήσεως) later (2002 271) seen as identical with the agent (τὸ πλῆττον) As a result of this linking (τὸ διrsquo οὗ ἡ πληγή = ἡ ἀρχὴ τῆς κινήσεως = τὸ πλῆττον) we fi nally fi nd two of the three factors equivalent to each other a confusion which makes it diffi cult to understand his interpreting the fourth diff erential cause of sounds (2002 270-5)


the latter43) By making both factors equal she contradicts herself44) since she describes (1980 216 218) the variations of pitch according to those of the bodily constitution of lsquothe item struckrsquo (here referring to the constitu-tion of τὸ διrsquo οὗ ἡ πληγή from the treatise) whereas on the contrary she considers (1980 217) the same bodily constitution of lsquothe object struckrsquo irrelevant to pitch (but such a constitution is now that of τὸ πληττόμενον from the Greek text)

All in all τὸ πληττόμενον is always the air in Ptolemyrsquos analysis inde-pendently of which other bodies are involved in the process those bodies instead are diff erent in each specifi c circumstance and we will identify them in specifi cally musical cases that is when related to musical prac-tice45) (human voice here included)

22 Τὸ πλῆττον

Th ere is no diffi culty in interpreting46) the fi rst factor τὸ πλῆττον also described as ἡ ἀρχὴ τῆς κινήσεως (cf sect 2) Th e agent triggering the process is undoubt-edly the drumstick in percussion instruments or the plectrum in chordo-phones47) As for aerophones and the human voice Porphyry identifi es48) it with the performerrsquos blow (πνεῦμα)49) Such identifi cation is not at all exclusive to him but can be found previously in Aristotle (de An 420b27-1a3) and many other authors (cf Archyt 47B1 DK Th phr fr 89 Ps-Arist

43) More precisely what we read in 76 is the plural τὰ διrsquo ὧν αἱ πληγαί later (77) made singular with σῶμα in the genitive44) Th is contradiction has already been pointed out by Raff a (2002 254-5)45) Allusions to instrumental practice are frequent in this chapter (cf eg 71-5 729-82 and 825-7)46) Th e striking agentrsquos activity is exemplifi ed (73-4) through the verbs φθέγγομαι ἐμπνέω and κρούω lsquoto utterrsquo (a voice) lsquoto blowrsquo (in an aerophone) and lsquoto strikersquo (eg with a plec-trum) respectively47) Th ose whose strings are struck as the kithara or the κανὼν ἁρμονικός when the string is plucked as in the lyre that role is played by the playerrsquos fi nger the agent which triggers the process by pulling at the string48) Cf in Harm 5427-9 for the voice and ibid 5516-7 for both together also cf ibid 5523 We should recall the above mentioned peculiarity of this case where Porphyry diff erentiates between the origin of the movement and the stricto sensu agent (cf n 27)49) As I have already said (cf n 33) the term πνεῦμα lsquoblowrsquo is usually preferred in that meaning to the less marked one ἀήρ lsquoairrsquo


Aud passim) and is justifi ed by Barker (2000 37) this way ldquoIn wind instruments the breath is usually conceived as a missile propelled down the pipe and causing sound when it strikes the air outside it through the nearest available aperture (normally a fi ngerhole)rdquo Some scholars50) how-ever on the grounds of this passage from Ptolemy

ὀξύτεροι γὰρ γίνονται πάντως τῶν ἄλλων διαμενόντων τῶν αὐτῶν ( ) κἀπὶ τῶν αὐλῶν οἱ διὰ τῶν ἐγγυτέρω τοῦ ὑφολμίου τουτέστι τοῦ πλήττοντος τρυπημάτων ἐκπίπτοντες τῶν διὰ τῶν ἀπωτέρω51) (827-94)

have assimilated τὸ πλῆττον in the αὐλός52) to the ὑφόλμιον herein men-tioned Now we cannot identify this ὑφόλμιον safely since few aremdashapart from the lattermdashthe appearances of the term Th e only one giving some information uncertain otherwise is Hesychiusrsquo53) this author identifi es the ὑφόλμιον either with the reed (γλωττίς)54) or with a part of the αὐλός near to its mouthpiece (στόμα) which according to the etymology of the term (literally lsquowhat is under the ὅλμοςrsquo) should be the part of the pipe where the ὅλμοι are inserted55) Whatever the truth is that part of the αὐλός seems to have been at the end of the pipe nearer to the mouthpiece and so both the proximity of a hole either to the ὑφόλμιον or to the focus issuing

50) Cf Levin 1980 213 and Redondo Reyes 2003 331 n 4751) lsquoIndeed provided that the other features remain identical ( ) in auloi the sounds coming out through the holes nearer to the hypholmion (that is to the striking agent) prove to be assuredly higher than those coming out through those further awayrsquo52) Th e αὐλός was a reed instrument whose main body was a pipe (βόμβυξ) of cylindrical or conical section (κοιλία) with several holes (τρυπήματα) In this pipe were inserted twomdashexceptionally threemdashmobile olive-shaped elements (ὅλμοι) in the fi rst of them (σῦριγξ) completing with it the mouthpiece was affi xed the (double) reed (γλωττίς) while the second one removable was an extension of the previous like the so-called lsquocrooksrsquo used in wind-metal instruments before the invention of the valves and serving to modify the instrumentrsquos pitch At the other end the pipe could be fi nished in an open slightly widened bell (κώδων) Cf Chailley 1979 62 and Michaelides 1978 42-353) ῾Υφόλμιον μέρος τι τοῦ αὐλοῦ πρὸς τῷ στόματι ἢ αἱ γλωττίδες καὶ ὑπόθεμά τι (Hsch υ 908)54) So Porphyry (in Harm 5422) seems to interpret it by replacing τοῦ ὑφολμίου with τῆς γλωσσίδος in his commentary55) Th is is indeed Michaelidesrsquo interpretation (1978 43 and 148) of the ὑφόλμιον and also Barkerrsquos (1989 282 n 34)


the blow become equivalent since the length of the vibrating air column is the same in both cases I do not think then that the ὑφόλμιον should be taken as the striking agent but as a reference to point at the greater or lesser proximity to the agent itself which is the blow emitted by the playerrsquos mouth (cf Barker 1989 282 n 34)

23 Τὸ διrsquo οὗ ἡ πληγή

Th is is the factor whose interpretation seems to have raised most contro-versy Many scholars have identifi ed it with τὸ πλῆττον56) maybe because they have attributed a causal agentive value to the expression with the genitive However the preposition διά in Ptolemy governs the accusative when it has a causal value57) while its use with the genitivemdashthe one we have heremdashregularly indicates means or place through58) If we accept thus the common value of διά with the genitive to express the means τὸ διrsquo οὗ ἡ πληγή should be a body intervening between agent and air which trans-mits to the latter the action triggered by the former the only one which can play this role is the instrument used at the moment basically the mem-brane or the metallic or wooden disk (in percussion instruments) the string (in chordophones) the pipe59) (in aerophones) or the windpipe60) (in

56) Cf Levin 1980 214 lsquothe agentrsquo Barker 2000 36 40 and 41-7 lsquothe agentrsquo lsquothe strikerrsquo Levin 1980 216 and 218 as I have already pointed out also confuses τὸ διrsquo οὗ ἡ πληγή with τὸ πληττόμενον57) Especially cf causal expressions such as διὰ τοῦτο (passim) or διὰ τό + infi nitive (626 1122 126 1633 186 188 2022 217 3412 3624 3634 3920 (4010) 5122 5927 6322 649 6517 6626 678 6714 8418 8427 904 964 966 10128 10212 10411 10416 10610 opposite to a single and peculiar διὰ τοῦ + infi nitive in 10116)58) Cf 426-7 54 65 710 717 815-6 917 etc and particularly the expressions διὰ πασῶν διὰ πέντε or διὰ τεσσάρων Th ere is a single case of διά with the accusative to express the means (διὰ [τὴν] τῆς λειότητος ἢ τραχύτητος ποιότητα 715) but the prob-lems it raises about both textual transmission and interpretation call for secluding the article in the accusative cf Alexanderson 1969 859) So in Ptolemy (96-9) Porph in Harm 5517-8 shows it as struck In fact it would be more exact to say that the medium is in this case the air column contained in the pipe whose length is determined by the blocked-up holes and which is set vibrating by action of the blow by selecting from it the oscillatory frequency which determines its corresponding pitch60) Th e windpipe (ἀρτηρία) considered as responsible for the sounds characteristic of the human voice cf Ptolemy (99-12) Porph in Harm 5517-8 shows it again as struck In


the human voice) besides the confusion between instrument and agent is obviously out of place61) since the attributes of the sound both of them determine are radically diff erent as we will see below Apart from that no other explanation for the medium fi ts since the instrument is the only onemdashof the three terms involved in the process (two lsquosolidrsquo bodies and the air)mdashwhose corresponding factor has not been established yet

I will try to show that my version of the three factors contributing to the sound-producing process fairly accounts for the four diff erential causes of sounds set out in 615-24 (cf sect 2)

3 Th e Four Diff erential Causes of Sounds

As for the air as the body struck only one characteristic is studied its bodily composition and as for the striking agent in turn also only one its striking force However as regards the intervening medium or instrument two are studied its bodily constitution (in which its form is also included) and its length62) as I will show at the proper time (cf sect 333) Th e two former are rejected as irrelevant to pitch while the two latter do justify it63)

fact the medium is in this case the vocal folds whose level of tension or tightness as well as their length establishes pitch though this fact seems to be unknown even to current authors For Greek musical writers anyway the voice pitch was controlled by the windpipe and Th eophrastus for example justifi es (fr 89 57-11) its varying pitches by modifi cation of its length while Ptolemy (96-15)mdashfollowed by Porphyry (in Harm 5518-26)mdashconceived a hybrid action for it halfway between αὐλός and monochord61) Moreover we should not forget that the terms lsquoagentrsquo and lsquomediumrsquo or lsquoinstrumentrsquo are the ones we use to interpret Ptolemyrsquos analytic expressions the only ones used by him Otherwise there is not a single point in chapter 13 where its author replaces τὸ διrsquo οὗ ἡ πληγή with τὸ πλῆττον or vice versa faithful to his own principles of terminological coher-ence expressed in Judic 101-1062) Th e analysis of the two latter is of the utmost importance since it is on them that the author places the quantitative character of pitch which is the explicit reason for tackling the study of sound attributes (cf sect 42)63) In 2616-272 again after excluding other instruments (as inadequate) from acoustic experimentation and accepting only the monochord and the like for this purpose Ptolemy points out three diff erential causes of sounds in the latter density cross section and length [sc between the movable bridges] of the string the fi rst two correspond to the bodily con-stitution of the string (ie the medium) and the third to the space between striker and thing struck as we will see


Th e establishment of them all is supposed to be empiric however the experimental method of diff erentiation proposedmdashwhich involves main-taining all the variables fi xed except for just one which is modifi ed to observe what the result is (622-4)mdashseems to have been hard to put into practice (cf Barker 2000 36-7) given the diffi culties raised by some of the observations proposed64) even in a modern laboratory As Barker loc cit suggests ldquoquite casual observation and refl ection will have seemed suffi -cient to confi rm that each of his factors can be responsible independently of the others for variations in the resulting pathē rdquo in fact according to what is shown in 1632-1726 (where the possibility of experimenting with instruments other than the ἁρμονικὸς κανών is rejected outright) we must think that the essential part of Ptolemyrsquos experiments would be carried out on chordophones whence the results would be expandedmdashsometimes in a somewhat forced waymdashto other instruments

Anyway even though Ptolemyrsquos conditions are not ideal for carrying out such experiments and though the results he obtained had thence a lesser scientifi c validity it would be completely unfair to ignore that Ptolemy isolates as determinant of pitch some characteristics (mass density ten-sion and length) which are just the ones modern acoustics recognizes as such65) Th e authorrsquos awareness of the signifi cance of these characteristics in strings can be seen here and also in other points of the treatise (cf 1727 ff and 2616-75) while his knowledge of it in pipes (except for the relevance of their length) is far less With regard to the understanding of the human

64) Such as verifying the invariability of some characteristics of the medium (thickness density smoothness etc) measuring the agentrsquos striking force studying the composition of the air as the body struck measuring the string tension accurately etc65) Sound pitch is in practice identifi ed with the frequency of the associate oscillatory motion In vibrant strings Mersennersquos (1588-1648) laws affi rm that such a frequency is inversely proportional to the string length (fi rst law) directly proportional to the square root of the tension to which it is subject (second law) and inversely proportional to the square root of its linear density (third law) In sounding pipes Bernoullirsquos (1667-1748) laws establish that it is inversely proportional to the length of the vibrating air column (second law) in aerophones also the rise in temperature implies a rise in vibratory frequency In the voice the fundamental sound frequency is that of the vocal folds vibrating primarily determined by their mass longitude and tension and secondarily by the speed and pressure of the air passing through them Cf Fernaacutendez de la Gaacutendara amp Lorente 1998 49 50 and 139 Donington 1986 112 and 199-200 and Michels 1982 23


voice it is really limited and leaves now a great deal to be desired in Ptol-emy as well as in Greek authors in general66)

31 Th e Bodily Constitution of the Air (ἡ σωματικὴ σύστασις τοῦ πληττομένου)

Although the only characteristic of the air as πληττόμενον (σῶμα) taken into account concerning sound attributes is its bodily constitution67) we are told (624-7) that such a characteristic does not constitute a diff erential cause of pitch or that at the very best the diff erence owing to it is imper-ceptible lsquosince also the variation of airs proves to be such [sc imperceptible] for the sensersquo68) It proves nevertheless to be surprising that such varia-tions can be left aside as imperceptible for the hearing when shortly before (415-9) we were acquainted with the need of rational instruments in order to compensate for the incapacity of the senses to appreciate lsquotrifl esrsquo individually irrelevant but whose accumulation can produce considerable errors We cannot but agree with Barker (2000 38) when he says that it is rather rash of Ptolemy to admit without demonstration that imperceptible causes have imperceptible eff ects

Otherwise Ptolemy by pointing out the bodily constitution of the air as a diff erential cause of sounds is perhaps likely to include an opinion present in previous authors according to which the air must fulfi l a minimum of requirements for sound to be produced Th is opinion can be found among the Pythagoreans (cf Adrastus apud Porph in Harm 81) and in Aristotle (de An 419b19-22) and Porphyry (in Harm 814-5) echoes it in his comments on Ptolemyrsquos defi nition of sound All in all the bodily constitution of the air could be a characteristic of it which

66) Which is not surprising for this instrument has always been treated in a peculiar way Even nowadays confusions of pitch with timbre for example can be traced in expressions such as lsquohead voicersquo or lsquochest voicersquomdashstill in usemdashwhich refer the voice high range to the head and the low one to the chest (cf NODE sv lsquohead voicersquo) also cf 10513-667) Th e lsquobodily constitutionsrsquo of the air become apparent according to Porphyry as lsquohot or cold foggy or pure airs and many other diff erences which go unnoticed by the sensersquo (in Harm 3929-30)68) Cf Porph in Harm 3928-401 According to Barker (2000 38) ldquomost Greek writers tacitly assume that this thesis is substantially true and at least one earlier theorist explicitly makes a comparable claimrdquo and quotes Ps-Arist Aud 800a


is held to be necessary for sound production though irrelevant as regards its attributes

32 Th e Force of the Striking Agent (ἡ τοῦ πλήττοντος βία)

Th is is in turn the only characteristic of the striking agent taken into account regarding sound attributes According to Ptolemy (627-75) the agentrsquos force is responsible only for loudness (μέγεθος) of the sound pro-duced69) not for its pitch In his comment on this cause Barker (2000 38-40) points out two questions the author overlooks Th e fi rst one of little relevance is the fact that blowing more intensely has the eff ect in some aerophones of a rise in pitch70) here lies probably the cause why that type of instrument is fi nally rejected (175-7) for harmonic experi-mentation Th e second onemdashnow more importantmdashis that the author is here ignoring Archytasrsquo doctrine (47B1 DK) according to which the agentrsquos force is cause of sound highness while its weakness is a cause of lowness in Barkerrsquos opinion (2000 40) such a doctrine had already been discarded by Plato (Ti 67b-c) and Aristotle (GA 786b28-7a22) but Ptolemy if seeming here (627-75) to agree with that rejection contra-dicts it later (83-5) by accepting that greater intensity determines greater sharpness in strokes Barker is not alone in pointing out that contradiction (cf n 89) although nobody seems to have noticed that the force (βία) corresponds in 627-75 to the agent while what is considered more intense (σφοδρότερον) in 83-5 is the medium and that identical properties in both factors do not necessarily have an identical eff ect in their correspond-ing sound attributes I will return to this below (cf sect 332)

33 Th e Characteristics of the Transmitting Medium

Th e medium is doubtless the most outstanding factor as far as pitch is concerned Its bodily constitution set out in 620-1 is complemented later by its form (σχῆμα) also understood as a diff erential cause of sounds Th e authorrsquos explanation goes this way

69) Similar idea in Nicom Harm 2435-870) Also cf Aristox Harm 5219 In my opinion Ptolemy is almost certainly thinking of chordophones in which pitch does not change however strong the impact made by the plectrum is unless the force of the stroke modifi es the tension of the string


Th e variation according to the media whereby the strokes happen in turn is here considered as variation according to the bodyrsquos primary constitutions71)mdashthat is those by which each body is rare or dense thin or thick and smooth or roughmdashand also as variation according to the forms What otherwise do the more aff ectable (παθητικαί)72) qualitiesmdashand I mean smells fl avours and coloursmdashhave in common with a stroke On one hand it73) provides sounds through the form (σχῆμα) in those which admit such a thing (as the tongue and the mouth) with conformations (σχηματισμοί) by way of patterns for them (corresponding to which names are coined such as lsquochatteringsrsquo lsquoclicksrsquo lsquoaccentsrsquo lsquowhistlesrsquo and thousands of the sort as we imitate each of such con-formations for man has a very rational skilful governing capacity) on the other hand through the smoothness or roughness it provides them only in turn with quality74) (according to which by homonymy some sounds are called lsquosmoothrsquo or lsquoroughrsquo) because also these are above all qualities while

71) Th at is those of τὸ διrsquo οὗ ἡ πληγή the particular body (σῶμα) which exemplifi es the media (τὰ διrsquo οὗ ἡ πληγή) here in question72) Cf Arist Cat 9a35-b7 lsquoTh ey are called lsquoaff ectable qualitiesrsquo not because the bodies which have received these qualities experience a certain aff ection themselves [ ] but because each of the aforementioned qualities is causative of sensory aff ection and that is why they are called lsquoaff ectable qualitiesrsquorsquo Th ere is a consensus among the diff erent scholars that Ptolemyrsquos use of παθητικαί is the Aristotelian one73) Th at is what a stroke has in common with smells fl avours and colours the capacity (cf n 72) to cause aff ections (sounds in the case of strokes remember again that in Ptolemyrsquos views sound is an lsquoaff ection of air struckrsquo) Th rough each of the aforesaid (75-8) mediumrsquos qualities such a capacity provides the corresponding sounds with a homonymous quality through the form (σχῆμα) it provides lsquoconformationsrsquo (σχηματισμοί) through the smoothness (λειότης) lsquosmoothrsquo (λεῖοι) sounds through the roughness (τραχύτης) lsquoroughrsquo (τραχεῖς) sounds etc Ptolemy is here following Ps-Arist Aud (803b26-9 802b10-3 b14-8 b18-21 etc) cf Barker 2000 4274) In contrast to the mediumrsquos form Ptolemy breaks its qualities down into two well-defi ned groups on the one hand λειότης (and its contrary τραχύτης) on the other μανότης (and πυκνότης) as well as παχύτης (and λεπτότης) Th ose of the fi rst group are lsquolegitimatersquo qualities since they are fi rst and above all (κυρίως) qualities those of the sec-ond group instead are lsquohybridrsquo qualities since theymdashthough qualitiesmdashare based on the quantity of the specifi c bodyrsquos substance (ἑκατέρα ποιότης οὖσα παρὰ τὸ ποσὸν γέγονε τῆς οὐσίας) As a result in the authorrsquos view those of the fi rst group provide sounds only with quality (ποιότητα μόνην) while the other ones provide them with quality and besides with highness and lowness (ποιότητας καὶ ἔτι βαρύτητας ἐνταῦθα καὶ ὀξύτητας) in this way is laid one of the foundations on which this author bases the quantitative character of pitch (cf sect 42)


through the rarity or density and the thickness or thinness it provides them with qualities (also according to which in turn by homonymy certain sounds are called lsquodensersquo or lsquoporousrsquo and lsquothickrsquo or lsquoleanrsquo)75) and here76) besides it provides them with lowness (βαρύτης) and highness (ὀξύτης)77) since also each of both aforementioned constitutions78) even though being quality is produced according to the quantity of the substance (because denser than other of similar constitution is the one which has more substance in the same volume and thicker is that which has more substance in the same length and the denser and the thinner ones are providers of the higher and the rarer and the thicker instead of the lower)79) (75-25)

Two features of the medium are thus considered its form and its primary bodily constitution

331 Th e Form of the Medium (τὸ τοῦ διrsquo οὗ ἡ πληγὴ σχῆμα)Sound takes on because of the transmitting mediumrsquos form80) certain lsquoconformationsrsquo (710-2) In the human voice specifi cally mentioned in 711 (οἷον τῶν γλωσσῶν καὶ τῶν στομάτων) the mediumrsquos form is that adopted by the buccal cavity81) as can be understood from the example

75) Neither lsquoleanrsquo (ἰσχνούς) nor lsquoporousrsquo (χαύνους) are the expected homonyms (lsquothinrsquo (λείους) and lsquorarersquo (μανούς) respectively)76) Th at is in the case of the qualities of rarity or density and thickness or thinness77) Th e terms used here (βαρύτης and ὀξύτης) have in fact the wider meaning of lsquoheavinessrsquo and lsquosharpnessrsquo which in Greek are inclusive of a specifi c sense for sounds lsquolownessrsquo and lsquohighnessrsquo respectively It cannot be rendered into English and I have then chosen the musical meaning though it could cause some problems in understanding the generaliza-tion 725-7 where the wider meaning is wanted78) Rarity or density and thickness or thinness respectively79) Text punctuation is mine80) If this one admits it specifi es the author So it is in pipes (by adding the so-called lsquobellrsquo) or in the human voice but not in chordophones81) Cf Barkerrsquos interpretation (2000 41) ldquothe conformation of the human mouth or tonguerdquo as an example of the most generic ldquoshape of the strikerrdquo However there is as a result a slight discrepancy in his interpreting ldquothe strikerrdquomdashhere ldquothe human mouth or tonguerdquo but before (2000 37) ldquothe breathrdquomdashdue to the agentive value of τὸ διrsquo οὗ ἡ πληγή (2000 41 ff ) put on an equal footing with τὸ πλῆττον Levin (1980 214) in her turn renders τῶν γλωσσῶν καὶ τῶν στομάτων as ldquoreeds and lipsrdquo referring to the aerophone mouthpiece such a translation though possible makes it diffi cult to understand which is then the lsquoformrsquo Ptolemy is talking about


712-5 (παρrsquo οὓς ὀνοματοποιοῦνται τὸν ἄνθρωπον) Now what are those lsquoconformationsrsquo which the instrumentrsquos form determines In the human voice (if we leave aside the eff ects made by the tonguemdashclicks or the likemdashas mere noises and not sounds) the modifi cation of the buccal cavityrsquos formmdashas that of any part of the vocal tractmdashhas as a result that of the formants and overtones of the sound emitted that is the variation of its timbre (cf Grove sv lsquoAcousticsrsquo 83-4 86 and also sv lsquoFormantrsquo 710-1 and Fernaacutendez de la Gaacutendara amp Lorente 1998 142-7) Something similar happens in aerophones with the form adopted by the pipe which if not responsible for pitch (this essentially depends on the pipersquos length) is determinant of timbre for the bell acts on them as a resonator as respon-sible for the harmonic resonances which it selects (cf Fernaacutendez de la Gaacutendara amp Lorente 1998 108)82) Consequently though we cannot deter-mine to what extent Ptolemy bore timbre in mind as a sound attribute or parameter since he does not explicitly mention it we might conjecture those sound conformations as diff erent timbres derived from the instru-mentrsquos form

332 Th e Constitution of the Medium (ἡ τοῦ διrsquo οὗ ἡ πληγὴ σύστασις)Th e primary bodily constitution of the medium is here seen as its smooth-ness or roughness rarity or density and thickness or thinness (77-8) Th ese lsquoqualitiesrsquo of the medium provide sound fi rst of all with homonymous qualities so that sounds are called lsquosmoothrsquo or lsquoroughrsquo lsquorarersquo or lsquodensersquo and lsquothickrsquo or lsquothinrsquo83) Besides rarity or density and thickness or thinness also determine sound highness (ὀξύτης) or lowness (βαρύτης) for lsquothe denser and the thinner ones are providers of the higher [sc sound] and the rarer and the thicker instead of the lower onersquo (723-5)84) Th is statement has

82) Also cf Paquette 1984 74 ldquoAgrave la diff eacuterence de lrsquoaulos la salpinx possegravede un pavillon amplifi cateur de forme varieacutee ( ) Agrave ces pavillons devaient correspondre des timbres dif-feacuterentsrdquo though about the αὐλός he says (1984 28) ldquoCet eacutevasement terminal [sc the bell] ne modifi e pas le timbre mais sert drsquoamplifi cateurrdquo meaning perhaps that the bell little testifi ed for this instrument (id ibid) does not off er the variety of forms we would associ-ate with a wish to change timbre83) As the aforesaid σχηματισμοί these lsquoqualitiesrsquo of sound are again among its character-istics of timbre84) So it is in chordophones according to Mersennersquos third law (cf n 65 density and thick-ness determine the string lineal density ie its mass per metre) but not in aerophones


general validity since lsquoalso in the other cases85) the sharper (ὀξύτερον) is called so because it is thinner as the blunter (ἀμβλύτερον) because it is thickerrsquo (725-7) and the greater sharpness is justifi ed because lsquothe thinner strikes more concentratedly (ἀθρούστερον)86) for it can penetrate quicker and the denser for it can penetrate deeperrsquo (727-8) hence lsquobronzersquo (χαλκός) produces a sharper sound than wood (ξύλον)87) and gut (χορδή) than linen thread (λίνον)88) because they are denser and of bronze pieces with similar density and equal size it is the thinner which produces it and of the guts of similar density and equal length it is the leaner and the hollow bodies more than the solid ones and of the windpipes in turn the denser and thinner are again those of sharper tonersquo (729-82) Th is happens since these bodies are tauter (εὐτονώτερα)

ὅτι τοῖς μὲν τοιούτοις εὐτονωτέροις εἶναι συμβέβηκεν τὸ δὲ εὐτονώτερον ἐν ταῖς πληγαῖς γίνεται σφοδρότερον τοῦτο δὲ ἀθρούστερον τοῦτο δrsquo ὀξύτερον (83-5)

where pitch depends essentially on the length of the vibrating air column (cf Michels 1982 47 and Donington 1986 199) It seems necessary to think of experimentation in chordophones here though the statement is also valid in percussion instruments (I mean those of defi ned tone of course)85) In which the adjective lsquosharprsquo (ὀξύς) can be used the metaphoric origin of its use referred to sounds is even in Arist de An 420a29-30 Also cf n 77 on the meaning of ὀξύτης and βαρύτης86) Barker (2000 44) suggests hesitantly ldquothat has greater compactnessrdquo to render ἀθρούστερον putting (p 45) lsquocompactnessrsquo on a level with lsquodensityrsquo (πυκνότης) Th e sense of this adjective here is I think rather lsquoconcentratedrsquo (cf LSJ sv for DH Comp 22144 about the state of the air in the implosionmdashτὸν ψόφον λαμβάνοντος ἀθροῦνmdashprevious to the explosion of the voiceless bilabial p) the same sense as it later (85) has referring to the way a string vibrates more concentrated when it is tenser than when less tense87) Bronze and wood could here refer to materials used to make both percussion instru-ments and aerophones although the material used for the latter is irrelevant concerning pitch and the statement is then false (cf n 84 and Paquette 1984 26)88) Χορδή here has its original value of lsquogutrsquo since gut was the commonly used material in ἐντατά (literally lsquocapable of being tensedrsquo) instruments which nowadays we just call lsquochor-dophonesrsquo On the other hand λίνον means lsquofl axrsquo and by extension the lsquolinen threadrsquo made out of this raw material and it seems to refer in turn to another material used tensely in an ἐντατόν even if the normal ones were gut and nerve (νεῦρα) (cf Suda κ 1415) and only in some scholia to Il 18570 is it a question of λίνον in that sense


Th erefore pitch derived from the mediumrsquos bodily constitution essentially depends on how taut (εὔτονον) this is because a tauter medium vibrates with greater intensity (σφοδρότερον) when struck thus its vibrating is more concentrated (ἀθρούστερον) and as a result the sound it produces is sharper (ὀξύτερον) Several scholars89) have criticized Ptolemyrsquos reasoning for it supposedly contradicts the previous (627-75) rejection of force (ἡ τοῦ πλήττοντος βία) as a cause of pitch However a fundamental error underlies these criticisms since the striking agent (whose force (βία) was postulated there) is mistaken for the medium transmitting the stroke (the cause of greater sharpness adduced here for its being more intense (σφοδρότερον) in the stroke)90) and no notice is taken of Ptolemy imme-diately insisting on this second referent

καὶ διὰ τοῦτο καθrsquo οἵαν ἂν δύναμιν εὐτονώτερον ἕκαστον ᾖ τῶν διrsquo ὧν αἱ πληγαί θᾶττόν τε καὶ ὀξύτερον91) ἀποτελεῖται92) (814-5)

with no room for doubt that the adjective εὐτονώτερον (and as a result σφοδρότερον) is attributed to τὸ διrsquo οὗ ἡ πληγή Nonetheless since τὸ διrsquo οὗ ἡ πληγή has been interpreted as agent and assimilated to τὸ πλῆττον (cf n 56) a fl agrant confl ict arises only due in my opinion to the confu-sion of two factors clearly distinguished by Ptolemy It is indeed obvious that identical characteristics of both the agent and the medium do not necessarily lead to identical properties of the resultant sound93) and that

89) Cf Porph in Harm 518-13 and Barker 2000 40 Redondo Reyes (2003 344-5 n 56) rejects the contradiction between both statements although his explanation after identify-ing the medium with the agent proves to be unsatisfactory90) Th us in a chordophone like the kithara the plectrumrsquos striking force would be being mistaken for the vibratory intensity of the string Th ough the former does not have any eff ect on sound highness or lowness it is not evident that it is so for the latter as long as it has not been clearly specifi ed what it consists of in fact such an intensity is intimately related to the oscillatory frequency as we will see91) Duumlring 1930 gives ἐλάττων τε καὶ ὀξύτερος here but I think it would be better to read the adverbial accusatives θᾶττόν τε καὶ ὀξύτερον with f (together with the ὀξύτερον from MEV ma ss W) which is also Wallisrsquo choice (1699 6) θάττων τε καὶ ὀξύτερος would also be possible although its testimony in the manuscripts is more doubtful92) lsquoAnd on this account [sc sound] is executed quicker and higher according to what extent each of those media whereby the strokes happen is tauterrsquo93) Nobody points out a contradiction in Ptolemy when he postulates the bodily constitu-tion of the transmitting medium as fundamental regarding pitch after deeming that of the


thence there is no matter of contradiction in Ptolemyrsquos opinion the strik-ing agentrsquos intensity leads to striking force (which only determines greater or lesser loudness) while the mediumrsquos intensity is the cause of the concen-tration (τὸ ἀθρόον) of the disturbance which in turn leads to greater or lesser highness In fact intensity as explanation for greater highness is used again later onmdashwith nobody talking there about contradictionsmdashwhen the author studies the relationship between the mediumrsquos length and sound highness

ταῖς μὲν ἐλάττοσι διαστάσεσιν ἑπομένης τῆς ὀξύτητος διὰ τὸ ἐκ τῆς ἐγγύτητος σφοδρόν (819-20)

It should moreover be underlined that the terms used in both cases are not the same in 627 it is a question of the striking agentrsquos lsquoforcersquo (βία) while in 84 it is its lsquocondition of more intensersquo (σφοδρότερον) which is postulated for the medium Th e adjective σφοδρός is very close in mean-ing in general to βιαῖος (lsquoforciblersquo) and Ptolemy himself uses them (71-5) as synonyms in that sense94) when stating that a σφοδρότερον striking agent leads to greater loudness But in 84 and 820 instead σφοδρότερον is used rather to allude to an implicit idea of speed (τάχος) concomitant with the intensity it represents such a meaning can be seen in Aristotle95) for example σφοδρός can also refer to ἄνεμος to express the intensity of the winds inseparable from their speed (AlexAphr Pr 3165 Gal 831611-2) or simply associated with the idea of speed (Pl Chrm 160c3-4 Plt 306e4) In this sense the adjective generally referring to motion entails a link with the Pythagorean theory of pitch (cf n 5) and is used as an explanation for sound highness in musical texts such as in this Peripa-tetic one

air as irrelevant for the same purpose though the characteristic is the same (the bodily constitution) its eff ects on the sound are nevertheless radically diff erent likewise the fac-tors which have that characteristic94) Cf ἠρεμαιότερον (73) ie ἀσθενέστερον (75) in contrast to ἢ σφοδρότερον ἢ ἁδρότερον (74) ie βιαιότερον (75)95) Cf Arist Mete 365b28-33 (esp 365b32-3 σφοδρότατον μὲν οὖν ἐξ ἀνάγκης τὸ τάχιστα φερόμενον πλήσσει γὰρ μάλιστα διὰ τὸ τάχος) and its comment in AlexAphr in Mete 11625 ff also cf Arist Mech 852b35-6 (τὰ δὲ φερόμενα θᾶττον ἐκ τοῦ ἴσου ἀποστήματος σφοδρότερον τύπτει)


ὅτε δὲ σφοδροτάτη ἡ κίνησις καὶ ἡ φωνὴ ὀξυτέρα τοῦ φερομένου διὸ καὶ χορδαὶ ἐπιτεινόμεναι ὀξύτερον φθέγγονται θᾶττον γὰρ ἡ κίνησις γίνεται96) (Ps-Arist Pr 920b1-4)

where we can appreciate to what extent the intensity (σφοδρότης) of a stringrsquos motion is indissolubly linked with highness of the corresponding sound while the speedmdashas the factor necessary for the formermdashis regarded as the cause of the latter Th us σφοδρός can be related with the loudness as much as with the pitch of the resulting sound and Ptolemy retains both possibili-ties already present in previous authors when he proposes a causal relation-ship between ἡ τοῦ πλήττοντος σφοδρότης and τὸ τοῦ ψόφου μέγεθος (71-5) and between τὸ τοῦ διrsquo οὗ ἡ πληγὴ σφοδρότερον and ἡ τοῦ ψόφου ὀξύτης (84-5)97) Nevertheless the author does not use (cf sect 41) the speed implicit in σφοδρός as an explanation for highness but a slightly diff erent substitute for it the mediumrsquos more concentrated way (ἀθρούστερον) of striking98) It is indeed easily verifi ed that a tauter string99) seems to vibrate more speedily and with a lesser distance from its position of restmdashthat is to do so in a more concentrated waymdashthan another which is less taut and seems to move with slower relaxed100) and wider motions thus the comparative ἀθρούστερον alludes to the more concentrated motion (in strings) which

96) lsquoWhen in turn the motion is most intense the voice of what shifts is also higher for this reason the tensest strings also emit notes with greater highness for their motion is quickerrsquo also cf id ibid 886b15-6 900a22 900a33-4 and 920b26-7 97) Th is is no doubt one of the empiric reasons which led him to diff erentiate both fac-tors agent and medium the fact that the same qualities cause diff erent sound attributes when shown by each one of them In his diff erentiating them this way lies the originality and brilliance of his analysis even though the material is pre-existent Ptolemy is alone in making use of it coherently 98) A similar explanation to Ptolemyrsquos and with identical purpose occurs in Ps-Euc SectCan 1489-11 Jan although the adjectives there used are πυκνός (not ἀθρόος) and ἀραιός also other adjectives (πυκνός and χαλαρός respectively) can be seen in Porph in Harm 3310-5 99) Again it is obvious that Ptolemyrsquos experimentation is not carried out on pipes where no examination can be made by just looking at them100) Th e contrast between lsquorelaxationrsquo (ἔκλυσις) and lsquointensityrsquo (τὸ σφοδρόν) as causes of lowness and highness respectively associated to a longer or shorter string length can be also seen in 819-21 (cf 333)


occurs quicker101) (θᾶττον 815) and together with greater highness (ὀξύτερον 85 815) Also Nicomachus (Harm 24320-41) more or less contemporary with Ptolemy appeals to intensity to justify greater highness in tauter strings whichmdashhe saysmdashlsquovibrate (ἀποκαθίστανται) quickest with much agitation (κραδασμός) and hitting the air around them in many parts as if being hurried by action of the intense tension itself (ἡ σφοδρὰ τάσις αὐτή)rsquo

One of the causes of pitch lies then in the mediumrsquos tautness since sound is nothing else but pitch-tension (τάσις)102) Porphyry explains the relationship between tautness of the medium and tension of the air because the latter by being lsquounifi edrsquo guaranteesmdashas if a stringmdashan uninterrupted transmission of the tension which the air between the sounding bodies receives from them (ie from the medium) when struck by them toward the outer air where such a tension is already understood as eff ective pitch (cf Porph in Harm 525-14 and the example Arist de An 419a25-7 therein mentioned) Air tension and the mediumrsquos tautness are thus identi-cal and the use of the terms τάσις and εὔτονοςmdashwhich obviously refl ects the synonymy (cf n 13) between τάσις and τόνοςmdashis not coincidental Nevertheless Ptolemy is wrong to propose as cause what instead is con-sequence Indeed pitch (inde sound once it has been identifi ed with it) was metaphorically called tension from strings (cf sect 12) since empirically the greater tension in the string (inde by extrapolation in the medium) the greater its sound highness instead the authormdashwithout justifying it from the initial defi nition of sound (32-3)mdashhere (812-4) identifi es pitch with tension just to explain why the greater mediumrsquos tautness results in greater sound highness

Otherwise once tautness has been assumed as cause of highness the author deduces that if one body is tauter for reasons diff erent from the previous ones (thinness and density) than another which is so for these two the fi rst one emits a higher sound than the second

lsquoTh ence also if a body is tauter in a diff erent way (for example by being harder (σκληρότερον) or as a whole larger (μεῖζον)) it produces a sharper

101) Th e lsquoquicknessrsquo or lsquospeedrsquo must here be read as lsquooscillatory frequencyrsquo (ie rate of oscil-lations per second) that is sound pitch (cf sect 41)102) Cf defi nition 812-4 in n 21


sound sincemdashover those two which have something productive of a similar eff ect103)mdashthe pre-eminence corresponding to the other ratio104) wins out (as when bronze produces a higher sound than lead since the former is harder than the latter to a greater extent than the latter is denser than the former and again the larger in its case and thicker bronze produces a higher sound than the smaller and thinner one when the ratio according to size is greater than that according to thickness)rsquo105) (85-12)

Tautness can be due according to Ptolemy not only to thinness and den-sity (83-4) but also to hardness and size (86 and examples from 88-12) Th e introduction of these two new reasonsmdashwhich certainly are not an evident cause of greater tensionmdashreceives from the commentator (Porph in Harm 5118-23) a harsh criticism Puzzling certainly is the lack of explanation fi rstly why a thinner or denser medium proved to be tauter and later how hardness or size can produce that same eff ect in fact a greater sizemdashif such is the mediumrsquos lengthmdashis cause of lowness not of highness as we are told immediately (cf sect 333) As for its hardness another passage should be taken into account wheremdashreviewing the stringsrsquo characteristics on which the highness of their corresponding sound dependsmdashthe author says

ἐπειδὴ γὰρ τρία ἐστὶν ἐπὶ τούτων τὰ αἴτια τῆς περὶ τὸ ὀξὺ καὶ τὸ βαρὺ διαφορᾶςmdashὧν τὸ μὲν ἐν τῇ πυκνότητι τῶν χορδῶν τὸ δὲ ἐν τῇ περιοχῇ τὸ δὲ ἐν τῇ διαστάσειmdashκαὶ ὀξύτερος γίνεται ὁ ψόφος ὑπό τε τῆς πυκνοτέρας καὶ τῆς λεπτοτέρας καὶ τῆς κατὰ τὴν ἐλάττονα διάστασιν παραλαμβάνεται δrsquo ἐπὶ αὐτῶν ἀντὶ τῆς πυκνώσεως ἡ τάσις (τονοῖ γὰρ καὶ σκληρύνει καὶ διὰ τοῦτο μᾶλλον ταῖς ἐν ἐλάττοσι διαστάσεσιν ὁμοιοῖ)106 ) 107) (2616-74)

103) lsquoTh ose tworsquo are greater thinness or density Since both of them produce a lsquosimilar eff ectrsquo (ie a greater sharpness as we have already seen) they must havemdashaccording to Ptol-emymdashlsquosomethingrsquo in commonmdashwhich is no doubt a greater tautnessmdashto account for it104) Th at is the ratiomdashdiff erent from the previous two ie those of thinness and den-sitymdashaccording to which one of the bodies is lsquotauter in a diff erent wayrsquo than the other one which will become clear in the following examples105) Text punctuation is mine106) I accept with certain reservations Alexandersonrsquos conjecture (1969 13) which I fi nd in any event preferable to Duumlringrsquos (1930) and to the diff erentmdashand for me incomprehen-siblemdashvariants in the manuscripts text punctuation is mine107) lsquoIndeed since in these [sc in strings] the causes of the diff erence concerning high and


So tension tones up and hardens strings and hardness is a necessary condi-tion for tension but is not evidently also suffi cient for it as we were told in 89 and we are again off ered as cause what in fact is consequence taut-ness is what determines the mediumrsquos thinness hardness etc where it occurs not the inverse

333 Th e Length Imposed by the Medium (ἡ ἀποχὴ τοῦ πληττομένου πρὸς τὴν ἀρχὴν τῆς κινήσεως)We have already seen (cf sect 2) that the fourth diff erential cause (ἡ ἀποχὴ τοῦ πληττομένου πρὸς τὴν ἀρχὴν τῆς κινήσεως) was lsquothe space between the thing struck and the striking agentrsquo (ἡ ἀποχὴ τοῦ τε πληττομένου καὶ τοῦ πλήττοντος) (817-8) Now by proposing the instrument as an intervening medium between striker and thing struck Ptolemy implicitly recognizes its length as the space between them the stroke whose source was the agent (τὸ πλῆττον) must lsquocoverrsquo the space which separates it from its fi nal term the air (τὸ πληττόμενον) a space which cannot be other than the length the medium between them imposes Th is length being variable (ie depending for example on the number of covered holes in a pipe on the point where the monochordrsquos movable bridge is placed etc) is not expressly framed in terms of τὸ διrsquo οὗ ἡ πληγή but it is the instrument which unquestionably imposes it in diff erent ways though depending on the diverse pitches within its range

Well then pitch also depends on length

since highness follows from the shorter distances owing to the intensity derived from proximity while lowness follows from the longer owing to the relaxation which occurs with being further away so sounds are aff ected inversely108) with distances for as the wider space from the origin is to the smaller so is the sound coming from the smaller to that from the wider (819-23)

low are threemdashone of which lies in their density another in their cross section and another in their lengthmdashand since sound proves to be higher when made by the denser as well as by the thinner and the one with the shorter length although tension is used in them instead of density (since it tones up and hardens them and for that reason makes them more similar to those with shorter lengths) rsquo108) lsquoInverselyrsquo since it is understood that highness is more than lowness and lowness is less than highness


though this explanationmdashas others in the treatisemdashis not original Ptol-emy109) Th e statement does not require any proof but is lsquopatentrsquo (ἐναργές 825) so that checking it is lsquoat handrsquo or lsquoreadily availablersquo (πρόχειρον 825) and it is shown that in fact this length is none other than that imposed by the instrument in the sound production process

And this [sc that sound highness corresponds to shorter length and lowness to longer] in turn since it is patent is readily available from sounds arising from a certain length as that of the strings auloi and windpipes Indeed provided that the other features remain identical in strings the sounds cap-tured in connection with smaller distances between the movable bridges prove to be assuredly higher than those in connection with greater and in auloi the sounds coming out through the holes nearer to the hypholmion (that is to the striking agent)110) are higher than those coming out through those further away and in windpipes the sounds having the origin of the stroke in the upper part and near to the thing struck are higher than those coming from their deep part111) (825-96)

since it is clear that the space (ἀποχή 817) in question is nothing else but the length (μῆκος 826) from which sounds arise (also expressed as lsquodis-tancersquo (διάστασις 91) or by means of adverbs (ἐγγυτέρω 93 ἀπωτέρω 94)) the length of the string the blocked-up length of the pipe or that of the windpipe112)

109) According to Barker (2000 36) the idea comes from Aristotle although the source is not mentioned Certainly the Peripatetic school justifi es it in diff erent ways Th us accord-ing to Ps-Arist Aud 800b6-7 the stroke can only be strong if the striking agent is separate enough from what it strikes Adrastus apud Th eo Sm 6524-63 on the other hand explains the greater lowness by the weakness (ἀσθένεια) caused by a greater resistance to the motion on the part of the longer or thicker string which results in a decrease of oscillations (ie of the oscillatory frequency) yet another reasoning can be seen in AlexAphr de An 5018-25 also cf Porph in Harm 5415-52 Th eophrastus (fr 89 64-8) on the other hand states that the longer the length of the pipe the more force (βία) needed to blow through it110) Th e hypholmion (ὑφόλμιον) should not be understood as striking agent but as the pipersquos point nearest to it which is the playerrsquos blow (cf sect 22)111) Th at is the voice would prove to be higher when produced near the head and lower when produced near the chest (cf n 66)112) A diff erent question is whether the lengths Ptolemy mentions are truly responsible for


4 Two Final Questions

41 Sound and Speed

As I have already pointed out (n 5) the Pythagorean explanation for pitch according to a certain speed (that of the motion determined by the stroke) was widely accepted in antiquity thus it is really surprising that Ptolemy who subscribes to almost every Pythagorean theory does not follow this one Now what in fact is that lsquospeed of the motionrsquo the diff erent authors refer to Truth to tell there are two diff erent kinds of motion involved in the sound process the oscillatory motion of the mediumrsquos and the airrsquos particles (which is not linear but waving around their position of rest without eff ectively moving forwards) and the advancing motion of the sound wave Th us two kinds of lsquospeedrsquo can be associated with such a pro-cess the oscillatory frequency of those particles (ie the number of oscilla-tions per second) and the speed of propagation of the wave (ie the space it covers per unit of time) but frequency varies with pitch and in fact constitutes sound pitch measure (in Hertz) while the speed of sound propagation in the air on the contrary can be regarded as constant113)

However if their texts are carefully analyzed Greek authors seem not to have been aware of the diff erence between both concepts Indeed Plato thinks (Ti 67a7-c3) of propagation speed while Xenocrates114) refers (apud Porph in Harm 317-17) to frequencies but only after saying (id ibid 3015-6) that lsquothe motion concerning the notes is a displacement from place to place in a straight line towards the organ of the hearingrsquo clearly alluding to propagation speed Aelianus the commentator on Plato talks (apud Porph in Harm 342-5) about propagation speed along the pipe and (id ibid 353-7) of oscillatory frequency of strings but refers (id ibid

the pitch of the resulting sound Th e same applies to strings (Mersennersquos fi rst law) and pipes (Bernoullirsquos second law) however concerning the voice to which the term ἀρτηρία refers the statement is false (cf n 60) and its mention here can only be understood as an extrap-olation from the other cases113) Its estimated value (in standard conditions of pressure and temperature and with an air density of 120 kgm3) is approximately 343 ms114) If it is truly him since the authorship of the fragment is doubtful and it could as well be Xenocratesrsquo as Pythagorasrsquo (read a Pythagorean) or Heraclidesrsquo a similar reference to frequencies is attributed to Pythagorean authors in Porph in Harm 337-10


369-12) to both of them as if it were something single Th e Peripatetic school discusses frequencies (τὰ τάχη τὰ τῆς πληγῆς Aud 803b32) while Th eophrastus (fr 89 115-7) thinking of propagation speed denies that the high note can be quicker than the low one reasoning that conso-nancesmdashin fact any type of intervalmdashcould not occur if this was the case since notes of diff erent pitch would not reach the hearing at the same time both (the high note and the low one) are as quick as each other since they arrive together Porphyry himself clearly refers to frequency in strings (τὸ τάχος τῆς περὶ τὴν πλῆξιν φορᾶς in Harm 4814 τὴν μετὰ τὴν πληγὴν ἀποκατάστασιν ibid 5418-9) but thinks of speed of propagation in pipes (ibid 5423-6 12027-30)

Th ere is no lack of logic in supposing that at fi rst the vibratory fre-quency115) of strings116) was seen to increase as their length decreasedmdashie as their sound became highermdashand that this result could later be extrapo-lated to pipesmdashwhere it was not possible to observe the oscillation of the air column and nobody was even aware of its existencemdashwith recourse to the only speed which seemed conceivable in them that of the wave along the pipe Th is confusion would certainly not contribute to separate the concepts of frequency and propagation speed and makes Th eophrastusrsquo remark extremely shrewd when he says that it is impossible for notes plucked simultaneously to be heard together if they propagate at a diff er-ent speed due to their diff erent pitch I would suggest that Ptolemy did not adduce speed as a cause of pitch just on account of the aporia resulting when frequency and propagation speed are regarded as the same thing (ie the variation of frequencies between high and low notes as opposed to the constancy of their propagation speed) which he could not but have realized117) Here possibly lies the reason why he expressed highness as a

115) Not always described as speed as is the case in Ptol Harm 83-5 Ps-Euc SectCan 1489-11 Jan and in Nicom Harm 24320-41116) As I have already said much of the acoustic experimentation must have been carried out on chordophones In practice sound production in strings is the only one which is observable by the naked eye117) Th ere is no proof that the constant value of the speed of sound propagation had been settled at the time (its fi rst measurements in the open air date from 1738 organized by the French Academy of Sciences) but that does not mean that sounds emitted together had not been noticed to reach the hearing at the same time (cf Th eophrastus supra) In fact that light and sound propagate at diff erent speeds was perfectly known even to Democritus


consequence of vibratory concentration (ie oscillatory frequency)118) without referring to speed which he only mentions secondarily119) and so rather as something concomitant with highness but not as its cause

42 Quantitative or Qualitative Nature of Pitch

I cannot fi nish this paper without referring to the question which in fact was Ptolemyrsquos prime objective in chapter 13 and ultimate reason for devel-oping a theory of sound and studying its attributes

Well then though the diff erence between sounds is constitutedmdashjust as also between all the other [sc sensibles]mdashin quality as well as in quantity in which of both the aforesaid genera [sc that of quantity and that of quality] must be placed the diff erence concerning highness and lowness is something which cannot be displayed off hand before the causes of such a characteristic have been researched causes which I fi nd to be somehow common also to the variations in the other strokes (614-9)

Quality and quantity are two of a variable number of categories distin-guished by Aristotle120) (ten in Top 103b21-3 eight in APo 83b15-7 and seven in Metaph 1068a8-9 and Ph 225b5-7) and which constitute the most generic way (whence their denomination of summa genera) in which a being can be described Th e Pythagoreans as they explained sound high-ness and lowness in terms of rapidity or slowness of motion (cf n 5 and sect 41) understood that the diff erence between both of them lay as a result in quantity (cf Porph in Harm 2927-33) Other thinkersmdashamong them Th eophrastus and Porphyrymdashconceived on the contrary both highness and lowness as qualities of sounds on the same plane as colour for the vis-ible and considered therefore the diff erence between them as being one of

(apud Porph in Harm 3210-6 this fragment has not been collected by Diels-Kranz) also cf Arist Mu 395a14-21 Chrysipp SVF II fr 703 and Epicur Ep [3] 1025-9 It is thus hard to admit that a scientist as important as Ptolemy could have been ignorant of such an aporia118) As Ps-Euc and Nicomachus do (cf n 115) with slight diff erences119) In 815 if the variant θᾶττόν τε καὶ ὀξύτερον is accepted and perhapsmdashas Barker (2000 46) suggestsmdashin 9425120) Although quality and quantity always fi gure among them in this philosopherrsquos lists


quality although they admitted a certain degree of quantifi cation for them To what extent the question was important is shown for example by the fact of Porphyry devoting about fi fty pagesmdashout of the 171 his work is composed of in Duumlring 1932mdashcommenting on it while the text com-mented consists of only three We must consider that the possibility of pitch being quantifi edmdashthat is measuredmdashdepends on which category such a diff erence is placed in Ptolemy as will be seen lines up with the Pythagoreans regarding this question but his explanation follows a diff er-ent course to theirs for he does not explicitly subscribe to the doctrine they used as a base as we have already seen in the previous section

Nearly all Greek authorsmdashwith the exception of Aristoxenus and his schoolmdashaccepted what was supposed to be a Pythagorean discovery the possibility of assigning numerical ratios to musical intervals whichmdashin the case of the consonants and the more important melodic onesmdashhad a very simple expression121) Now in Ptolemyrsquos opinion it was surely inconceivable that the musical intervals seen as lsquointermediate between two given notesrsquo122) could be expressed by means of a numerical proportion while the notes determining them on the contrary lacked numerical expression and this represents just one of the stronger criticisms he makes (219-11) of the Aristoxenians Th ence together with the obvious gradability of pitch123) Ptolemy could not but think that pitch should so as to say necessarily have a quantitative nature Th e simplest procedure to prove such a thing would have certainly entailed providing a measurement system which explained how to proceed to its quantifi cation124) However as sound pitch is

121) And duple ratio (21) was assigned to the octave sesquialtera (32) to the fi fth sesquiter-tia (43) to the fourth and so on122) Cf 207 where he refers to them as τῶν μεταξύ [sc τῶν φθόγγων]123) Th at is the possibility of a sound having more or less pitch ie of being higher or lower Th e question whether highness was considered to be more or less than lowness is not that clear as examples in both senses can easily be found124) Th us does Aristotle explain what quantity entails by way of example (ποσὸν δὲ οἷον δίπηχυ τρίπηχυ Cat 1b28-9) Barker (2000 34) assumes that ldquothey [sc diff erences between sounds] are in principle measurable (though not necessarily in practice) on some appropriate scalerdquo but he adds (p 35) ldquoIf the perceptive attributes diff er quantitatively even though this is not how their diff erences are actually perceived (as it is not) there must be some true description of them which will represent them in a quantitative mannerrdquo


intimately linked to sound oscillatory frequency and as it is evident that he at the time did not have an appropriate procedure at his disposal to determine the latter his scientifi c certainty of the possibility of measuring the former could not materialize in a quantitative expression of it Instead we are surprised to see how Ptolemy to express notes by means of numbers has recourse in a completely artifi cial and arbitrary way to those that make the subsequent operations as simple as possible (cf 233-13 316-18 etc)

In fact what is more his demonstration of the quantitative nature of pitch is itself somehow fallacious as Porphyry clearly established Indeed the possibility of quantifying pitch is evident according to Ptolemy for two diff erent reasons Now the fi rst one (815-7) rests on the basis that sound highness and lowness are determined by the quantity of substance because it is the mediumrsquos greater or lesser thinness or density which actu-ally determines them and thinness and density depend on the quantity of matter (720-3 cf sect 332) ie they are quantifi able however it seems to be incongruousmdashas Porphyry rightly points out (in Harm 4411-5)mdashthat thinness and density (as well as their contraries thickness and rarity) are accepted as qualities although based on quantity while highness and lowness are expected not to be qualities but quantities because they are based on quantity Likewise the second reason given (817-8) to justify the quantitative nature of pitch rests on the measurability of the length imposed by the medium on which pitch directly depends nevertheless as there are several characteristics of the medium (length thickness density) having a concomitant infl uence on pitch and as it is impossible to quan-tify the latter by restricting it to the consideration of only one of them it again proves to be impossible to carry out such a quantifi cation using the means proposed

Consequently even though Ptolemyrsquos analysis of sound production is both original and faithful to the experimental data and despite his reason-able certainty about the quantitative status of pitch we must recognize that neither the former nor the latter provided him with the requisite means to measure the highness and lowness of sound on account of the technical impossibility of observing at the time the diff erent aspects involved in such a measurement


Bibliography

Alexanderson B 1969 Textual Remarks on Ptolemyrsquos Harmonica and Porphyryrsquos Commentary (Gothenburg)

Barker A 1989 Ptolemy in id Greek Musical Writings II (Cambridge) 270-391mdashmdashmdash 2000 Scientifi c Method in Ptolemyrsquos lsquoHarmonicsrsquo (Cambridge)Beacutelis A 1986 Aristoxegravene de Tarente et Aristote le Traiteacute drsquoharmonique (Paris)Chailley J 1979 La musique grecque antique (Paris)Chantraine P 1999 (= 21968) Dictionnaire eacutetymologique de la langue grecque (Paris)Donington R 1986 La muacutesica y sus instrumentos (Madrid) (Spanish tr of id 1982 Music

and Its Instruments [London])Duumlring I (ed) 1930 Die Harmonielehre des Klaudios Ptolemaios (Gothenburg)mdashmdashmdash 1932 Porphyriosrsquo Kommentar zur Harmonielehre des Ptolemaios (Gothenburg)Fernaacutendez de la Gaacutendara G Lorente M 1998 Acuacutestica musical (Madrid)Grove = Sadie S (ed) 1988 (= 71980) Th e New Grove Dictionary of Music and Musicians

I-XX (London)Hagel S 2005 Twenty-four in auloi Aristotle Met 1093b the Harmony of the Spheres and

the Formation of the Perfect System in Hagel S Harrauer Ch (eds) Ancient Greek Music in Performance (Vienna) 51-91

Levin FR 1980 πληγή and τάσις in the Harmonika of Klaudios Ptolemaios Hermes 108 205-29

Long AA 1988 Ptolemy on the Criterion An Epistemology for the Practising Scientist in Dillon JM Long AA (eds) Th e Question of ldquoEclecticismrdquo Studies in Later Greek Philosophy (Berkeley) 176-207

Michaelides S 1978 Th e Music of Ancient Greece An Encyclopaedia (London)Michels U 1982 Atlas de muacutesica I (Madrid) (Spanish tr of id 1977 Atlas zur Musik I

[Munich])NODE = 2003 New Oxford Dictionary of English (Oxford)Paquette D 1984 Lrsquoinstrument de musique dans la ceacuteramique de la Gregravece antique (Paris)Raff a M 2002 La Scienza Armonica di Claudio Tolemeo (Messina)Redondo Reyes P 2003 La Harmoacutenica de Claudio Ptolomeo edicioacuten criacutetica con introduc-

cioacuten traduccioacuten y comentario (Murcia) (doctoral thesis available on CD)Wallis J (ed) 1699 Claudii Ptolemaei Harmonicorum libri tres in id Operum Mathema-

ticorum III (Oxford = 1972 HildesheimNew York)West ML 1992 Ancient Greek Music (Oxford)


1 Ptolemyrsquos Concept of Sound

Ptolemyrsquos theory of sound as appears in his Harmonics1) is based on an original re-arranging of previous notions so that they concur with practi-cal experience I will thus start by setting them out

11 Background

Th e study of the production transmission and properties of sound is ancient even within Greek tradition Some aspects of such a study were not in my opinion diffi cult to establish For example it is easy to check experimentally that sound can be produced by some kind of impact it is enough to clap the hands or to consider the beat of a drumstick on a drum or that of the plectrum on the string it strikes once this has been stated the same production mechanism can after thoughtful refl ection be recog-nized in other cases Likewise when the transmission of sound is consid-ered the channel between the impact produced and our hearing it is obviously air (sometimes water) so it was easy to realize that somehow such an impact interacts with the air and the latter in turn with the hear-ing Th ese common-sense considerations could not but be familiar to ancient philosophers and in fact we fi nd them expressed by authors such as Anaxagoras (56A106 DK) Empedocles (31A93 DK) Alcmaeon (24A6 DK) Diogenes of Apollonia (19A21 DK) and Democritus (68A135 DK) they all include the idea of an impact though placing its signifi cance in a diff erent stage of the process

Th e concept of sound in terms of a stroke (πληγή)2) can be found early in Pythagorean acousticsmdashfi rstly in Archytas (47B1 DK) though he refers to previous possibly Pythagorean authorsmdashand the description of sound

1) Ptolemyrsquos Harmonics (Ptol Harm) is quoted by page and line number from Duumlring 1930 without any other indication though occasionally Wallis 1699 is used Also Porphy-ryrsquos commentary (Porph in Harm) is quoted by page and line number from Duumlring 19322) Th e noun πληγή (sometimes πλῆξις) lsquostrokersquo and its corresponding verb πλήττω (πλήσσω in the Ionian variant which can be found once [32] in Ptolemy) lsquoto strikersquo are commonly used to refer to the knock or impact other verbs as τύπτω or προσπίπτω appear sporadically


























τινος


































































































































41 Sound and Speed
























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τινος


































































































































41 Sound and Speed
























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41 Sound and Speed
























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ptolemy on sound_ harmonics 1.3 (6.14-9.15 düring)

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