Order 804857: Mathematics vs Music
ATHANASE PAPADOPOULOS
Mathematics and Music Theory: I-tom Pythagoras to Rameau
~ n usic theory is a wide and beautiful subject, and some basic math-
ematical ideas are inherent in it. Some of these ideas were intro-
duced in music theory by mathematicians, and others by musi-
cians with no special mathematical skill. This paper describes some
of the connections between music theory and mathemat- ics. The examples are chosen mainly from the works of Pythagoras and of J. Ph. Rameau, who both were impor- tant music theorists, although the former is usually known as a mathematician, and the latter as a composer.
Before going into the works of Pythagoras and Rameau, I present, in the next section, a summary history of the re- lation between music and mathematics.
A Few Histor ical Markers I start wi th Greek antiquity.
It is we]] known that the schools of Pythagoras, Plato, and Aristotle considered music as part of mathematics, and a
Greek mathematical treatise from the beginning of our era would usually contain four sections: Number Theory, Geom- etry, Music, and Astronomy. This dix4sion of mathematics, which has been called the quadrivium 1 (the "four ways"), lasted in European culture until the end of the middle ages (ca. 1500). One can see bas-reliefs and paintings represent- ing the four branches of the quadrivium on the walls or pil- lars of cathedrals in several places in Europe (see for instance the pictures in [1]). The situation changed with the Renais- sance, when theoretical music became an independent field, but strong links with mathematics were maintained. 2
Several important mathematicians of the seventeenth and eighteenth centuries were also music theorists. For in-
1This terminology is due to Boethius (ca. 480-524 AD), who worked on the translation and the diffusion of Greek science and philosophy in the Latin world. He is re-
sponsible in particular for a Latin translation and a commentary of the mathematical treatise of Nichomachus. Boethius 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.
2The AMS subject classification ilas a section called Astronomy, but none called Music.
�9 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 1, 2002
stance, the first book that Ren~ Descartes wrote is on mu- sic (Compendium Musicae, 1618). Marin Mersenne wrote several treatises on music, among them the Harmonico- rum Libri (1635) and the Traitd de l 'harmonie universeUe (1636), and he had an important correspondence on that subject with Descartes, Isaac Beekman, Constantijn Huy- gens, and others. John Wallis published critical editions of the Harmonics of Ptolemy (2d c. AD), of Porhyrius (3d c. AD), and of Bryennius (a Byzantine musicologist of the four- teenth century). Leonhard Euler published in 1731 his Ten- tamen novae theoriae musicae ex cer t i ss imis harmoniae
pr inc ip i i s dilucide e:tTositae. Jean d'Alembert wrote in 1752 his Eldments de musique thdorique et pratique suiv- ant les principes de M. Rameau and in 1754 his Rdfiexions sur la musique; and there are many other examples.
Until well into the Renaissance, the term "musician" re- ferred to music theorists rather than to mu- sicai performers. Research and teaching in music theory were much more prestigious occupations than musical composition or performance. Some famous mathemati- cians were also composers or performers, but this is another subject. 3
J. Ph. Rameau, who is certainly the great- est French musicologist of the eighteenth century, wrote in his Traitd de l 'harmonie rdduite d ses principes naturels (1722):
Music theory as well as musical composition requires a certain abstract way of thinking and contemplation which are very close to mathematical pure thought. Music makes use of a symbolic language, together with a rich system of notation, including diagrams which, starting from the eleventh century (in the case of Western European music), are similar to mathematical graphs of discrete functions in two-dimensional cartesian coordinates (the x-coordinate representing time and the y-coordinate representing pitch). Music theorists used these "cartesian" diagrams long be- fore they were introduced in geometry. Musical scores from the twentieth century have a variety of forms which are close to all sorts of diagrams used in mathematics. Besides abstract language and notation, mathematical notions like symmetry, periodicity, proportion, discreteness, and conti- nuity, among others, are omnipresent in music. Lengths of
La musique est une science qui doit avoir des r~gles certaines; ces r~gles doivent ~tre tirdes d'un principe dvident, et ce principe
ne peut gu~re nous ~tre connu sans le sec- ours des mathdmatiques. Auss i dois-je avouer que, nonobstant toute l' expdrience
que j e pouvais m'~tre acquise clans la musique pour l'avoir pratiqude pendant
une assez longue suite de temps, ee n'est eependant que par le secours des mathd- matiques que rues iddes se sont ddbrouil-
ldes, et que la lumi~re y a succddd d u n e certaine obscuritd dont j e ne m'apercevais pas auparavant.
Music is a science which must have deter- mined rules. These rules must be drawn from a principle which should be evident, and this principle cannot be known without the help of mathematics. I must confess that in spite of all the experience which I have acquired in music by practising it for a fairly long period, it is nevertheless only with the help of math- ematics that my ideas became disentangled and that light has succeeded to a certain dark- ness of which I was not aware before.
3For instance, Pythagoras, according to his biographers, be-
sides being a geometer, a number-theorist, and a musicol-
ogist, was a composer, and he also played several instru-
ments; see e.g., [7], Chapter XV, p. 32.
Figure 1. Glowing words from Rameau's Traitd de I'harmonie rdduite a ses principes
naturels.
66 THE MATHEMATICAL INTELLIGENCER
musical intervals, rhythm, duration, tempi, and several other musical notions are naturally expressed by num- bers. The mathematical use of the word "harmonic" (for instance in "harmonic series" or "harmonic analy- sis") has its origin in music theory. The composer Mil- ton Babbitt, who taught mathematics and music the- ory at Princeton University, writes in [2] that a musical theory should be "statable as a connected set of ax- ioms, definitions and theorems, the proofs of which are derived by means of an appropriate logic."
It is important to realize that there are contribu- tions in both directions. On the one hand, mathe- matical language and mathematical ideas have shaped the language and the concepts of music the- ory. This is illustrated in the work of Rameau dis- cussed 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 "sym- metric permutations." Some pieces of Iannis Xenakis are based on game theory, others on probability the- ory; and so on. 4 On the other hand, questions and problems arising in music theory have constituted, at several points in history, strong motivation for in- vestigations in mathematics (and of course in physics). For example, phenomena like the produc- tion of beats or the production of the harmonic fre- quencies were noticed and discussed by music theo- rists several decades before they were explained by mathematical and physical theories. Some of the the- ories developed in the seventeenth century by Wal- lis, 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 arti- cle. It is also fair to acknowledge that there are in- Figure
stances where music theorists have used mathemat- ical notions in an intuitive manner, before these notions had been shaped and refined by mathematicians. One such example is the use of logarithms, also discussed below.
Now let us start from the beginning, that is, with Pythagoras.
Pythagoras and the Theory of Musical Intervals Historians of science usually agree that Pythagoras (sixth c. BC) is at the origin of mathematics as a purely theoreti- cai science. 5 At the same time, Pythagoras is regarded as the first music theorist (from the point of view of European music). The major musical discovery of Pythagoras is the relation of musical intervals with ratios of integers. This is described by Jamblichus ([7], Chap. XXVI, p. 62) in these terms: Pythagoras was "reasoning with himself, whether it
2. The hammers of Pythagoras, accord ing to Gafur ius (1492).
would be possible to devise instrumental assistance to the hearing, which could be firm and unerring, such as the sight 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 by a particular hammer, depended only on the weight of the hammer and not on the particular place where the ham- mer hit the anvil, or on the magnitude of the stroke. Pythagoras realized also that the compass of a musical in- terval between two notes produced by two different ham- mers depended only on the relative weights of the ham- mers, and in particular that the c o n s o n a n t musical intervals, which in classical Greek music were the intervals of octave, of fifth and of fourth, correspond, in terms of
4The 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 mathemat- ical 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 mem-
bers of the Pythagorean school to earn money by teaching mathematics, and the exceptions confirm the rule: Jamblichus (see [7], Chap. XXlV, 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 con-
sequence of this misfortune, he was permitted to enrich himself from geometry."
VOLUME 24, NUMBER 1, 2002 67
I I I
I , I I I I I I
fourth-- ' t I I ! ! I I
, ~ l I I< f i f t h . - i !
, o c t a v e . . . . > ' Figure 3. The c lass ica l ly " c o n s o n a n t " intervals.
weights, to the numerical fraction 2/1, 3/2, and 4/3, respec- tively. Thus, Pythagoras thought that the relative weights of two hammers producing an octave is 2/1, and so on. As soon as this idea occurred to him, Pythagoras went home and performed several experiments using different kinds of instruments, which confirmed the relationship between musical intervals and numerical fractions. Some of these experiments consisted of listening to the pitch produced by the vibrations of strings that have the same length; he had suspended the strings from one end and attached dif- ferent weights to the other end. Other experiments involved strings of different lengths, which he had stretched end-to- end, as in musical instruments. He also did experiments on pipes and other wind instruments, and all these experi- ments confn-med him in his idea that musical intervals cor- respond in an immutable way to definite ratios of integers, whether these are ratios of lengths of pipes, lengths of strings, weights, etc. 6
Theon of Smyrna, in Part 2, Chapter XIII of his mathe- matics treatise [12], describes other experiments which il- lustrate this relation between musical intervals and quo- tients of integers. He relates, for instance, that the Pythagoreans considered a collection of vases, filled par- tially with different quantities of the same liquid, and ob- served on them the "rapidity and the slowness of the move- ments of air vibrations." By hitting these vases in pairs and
listening to the harmonies produced, they were able to as- sociate numbers to consonances. The result is again that the octaves, fifths, and fourths correspond respectively to the fractions 2/1, 3/2 and 4/3, 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.
L o g a r i t h m s The arithmetic of musical intervals involves in a very nat- ural way the theory of logarithms. For an example, we re- turn for a moment to Jamblichus, who relates in Section XV of [7] that Pythagoras defined the tone as the dif ference between the intervals of fifth and of fourth. (The definition may seem circuitous, but it becomes natural if we recall that the defmitious of musical intervals had to be based on those of consonant intervals, which are naturally recog- nisable by the ear.) The point now is that the fraction as- sociated to the tone interval is not the difference 3/2 - 4/3, but the quot ient (3/2)/(4/3) = 9/8.
It is natural to define the compass of a musical interval as the number (or the fractions of) octaves it contains. Thus, when we say that two notes are n octaves apart, the fraction associated to the interval that they define is 2 n. The definition of the compass can be made in terms of fre- quency, and in fact one usually defines the p i t ch as the log- arithm in base 2 of the frequency. (Of course, the notion of frequency did not exist as such in antiquity, but it is clear that the ancient Greek musicologists were aware that the lowness or the highness of pitch depends on the slowness or rapidity of the air vibration that produces it, as explained in Theon's treatise [12], Chapter XIII.) The relation of mu- sical intervals with logarithms can also be seen by consid- ering the lengths of strings (which in fact are inversely pro- portional to the frequency). For instance, if a violinist (or a lyre player in antiquity) wants to produce a note which is an octave higher than the note produced by a certain string, he must divide the length of the string by two.
Thus, music theorists dealt intuitively with logarithms long before these were defined as an abstract mathemati- cal notion. (It was only in the seventeenth century that log- arithms were formally introduced in music theory, by Isaac Newton, and then by Leonhard Euier and Jacques Lam- bert.) The theory of musical intervals is a natural example of the practical use of logarithms, an example easily ex- plained to children, provided they have some acquaintance with musical intervals.
6We 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 mis-
interpretation of what Pythagoras really said. This mistake was noticed by Vincenzo Galilei (the father of Galileo GalileO. Vincenzo was a most cultivated person, in par- ticular 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. Galilei 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 intomo 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).
~8 THE MATHEMATICAL INTELLIGENCER
Music in the Mathematical Treatise of Theon of Smyrna It is interesting to go through the music theory part of a mathematics treatise of the classical Greek era. I consider here the section on Music (Part 2) of Theon's treatise [12]. This section deals with the definition and the combinations of musical intervals, with proportions, musical units, and so on. It involves non-trivial arithmetic, and Theon, in this section, often refers to the discoveries made by Pythago- ras and the Pythagoreans.
The title of Part 2 of Theon's mathematical treatise is "A book containing the numeric laws of music." In the intro- duction, he says, "Harmony is spread in the world, and of- fers itself to those who seek it only if it is revealed by num- bers." The first part of this sentence, that "Harmony is spread in the world," has been repeated throughout the ages, and it was at the ba- sis of a strong feeling of cosmic structure and or- der. There are important philosophical and esoteric traditions behind this idea, which led eventually to explanations of physical phenomena, like the mo- tion of planets. Famous adepts and advocates of such tra- ditions include, after Pythagoras himself, Plato, Bo~thius, Copernicus, and Kepler (see for instance [8], Book V, where Kepler gives a relation between the eccentricities of the or- bits of the planets and musical intervals). The second part of Theon's sentence, that "harmony is revealed by num- bers," has also been repeated throughout the ages, for in- stance in the citation of Rameau mentioned earlier and in the following citation of Gottfried Wilhelm Leibniz, from his Pr inc ip les o f na ture and o f grace (1712): "Musica est exerc i t ium ar i thmet icae occul tum . . . " (Music is a secret exercise in arithmetic).
Let us look at the treatment of semi tones in Theon's treatise. There are several kinds of semitones used in an- cient Greek music, two of which are the "diatonic semi- tone" and the "chromatic semitone," the values of which are, respectively, 16/15 and 25/24. One could expect that there is a semitone whose value is equal to half of the value of a tone, in the sense that if we concatenate two such semi- tones, we obtain a tone. This is not the case for any of the semitones used by the Pythagoreans, however. Indeed, by the discussion on logarithms above, we know that if the semitone were half of the tone, then its numerical value should have been X/-9~, which is an irrational number. For the Pythagoreans, dealing with irrational numbers would have been incompatible with their philosophy. Theon writes in w of Part 2 that "one can prove that" the tone, the value of which is 9/8, cannot be divided into two equal parts, "because 9 is not divisible by 2." Of course, this is nonsense: the point is not to divide 9 by 2, but to take the square root of 9/8. Although Pythagoras and his school were aware of the existence of irrational numbers, they consid-
For the Pythagoreans, deal- ing with irrational numbers would have been incompati- ble with their philosophy.
ered them unnatural and a threat to their philosophical sys- tem, based on positive integers. The adjective "irrational" which they introduced clearly indicates this. It is also well known that the Pythagoreans wanted to keep the existence of irrational numbers (the discovery of which is attributed to Pythagoras himself) a secret. Jamblichus relates in [7] Chapter XXIX (p. 126) that "he who first divulgated the the- ory of commensurable and incommensurable quantities, to those who were unworthy to receive it, was so hated by the Pythagoreans that they not only expelled him from their common association, and from living with them, but also constructed a tomb for him."
The reasons why ancient Greek music used semitones of 16/15 or 25/24 are certainly related to the fact that these intervals are acceptable by the ear. But it is also a fact that the ancient Greek musicologists liked to deal with su-
perpar t icu lar ratios de-
r ived f r o m 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 num- ber symbolism is involved here, but that subject is
beyond the scope of this paper. The following is a list of "useful" musical intervals, which was known to Gioseffo Zarlino and Descartes:
2/1 octave
3/2 fifth
4/3 fourth
5/4 major third
6/5 minor third
9/8 major tone
10/9 minor tone
16/15 diatonic semitone
25/24 chromatic semitone
81/80 comma of Didymus.
There is a discussion of this list in both [6] and [9]. Many years after this list was known to music theorists, C. Stormer proved that this is a complete list of the superparticular ra- tios derived from the prime numbers 2, 3, and 5 [11].
Scales Scales are building blocks for musical compositions. (This is true at least in tonal music, that is, in almost all pre-twen- tieth-centtu~ European music.) I shall talk in this section about the arithmetic of scales, and I remark by the way that in addition to this arithmetic, there is a more abstract re- lation between scales and mathematics, namely in the con- text of formal languages. Classical musical compositions are based on scales, fragments of which appear within a piece in various forms, constituting a family of privileged sequences of musical motives. This fact has been exploited
VOLUME 24, NUMBER 1, 2002 ~ 9
and systematically generalized in certain twentieth-century compositional techniques (for instance, serial music), which are related to mathematics, but which are beyond the subject matter of this paper.
The major part of post-Renaissance Western European classical music uses a very limited number of scales; in fact, since the general acceptance of the tempered scale in the eighteenth century, there are basically two scales, the ma- jor and the minor scale. The tempered scale (the one we play on a piano keyboard), is based on the division of the octave into 12 equal intervals, the unit being the tempered semitone, the value of which is equal therefore to t 2 ~ . Any two major (respectively, minor) tempered scales are translations of each other on the set of pitches. (In musi- cal terms, these translations are called transpositions.) This was not the case in pre-Renaissance music.
In contrast, the theory of harmony in classical Greece included a complicated and very subtle system of scales. Greek mathematical treatises usually contain a descrip- tion of scales in terms of fractions, with a discussion of 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 ra- tio of a pair of strings having the same section and stretched at the same tension, producing the interval between the first and the nth note. Thus, for instance, the interval be- tween 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 ex- pected, since the Pythagorean scale needs to contain these three conso- nant intervals. The intervals between consecutive notes, except those be- tween 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. The scale of Pythagoras sounds approximately, but not exactly, like our tempered major scale. The semitone which is used in our tempered scale, 12~/~, is closer to the diatonic semitone, i6/15, than to the other two which we encountered.
There is a logic behind the defini-
tion of the scale of Pythagoras. One starts by assigning the values 2, 3/2, and 4/3, respectively, to the eighth, fifth, and fourth notes in the list. The rest of the values are obtained by an iterative process involving fifths whose values are 3/2 (such fifths are called pure fifths). Thus, for example, if we start from the first note (with value 1) and concatenate two pure fifths, we obtain an interval of ninth, with value 3/2 x 3/2 = 9/4, which is greater than 2 (as expected, since this interval is larger than an octave). To come back inside our octave, we divide by two, obtaining the value 9/8. In the same way, the value 27/16 is found as (3/2) 3 divided by 2, and so on. Unfortunately the process gives an infinite num- ber of notes, but it is reasonable to stop after the octave has been divided into these seven intervals.
The scale of Pythagoras has beautiful properties. One is that all fifths and all fourths are pure, their common val- ues being 3/2 and 4/3. For instance, the value of the inter- val between the second and the fifth note is (3/2)/(9/8) = 4/3. This is a remarkable property which does not follow obviously from the construction.
Figure 4. Jean-Philippe Rameau. Portrait by Jean Bernard Restout, titled "The inspired poet."
7 0 THE MATHEMATICAL INTELLIGENCER
Providing a scale with the max imum number of pure in-
tervals was a domain of research of ear ly music theory. In s ix teenth-century Western European music, the intervals
of minor and major third began to be cons idered as con- sonant , and the scale of Pythagoras was less sui table for
new harmonies that involved many o f the new intervals. (The value of a pure major third interval is 5/4, whereas in the scale of Pythagoras the value of the interval be tween the first and the third notes is 81/64, which is a little bi t
grea ter than 5/4). A scale which was useful in tha t r espec t is the one named after Gioseffo Zarlino, a famous s ixteenth-
century Venetian musicologist . Zarl ino 's scale makes a compromise be tween 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.
Some of the fifths in this scale are pure, but not all of them. Fo r instance, the value of the interval be tween the second and the s ix th note is 40/27, which is s tr ict ly 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 imposs ib le to have only pure intervals in a scale, unless the scale is short. Ar i s toxenus (fourth c. sc) made
a sys temat ic theory of scales based on "tetrachords," scales consis t ing o f four notes cor respond ing to different divi- s ions of the fourth by tones and semitones. A long scale
would be ob ta ined by conca tena t ing te t rachords . Let us re turn for a momen t to the scale of Pythagoras.
P rob lems are encounte red as soon as one needs to con-
catenate several such scales, for ins tance in order to play musical ins t ruments whose ranges cover several octaves. Fo r example , one would expec t that the conca tena t ion of 12 fifths gives 7 octaves (as is the case for instr tmlents l ike
the guitar or the harps ichord) . This cannot be the case if one uses the scale of Pythagoras, s ince (3/2) 12 is not equal to (2/1) 7 . The interval with value (3/2) 12 is larger than the
one with value (2/1) 7. The difference is a small (but never- the less audible) interval, (3/2)12/2 7 . This small interval is
cal led a "Pythagorean comma." Similar p rob l ems occur in all the o ther scales based on
pure intervals. Fo r instance, we would expec t that the con-
ca tenat ion of 4 fifths gives an interval of 2 octaves and one major third. If we do the computa t ion in Zarl ino's scale, we find that this is not the case, and the difference is the Didy-
mus c o m m a (81/80). It is wor thwhi le to ment ion here that music theor is ts in
ancient China encounte red s imilar ar i thmet ical p rob l ems
in their t heo ry of scales. It should be c lear now that the definit ion of a scale in-
volves some arbi t rar iness and depends s t rongly on which intervals we insist be pure. One solut ion to the p rob lem was, ins tead of making a res t r ic ted choice, to keep differ-
ent possibi l i t ies . This is one of the reasons why there are so many scales in antique Greek music. In this music, dif- ferent sca les were adap ted to different melodies and dif- ferent types of instruments . The choice of scale for a mu- sical p iece de te rmined much of the charac te r of the p iece
and of its psychologica l effects on the listener. (This is also
re la ted in Jambl ichus [7].) This subtle dependence of the p iece upon the scale las ted in European music until the
adopt ion of the t e m p e r e d scale. For instance, Rameau gives a list of charac ter i s t ics of different tonalities in his Traitd de l'harmonie rdduite d ses principes naturels, Book II, Chapter 24 (Vol. 1 of [10]).
Rameau and the Harmonic Sequence Like Pythagoras 2000 years before him, the composer and
theoret ician Jean-Phil ippe Rameau made a real synthesis be- tween music as an art whose aim is to express and to cre-
ate emotions, and music as a mathemat ica l science with a deductive approach and r igorous rules. Pythagoras estab- lished the important relat ion be tween musical intervals and
pairs of integers, Ramean went a step further and gave a mu- sical content to the whole sequence of posit ive integers.
One of the main ideas for Rameau is tha t the infinite se-
quence of integers is contained, in a beaut i ful way, in na-
ture, as a sequence of frequencies. When a sonorous body (Ramean 's terminology: "corps
sonore') vibrates, it c rea tes a local pe r iod ic var ia t ion of the p ressure of air. This v ibra t ion p ropaga tes as an acoust ic wave. It hits our ea r drums, and we hea r a musical note. The musical note p r o d u c e d by a vibrat ing str ing (bowed or
plucked) , consis ts usua l ly in a superpos i t ion of a funda- menta l tone and overtones. The frequencies of the over- tones, which are cal led the harmonic frequencies, are in- tegral mul t ip les of the f requency of the fundamenta l tone. The sequence of harmonic frequencies is natural ly param- etr ized by the posi t ive integers. For instance, the f requency
of the note C1 (which co r re sponds to the lowest C key on a p iano keyboard) is ( a p p r o x i m a t e l y ) f = 33 Hz (cycles pe r second). The frequencies of the cor respond ing over tones
are therefore
f , 2f, 3f, 4f, 5f, 6 f , . . .
whose values in Hz are
33, 66, 99, 132, 165, 1 9 8 , . . .
The cor responding sequence of notes is
C1, C2, G2, C3, E3, G3, . �9 .
In principle, one can hear the first four or five over tones on an instrument like an organ. (Mersenne, in his Harmonie UniverseUe, says that he can hear the fwst nine overtones.)
Rameau 's theore t ica l work is based on scientif ic dis- cover ies in acous t ics which were made in the seventeenth century, in par t icu la r by the mathemat ic ian Joseph
Sauveur. The p h e n o m e n o n of "harmonics" in music had been no t iced long before Rameau, but Rameau was the one who used it as the basis of a coheren t theore t ica l teaching of music, in par t icu la r in his Traitd de l'harmonie rdduite
ses principes naturels. Rameau ' s t e x t b o o k s on mus ic t heo ry ( abou t 2000
pages ) inc lude the bas i c s of f igured bass , a ccompan i - ment, chords , modula t ion , and c o m p o s i t i o n techniques . All the theo r i e s he deve loped are b a s e d on s imple rules
VOLUME 24, NUMBER 1, 2002 71
38 T R A I T E " DE L ' H A K M O N I E ,
DE'MO N S T R A T I O N .
ay p1o=~v
raifo.s ~e
cellocy, o~ ', ce dernier
.qo. zoS.~ p Mi~ Sol.~
j ~A.oZa zon~a.llleng,~| & L~ Ccpd~me
!Son ~rave de l 'A~ord de la'Quinte-fuFcr tb,e.
Figure 5. A diagram in Rameau's Trait6, discussing ratios of frequencies of a dissonant
chord (containing a minor seventh).
derived from the existence and the properties of the har- monic sequence. For instance, in his analysis of chords, the root of a triad is treated as a unit, in a mathematical sense, and this point of view makes things simple and ev- ident. The theory of triads (consisting of three notes, like C, E, G) had already been derived from the harmonic se- quence by Zarlino and Descartes, but Rameau worked on a complete theory of dissonant chords. The diagram in Figure 5 is one of Rameau's pictures in the Traitd de l 'harmonie rdduite ~ ses pr incipes naturels, in which he represents the dissonant chord La, Do~, Mi, Sol (that is, A, Ct, E, G), with four other derived chords. The num- bers below the notes are the corresponding elements of the harmonic sequence.
Rameau liked to consider the har- 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 Nou- velles rdflexions sur le principe sonore, Rameau argued that since the funda- mental objects of mathematics are de- rived from the sequence of positive in- tegers, and since this sequence is contained in music, then mathematics it- self is part of music. These reflections provoked a dispute between Rameau and eighteenth-century French mathe- maticians, like L. B. Castel and J. d'Alembert, and with the ency- clopaedists, like Denis Diderot, Jean- Jacques Rousseau, and Friedrich von 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 Rameau and d'Alembert; do not fall into the facile conclusion that the interaction between music theorists and mathe- maticians was always friendly. Still the interaction was there.
In this report I have concen- trated on examples, starting with Pythagoras and ending with Rameau. To support the choice of Pythagoras and Rameau, let me conclude by citing Jacques Chailley [4] 7
En 2500 ans d'histoire dcrite, la musique n'a peut-~tre connu que deux v~ritables thdoriciens, dont les autres
n'ont gu~re f a i t qu'amdnager ou rapetasser les proposi- tions. L'un, au Vie si~cle avant notre #re, f u t le fabuleux Pythagore. L'autre mourut ~ Paris en 1764: ce f u t Jean- Philippe Rameau.
In 2500 years of written history, music has perhaps known only two genuine theoreticians, and what the others did was only to repackage or patch up their propositions. The first one, in the VIth century before our era, was the fabu- lous Pythagoras. The other one died in Paris in 1764: this was Jean-Philippe Rameau.
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 Superieur de Musique de Paris and at the University of Paris. I borrowed this quota-
tion from the Introduction to the collected works of Rameau [10].
72 THE MATHEMATICAL INTELLIGENCER
[2] Milton Babbitt, Past and present concepts of the nature and lim-
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 theorie musicale", La Revue Mu-
sicale, Numero special 260, 1964.
[5] Galileo Galilei, Discorsi e dimostrazioni matematiche intorno a due
nuove scienze, in Vol XII of the Complete Works, Societ~ 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-
tion by Thomas Taylor, London, John M. Watkins, 1965.
[8] Johannes Kepler, Harmonices Mundi. (I have used the French
translation with comments by J. Peyroux, Librairie A. Blanchard, 9
rue de Medicis, Paris, 1977.)
[9] A. L. Leigh Silver, Musimatics or the nun's fiddle, Amer. Math.
Monthly 78 (1971), 351-357.
[10] J. Ph. Rameau, Complete Theoretical Writings, edited by R. Ja-
cobi, a facsimile of original editions, published by the American In
stitute of Musicology, 1967.
[11 ] C. Stormer, Sur une inequation indetermin6e, 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 bilin-
gual (Greek-French) edition due to J. Dupuis (Paris 1892) is
reprinted by Culture et Civilisation, 115 Av. Gabriel Lebon, Brus-
sels, 1966.
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