‘Mathematics’ and ‘physics’ in the science of

harmonics

Stefano Isola

Scuola di Scienze e Tecnologie

Universita di Camerino

Abstract

Some aspects of the role that the science of harmonics has played inthe history of science are discussed in the light of Russo’s investigationabout the history of the concepts of ‘mathematics’ and ‘physics’.

1 The rambling route of the ancient sci-

entific method

In several places in Russo’s writings on history of science one can findenlightening discussions about the meaning of the concepts of ‘physics’and ‘mathematics’, along with the particular notions of truth involvedin them1. Both terms derive from the Greek: the original meaning ofthe former was the investigation of everything that lives, grows or, moregenerally, comes into existence, whereas the latter referred to all thatis studied, thus deriving its meaning not from its content but from itsmethod. In the Hellenistic period the term ‘physics’ continued to be usedto indicate that sector of philosophy that addressed nature (the othersectors being ethics and logic), thus corresponding to what came to becalled ‘natural philosophy’ in modern times. On the other hand, the term‘mathematics’ was used to indicate all the disciplines (including geom-etry, arithmetic, harmonics, astronomy, optics, mechanics, hydrostatics,pneumatics, geodesy and mathematical geography) that shared the samemethod of investigation, based on the construction of theories by which

1E.g. [42], chap. 6.6; [43], chap. 15; [44]; [45].

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‘theorems’ are proved, leaning on explicitly stated initial assumptions. Itsmeaning thus corresponded to what we call ‘exact sciences’, and refersto a unitary body of scientific disciplines alien to the modern distinctionbetween physical and mathematical sciences.In antiquity, how the scope of mathematics contrasted with that of physicswas a topic of much debate. According to some key testimonials2 − inparticular that due to Geminus in the first century B.C.3, and, much later,that of Thomas Aquinas4 − the ‘physicist’ would be able to grasp the‘substance’ of reality using philosophical categories, whereas a charac-teristic feature of the work of the astronomer, viz the ‘mathematician’,is its incapability to assert absolute truths, in that he is able to rigor-ously deduce/construct a number of consequences from previously statedhypotheses whose ultimate validity remains however out of control.

To better understand this discrepancy, let’s step back again to highlightanother important methodological difference between natural philosophyand exact science in antiquity, in that the former operates on a single levelof discourse, where data from experience and thoughts are organized soas to produce ‘directly’ a rational account of the perceptions themselves5.

2Reported and discussed in [44].3“The physicist will prove each fact by considerations of essence or substance, of force, of

its being better that things should be as they are, or of coming into being and change; theastronomer will prove them by the properties of figures or magnitudes, or by the amount ofmovement and the time that is appropriate to it. Again, the physicist will in many cases reachthe cause by looking to creative force; but the astronomer, when he proves facts from externalconditions, is not qualified to judge of the cause... sometimes he invents by way of hypothesis,and states certain expedients by the assumption of which the phenomena will be saved. Forexample, why do the sun, the moon, and the planets appear to move irregularly? We mayanswer that, if we assume that their orbits are eccentric circles or that the stars describe anepicycle, their apparent irregularity will be saved; and it will be necessary to go further andexamine in how many different ways it is possible for these phenomena to be brought about...’’(reported in Simplicius, Commentary on Aristotle’s Physics, vol. 9, 291.21-292.19).

4“Reason may be employed in two ways to establish a point: firstly, for the purpose offurnishing sufficient proof of some principle, as in natural science, where sufficient proof canbe brought to show that the movement of the heavens is always of uniform velocity. Reason isemployed in another way, not as furnishing a sufficient proof of a principle, but as confirmingan already established principle, by showing the congruity of its results, as the theory ofeccentrics and epicycles is considered as established in astronomy, because thereby the sensibleappearances of the heavenly movements can be explained; not, however, as if this proof weresufficient, forasmuch as some other theory might explain them” (Thomas Aquinas, SummaTheologica, The First Part, Question 32).

5 In criticizing Protagora’ statement that man is the measure of all things, Aristotle says:“We say that knowledge and sense-perception are the measure of things because our recognition

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In particular, natural philosophy starts “from the things which are moreknowable and obvious to us and proceeds towards those which are clearerand more knowable by nature”6, thus revealing the alleged genuine, mind-independent nature of things. This is also reflected in the use of language.As reported by the 5th-century Alexandrian scholar Ammonius, “... Aris-totle teaches what the things principally and immediately signified bysounds [e.g. names and verbs] are, and these are thoughts. Throughthese as means we signify things; and it is not necessary to consider any-thing else as intermediate between the thought and the thing, as theStoics do, who assume what they name to be the utterable (lekton)”7.Thus, at variance with the Aristotelian point of view, the early Stoicsconsidered it necessary to distinguish between the pronounced sound andthe meaning of what is pronounced as an intermediate link between athought and a sound. The same kind of epistemological attitude charac-terized the exact sciences − which flourished in the same period of theearly Stoic school − with their specific effort to overcome the illusionof being able to build intellectual schemes based directly on perceptiblereality and the elaboration of abstract languages capable of describingnot only aspects of the sensible world but also other designable realities8.The existence of a double level of discourse seems therefore an essentialfeature of exact sciences, in that their assertions do not directly concernthe things of the natural world, but rather theoretical entities which areobtained by a procedure of ‘pruning’ which allows one to focus on certainaspects of the phainomena − that is what appears to the senses and callsfor an explanation − and to ignore those considered unessential. In brief,the methodological mark of exact sciences consists in the construction ofsimplified models of aspects of reality which, starting from suitable but‘unjustified’ hypotheses, operate on their internal entities in a logicallyrigorous way and then move back to the real world. Note that, by its verynature, every hypothesis is somehow ‘false’, so nothing prevents different

of something is due to them” (Metaph. 1053a, 32-35). To him, therefore, sense-perceptionand knowledge are the faculties that furnish all our understanding of things and thus exhaustedall possible meanings of the expression ‘criterion of truth’. In Hellenistic practice, however,other meanings of this expression were put forward (of which Protagoras’s dictum could beconsidered a likely precursor) including Stoic’s infallible act of cognition based on katalepticimpressions (self-certifying acts of sense-perceptions) as well as the hypothetico-deductivemethod of exact sciences.

6Aristotle, Physics, 184a16-20.7SVF II, 168.8[42], in particular Chap. 6.

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models based on different hypotheses being capable of ‘saving’ the samephenomena. In addition, while the assertions obtained at the theoreticallevel are ‘objective’ and universally valid, the correspondence rules whichtransform the entities involved in the real world and the claims aboutthem into theoretical entities and theoretical statements are instead his-torically determined. For example, in Hellenistic scientific theories dealingwith phenomena related to the sense of sight, devices such as ruler andcompass, designed to assist in the construction of the straight line andthe circle, as well as in the measurement of their parts, incorporate thecorrespondence rules relating theoretical statements of geometry or opticsto concrete objects. In theories of acoustic-musical phenomena, this rolewas played instead by the canon (see below). In both cases, the ‘concreteobjects’ − drawings with ruler and compass, and pitches produced by aplucked string, respectively − are not rough natural data: rather, they arethe result of a somewhat refined human activity, which in turn is rootedinto the historical and cultural context.

Although rarely acknowledged, the scientific method, as a cultural prod-uct of earlier Hellenistic times, underwent a rapid decline in the context ofa more general cultural collapse that occurred during the second centuryB.C.9 Notwithstanding the loss of a major part of ancient knowledge, thememory of Hellenistic science survived thanks to a series of geographicallylocalized revival periods10. On the other hand, a peculiar feature of theserevivals was the plunging of individual contents recovered from ancientscience or derived from it, into foreign overarching systems of thoughtwhich provided their main motivating framework11. In particular, “inthe Age of Galileo”, Russo says, “the exact science preserved the unitythat distinguished the Greek models, from which it drew the terminology,but the ancient method was rarely understood. Not that the explana-tion reported by Simplicus had been forgotten, but few, as Stevin, used

9Particularly dramatic were the years 146-145 B.C., with the sharp hardening of the Romanpolicy in the Mediterranean that had among its consequences the reduction of Macedonia toa Roman province, the razing of Carthage and Corinth, and the heavy political interference inEgypt with persecution and extermination of the Greek intellectual class (see [46], Chap. 5).

10The first of them was the resumption of scientific studies in imperial times, whose mainprotagonists were Heron, Ptolemy and Galen. The next ones occurred in the 6th centuryByzantine world, then in the medieval Islamic world (8th-9th cent.), and finally in WesternEurope, from the ‘twelfth century Renaissance’ until the Renaissance par excellence of earlymodern times (see [42], Chap. 11).

11We shall discuss below an example which illustrates this fact in connection with Ptolemy’swork.

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the freedom of choice of the hypotheses to build models; much morefrequently the relative arbitrariness of the initial assumptions appeared(as it had appeared to Simplicius and Thomas Aquinas) as a particular-ity (as well as an oddity) of the method of the ‘mathematician’, whichdetermined its inferiority with respect to philosophers and theologians,who knew how to distinguish ‘truth’ from ‘falsehood’ ”.12 The idea thatthe hypothetico-deductive method was mostly a limit preventing fromapproaching the absolute truth reached a summit with Newton. In thewell known General Scholium added to the Principia in 1713, he writes:But hitherto I have not been able to discover the cause of those proper-ties of gravity from phenomena, and I frame no hypotheses (hypothesesnon fingo). For whatever is not deduced from the phenomena, is to becalled an hypothesis; and hypotheses, whether metaphysical or physical,whether of occult qualities or mechanical, have no place in experimentalphilosophy. In this philosophy particular propositions are inferred fromthe phenomena, and afterwards rendered general by induction13.It is worth stressing that the term ‘phenomena’ is used here with a mean-ing which differs considerably from the ancient one, in that it refers tosomething which lies beyond our perception14. Likewise, the term ‘hy-pothesis’ was given the new meaning − still in use − of a statement lyingat the beginning of our interpretation of the external world but wait-ing to be corroborated or refuted as soon as the ‘facts’ are known withsufficient detail. Thus, in every genuine search for the truth, hypothesescannot be anything but a hindrance15. As it is well known, Newtonianism

12[44], p. 37 (eng. transl. mine).13Eng. transl. by Andrew Motte (1729).14Think of the absolute motions of material bodies with respect to the immovable space

which, coexisting with Aristarchan heliocentrism, cannot correspond to any observable datum(see the discussion given in [42], 11.7). In a letter of 1698 Newton affirmed: “I am inclined tobelieve some general laws of the Creator prevailed with respect to the agreeable or unpleasingaffections of all our senses” (see [19], letter XXIX).

15As D’Alembert wrote in his Preliminary Discourse to the Encyclopedia of Diderot, “it is notat all by vague and arbitrary hypotheses that we can hope to know nature; it is by thoughtfulstudy of phenomena, by the comparisons we make among them, by the art of reducing, as muchas that may be possible, a large number of phenomena to a single one that can be regardedas their principle” (29), and, a little further, “let us conclude that the single true method ofphilosophizing as physical scientists consists either in the application of mathematical analysisto experiments, or in observation alone, enlightened by the spirit of method, aided sometimesby conjectures when they can furnish some insights, but rigidly dissociated from any arbitraryhypotheses” (32). As an aside, this semantic transformation may have played a role in claiminga ‘historic mission’ to the human knowledge, from the naivety of the myth towards the final

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was presented in the European continent as the philosophy of progress.The most famous of his supporters was Voltaire, who in the preface ofthe french translation of the Principia liquidated as ‘foolish’ the follow-ers of vortices formed by the ‘thin matter’ of Descartes and Leibniz, andaffirmed that only a follower of Newton could be truly called a ‘physi-cist’. Indeed, according to Russo, the spread of Newtonian mechanicshas brought with it the way of reasoning on the basis of which ”the exactscience got broken into two stumps: ‘mathematics’ and ‘physics’. Bothof them inherited from the ancient ‘mathematics’ the quantitative ap-proach and several technical results, and from the ancient ‘physics’ (thatis from natural philosophy) the idea of producing statements which areabsolutely ‘true’. The essential difference was lying in the nature of suchtruth. While the truth of the assumptions of ‘mathematics’ (called pos-tulates) was considered immediately evident, the assumptions of ‘physics’(called principles) were regarded true inasmuch as they are ‘proven by thephenomena’ [...] It is plain that these differences were strictly connectedto the diverse nature attributed to the entities studied by the two disci-plines: the ‘mathematical’ entities, although usable to describe concreteobjects, were considered abstract, whereas the ‘physical’ entities wereconsidered as concrete as the objects they were referring to.”16 In bothcases, the ‘truth’ of a scientific theory (e.g. a theory of the planetarymotions) does not lie in its capability to ‘save the phenomena’ (e.g. todetermine with some accuracy the observable position of a planet at anytime), but becomes something that one can ‘prove’ by means of its owninstruments, in the same way in which one can prove a statement on theentities internal to the theory itself. If so, a scientific theory would ceaseto be a theoretical model, instead becoming a system of statements setto describe the true nature of the real world17.Following this reshaping of the scientific enterprise, some disciplines havebeen counted on one side, others on the other, still others have been

enlightenment, passing through an increasing control of the sources of error which allows toprogressively overcome all ‘false hypotheses’ (viz ‘prejudices’): a kind of secularized version ofthe medieval millenarianism, of which, amongst others, Newton was an ardent supporter.

16[44], pp. 42-43 (eng. transl. mine).17Such a pre-scientific position is proudly maintained in the entry System of Voltaire’s

Philosophical Dictionary, which starts stating: “We understand by system a supposition; forif a system can be proved, it is no longer a system, but a truth. In the meantime, led byhabit, we say the celestial system, although we understand by it the real position of thestars”. By the way, and not surprisingly, this entry proceeds with a tentative to strengthen theidea of a necessary progression of the knowledge by denying the paternity of Aristarchus onheliocentrism.

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somehow internally divided or eventually disappeared, as we shall see be-low for a particular example. Referring to the already-cited Russo’s writ-ings for a discussion of the splitting between ‘mathematics’ and ‘physics’in 19th and 20th centuries, let us just remark that the final failure ofthe efforts towards a methodological reunification of the exact science, ofwhich Poincare was a prominent exponent, and the prevailing of powerfultrends towards specialization and fragmentation of the scientific disci-plines, if on one side has led some to wonder what mystery lies behindthe ‘unreasonable effectiveness’ of mathematics in providing accurate de-scriptions of the phenomena18, on the other side prompted one of thegreatest contemporary mathematicians to grasp in this trend a severecrisis of science itself: “In the middle of the twentieth century it wasattempted to divide physics and mathematics. The consequences turnedout to be catastrophic. Whole generations of mathematicians grew upwithout knowing half of their science and, of course, in total ignoranceof any other sciences. They first began teaching their ugly scholasticpseudo-mathematics to their students, then to schoolchildren (forgettingHardy’s warning that ugly mathematics has no permanent place underthe Sun)”19.

2 Acoustic-musical phenomena

Ho detto che la nostra scienza o arte musicale fu dettata dallamatematica. Doveva dire costruita. Essa scienza non nacquedalla natura, [...] ma ebbe origine ed ha il suo fondamento inquello che e giustamente chiamato seconda natura, ma che al-trettanto a torto quanto facilmente e spesso e confuso e scam-biato [...] colla natura medesima, voglio dire nell’assuefazione.Le antiche assuefazioni de’ greci [...] furono l’origine e il fonda-mento della scienza musicale da’ greci determinata, fabbricatae a noi ne’ libri e nell’uso tramandata, dalla qual greca scienzavien per comun consenso e confessione la nostra europea. (G.Leopardi, Zibaldone, 3125-3126)

Today musical theory is mainly ‘the study of the structure of music’,whereas originally it was part of mathematics. What happened in themeantime? In order to get an idea it is necessary to go back again to

18See [55].19See [2]; also [45] for a further discussion.

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the rambling route of the ancient scientific method through the sub-sequent history. Resuming what was said in the previous section in aconcise albeit vague way, we can say that the general objects of Greekscience were not so much the ‘laws’ of the natural world viewed as anentity independent from the man who observes it, but rather those indu-bitable epistemological data provided by the phainomena resulting fromthe interaction between subject and object through active perception. Inparticular, the models of Hellenistic exact science were primarily suitedfor that purpose: the creation of theoretical entities as intermediate ut-terances between the real objects and abstract truths has the effect ofmaking that interaction available to conscious manipulation. This gets apeculiar meaning within the context of music theory which, as such, es-tablishes sound, the material aspect of music, as something which can beknowingly investigated in connection with human experience. Althoughmusic − perhaps the most unfathomable expression of psychic activity− might seem not properly suited to scientific analysis, nonetheless theinvestigation of acoustic-musical phenomena provides an example wherethe epistemological opposition sketched in the previous section occurredwith a striking character within the same domain, as we shall now brieflyoutline.It is rather well known that Pythagorean music theory − as a part oftheir program of liberation of the soul by means of the intellectual per-ception of proportions in all things − starts from the recognition thatthe harmonic intervals can be expressed as simple numerical ratios. Thefollowing ‘Pythagorean principle’ has been viewed as the first ‘naturallaw’ expressed in terms of numerical entities20: if two sounding bodies,such as stretched strings or sounding pipes, have lengths which are insimple proportions, and all other aspects are kept fixed, together theywill produce musical intervals which are judged by the ear to be in har-monious agreement, or ‘consonant’. Conversely, all intervals that the earaccepts as consonant can be represented as ratios of numbers from thetetrad 1, 2, 3, 421 . The harmonic system of Philolaus22, for example, is astructure of intervals externally limited by the octave (diapason), whoseratio is 2:1, and internally articulated by intervals of fifth (diapente), withratio 3:2, and fourth (diatessaron), with ratio 4:3. If we want to find fourquantities − for instance the lengths of the strings of a four-string lyre −

20See e.g. [6].21The question of which observations lay behind the detection of these ratios and when this

happened is hard to answer, see [9].22See e.g. [14].

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such that, taken in pairs, reproduce these ratios, then we can choose aunity of measure so that the longest string is 12 units, the intermediateones 9 and 8, and the shortest 6. It is plain that the system of reciprocalratios, and therefore the whole harmonic structure, does not change ifthe strings have length 12, 9, 8 and 6 meters, centimeters, stadiums etc.Finally, observing that (3:2):(4:3) = 9:8, the interval of a tone, equivalentto the difference between a fifth and a fourth, is represented by the ratio9:8. The octave is thus ‘harmonically’ divided into two fourths spaced bya tone23.Let us point out that in the transition from the Hellenic to the Hellenis-tic period mathematics becomes an exact science, in the sense specifiedabove, not only by distinguishing theoretical entities from concrete ob-jects, but also from pure abstractions in the ‘platonic’ sense. While dis-cussing about the subjects for the education of the ‘Guardians’ of theRepublic, Plato lets Socrates conceive that “as the eyes are designed tolook up at the stars, so are the ears to hear harmonic motions”, there-fore agreeing that astronomy and music theory are sister sciences, as thePythagoreans said (Republic, 530d). On the other hand, those scholarsare judged inadequate to reach the ‘true knowledge’ beyond the sensibleworld in that “their method exactly corresponds to that of the astronomer;for the numbers they seek are those found in these heard concords, butthey do not ascend to generalized problems and the consideration whichnumbers are inherently concordant and which not and why in each case”(Republic, 531c). Clearly, the just mentioned ‘ascension’ above experi-ence does not need to be embedded in a theory. Rather, it would rely on‘evidences’ per se.In a different direction, music theory, or at least that part of it dealingwith tuning systems, was set to become a scientific discipline by puttingtogether the arithmetic theory of proportions and the recognition of theproportionality between the pitch of the sounds and the speed of the vi-brations that produce them24, a conceptual step that according to somesources had been made in the circle of Archytas about 400 B.C.25. The

23Note that 6 · 12 = 8 · 9, i.e. the four numbers are in geometrical proportion. Moreover8 = 2 : (1

6+ 1

12) and 9 = (6 + 12) : 2, namely 8 and 9 are respectively the harmonic mean

and the arithmetic mean of the extremes 6 and 12. The exclusion of the geometric mean in‘Pythagorean’ music theory is justified by an impossibility result due to Archytas (see below).

24The recognition of the nature of sound as vibration of air, with alternation of rarefactionand condensation, can be already found in the Aristotelian Problemata as well as in PeripateticDe audibilibus, whereas the idea of a sound wave is attested at least as early as in the Stoa.

25See [9], [28].

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‘experimental device’ enabling the establishment of a correspondence be-tween concords and numerical ratios was the canon (kanon harmonikos),an instrument that in its simplest form is made by a single string stretchedbetween two bridges fixed on a rigid base, and equipped with another mov-able bridge by which one may divide the string into two parts, yieldingsounds of variable pitch. One can further imagine a row fixed at its base onwhich the positions of the movable bridge corresponding to the notes canbe marked. The name of the entire device is then transmitted metonymi-cally to the row and from there to the line segment that represents it as atheoretical entity. The theory outlined in the Section Canonis, attributedto Euclid, deals precisely with the harmonic divisions of this segment,i.e., with those divisions corresponding to musical intervals judged to beconsonant.26 In this work, far away from any mystical efflorescence aboutthe music of the cosmos, a scheme of division of the octave by means oftheory of proportions contained in the Elements is proposed with the aimof producing patterns of consonant intervals adoptable in practice, e.g.when tuning musical instruments. Along this path, the branch of Greekmusic theory referred to as science of harmonics entered the unitary bodyof Hellenistic mathematics, along with astronomy, arithmetics, geometry,optics, topography, pneumatics, mechanics, and other disciplines27.In the short introduction of the Sectio Canonis the author establishes acorrespondence between musical intervals and numerical ratios and statesthe main hypothesis underlying the model: consonant intervals corre-spond to multiple or epimoric ratios28. The rationale of this postulaterelies on the observation that as consonant intervals produce a percep-tion of unity or tonal fusion between the notes, they must correspondto numbers which are spoken of under a ‘single name’ in relation to oneanother29. On the other hand, under the form of simple implication this

26See [7], chap. 8; also [20].27See [42], chap. 3.28 Greek arithmetic classified ratios into three basic types, which reduced to lowest terms

correspond to n : 1 (multiple), (n+ 1) : n (epimoric or superparticular), or (n+m) : n, n >m > 1 (epimeric or superpartient). Note that the first two are in a one-to-one correspondence:p : q is multiple if and only if p : (p − q) is epimoric, so that q is the g.c.d. of both p andp− q. In particular the octave 2:1, participating to both consonant classes, is the ‘consonanceof the consonances’.

29The interpretation of this seemingly arbitrary correspondence is controversial. According tosome scholars [6], the ‘single name’ has to be ascribed to the the fact that, unlike epimerics,multiple and epimoric ratios, were indicated with a one-word name, like epitritos, ‘third inaddition’, for 4:3. According to a different interpretation [20] (based on [39], I.5), the ‘singlename’ is not a linguistic unity, but a numerical one, corresponding to the greatest common

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postulate is plainly false, not only because it includes among the con-cords also intervals considered dissonant by the Pythagorean principlestated above, such as the tone 9:8 or the ratio 5:4 (natural major third),but also because it counts as dissonant the interval composed by an oc-tave plus a fourth, represented by the ratio 8:3, unanimously recognizedas consonant by the music theorists of antiquity (exactly as an octaveplus a fifth, that is 3:1). However this is not a problem in itself, for allhypotheses are somehow ‘false’: what matters is that the theory basedon them be consistent and suited to save the phenomena which aims tomodelize. The introduction is then followed by twenty propositions, thefirst nine, of pure ‘number theory’, provide a deductive construction of thePhilolaus harmonic system sketched above, whereas the remaining onesform the part properly relevant to tuning systems. Of particular interestis the third proposition, which states that neither one nor more meanproportionals can be inserted within an epimoric interval. In particular,it is not possible to divide the octave at equal parts that form rationalrelationship with the octave itself30.The consequences of this simple result have been the subject of a contro-versy which has lasted for over two millennia, at the basis of which there isthe distinction between natural and tempered tunings31. We’ll not dwellhere on the ways in which the different music theorists have conceived thedivision of the tonal continuum32. Rather, we shall briefly discuss howin this domain the epistemological opposition between natural philosophyand mathematics manifested itself. To this end we recall that the firstwritings of some importance dealing with Greek music theory are thoseof Aristoxenus and Theophrastus, both students of Aristotle, and bothharshly critical of the doctrine according to which the pitch can be con-ceived as a quantitative attribute of sound, representable by numbers. Asrepresentatives of the Peripatetic school they were mainly interested in the‘natural qualities’ of the object of investigation. For instance Theoprastusclaims that differences of pitch are due to differences in the ‘shape’ of thesounds’ movement, not to differences of velocity, frequency of impact, orthe like, inasmuch as high notes travel in a straight line from the object

‘part’ which composes the notes in both multiple and epimoric ratios (see above, n. 28).30Nor would it be possible to divide in this way the fifth, the fourth or the whole tone. This

seems the first impossibility result surely ascribable to an author, as Boethius reports a proofof it given by Archytas ([8], III.11).

31See [5], [29]32See e.g. [4], [10], [15]

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to the ear, while low ones spread more evenly all round the object33. Incriticizing the ‘Pythagorean approach’ he maintains that “if every intervalwere a quantity, and if melody arose from differences between notes, themelody would be as it is because it is a number. But if it were nothingbut a number, everything numerable would participate in melody too,to the extent that it does in number”. This illustrates in some way thesingle level of discourse maintained by natural philosophy we were talkingabout in the previous section, where there is no space for intermediateentities in between the concrete objects and the abstract thoughts aboutthem. A rather similar position is held by Aristoxenus, the leading musicaltheorist of antiquity. His Harmonic Elements opens with the subject ofvocal motion within the musical topos in which it moves, which is thecontinuum whose maximal compass and minimal internal intervals are de-fined solely by what the human voice is capable of doing and by what thehuman ear can apprehend the moving voice to be doing. In particular, heclaims that harmonic properties such as consonance are firstly subject ofexperience by a musically trained ear and cannot be traced to numericalratios. He wrote indeed: “we endeavour to supply proofs that will be inagreement with the phenomena, in this unlike our predecessors. For someof these introduced extraneous reasoning, and rejecting the senses as in-accurate fabricated rational principles, asserting that height and depthof pitch consist in certain numerical ratios and relative rates of vibration− a theory utterly extraneous to the subject and quite at variance withthe phenomena”34. Although one may argue that the polemical targethere is mostly the Platonic treatment of Pythagorean music theory, inthe rejection of rational arguments based on assumptions external to themusical experience itself we can see the demand that every assumptionmust be justified by the phenomena, that is a substantial disclaiming ofthe scientific method35.

33Theophrasti de musica fragmentum, in Porphyry, Commentary on Ptolemy’s Harmonica,pp. 61.22-65.15, During. Also [49].

34[1], 32-33.35In particular, Aristoxenus and his followers ‘admitted’ that it was possibile to divide the

tone into two equal parts. This, of course, does not ‘contradicts’ the third proposition of theSectio Canonis − implying that no (rational) mean proportional can be inserted between 9and 8 − which characterizes intervals as elements of a theoretical model (hence defined solelyby the hypotheses underlying the model itself). Rather, it results from the direct experienceof placing one’s finger on the string at the point corresponding to the division into equalsemitones. A theoretical construction of the equal temperament has been made almost twomillennia later by Stevin (see below).

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Be that as it may, with the crisis of Hellenistic civilization and in partic-ular after the dramatic cultural collapse which occurred midway throughthe second century B.C., the scientific methodology rapidly disappearedtogether with the very possibility of understanding the need of the lekta− the conceptual constructions intermediate between the thoughts andthe things − in the devising of meaningful representations of the rela-tionships between human activity and natural world. As Russo says, “inthe imperial age, when the notion of theoretical models had been lost,such entities were conceivable only as real objects: the alternative be-tween ‘bodies’ and ‘incorporeal beings’ thus became ineluctable. Somesuch entities were indeed made corporeal − witness the crystalline ce-lestial spheres which replaced the spheres of Eudoxus of Cnidus and theepicycles of Apollonius of Perga. Likewise, the ‘visual rays’ of opticsreacquired the character of physical objects emitted by the eyes, whichwas not present in Euclid’s theory. [...] Other entities, such as those ofgeometry, were given an incorporeal reality. This placed geometry in therealm of Platonic thought, a position that Hellenistic mathematics hadleft behind”36.A similar fate befell to the basic entities of the science of harmonics, suchas the musical intervals, which lost the character of theoretical entitiesgained within Hellenistic science to be identified (again) either with cor-poreal items − such as the discrete movements of a melodic voice − orelse with purely ideal abstractions, entities considered as much real asthey are not attainable − such as the harmonic ratios composing the Zo-diac or the human soul − both deemed to possess quantitative featuresto which reason can be ‘directly’ applied by assigning them appropriatenumbers.

An important example is provided by Ptolemy in the first book of hisHarmonica, where, no longer being able to grasp the methodologicaltenets maintained by his Hellenistic sources, falls back upon epistemo-logical bases close to the Peripatetic ones, without thereby giving up theclaim to employ refined mathematical tools inherited from his predeces-sors (yet conceived in the Platonic sense). For example, with a kind ofinversion of the Hellenistic rule that requires a theory of saving the phain-omena, he says: “The purpose of the harmonicist would be to preservein every way the reasoned hypotheses of the canon which do not in anyway at all conflict with the perceptions as most people interpret them,just as the purpose of the astronomer is to preserve the hypotheses of

36[42], p. 232.

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the heavenly movements concordant with observable paths. Even thesehypotheses are themselves assumed from what is clear and roughly ap-parent, but with the help of reason they discover detail with as muchaccuracy as is possible. For in every subject it is inherent in observationand knowledge to demonstrate that the works of nature have been craftedwith some reason and prearranged cause and completed not at all in ran-dom”37. Like other scholars of his time, Ptolemy’s ‘criterion of truth’ isdictated by a strange kind of ‘concord’ between theory and observation,where the model, although highly mathematized, has lost its meaning asa theoretical entity and taken over the former pre-scientific meaning ofdirect representation of the known reality. In this regard, he seems towant to frame Hellenistic scientific results within philosophical argumentsof the classical period. Although one might regard this as a dialecticalstrategy to give maximum credibility to the position he wants to hold, inthis way he ends up denying the method of his Hellenistic predecessors,deeming legitimate only one theory: that whose assumptions are entirelyjustified by the phenomena and at the same time reflects the ‘rationality’of nature’s works38.Akin to Galen’s craving to strike a balance between the ‘rationalist’ and‘empiricist’ schools of medicine39, Ptolemy loudly distinguishes his ap-proach to the study of consonances from that of the ‘excessively rational-ist’ Pythagoreans − accused of accepting rationally justifiable statementseven when they are contradicted by the senses40 − and that of the ‘overlyempirical’ Aristoxeneans, for whom audible harmonies are not subject tomathematical analysis at all. Since for him the objects of sense-perceptionand thought are (again) identical41, though apprehended in different ways,he feels entitled to set his ‘hypotheses’ in the form of alleged mathematical

37[39], I.2, 5.13-21. This goes hand in hand with some passages of the Almagest, for examplewhere he says: “Now it is our purpose to demonstrate for the five planets, just as we did forthe sun and moon, that all their anomalies can be represented by uniform circular motion,since these are proper to the nature of divine beings, while disorder and non-uniformity arealien [to such beings]” [40], 9.2.

38One should perhaps also put in evidence how these narrow epistemological bases willactively operate in the development of modern science.

39See [22]. A discussion on Ptolemy’s epistemological affinity to his contemporary Galencan be found in [32].

40His concern focused in particular on the fact − already mentioned above− that the intervalcorresponding to the ratio 8:3 (diapason plus diatessaron), although unanimously recognizedas consonant, was ‘deductively’ counted as dissonant on the basis of the postulate openingthe Sectio Canonis ([39], I.5, 12.4-8).

41Cf. above, n. 5.

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counterparts of the relevant perceptual impressions. In so doing, Ptolemyassigns to the science of harmonics the task of explaining the audible andinaudible harmonies by reference to the formal, quantitative attributes ofthe different pitches, as to astronomy that of explaining the movementsof the observable heavenly bodies by reference to the formal features ofthe spheres or other bodies on which they are physically carried. In thisregard, the use of mathematics would serve mostly as a ‘rational criterion’to assist the senses in making fine discriminations.As an aside, in the Alexandrian milieu of the imperial age, where the lin-gua franca was still Greek, but life and thought were dominated by a co-hort consisting of astrological fatalism, gnostic dualism and transcendentmonotheism, even mathematics was mostly plunged into an atmosphereof irrationalism, with the distancing from the deductive method and thereturn of numerology. At the same time, among the objects of musical‘perception’ the prototype was considered the ‘music of the spheres’, anold conception dating back at least to Plato’s Timaeus and Republic,resumed by Nicomachus42 and subsequently by Ptolemy himself in thethird book of his Harmonica43. Thereafter, the Platonic connection be-tween planetary motion and music became a cornerstone of the musicaspeculativa − which together with musica poetica and musica practicaconstituted the quadrivial discipline of musica44. The regaining of a cor-poreal nature of the crystalline spheres to which the heavenly bodies weresaid to be attached goes hand in hand with the resumption of the celestialharmony as a ‘perceptible’ datum, although emanating from an incorpo-real and inaudible reality45. Both subjects were then transmitted through

42See [36].43After comparing the various harmonic functions with several aspects of the human soul

([39], III.4-7), he proceeds by regarding the zodiacal circle as a vibrating string and compar-ing the principal astrological ‘aspects’ (angles between heavenly bodies that were believed tomodify their degree of influence) with musical consonances, thereby explaining their different‘effectiveness’ ([39], III.8-9).

44This tripartition of music reverberated Aristotle’s division of knowledge (episteme) intotheoretike, poietike and praktike, and was codified by the 6th century roman philosopherBoethius as musica mundana, musica humana and musica instrumentalis [8]. More generally,the resumption of a pre-Hellenistic classification of knowledge (in particular that outlined inPlato’s Republic) becomes glaring with the reduction of the manifold Hellenistic science tothe quadrivium, consisting of arithmetic, geometry, music and astronomy, which, along withthe trivium (made by grammar, logic and rhetoric) constituted the seven ‘liberal arts’ thatarticulated the preparatory training for the study of theology in medieval times.

45Note that Aristotle, who believed in the existence of the rigid sidereal sphere (De Caelo,II, 6), refuted the conception of the celestial music on the basis of physical arguments (De

15

centuries by sheer copying46 until they were taken seriously again in earlymodern times. For example, Kepler’s ‘estimate’ of the thickness of thecrystal sphere of fixed stars47 went together with his tentative attemptsto improve Ptolemy’s harmonic investigations by searching for musicalproportions in various quantities in the Solar system, such as the periodsof the (heliocentric) planetary motions48. Note however that althoughthe faith in the ‘harmony of the world’ had played an indubitable role inthe reappearance of mathematics as the pivotal language of the resur-gent sciences in early modern times, when at the time of Newton theterms ‘physics’ and ‘mathematics’ got the new meanings we have dis-cussed above, science has begun to need different images, hence celestialmusic became old-fashioned as a scientific subject and eventually becamea purely literary metaphor49.

Altogether, the early modern resumption of studies on the science of har-monics took different veins, sometimes in open conflict with each other,often revealing with particular vividness the prevailing beliefs on the moregeneral meaning of the scientific enterprise50. A well known example isthe harsh controversy between Vincenzo Galilei (the father of Galileo)and Zarlino, where, among other things, to the ‘well-ordered’ Nature ofZarlino, which whispers to the human ear the true consonances, Galileiopposed the image of a Nature which proceeds ‘without cognition’ (senzacognitione), with principles and purposes unrelated to man, and against

Caelo, II, 9). On the other hand, in the cultural context we are referring to it would haveseemed vain to refute on physical basis such kind of Platonic mythological representations.

46Or else by anthological syntheses of the prisca sapientia, such as the commentary on Ci-cero’s Somnium Scipionis by the 5th century neoplatonist Macrobius [34], where he drew com-prehensively on the whole body of Pythagorean, Orphic, and Platonic teachings and cosmology.How deep was the decline of science at the end of the ancient world can be grasped from thefact that, although Macrobius faithfully reports the ratios corresponding to the Pythagoreanconsonances, he does not even understand that they are ratios. For instance, he justifies thefact that the tone 9:8 cannot be divided at equal parts (a consequence of Archytas impossibilityresult discussed above) not with the non-existence of a rational square root of 2 (and henceof 8), but because 9 is not divisible by 2 ([34], II, 1, 21-23).

47Of about two German miles ([30], p. 288).48We have to recognize that, unlike the first, the second concern gave good fruit, as it is

well known that the search for an harmonic correspondence between the periods of revolutionand the radii of planets’ orbits eventually led to the celebrated ‘third law’ [31].

49Nevertheless, Newton himself had imagined to recover the lost prisca sapientia in which,among other things, the inverse square law of gravitational attraction between the planetswould have been encrypted within Pythagoras’ music of the spheres [33].

50See [12], [26], [13].

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which man takes advantage of the mechanical arts to an end that naturecannot achieve51. Among the seventeenth century scholars who tookan active role in producing musical theories, like Simon Stevin, Kepler,Isaac Beeckman, Descartes, Mersenne, Francis Bacon, Galileo Galilei,Lord Brouncker, John Wallis, Christian Huygens, Robert Hooke and oth-ers, only the first one seems to have retained the option to build a modelbased on a free choice of hypotheses52.

Somewhat later, in the age of Lights, and thus after the splitting between‘physics’ and ‘mathematics’, an interesting confrontation about the sci-ence of harmonics took place between Euler and d’Alembert53. The swissmathematician − perhaps the last representative of the conception of mu-sic as a part of mathematics − at age 24 finished the writing of his majorwork on the subject, the Tentamen novae theoriae musicae ex certissimisharmoniae principiis dilucide expositae (1731), whose main goal was togive an answer to the old question of why certain sounds are pleasant andothers are not, an answer which would feature not only the perception ofsingle intervals, but also sequences of chords or even of a complete musi-cal piece. He pursued this goal by assuming that any pleasure comes fromthe perception of a ‘perfection’, which in turn is embodied in a notion oforder that can be measured by an exponent calculated only in terms of thearithmetic proportions associated to the pitches of the tones involved54.As a consequence, the fact that some people appreciate the use of somechords and others not, is explained by saying that the latters’ ear is nottrained enough to perceive the order hidden in them. As we have pointedout previously about the Sectio Canonis, also the assumptions underlyingthis construction can not exempt themselves from being somehow ‘false’:for instance, the same ‘exponent’, and thus the same degree of pleasant-ness, is associated to a musical piece regardless if it is played forward or

51See [25] and [53]; also [54], chap. 2, and [37].52Which is the subject of a short treatise written in Dutch where, among other things, the

equal temperament (i.e. the geometrical division of the octave into twelve equal parts, eachcorresponding to a ratio 1 : 12

√

2) is constructed on the basis of two postulates. The first saysthat as one part of a string is to another, so is the coarseness of the sound of the one to thatof the other. The second says that natural singing is in the major diatonic scale, and in thisscale all whole tones are equal and so are the semitones [48].

53See, e.g., [3], chap. 4.54See ([21], Series III, Vol.1, pp. 197-427), where he starts from a masterful generalization

of some previous ideas of Galileo and Mersenne ([23], [35]) according to which a chord of twosounds is all the more consonant when the ‘coinciding’ blows resulting from the two soundsare in higher proportion in the whole of the produced blows. An example of calculation of thisexponent is given below.

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backward, which is in general something far from usual experience. But,as we have seen, this is not a problem in itself, at least as long as thingsremain consistent with the ancient scientific method. It is not clear (tome) whether Euler considered the problem dealt with in his treatise asone of defining the value of something which was not defined before, orelse as a Platonic search for some ‘true’ value. Be that as it may, evenonly in his ambition to modelize far more than just a system of tuning heexposed himself to the critics that Johann Bernoulli levelled against him:“... you have derived the rule which establishes how the notes are to becombined, so that an intelligent ear can take delight in them. I thinkthat this is appropriate for a musician who is more concerned about theaccuracy of a piece of music than its effect, which satisfies the listener;a person of this kind will undoubtedly find enjoyment and delight, if youhave written this down and examine it and find that it is well composedin accordance with the fundamental rules; but as a piece of music is usu-ally payed to ears that are devoid of understanding, and are not able torecognize the ratio between the beats of the intervals produced by thestrings, and are even less able to count, then I believe that the same earswill appreciate or refuse the same piece of music, depending on whetherthey are used to this or that kind of music”55.But a critique of a different tenor was put forward by d’Alembert who,having in mind the example, bad for him, of the ‘mathematician’ Euler,embodied the role of the ‘physicist’, in the sense advocated by his mentor,Voltaire. In particular, he maintained that in music theory there is no placefor ‘demonstration’ − insofar as that term is reserved to ‘mathematics’− and one should adopt an ‘empirical-deductive’ methodology modeledon that of Newton. As he made clear in the Preface of his widely-readtreatise on music theory56, issued in 1752, his main purpose in writing thework was “to show how one may deduce from a single experiment the lawsof harmony which artists had arrived at only, so to speak, by groping”57.The single experiment had to do in this case with Rameau’s corps sonore,that is any resonating system which, besides the fundamental frequency(sounding pitch), generates also a series of harmonically related overtones,such as the octave, the perfect twelfth (the octave of the perfect fifth),the major seventeenth (the double octave of the major third), and so

55Johann Bernoulli to Leonhard Euler (1731), in [21], Ser. IV A, Vol. II, pp. 146-150.56A kind of resume of the music theoretic writings of the great composer Jean-Philippe

Rameau [41], where he thought to find a paradigm of systematic method and synthetic struc-ture which somehow confirmed his own scientific ideas ([17], see also [3] and [11]).

57[17], vi.

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on. To d’Alembert, the resonance of the sonorous body was the “mostprobable origin of harmony, and the cause of that pleasure which wereceive from it”. He thus strove to structure music as a science based ona single ‘principle’ which is somehow ‘dictated by nature’ and from whichone should deduce “by an easy operation of reason, the chief and mostessential laws of harmony”.The different positions embodied by the two scholars resulted in severalcontroversies, among which the one about the possible solutions of thewave equation is perhaps the best known, although its current reconstruc-tions usually neglect musical motivations and implications. In particular,the last account on the subject written by d’Alembert ends with a polem-ical stance against the music theory maintained by Euler, in which thevery possibility to deal with a musical phenomenon in terms of theoreticalentities seems denied: “It is clear from the preceding formulae that, givenan equal tension and thickness, the number of vibrations in the sametime is inversely proportional to the length of the strings. As the higheror lower sound of the strings depends on their larger or smaller number ofvibrations in the given time, it is undoubtedly for this reason that somevery capable modern authors have considered it possible to represent thesounds by means of the logarithms of the ratios between the lengths of thestrings. This idea is ingenious, and would appear to be based equally onfigures of speech in acoustics and music, when we say that if four stringsa, b, c, d are geometrically proportional, the interval formed by soundsa and b will be equal to the interval formed by c and d; hence it wasconsidered possible to conclude that the logarithms of the relationshipsa:b and c:d represented the intervals between the sounds. But undoubt-edly this conclusion was not claimed to be anything more than a purelyarbitrary supposition; the words interval between sounds, equality and dif-ference of intervals are only abbreviated figures of speech, which shouldnot be given a wider meaning than they really have. Sounds are merelysensations, and consequently they do not in reality have any ratio withone another; sounds cannot be compared, any more than colours can; allthat is needed is a little attention to hear this...”58. A further interestingconfrontation between Euler and d’Alembert’s methodologies concernedthe interpretation of the (widely used) dominant seventh chord, namelya chord made out of a root, a major third, a perfect fifth, and a minorseventh. Its name comes from the fact that it occurs naturally in theseventh chord built upon the fifth degree − the dominant − of a given

58Addition au memoire sur la courbe que forme une corde tendue mise en vibration, Mem.Acad. Sci. Berlin, 6 (1750), 355-366; cited in [50].

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major diatonic scale. For example, in the case of C-major diatonic scalewe get the aggregate G-B-d-f . To d’Alembert, this aggregate was a nicemajor triad G-B-d to which the dissonant seventh f is added to markunambiguously the root tone. Differently said, the dissonance G-B-d-fis there just to indicate to the listeners that the piece being played mustbe in the key of C. Euler discussed this topic in one of the three or fourarticles that he devoted to music theory in his later years59. To brieflyretrieve Euler’s argument, we start by recalling that he worked with ‘justintonation’, i.e. the system of ratios described by Ptolemy and revivedin the sixteenth century by Zarlino to account for the intervals used inpolyphonic music60. A portion of this system, covering a perfect twelfthinterval, and reduced to a series of whole numbers, is presented in thefollowing table:

C D E F G A B c d e f g

24 27 30 32 36 40 45 48 54 60 64 72

The seventh chord G-B-d-f is then expressed by the ratios 36 : 45 :54 : 64, to which Euler assigns the exponent given by their least commonmultiple, that is 26 · 33 · 5 = 8640. One recognizes that it is the tonef that troubles this chord. Indeed, if we omit this tone, we obtain themuch simpler ratios 4 : 5 : 6, whose exponent is 22 · 3 · 5 = 60. Fromthis “it seems that the addition of the note f ruins the harmony of thisconsonance too much for it to have a place in music. However, to theear’s judgment, this dissonance is at worst disagreeable and has beenused in music with great success. It even seems that musical compositionacquires a certain force from it, and without it would bee too smooth anddull. Here we have quite a paradox, where the theory seems to be in con-tradiction with the practice, to which I will try to give an explanation”61.The explanation of Euler is based on the following hypothesis: the organof hearing is accustomed to taking as simple proportion all proportions

59Conjectures sur la raison de quelques dissonances generalement recues dans la musique(1766), in [21], Ser. III, Vol. I, pp. 508-515. We shall use the english translation available athttp://eulerarchive.maa.org/docs/translations/E314en.pdf.

60Thus replacing the so called Pythagorean system − constructed with the perfect fifth 3:2as the only reference interval besides the octave − which remained in use until the late MiddleAges, meeting the needs of the monophonic composition and medieval parallel singing. Thesubsequent invention of the polyphony claimed an increasingly frequent use of intervals of thirdand sixth, which in the Pythagorean scale are not very consonant (see [52],[5]).

61Ibid, 4.

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that differ very little from it, so that the difference is almost impercep-tible. For example, in equal temperament the fifth is expressed by the(irrational) ratio 1 : 12

√

27, which hardly differs from the proportion of2 : 3, but the ear is not bothered too much by this small discrepancy62

and in hearing the interval C : G one may safely ‘think’ the ratio 2 : 3.More generally, if the proportions expressing a combination of tones aretoo complicated, the ear will ‘substitute’ a close approximation that issimpler. “Thus the heard proportions are different than the true, and it isfrom them that we must judge the true harmony and not from the actualnumbers”63. According to this assumption, the effect of listening to thedominant seventh chord, which corresponds to the tones 36, 45, 54, 64, isabsolutely the same as listening the tones 36, 45, 54, 63, which yield theproportion 4 : 5 : 6 : 7, whose exponent is 22 ·3 ·5 ·7 = 420, about twentytimes smaller than the ‘true’ one.In the perspective of the present work, we can recognize in this con-struction a way of ‘saving the phenomena’ (the strange acceptation bythe ear of a ‘dissonant’ acoustic aggregate) in the same spirit as the an-cient exact sciences, and therefore the product of the scientific activityof someone who has not yet introjected the division between ‘mathe-matics’ and ‘physics’ in vogue at his time. Euler’s explanation is oftenpresented as the legacy of an outdated attitude, still attached to a cal-cified ‘Pythagorean tradition’, whereas d’Alembert would belong to ‘theright side of History’64. Indeed, unlike Newtonian mechanics, which, al-though leaning on outlandish foundations, rather quickly has developedinto a true scientific theory, the science of harmonics eventually has fallenapart, in that the subsequent evolution has gradually ousted music theoryfrom the field of direct interest of the majority of scientists: on the onehand, musica theorica has been largely absorbed into musica practica,written by musicians for musicians and mainly focused on the empiricalproblems of harmony and counterpoint; on the other hand, the theoreti-cal work on the phenomena regarding musical perception was broken upinto several branches, with the result that the acoustic problems thattraditionally were part of music theory were detached from their musicalcontext to become subjects treated separately by the ‘natural sciences’,

62Adopting the tempered fifth amounts to using the convergent 7/12 of (the continuedfraction expansion of) log

2(3/2). The next convergent being 24/41, the error is smaller than

(12 · 41)−1, that is about a hundredth of a tone.63Ibid, 12.64See e.g. [50] p. 289; [3], pp. 139-141.

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such as physics65, physiology66 and psychology67. Nowadays the scenariois rather involved, with the coexistence of several tendencies which mostlyignore each other. On the one side, we witness a significant renewal ofinterest on mathematical modeling of some aspects of music theory, withthe search for structural, and to some extent universal, principles in theformation of musical scales68. In other directions, the massive adventof new information technologies in last decades has created an unprece-dented situation in which quantitative methods based on the automaticprocessing of large masses of data invade all fields, including music. Be-sides the indubitable enrichment with new sound media and new com-posing techniques (often directly inspired by mathematical constructionssuch as probability theory or game theory), as far as the new quantitativetreatments of musical-acoustical phenomena are concerned − with therelated conceptualizations and cultural trends − we have to say that theaims and the methodologies adopted in this context are often placed quitefar from those embodied by the exact sciences. This calls for a criticalanalysis which is still to come.

Acknowledgments. I wish to thank Lucio Russo for his warm adviceand Sara Munday for her careful reading of the manuscript and the manyvaluable corrections.

65Mainly forwarded towards investigations on the ‘nature of sound’, in a way to some aspectsanalogous to that in which the ancient optics, a science of vision, became the study of the‘nature of light’.

66With major achievement on the physiological analysis of the tonotopic structure of theinner ear, starting with the important work of Helmholtz [27] until more recent findings on theband structure of the basilar membrane [38], and the study of auditory cortex by the so called‘neurosciences’. By the way, these researches can be viewed as providing a physiological basisboth to the ancient principle of consonance and to the Euler principle mentioned above.

67Mainly focused on the empirical study of perceptive and cognitive aspects as well as socialand therapeutic applications (see, e.g., [18]).

68See, e.g., [29] and references therein.

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