Mathematics and Music Theory: I-tom Pythagoras to Rameau

ATHANASE PAPADOPOULOS athematics and usic Theory. ED) -rom nagoras to Rameau usic theory is a wide and beautiful subject, and some basic mathematical ideas are inherent in it. Some of these ideas were introduced in music theory by mathematicians, and others by musicians with no special mathematical skill. This paper describes some of the connections between music theory and mathemat- Greek mathematical treatise from the beginning of our era ics. The examples are chosen mainly from the works of would usually contain four sections: Number Theory, Geom- Pythagoras and of J. Ph. Rameau, who both were imporetry, Music, and Astronomy. This division of mathematics, tant music theorists, although the former is usually known which has been called the quadrivium! (the “four ways”), as a mathematician, and the latter as a composer. lasted in European culture until the end of the middle ages Before going into the works of Pythagoras and Rameau, (ca. 1500). One can see bas-reliefs and paintings represent- I present, in the next section, a summary history of the reing the four branches of the quadrivium on the walls or pillation between music and mathematics. lars of cathedrals in several places in Europe (see for instance the pictures in [1]). The situation changed with the Renais- A Few Historical Markers sance, when theoretical music became an independent field, I start with Greek antiquity. but strong links with mathematics were maintained.” It is well known that the schools of Pythagoras, Plato, and Several important mathematicians of the seventeenth Aristotle considered music as part of mathematics, and a and eighteenth centuries were also music theorists. For in- ‘This terminology is due to Boéthius (ca. 480-524 ap), who worked on the translation and the diffusion of Greek science and philosophy in the Latin world. He is responsible in particular for a Latin translation and a commentary of the mathematical treatise of Nichomachus. Boéthius considered the study of the quadrivium to be a prerequisite for philosophy, and this idea was at the basis of Western European curricula for almost ten centuries. ?The AMS subject classification has a section called Astronomy, but none called Music. © 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 1, 2002

stance, the first book that René Descartes wrote is on mu- Music theory as well as musical composition requires a sic (Compendium Musicae, 1618). Marin Mersenne wrote certain abstract way of thinking and contemplation which several treatises on music, among them the Harmonicoare very close to mathematical pure thought. Music makes rum Libri (1635) and the Traité de l'harmonie universelle use of a symbolic language, together with a rich system of (1636), and he had an important correspondence on that notation, subject with Descartes, Isaac Beekman, Constantijn Huyeleventh century (in the case of Western European music), gens, and others. John Wallis published critical editions of are similar to mathematical graphs of discrete functions in including diagrams which, starting from the the Harmonics of Ptolemy (2d c. AD), of Porhyrius (3d c. two-dimensional cartesian coordinates (the x-coordinate AD), and of Bryennius (a Byzantine musicologist of the fourrepresenting time and the y-coordinate representing pitch). teenth century). Leonhard Euler published in 1731 his Ten- Music theorists used these “cartesian” diagrams long betamen novae theoriae musicae ex certissimis harmoniae fore they were introduced in geometry. Musical scores from principiis dilucide expositae. Jean d’Alembert wrote in the twentieth century have a variety of forms which are 1752 his Eléments de musique théorique et pratique suivclose to all sorts of diagrams used in mathematics. Besides ant les principes de M. Rameau and in 1754 his Réflexions abstract language and notation, mathematical notions like sur la musique; and there are many other examples. Until well into the Renaissance, the term “musician” resymmetry, periodicity, proportion, discreteness, and continuity, among others, are omnipresent in music. Lengths of ferred to music theorists rather than to musical performers. Research and teaching in music theory were much more prestigious occupations than musical composition or performance. Some famous mathematicians were also composers or performers, but this is another subject.? J. Ph. Rameau, who is certainly the greatest French musicologist of the eighteenth century, wrote in his Traité de l'harmonie réduite à ses principes naturels (1722): PAR EF A CE auffi dui eft naturelle , afin que l'efprit en congoive les proprietez,, acilement que l'oreille les fene. Un feul homme n'eff pas capable d'épuifèr une matiere auf; profonde que celle-cy ; il eft prefque impofjible qu'il n'y oublie toujours quelque chofè, malgre tous fes foins; mais du moins, les nouvelles découvertes qu'il peut joindre à ce qui a déja paru für le même füjet, font autant de routes frayées pour ceux qui peuvent aller plus loin. La Mufique eft une fcience qui doit avoir des regles certaines; tirées d'un principe évident, et ce principe ces regles doivent être tirées d'un principe évident , & ce principe ne peut gueres nous être connu [ans le fecours des Mathematiques: Auffi dois-je avoñer que , nonobftant toute l'experience que je poune peut guère nous être connu sans le secvois m'être acquife dans la Mufique, pour l'avoir pratiquée La musique est une science qui doit avoir des règles certaines; ces règles doivent être dois-je pendant une a longue fuite de temps, ce n'eft cependant que avouer que, nonobstant toute l'expérience par le fecours des Mathematiques que mes idées fe font débroüil. que je pouvais m'être acquise dans la lees, & que la lumiere y a fuccedé à une certaine obfiurité , dons je ne m'appercevois pas auparavant. Si je ne Jçavois pas faire la difference du principe à la regle , bien tôt ce principe s'eft ours des mathématiques. Aussi musique pour l'avoir pratiquée pendant une assez longue suite de temps, ce n'est cependant que par le secours des mathématiques que mes idées se sont debrouiloffert à moi avec autant de fimplicité que d'évidence ; les conSequences qu'il m'a fournies enfuite, m'ont fait connoftre en elles lées, et que la lumière y a succédé à une autant de regles , qui devoient fe rapporter par confequent à ce certaine obscurité dont je ne m'apercevais principe ; le veritable fens de ces regles, leur jufte application , pas auparavant. Leur rapport, @ l'ordre qu elles doivent tenir entr'elles (la plus Music is a science which must have deter- Simple y fervant tohjours d'introdultion à la moins fimple , & mined rules. These rules must be drawn from ainfipar degrez) enfin le choix des termes ;tout cela, dis-je, que a principle which should be evident, and this 7 ignorois auparavant,s ef? développé dans mon efprit avec tant de of all the experience which I have acquired in netteté 6 de precifion, que je n'aipú m'empécher de convenir qu il Seroit à foubaiter (comme on me le diföit, un jour que 7 applaudiffois à la perfettion de nôtre Mufique moderne ) que les connoi/- music by practising it for a fairly long period, fances des Muficiens de ce fiecle répondiffent aux principle cannot be known without the help of mathematics. I must confess that in spite it is nevertheless only with the help of mathbeautez de leurs Compofitions. Il nefuffit donc pas de fentir les ‘effets d'une Science ematics that my ideas became disentangled ou d'un Art, ilfaut de plus les concevoir de façon qu'on puiffe and that light has succeeded to a certain darkles rendre inselligibles'; es c'eft à quoi je me fuis principalement appliqué dans le corps de cet Ouvrage, que j'ai difiribué ex ness of which I was not aware before. quatre Livres. SFor instance, Pythagoras, according to his biographers, besides being a geometer, a number-theorist, and a musicologist, was a composer, and he also played several instru- Figure 1. Glowing words from Rameau’s Traité de l'harmonie réduite à ses principes ments; see e.g., [7], Chapter XV, p. 32. naturels.

musical intervals, rhythm, duration, tempi, and several other musical notions are naturally expressed by numbers. The mathematical use of the word “harmonic” (for instance in “harmonic series” or “harmonic analysis”) has its origin in music theory. The composer Milton Babbitt, who taught mathematics and music theory at Princeton University, writes in [2] that a musical theory should be “statable as a connected set of axioms, definitions and theorems, the proofs of which are derived by means of an appropriate logic.” It is important to realize that there are contribu- EITAS tions in both directions. On the one hand, mathematical language and mathematical ideas have shaped the language and the concepts of music theory. This is illustrated in the work of Rameau discussed below, but there are several other instances. For example, Milton Babbitt uses group theory and set theory in his theoretical musical teaching and in his compositions. Olivier Messiaen speaks of “symmetric permutations.” Some pieces of Iannis Xenakis are based on game theory, others on probability theory; and so on.‘ On the other hand, questions and problems arising in music theory have constituted, at several points in history, strong motivation for investigations in mathematics (and of course in physics). For example, phenomena like the production of beats or the production of the harmonic frequencies were noticed and discussed by music theorists several decades before they were explained by mathematical and physical theories. Some of the theories developed in the seventeenth century by Wallis, J. Sauveur, and others were essentially motivated by these phenomena. I shall discuss the question of the harmonic frequencies in the last part of this article. It is also fair to acknowledge that there are in- Figure 2. The hammers of Pythagoras, according to Gafurius (1492). stances where music theorists have used mathematical notions in an intuitive manner, before these notions would be possible to devise instrumental assistance to the had been shaped and refined by mathematicians. One such hearing, which could be firm and unerring, such as the sight example is the use of logarithms, also discussed below. Now let us start from the beginning, that is, with Pythagoras. obtains through the compass and the rule.” Walking through a brazier’s shop, Pythagoras heard the different sounds produced by hammers beating an anvil. He realized that the pitch, that is, the musical note, that was produced Pythagoras and the Theory of Musical Intervals by a particular hammer, depended only on the weight of Historians of science usually agree that Pythagoras (sixth the hammer and not on the particular place where the hamc. BC) is at the origin of mathematics as a purely theoretimer hit the anvil, or on the magnitude of the stroke. cal science.? At the same time, Pythagoras is regarded as Pythagoras realized also that the compass of a musical inthe first music theorist (from the point of view of European terval between two notes produced by two different hammusic). The major musical discovery of Pythagoras is the mers depended only on the relative weights of the hamrelation of musical intervals with ratios of integers. This is mers, described by Jamblichus ([7], Chap. XXVI, p. 62) in these intervals, which in classical Greek music were the intervals terms: Pythagoras was “reasoning with himself, whether it of octave, of fifth and of fourth, correspond, in terms of and in particular that the consonant musical “The idea of using mathematical theories in musical composition is not new. Athanasius Kircher, a seventeenth-century mathematician at the Court of Vienna, wrote a treatise on musicology, Misurgia Universalis (1622) in which he described a machine, Arca Musicarithma, which produces musical compositions based on mathematical structures. 5The theories and results which Pythagoras and his school developed were not intended for practical use or for applications, and it was even forbidden for the members of the Pythagorean school to earn money by teaching mathematics, and the exceptions confirm the rule: Jamblichus (see [7], Chap. XXIV, p. 48) relates that “the Pythagoreans say that geometry was divulgated from the following circumstances: A certain Pythagorean happened to lose the wealth that he possessed; and in consequence of this misfortune, he was permitted to enrich himself from geometry.”

listening to the harmonies produced, they were able to associate numbers to consonances. The result is again that the octaves, fifths, and fourths correspond respectively to the fractions 2/1, 3/2 and 43, in terms of the quotients of levels of the liquid. These experiments were repeated and reinterpreted by the acousticians of the seventeenth century. The ideas and observations of Pythagoras and his school established the relation between musical intervals and ratios of integers. \f Na f \ | S Logarithms The arithmetic of musical intervals involves in a very natural way the theory of logarithms. For an example, we re- I 1<—fourth—>: | y ' ' turn for a moment to Jamblichus, who relates in Section XV of [7] that Pythagoras defined the tone as the difference between the intervals of fifth and of fourth. (The definition may seem circuitous, but it becomes natural if we recall 1<—_—— fifth —>' ' ' ’ té octave >" that the definitions of musical intervals had to be based on those of consonant intervals, which are naturally recognisable by the ear.) The point now is that the fraction associated to the tone interval is not the difference 3/2 — 4/3, Figure 3. The classically “consonant” intervals. but the quotient (3/2)/(4/3) = 9/8. It is natural to define the compass of a musical interval weights, to the numerical fraction 2/1, 3/2, and 4/3, respecas the number (or the fractions of) octaves it contains. tively. Thus, Pythagoras thought that the relative weights Thus, when we say that two notes are n octaves apart, the of two hammers producing an octave is 2/1, and so on. As fraction associated to the interval that they define is 2”. The soon as this idea occurred to him, Pythagoras went home definition of the compass can be made in terms of freand performed several experiments using different kinds quency, and in fact one usually defines the pitch as the logof instruments, which confirmed the relationship between arithm in base 2 of the frequency. (Of course, the notion of musical intervals and numerical fractions. Some of these frequency did not exist as such in antiquity, but it is clear experiments consisted of listening to the pitch produced that the ancient Greek musicologists were aware that the by the vibrations of strings that have the same length; he lowness or the highness of pitch depends on the slowness had suspended the strings from one end and attached difor rapidity of the air vibration that produces it, as explained ferent weights to the other end. Other experiments involved in Theon's treatise [12], Chapter XIII.) The relation of mustrings of different lengths, which he had stretched end-tosical intervals with logarithms can also be seen by considend, as in musical instruments. He also did experiments on ering the lengths of strings (which in fact are inversely propipes and other wind instruments, and all these experiportional to the frequency). For instance, if a violinist (or ments confirmed him in his idea that musical intervals cora lyre player in antiquity) wants to produce a note which respond in an immutable way to definite ratios of integers, is an octave higher than the note produced by a certain whether these are ratios of lengths of pipes, lengths of string, he must divide the length of the string by two. strings, weights, etc.® Thus, music theorists dealt intuitively with logarithms Theon of Smyrna, in Part 2, Chapter XIII of his mathelong before these were defined as an abstract mathematimatics treatise [12], describes other experiments which ilcal notion. (It was only in the seventeenth century that loglustrate this relation between musical intervals and quoarithms were formally introduced in music theory, by Isaac tients of integers. He relates, for instance, that the Newton, and then by Leonhard Euler and Jacques Lam- Pythagoreans considered a collection of vases, filled parbert.) The theory of musical intervals is a natural example tially with different quantities of the same liquid, and obof the practical use of logarithms, an example easily exserved on them the “rapidity and the slowness of the moveplained to children, provided they have some acquaintance ments of air vibrations.” By hitting these vases in pairs and with musical intervals. $We must note that the experiment with the hanging weights is considered to be a mistake of Pythagoras, or an extrapolation due to Pythagoras’s disciples, or a misinterpretation of what Pythagoras really said. This mistake was noticed by Vincenzo Galilei (the father of Galileo Galilei). Vincenzo was a most cultivated person, in particular a music theorist and a music composer. He did the experiment with the hanging weights and realized that to produce the intervals of octave, fifth, and fourth, the ratios of the pairs of weights should be respectively 4/1, 9/4, and 16/9, which are the squares of the numbers which occur in the experiments involving the lengths of strings. Galilel was proud of that discovery {and of the discovery of a mistake in the theory of Pythagoras), and he published it in his famous musical treatise, the Discorso intorno alle opere de Gioseffo Zarlino. The physical reason behind this fact is that the frequency of a vibrating string, while it is proportional to the length of the string, is proportional to the square root of the tension. Nonetheless, the relation between musical intervals and ratios of integers is still there, even though it is not so direct in all cases. We note too that the same experience with the hanging weights is described by Vincenzo’s son, Galileo (see [5], p. 98 to 110).

Music in the Mathematical Treatise ered them unnatural and a threat to their philosophical sysof Theon of Smyrna tem, based on positive integers. The adjective “irrational” It is interesting to go through the music theory part of a which they introduced clearly indicates this. It is also well mathematics treatise of the classical Greek era. I consider known that the Pythagoreans wanted to keep the existence here the section on Music (Part 2) of Theon’s treatise [12]. of irrational numbers (the discovery of which is attributed This section deals with the definition and the combinations to Pythagoras himself) a secret. Jamblichus relates in [7] of musical intervals, with proportions, musical units, and Chapter XXIX (p. 126) that “he who first divulgated the theso on. It involves non-trivial arithmetic, and Theon, in this ory of commensurable and incommensurable quantities, to section, often refers to the discoveries made by Pythagothose who were unworthy to receive it, was so hated by ras and the Pythagoreans. the Pythagoreans that they not only expelled him from their The title of Part 2 of Theon’s mathematical treatise is “A book containing the numeric laws of music.” In the introcommon association, and from living with them, but also constructed a tomb for him.” duction, he says, “Harmony is spread in the world, and of- The reasons why ancient Greek music used semitones fers itself to those who seek it only if it is revealed by numof 16/15 or 25/24 are certainly related to the fact that these bers.” The first part of this sentence, that “Harmony is intervals are acceptable by the ear. But it is also a fact spread in the world,” has been repeated throughout the that the ancient Greek musicologists liked to deal with suages, and it was at the basis of a strong feeling of cosmic structure and order. There are important philosophical and esoteric traditions behind this idea, which led eventually to explanations of physical For the Pythagoreans, dealing with irrational numbers would have been incompatible with their philosophy. ratios derived from 2, 3, and 5, that is, fractions of the form (n + 1)/n with numerator and denominator having only 2, 3, and 5 as prime factors. Pythagorean number symbolism is involved here, but that subject is phenomena, like the motion of planets. Famous adepts and advocates of such traperparticular beyond the scope of this paper. The following is a list of ditions include, after Pythagoras himself, Plato, Boéthius, “useful” musical intervals, which was known to Gioseffo Copernicus, and Kepler (see for instance [8], Book V, where Zarlino and Descartes: Kepler gives a relation between the eccentricities of the orbits of the planets and musical intervals). The second part 2/1 octave of Theon’s sentence, that “harmony is revealed by num- 3/2 fifth bers,” has also been repeated throughout the ages, for in- 4/3 fourth stance in the citation of Rameau mentioned earlier and in 5/4 major third the following citation of Gottfried Wilhelm Leibniz, from 6/5 minor third his Principles of nature and of grace (1712): “Musica est exercitium arithmeticae occultum ...” (Music is a secret exercise in arithmetic). Let us look at the treatment of semitones in Theon’s treatise. There are several kinds of semitones used in ancient Greek music, two of which are the “diatonic semitone” and the “chromatic semitone,” the values of which 9/8 major tone 10/9 minor tone 16/15 diatonic semitone 25/24 chromatic semitone 81/80 comma of Didymus. are, respectively, 16/15 and 25/24. One could expect that There is a discussion of this list in both [6] and [9]. Many there is a semitone whose value is equal to half of the value years after this list was known to music theorists, C. Stormer of atone, in the sense that if we concatenate two such semiproved that this is a complete list of the superparticular ratones, we obtain a tone. This is not the case for any of the tios derived from the prime numbers 2, 3, and 5 [11]. semitones used by the Pythagoreans, however. Indeed, by the discussion on logarithms above, we know that if the Scales semitone were half of the tone, then its numerical value Scales are building blocks for musical compositions. (This should have been V9, which is an irrational number. For is true at least in tonal music, that is, in almost all pre-twenthe Pythagoreans, dealing with irrational numbers would tieth-century European music.) I shall talk in this section have been incompatible with their philosophy. Theon about the arithmetic of scales, and I remark by the way that writes in §VIII of Part 2 that “one can prove that” the tone, in addition to this arithmetic, there is a more abstract rethe value of which is 9/8, cannot be divided into two equal lation between scales and mathematics, namely in the conparts, “because 9 is not divisible by 2.” Of course, this is text of formal languages. Classical musical compositions nonsense: the point is not to divide 9 by 2, but to take the are based on scales, fragments of which appear within a square root of 9/8. Although Pythagoras and his school were piece in various forms, constituting a family of privileged aware of the existence of irrational numbers, they considsequences of musical motives. This fact has been exploited

and systematically generalized in certain twentieth-century tion of the scale of Pythagoras. One starts by assigning the compositional techniques values 2, 3/2, and 4/3, respectively, to the eighth, fifth, and (for instance, serial music), which are related to mathematics, but which are beyond fourth notes in the list. The rest of the values are obtained the subject matter of this paper. by an iterative process involving fifths whose values are 3/2 The major part of post-Renaissance Western European (such fifths are called pure fifths). Thus, for example, if we classical music uses a very limited number of scales; in fact, start from the first note (with value 1) and concatenate two since the general acceptance of the tempered scale in the pure fifths, we obtain an interval of ninth, with value 3/2 x eighteenth century, there are basically two scales, the ma- 3/2 = 9/4, which is greater than 2 (as expected, since this jor and the minor scale. The tempered scale (the one we interval is larger than an octave). To come back inside our play on a piano keyboard), is based on the division of the octave, we divide by two, obtaining the value 9/8. In the octave into 12 equal intervals, the unit being the tempered same way, the value 27/16 is found as (3/2)? divided by 2, semitone, the value of which is equal therefore to 12V2. and so on. Unfortunately the process gives an infinite num- Any two major (respectively, minor) tempered scales are ber of notes, but it is reasonable to stop after the octave translations of each other on the set of pitches. (In musihas been divided into these seven intervals. cal terms, these translations are called transpositions.) This was not the case in pre-Renaissance music. The scale of Pythagoras has beautiful properties. One is that all fifths and all fourths are pure, their common val- In contrast, the theory of harmony in classical Greece ues being 3/2 and 4/3. For instance, the value of the interincluded a complicated and very subtle system of scales. val between the second and the fifth note is (8/2)/(9/8) = Greek mathematical treatises usually contain a descrip- 4/3. This is a remarkable property which does not follow tion of scales in terms of fractions, with a discussion of obviously from the construction. the logic behind the definitions. For instance, the scale which is known today as the “scale of Pythagoras” is defined by the following sequence of numbers: 1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2. These numbers can be regarded as representing ratios of lengths of strings, the nth number being the ratio of a pair of strings having the same section and stretched at the same tension, producing the interval between the first and the mth note. Thus, for instance, the interval between the first and the last note in the list is an octave, the interval between the first and the fourth note is a fourth and the interval between the first and the fifth note is a fifth, as expected, since the Pythagorean scale needs to contain these three consonant intervals. The intervals between consecutive notes, except those between the third and the fourth and the seventh and the eighth, have the value 9/8. The intervals which we have excluded have the common value 256/243, which corresponds to another semitone. Pythagoras sounds The scale of approximately, but not exactly, like our tempered major scale. The semitone which is used in our tempered scale, !2\/2, is closer to the diatonic semitone, 16/15, than to the other two which we encountered. There is a logic behind the defini- Figure 4. Jean-Philippe Rameau. Portrait by Jean Bernard Restout, titled “The inspired poet.”

Providing a scale with the maximum number of pure inrelated in Jamblichus [7].) This subtle dependence of the tervals was a domain of research of early music theory. In piece upon the scale lasted in European music until the sixteenth-century Western European music, the intervals adoption of the tempered scale. For instance, Rameau gives of minor and major third began to be considered as cona list of characteristics of different tonalities in his Traité sonant, and the scale of Pythagoras was less suitable for de l'harmonie réduite à ses principes naturels, Book II, new harmonies that involved many of the new intervals. Chapter 24 (Vol. 1 of [10]). (The value of a pure major third interval is 5/4, whereas in the scale of Pythagoras the value of the interval between the first and the third notes is 81/64, which is a little bit greater than 5/4). A scale which was useful in that respect is the one named after Gioseffo Zarlino, a famous sixteenthcentury Venetian musicologist. Zarlino’s scale makes a compromise between pure thirds, pure fourths, and pure fifths. The sequence of numbers is 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2. Rameau and the Harmonic Sequence Like Pythagoras 2000 years before him, the composer and theoretician Jean-Philippe Rameau made a real synthesis between music as an art whose aim is to express and to create emotions, and music as a mathematical science with a deductive approach and rigorous rules. Pythagoras established the important relation between musical intervals and pairs of integers, Rameau went a step further and gave a musical content to the whole sequence of positive integers. Some of the fifths in this scale are pure, but not all of them. For instance, the value of the interval between the second and the sixth note is 40/27, which is strictly less than 3/2. The value of the difference is (3/2)/(40/27) = 81/80, the Didymus comma, which is an audible interval. It is impossible to have only pure intervals in a scale, unless the scale is short. Aristoxenus (fourth c. Bc) made a systematic theory of scales based on “tetrachords,” scales consisting of four notes corresponding to different divisions of the fourth by tones and semitones. A long scale would be obtained by concatenating tetrachords. Let us return for a moment to the scale of Pythagoras. Problems are encountered as soon as one needs to concatenate several such scales, for instance in order to play musical instruments whose ranges cover several octaves. For example, one would expect that the concatenation of 12 fifths gives 7 octaves (as is the case for instruments like the guitar or the harpsichord). This cannot be the case if one uses the scale of Pythagoras, since (3/2)! is not equal to (2/1)”. The interval with value (3/2)!* is larger than the one with value (2/1)7. The difference is a small (but nevertheless audible) interval, (3/2)12/27. This small interval is One of the main ideas for Rameau is that the infinite sequence of integers is contained, in a beautiful way, in nature, as a sequence of frequencies. When a sonorous body (Rameau’s terminology: “corps sonore”) vibrates, it creates a local periodic variation of the pressure of air. This vibration propagates as an acoustic wave. It hits our ear drums, and we hear a musical note. The musical note produced by a vibrating string (bowed or plucked), consists usually in a superposition of a fundamental tone and overtones. The frequencies of the overtones, which are called the harmonic frequencies, are integral multiples of the frequency of the fundamental tone. The sequence of harmonic frequencies is naturally parametrized by the positive integers. For instance, the frequency of the note C, (which corresponds to the lowest C key on a piano keyboard) is (approximately) f = 33 Hz (cycles per second). The frequencies of the corresponding overtones are therefore S, 2F, 3F, Af, 5f, OF... whose values in Hz are called a “Pythagorean comma.” 33, 66, 99, 132, 165, 198, . . . Similar problems occur in all the other scales based on pure intervals. For instance, we would expect that the con- The corresponding sequence of notes is catenation of 4 fifths gives an interval of 2 octaves and one Ci, Ca, Go, Cs, Ez, G3,... major third. If we do the computation in Zarlino’s scale, we find that this is not the case, and the difference is the Didy- In principle, one can hear the first four or five overtones on mus comma (81/80). an instrument like an organ. (Mersenne, in his Harmonie It is worthwhile to mention here that music theorists in ancient China encountered similar arithmetical problems in their theory of scales. It should be clear now that the definition of a scale in- Universelle, says that he can hear the first nine overtones.) Rameau’s theoretical work is based on scientific discoveries in acoustics which were made in the seventeenth century, in particular by the mathematician Joseph volves some arbitrariness and depends strongly on which Sauveur. The phenomenon of “harmonics” in music had intervals we insist be pure. One solution to the problem been noticed long before Rameau, but Rameau was the one was, instead of making a restricted choice, to keep differwho used it as the basis of a coherent theoretical teaching ent possibilities. This is one of the reasons why there are of music, in particular in his Traité de Vharmonie réduite so many scales in antique Greek music. In this music, difà ses principes naturels. ferent scales were adapted to different melodies and dif- Rameau’s textbooks on music theory (about 2000 ferent types of instruments. The choice of scale for a mupages) include the basics of figured bass, accompanisical piece determined much of the character of the piece ment, chords, modulation, and composition techniques. and of its psychological effects on the listener. (This is also All the theories he developed are based on simple rules

Rameau liked to consider the har- DE LHARMONIE, monic sequence of frequencies emitted by a sonorous body as a proof that the principles of music theory are contained in nature. Later on (starting from the year 1750), and especially in his Nouvelles réflexions sur le principe sonore, Rameau argued that since the fundamental objects of mathematics are derived from the sequence of positive integers, and since this sequence is contained in music, then mathematics itself is part of music. These reflections provoked a dispute between Rameau and eighteenth-century French mathematicians, “n401p3, P:059V u é like d'Alembert, L. B. and Castel with and the J. ency- I clopaedists, like Denis Diderot, Jean- $è Jacques Rousseau, and Friedrich von ~~ . A Grimm. The details of the controversy <& are worth studying, but they cannot be included in this short report. A very strong hostility followed several years of friendship and mutual praise between ron, y Accord trouver de la les raifons feptieme - de ¿ il faut tripler les nombres de cile, 40. donnera le Son grave de ce dernier Accord aint § 40: 60, 75; 90. 108. Re,La, UL X Mi, Sol. Rameau and d’Alembert; do not fall into the facile conclusion that the interaction between music theorists and mathematicians was always friendly. Still the interaction was there. In this report I have concen- ; Accord fondamental | de la fepriéme trated on examples, starting with Pythagoras and ending with Rameau. To Fa support the choice of Pythagoras and ! Son grave de l’Accord | Rameau, de la Quinte-luperfiuè. let me conclude by citing Jacques Chailley [4]? Figure 5. A diagram in Rameau's Traité, discussing ratios of frequencies of a dissonant En 2500 ans d'histoire écrite, la musique n'a peut-être connu que deux chord (containing a minor seventh). veritables théoriciens, dont les autres n'ont guère fait qu'aménager ou rapetasser les proposiderived from the existence and the properties of the hartions. L'un, au VIe siècle avant notre ère, fut le fabuleux monic sequence. For instance, in his analysis of chords, Pythagore. L'autre mourut à Paris en 1764: ce fut Jeanthe root of a triad is treated as a unit, in a mathematical Philippe Rameau. sense, and this point of view makes things simple and evident. The theory of triads (consisting of three notes, like In 2500 years of written history, music has perhaps known C, E, G) had already been derived from the harmonic seonly two genuine theoreticians, and what the others did quence by Zarlino and Descartes, but Rameau worked on was only to repackage or patch up their propositions. The a complete theory of dissonant chords. The diagram in first one, in the VIth century before our era, was the fabu- Figure 5 is one of Rameau’s pictures in the Traité de lous Pythagoras. The other one died in Paris in 1764: this Vharmonie réduite à ses principes naturels, in which he was Jean-Philippe Rameau. represents the dissonant chord La, Dot, Mi, Sol (that is, A, Ct, E, G), with four other derived chords. The numbers below the notes are the corresponding elements of the harmonic sequence. REFERENCES [1] Benno Artmann, The liberal arts, Math. Intelligencer 20 (1988), no. 3, 40-41. 7J. Chailley was a famous musicologist, professor at the Conservatoire National Supérieur de Musique de Paris and at the University of Paris. | borrowed this quotation from the Introduction to the collected works of Rameau [10].

[2] Milton Babbitt, Past and present concepts of the nature and lim- AUTHOR | its of music, international Musical Society Congress Reports 8 (1961), no. 1, 399. [3] J. M. Barbour, Music and ternary continued fractions, Amer. Math. Monthly 55 (1948), 545-555. [4] Jacques Chailley, “Rameau et la théorie musicale”, La Revue Musicale, Numéro spécial 260, 1964. [5] Galileo Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze, in Vol XI of the Complete Works, Società Editrice Fiorentina, 1855. [6] G. D. Hasley and Edwin Hewitt, More on the superparticular ratios in music, Amer. Math. Monthly 79 (1972), 1096-1100. [7] Jamblichus (ca. 240 AD), The Life of Pythagoras, English transla- ATHANASE PAPADOPOULOS tion by Thomas Taylor, London, John M. Watkins, 1965. nstitute de Mathématiques, CNRS [8] Johannes Kepler, Harmonices Mundi. (| have used the French ” rue Rene translation witn comments by J. Peyroux, Librairie A. Blanchard, 9 > 6708 rue de Medicis, Paris, 1977.) [9] A. L. Leigh Silver, Musimatics or the nun’s fiddie, Amer. Math. e-mail: papadopoulos@math.u-strasbg.fr Monthly 78 (1971), 351-857. [10] J. Ph. Rameau, Complete Theoretical Writings, edited by R. Jacobi, a facsimile of original editions, published by the American In- Athanase Papadopoulos graduated as an engineer from the Ec ventrale a 981 3 doct¢ stitute of Musicology, 1967. [11] C. Stormer, Sur une inéquation indéterminée, C. R. Acad. Sci. Paris 127 (1898), 752-754. [12] Theon of Smyrna (beginning of the second c. AD), Exposition of the mathematical knowledge useful for the reading of Plato. A bilingual (Greek-French) edition due to J. Dupuis (Paris 1892) is reprinted by Culture et Civilisation, 115 Av. Gabriel Lebon, Brussels, 1966. STATEMENT OF OWNERSHIP, MANAGEMENT, AND CIRCULATION (Required by 39 U.S.C. 3685). (1) Publication title: Mathematical Intelligencer. (2) Publication No. 001-656. (3) Filing Date: 10/01. (4) Issue Frequency: Quarterly. 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