Mathematicians and Music 2.1

We give below a version of the beginning of the second section of Raymond Clare Archibald's Presidential Address to the Mathematical Association of America on Mathematicians and Music.

The first part of Archibald's talk is here

Mathematicians and Music

by R C Archibald, Brown University.

2.1

Turning now to the theory of music, it is natural to inquire: What are the relations of mathematics to music? What have mathematicians written about music or its theory? Even on the part of one fully informed and competent, to answer these questions with any degree of completeness would require not one hour only, but many hours. I shall therefore limit myself to brief statements, with references to only a score or so of the better-known mathematicians.

In any consideration of the history of music and its relation to mathematics it is important to have in mind the general character of music of different periods. With Helmholtz these may be stated as follows:

(a) The Homophonic or Unison Music of the ancients, including the music of the Christian era up to the eleventh century, to which also belongs the existing music of Oriental and Asiatic nations.

(b) Polyphonic Music of the middle ages, with several parts, but without regard to independent musical significance of the harmonies, extending from the tenth to the seventeenth century, when it passes into

(c) Harmonic or Modern Music, characterized by the independent significance attributed to the harmonies as such.

Our first consideration is therefore to be given to the homophonic music of the Greeks: for in music as in mathematics the period of real development began in the sixth century B.C. with Pythagoras. Before his time, tones an octave or a fifth apart, above and below, were regarded as consonant and as the basis of ordinary needs in declamation. If the c be taken as a point of departure, its fifth is g, and its fifth below is f. If this last note f be raised an octave so as to bring it nearer to the other notes, and if the octave of c be also added, the following four notes are obtained: c, f, g, c. Tradition affirms that these four notes constituted the range of the lyre of Orpheus. As Blaserna remarks:-

Musically speaking it is certainly poor, but the observation is interesting that it contains the most important musical intervals of declamation. In fact, when an interrogation is made, the voice rises a fourth. To emphasize a word, it rises another tone and goes to the fifth. In ending a story, it falls a fifth, etc. Thus it may be understood that Orpheus' lyre, notwithstanding its poverty, was well suited to a sort of musical declamation.
The notable contribution of Pythagoras was his enunciation of the law governing such sounds that are found in all the musical scales known:-
He proclaimed the remarkable fact, of which the proof existed in his famous experiments with stretched strings of different lengths, that the ratios of the intervals perceived as consonant could all be expressed by the numbers 1, 2, 3, 4. His method of demonstration was afterward improved and rendered more exact by the invention of the monochord, and his law may now be stated as follows:

If a string be divided into two parts by a bridge, in such a manner as to give two consonant sounds when struck, the lengths of those parts will be in the ratio of two of the first four positive integers. If the bridge be so placed that two thirds of the string lie to the right and one third to the left, so that the two lengths are in the ratio of 1 : 2, they produce the interval of the octave, the greater length being given to the deeper note. If the bridge be so placed that three fifths of the string lie to the right and two fifths to the left, the ratio of the two lengths is 2 : 3 and the interval produced is the fifth. If the bridge be again shifted to a position which gives four sevenths on the right and three sevenths on the left, the ratio is 3 : 4 and the interval is the fourth.

Thus corresponding to the successively higher notes c, f, g and c we have the numbers 1 , 3/4 , 2/3 , and 1/2 for the relative lengths of the strings corresponding to the different notes.

The fourth and fifth gave the means of fixing a much smaller interval, called a tone, corresponding to which is the number 8/9 = 2/3 ÷ 3/4. Starting with a fundamental c and inserting two tones between it and its fourth, two more between its fifth and its octave, the corresponding numbers for the succession c d e f g a b c would be 1 , 8/9 , 64/81 , 3/4 , 2/3 , 16/27 , 128/243 , 1/2 . The numbers corresponding to successive pairs of notes would be 8/9 , 8/9 , 243/256 , 8/9 , 8/9 , 8/9 , 243/256 , the 243/256 being that number by which it is necessary to multiply into 8/9 × 8/9 in order to give 3/4 .

Pythagoras looked upon the diatonic scale to which we have just referred in quite a different manner, namely, as derived from a succession of fifths. Thus starting from a prime c we have

c g d a e b.
Reducing d an octave, a an octave, e two octaves, and b two octaves, we have the series
c d e g a b.
To obtain the f missing in this series and to fill up the wide interval between e and g it appears that c as a fifth below the prime was raised an octave. It may be readily verified that we are thus led to the same results as before; for example, d, the second fifth above the prime, is given by 2/3 × 2/3 ; to the d an octave lower corresponds 2 × 2/3 × 2/3 = 8/9 .

Pythagoras proposed to find in the order of the universe, where whole numbers and simple ratios prevail, an answer to the question: Why is consonance (the beautiful in sound) determined by the ratio of small whole numbers? The correct numerical ratios existing between the seven tones of the diatonic scale corresponded, according to Pythagoras, to the sun, moon and five planets, and the distances of the celestial bodies from the central fire, etc.

It was the elaboration of these figments of philosophy, and because the fifth as the central tone of the octave corresponded to the astronomical order in which the Samian sage ranged the sun and planets, that he laid such a deep stress upon the c scale obtained from fifths only.
Pythagoras limited himself to the insertion of seven notes within the octave. But from the primal scale he evolved six others. This was done not by setting up a new succession of fifths on the several notes of the primal scale but by making the second note of his first scale the prime of his second and so for each of five remaining notes. In this way, for example, we get the scale d, e, f, g, a, b, c, d with the corresponding numbers 1 , 8/9 , 27/32 , 3/4 , 2/3 , 16/27 , 9/16 , 1/2 . To the succession b, c, d, e, f, g, a, b corresponds 1 , 243/256 , 27/32 , 3/4 , 729/1024 , 81/128 , 9/16 , 1/2 .

It is not apparent in this latter scale that the method of Pythagoras can be said to illustrate the principle that the beautiful in sound must depend upon a succession of notes related to each other and a prime, by the simplest possible ratios.

The most noted of all the musical theorists of antiquity was Aristoxenus of Tarantum, a contemporary and pupil of Aristotle. To him as author have been assigned no less than 453 works but of these none now remain except the Harmonics, portions of a treatise on rhythm, and some fragments recently found in Egypt. According to Macran, his great service was rendered:-

... firstly, in the accurate determination of the scope of musical science, lest on the one hand it should degenerate into empiricism, or on the other hand lose itself in mathematical physics; and secondly, in the application to all questions and problems of music of a deeper and truer conception of the ultimate nature of music itself.
Of two treatises on music attributed to Euclid, only the Theory of Intervals or Section of the Canon, as it is sometimes called, may be regarded as genuine. It is based on the Pythagorean theory of music:-
... is mathematical, and clearly and well written, the style and form of the propositions agreeing well with what we find in the Elements.
The way in which the work starts out seems somewhat remarkable when we remember that it was written about three hundred years before Christ. It commences as follows:-
If all things were at rest, and nothing moved, there must be perfect silence in the world; in such a state of absolute quiescence nothing could be heard. For motion and percussion must precede sound; so that as the immediate cause of sound is some percussion, the immediate cause of all percussion must be motion. And whereas of vibratory impulses or motions causing a percussion on the ear, some there be returning with a greater quickness which consequently have a greater number of vibrations in a given time, whilst others are repeated slowly and of consequence are fewer in an assigned time, the quick returns and greater number of such impulses produce the higher sounds, whilst the slower which have fewer courses and returns, produce the lower. Hence it follows, that if sounds are too high they may be rendered lower by a diminution of the number of such impulses in a given time, and that sounds which are too low, by adding to the number of their impulses in a given time, may be made as high as we choose. The notes of music may be said then to consist of parts, inasmuch as they are capable of being rendered precisely and exactly tuneable, either by increasing or diminishing the number of the vibratory motions which excite them. But all things which consist of numerical parts when compared together, are subject to the ratios of numbers, so that musical sounds or notes compared together, must consequently be in some numerical ratio to each other.
Nearly two thousand years passed before Galileo went one step further, and proved that the lengths of strings of the same size and tension were in the inverse ratios of the numbers of the vibrations of the tones they produced. It was not for another seventy years that the actual number of vibrations corresponding to a given tone was determined; but we shall return to this a little later.

Euclid's work contains 19 theorems. They are mostly concerned with results that may be obtained by the division of a monochord, or string to be experimented upon, which Euclid calls Proslambanomenos. Let this be named A.

This string A was first divided into four parts; three parts were taken and the perfect fourth established with the ratio 3 : 4; two parts were taken and the sound of the octave established; one part was taken and the sound of the double octave A was given.

The next experiment was to divide the length that produced the fourth of the prime into two equal parts, when the sound, the octave of the fourth, was established.

Proslambanomenos was then divided into two equal parts, and one of these being again divided into three parts, two parts were taken and the octave of the fifth was established.

And so till all the tones in two octaves were determined. By beginning with different letters in the series thus determined, Euclid got the seven Pythagorean scales covering two octaves instead of one. Euclid arrived at these sounds by the division of the monochord instead of by successions of fifths employed by Pythagoras.

Two of Euclid's theorems prove that an octave is less than six tones, the ratio of the interval being (8/9)6 ÷ (1/2) = 524288/531441, or nearly 80 : 81.0915. This same ratio is got from (2/3)12 ÷ (1/2)7. In other words it is the ratio determined by the difference of tones derived by counting 12 fifths and 7 octaves from a fundamental. This interval, between notes theoretically the same, was noted by Pythagoras and is called a Pythagorean comma.

The scales of Pythagoras and Euclid differ in two important respects from our major scales, namely, in the ratios for the intervals of a third and a sixth. In the scale of c, the interval of a major third from the tonic is now 4/5 = 64/80 instead of the Pythagorean 64/81 = (8/9)(8/9). This substitution of 4/5, even though not mentioned by Euclid, is not modern, but was already suggested in the late Pythagorean school. The second substitution of 3/5 for the major sixth interval from the tonic naturally followed from this, since it is the octave of the fifth below the third. In this way the ratios of the intervals of the major scale became 1 , 8/9 , 4/5 , 3/4 , 2/3 , 3/5 , 8/15 , 1/2 , while the intervals between successive pairs of notes became 8/9 , 9/10 , 15/16 , 8/9 , 9/10 , 8/9 , 15/16 .

In such a scale if we tune up four perfect fifths on the one hand and two octaves and a major third on the other, we ought to arrive at the same note. The resulting comma here is 80/81 instead of the 80/81.0915 already referred to. It is the distribution of this comma that is ordinarily carried through in our equal-tempered scale. This temperament is said to have been proposed by Aristoxenus.

And last among the Greeks to whom we shall refer is the celebrated mathematician, astronomer and geographer, Claudius Ptolemy, who flourished in the second century of the Christian era. Apart from the Almagest, works on optics and mechanics, a book on stereographic projection, a book in which he tried to show that the possible number of dimensions is limited to three, and other works, Ptolemy wrote a remarkable treatise on music. In it he discusses critically the earlier Pythagorean and Aristoxenean modes and tonalities, and presents new developments. But the restrictions made in connection with the music seem to indicate the beginning of a decline.

Some interesting suggestions have been made by Paul Tannery as to the possible role of Greek music in the development of pure mathematics. One of these is to the effect that the idea of logarithms may have been suggested by such mathematical relations as the following going back to Pythagoras:

1/2 = 2/3 × 3/4    1/2 = (3/4)2 × 8/9
being immediately interpreted in music by: The octave is composed of a fifth and a fourth; the octave is composed of two fourths and of a major tone. Thus mathematical multiplication is changed into musical addition.

Another of Paul Tannery's suggestions involves finding solutions of a Diophantine equation in three variables. In the first four notes of the major scale we had the relation

8/9 × 9/10 × 15/16 = 3/4 .
Ptolemy derived many scales in which the relations were similar; for example.
7/8 × 9/10 × 20/21 = 8/9 × 7/8 × 27/28 = 9/10 × 1O/11 × 11/12 = 3/4 .
In other words the question of the composition of the tetrachord reduces to the following mathematical problem:-
Determine all possible ways of decomposing the ratio 3/4 into a product of three ratios of the form n/(n+1).
From these results, those were finally selected which seemed practicable after trial with the monochord.

In my brief sketch of the work done by the Greeks, I have not intended to give you any idea of their music, but merely to select a few illustrations of the manner in which their music is connected with mathematics. On the varieties of their scales and their colouring through chromatics (as the name implies) and quarter-tones, I have not touched. Nor have I commented on the great beauties of the music even though it was homophonic. Authorities agree with the following summing up of Helmholtz:-

Of course where delicacy in any artistic observations made with the senses come into consideration, moderns must look upon the Greeks in general as unsurpassed masters. And in this particular case they had very good reason and abundance of opportunity for cultivating their ears better than ours. From youth upwards we are accustomed to accommodate our ears to the inaccuracies of equal temperament, and the whole of the former variety of tonal modes, with their different expression, has reduced itself to such an easily apprehended difference as that between major and minor. But the varied gradations of expression, which moderns attain by harmony and modulation, had to be effected by the Greeks and other nations that used homophonic music by a more delicate and varied gradation of tonal modes. Can we be surprised, then, if their ear became much more finely cultivated for differences of this kind than it is possible for ours to be?
The next part of Archibald's talk is here


JOC/EFR August 2006

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