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Mathematicians and Music
On 6 September 1923 Raymond Clare Archibald of Brown University gave his Presidential Address to the Mathematical Association of America on Mathematicians and Music. His talk, which was delivered at Vassar College, Poughkeepsie, to a joint meeting of the Mathematical Association of America and the American Mathematical Society, was published in the American Mathematical Monthly in the January 1924 part. We give below a version of Archibald's talk:-
Mathematicians and Music
by R C Archibald, Brown University.
Mathematics and Music, the most sharply contrasted fields of intellectual activity which one can discover, and yet bound together, supporting one another as if they 'would demonstrate the hidden bond which draws together all activities of our mind, and which also in the revelations of artistic genius leads us to surmise unconscious expressions of a mysteriously active intelligence.In such wise wrote [Helmholtz] one supremely competent to represent both musicians and mathematicians, the author of that monumental work, On the Sensations of Tone as a physiological basis for the Theory of Music.
"Bound together?" Yes! in regularity of vibrations, in relations of tones to one another in melodies and harmonies, in tone-colour, in rhythm, in the many varieties of musical form, in Fourier's series arising in discussion of vibrating strings and development of arbitrary functions, and in modern discussions of acoustics.
This suggests that the famous affirmation of Leibniz, "Music is a hidden exercise in arithmetic, of a mind unconscious of dealing with numbers," must be far from true if taken literally. But, in a very general conception of art and science, its verity may well be granted; for, in creating as in listening to music, there is no realization possible except by immediate and spontaneous appreciation of a multitude of relations of sound.
Other modes of expression and points of view were suggested by that great enthusiast to whom America owed much [Sylvester], him who called himself "the Mathematical Adam" because of the many mathematical terms he invented; for example, mathematic - to denote the science itself in the same way as we speak of logic, rhetoric or music, while the ordinary form is reserved for the applications of the science. He referred to the cultures of mathematics and music "not merely as having arithmetic for their common parent but as similar in their habits and affections." "May not Music be described," he wrote, "as the Mathematic of Sense, Mathematic as the Music of reason? the soul of each the same! Thus the musician feels Mathematic, the mathematician thinks Music, - Music the dream, Mathematic the working life, - each to receive its consummation from the other when the human intelligence, elevated to the perfect type, shall shine forth glorified in some future Mozart-Dirichlet, or Beethoven-Gauss - a union already not indistinctly foreshadowed in the genius and labours of a Helmholtz"!
But such intimacies in these cultures are not discoveries and imaginings of a later day. For two thousand years music was regarded as a mathematical science. Even in more recent times the mathematical dictionaries of Ozanam, Savérien, and Hutton, contain long articles on music and considerable space is devoted to the subject in Montucla's revised history, - which brings us to the threshold of the nineteenth century. It is, therefore, not surprising that many mathematicians wrote on musical matters. I shall presently consider these at some length. But certain other facts may first be reviewed.
The manner in which music, as an art, has played a part in the lives of some mathematicians is recorded in widely scattered sources. A few instances are as follows.
Maupertuis was a player on the flageolet and German guitar and won applause in the concert room for performance on the former. At different times William Herschel served as violinist, hautboyist, organist, conductor, and composer (one of his symphonies was published) before he gave himself up wholly to astronomy. Jacobi had a thorough appreciation of music. Grassmann was a piano player and composer, some of his three-part arrangements of Pomeranian folk-songs having been published; he was also a good singer and conducted a men's chorus for many years. János Bolyai's gifts as a violinist were exceptional and he is known to have been victorious in 13 consecutive duels where, in accordance with his stipulation, he had been allowed to play a violin solo after every two duels. As a flute player De Morgan excelled. The late G B Mathews knew music as thoroughly as most professional musicians; his copies of Gauss and Bach were placed together on the same shelf. It was with good music that Poincaré best liked to occupy his periods of leisure. The famous concerts of chamber music held at the home of Emile Lemoine during half a century exerted a great influence on the musical life of Paris. And in America we have only to recall colleagues in the mathematics departments of the Universities of California, Chicago and Iowa, and of Cornell University, who are, to use Shakespeare's phrase, "cunning in music and mathematics."
While Friedrich T Schubert, the Russian astronomer and mathematician, played the piano, flute, and violin in an equally masterly fashion, his great-grand-daughter Sophie Kovalevskaya was devoid of musical talent; but she is said to have expressed her willingness to part with her talent for mathematics could she thereby become able to sing. Abel had no interest in music as such, but only for the mathematical problems it suggested. His close attention to a performer at a piano was once explained by the fact that he sought to find a relation between the number of times that each key was struck by each finger of the player. Lagrange welcomed music at a reception because he could by the fourth measure become oblivious to his surroundings and thus work out mathematical problems; for him the most beautiful musical work was that to which he owed the happiest mathematical inspirations. Dirichlet seemed to be sensible to the charms of music in a similar manner.
Such are a few instances, which could be considerably multiplied, of the relation of mathematicians to the art of music:-
that gentlier on the spirit liesThey suggest the accuracy of at least a part of the following observations of Möbius in his book on mathematical abilities:-
Than tir'd eyelids upon tir'd eyes.
Musical mathematicians are frequent ... but there are wholly unmusical mathematicians and many more musicians without any mathematical capability.That there are musicians with some mathematical ability will be granted when we recall, not only that Henderson, the prominent New York music critic and the author of many works on musical topics, has written a little book on navigation, but also that the late Sergei Tanaieff, pupil of Rubenstein and Tchaikovsky, successor of the latter as professor of composition and instrumentation at the Moscow Conservatory of Music, and one of the most prominent of modern Russian composers, found algebraic symbolism and formulae of fundamental importance in his lectures and work on counterpoint.
A question which has interested more than one group of inquirers is: Can one establish any relationship between mathematical and musical abilities? Within the past year two Jena professors, Haecker and Ziehen, published the results of an elaborate inquiry as to the inheritance of musical abilities in musical families. As a by-product of the inquiry they arrived at the result that in only about 2 per cent of the cases considered was there any appreciable correlation between talent for music and talent for mathematics; they found also that the percentage of males lacking in talent for music but showing a talent for mathematics was comparatively high, about 13 per cent. At the Eugenics Record Office of Cold Spring Harbor, Long Island, there has been collected a considerable body of data upon which a study of the correlation of mathematical and musical abilities could be based. It will be interesting to see if the conclusions of Haecker and Ziehen are here checked, and also if some results are found as to the extent to which musical abilities are present in a group of mathematicians.
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:Thus corresponding to the successively higher notes c, f, g and c we have the numbers , and for the relative lengths of the strings corresponding to the different notes.
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.
The fourth and fifth gave the means of fixing a much smaller interval, called a tone, corresponding to which is the number . 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 . The numbers corresponding to successive pairs of notes would be , the being that number by which it is necessary to multiply into in order to give .
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 ; to the d an octave lower corresponds .
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 . To the succession b, c, d, e, f, g, a, b corresponds .
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 , or nearly 80 : 81.0915. This same ratio is got from . 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 instead of the Pythagorean . This substitution of , even though not mentioned by Euclid, is not modern, but was already suggested in the late Pythagorean school. The second substitution of 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 , while the intervals between successive pairs of notes became .
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 instead of the 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:
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
.
Ptolemy derived many scales in which the relations were similar; for example.
.
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 into a product of three ratios of the form .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?In the eighteenth century when calculus had become a tool, there was a notable series of theoretical discussions of vibrating strings. But before considering these I wish to draw special attention to the first English scientific treatment of harmony, a work of high order, by Robert Smith. It was entitled Harmonics or the Philosophy of Musical Sounds and was first published in 1749. The theory of intervals and various systems of temperament are discussed in a manner very attractive even for a reader in the present day. Smith held the Plumian chair at Cambridge, the one of which A S Eddington is the present incumbent, and his work on harmonics contained the substance of lectures he had delivered for many years. It was he who was the author of the notable work on Optics which has been translated into several languages. He was also the founder of the well-known Smith's Prizes:-
... annually awarded to those candidates who present the essays of greatest merit on any subject in mathematics or natural philosophy.First in the series of theoretical discussions to which I have referred are those of Brook Taylor, who, according to his biographer, "possessed considerable ability as a musician and an artist." His discussions appeared in the Philosophical Transactions for 1713 and 1715 and in his book Methodus Incrementa Directa et Inversa, the first treatise dealing with finite differences, and the one which contains the celebrated theorem regarding expansions, now connected with Taylor's name. He solved the following problem which he believed to be entirely new:-
To find the number of vibrations that a string will make in a certain time having given its length, its weight, and the weight that stretches it.In discussing the form of the vibrating string, his suppositions regarding initial conditions, including that it vibrated only as a whole, led to a differential equation whose integral gave a sine curve. Thus started a discussion that was to culminate a century later in the work of Fourier.
I have already referred to the discovery of upper partial tones by Rameau and how he made this the basis of a system of harmony; his first work on this subject was published in 1726, but the first mathematician who seemed to take account of the fact was Daniel Bernoulli in a memoir of 1741-43 though not published till 1751. About this time D'Alembert's thorough acquaintance with Rameau's theories was shown by his publication in 1752 of a volume entitled "Elements of theoretical and practical music according to the principles of Monsieur Rameau, clarified, developed, and simplified." Of this work six French editions and one in German were published. Helmholtz remarks that D'Alembert's book:-
... is an extremely clear and masterly performance, such as was to be expected from a sharp and exact thinker, who was at the same time one of the greatest physicists and mathematicians of his time. Rameau and D'Alembert lay down two facts as the foundation of their system. The first is that every resonant body audibly produces at the same time as the prime its twelfth and next higher third as upper partials. The second is that the resemblance between any tone and its octave is generally apparent. The first fact is used to show that the major chord is the most natural of all chords, and the second to establish the possibility of lowering the fifth and the third by one or two octaves without altering the nature of the chord, and hence to obtain the major triad in all its different inversions and positions.D'Alembert wrote also a long essay on the liberty of music and articles of musical interest in the great Encyclopédie Methodique. But from a mathematical point of view, his memoir of 1747 dealing with Taylor's problem of the vibrating string, and taking account of matters previously overlooked, is very notable. He was led to the differential equation (with , a constant, equal to unity)
,
where the origin of coordinates was at the end of the chord whose length is , the axis of x in the direction of the chord, and y the displacement at any time . Of this equation he found the solution
,
where represents any function such that . He then found certain equations for determining the functions satisfying this relation of periodicity.
Euler immediately raised the question of the generality of the solution and set forth his interpretation. D'Alembert had supposed the initial form of the string to be given by a single analytical expression, while Euler regarded it as lying along any arbitrary continuous curve, different parts of which might be given by different analytical expressions. Lagrange joined in the discussion, to which Daniel Bernoulli contributed chiefly from physical rather than mathematical considerations. He started with Taylor's particular solution and found, in effect, that the function for determining the position of the string after starting from rest could naturally be expressed in a form later called a Fourier series. Thus were such series first introduced into mathematical physics. Bernoulli remarked that since his solution was perfectly general it should include those of Euler and D'Alembert. In this way mathematicians were led to consideration of the famous problem of expanding an arbitrary function as a trigonometric series. No mathematician would admit even the possibility of its solution till this was thoroughly demonstrated, in connection with certain problems in the flow of heat, by Fourier who gives due credit to the suggestiveness of the work of those in the previous century to whom I have referred. Fourier's results were contained in a memoir crowned by the French Academy in 1812 but not printed till more than a decade later. It is sometimes asserted that the first mathematical proof of Fourier's results, with the limits of arbitrariness of the function carefully stated, was given by Dirichlet in his classic memoirs of 1829 and 1837. So far as the limits of arbitrariness are concerned this is correct; but that Fourier rigorously established his expansion of an arbitrary function seems to admit of no denial or qualification.
One of Euler's most notable papers connected with the history of Fourier's series did not appear in print till 1793, ten years after his death. Thus for eighty years, from Taylor to Euler and Lagrange, mathematicians were occupied with the problem of the vibrating string and allied problems including the vibration of a column of air and of an elastic rod. Then thirty years of silence and the great advance by Fourier.
I have indicated only a few bald facts since details in this regard are readily available elsewhere.
Although more than twenty years Fourier's senior, Gaspard Monge, so well known as an expounder of the applications of analysis to geometry, and of descriptive geometry, was associated with him in more than one undertaking. They were professors at the École Polytechnique in Paris, which Monge was largely instrumental in founding. They both accompanied Napoleon to Egypt where Monge was the first president of the Institute of Egypt and Fourier its secretary. Monge was a passionate devotee of music and made a journey to Italy in order to procure copies of all of the musical works in the chapel of St Mark's, Venice. He was also an ardent republican and, according to Arago, an enthusiast for the "Marseillaise" which he sang every day at the top of his voice before seating himself at the table. He, too, occupied himself with the problem of the vibrating string and constructed a model of a surface, certain parallel sections of which give the form of the curve of the vibrating string at any time under conditions which Monge states. This model which was made in 1794 is still preserved in the École Polytechnique.
And finally in connection with great mathematicians of the eighteenth century, the extent of Euler's contributions to the theory of vibrating bodies, acoustics, and music, may be indicated somewhat further. About 30 of his published memoirs, and a treatise, Tentamen novae theoriae musicae, not to speak of letters in his Letters to a German Princess, deal with such subjects. They appeared during about 60 years 4 from the first, a dissertation on sound, published in 1727, when he was 20 years old. Among the topics of memoirs not already referred to are: On the sound of bells, Conjectures as to the reason of some dissonances generally accepted in music, The true character of modern music, and On the vibratory motion of drums. It is in this last mentioned memoir of 1766 that the general so-called Bessel's functions of integral order first occur.
Euler's treatise on music was first published in 1739, but we learn from a letter Euler wrote to Daniel Bernoulli in May, 1731, that he had already almost completed the manuscript of the work. This letter describing the ideals of the work in some detail, as well as Bernoulli's reply in the following August, are readily accessible. I shall therefore make but brief extracts from the early parts of the letters. Euler explains:-
My main purpose was that I should study music as a part of mathematics and deduce in an orderly manner, from correct principles, everything which can make a fitting together and mingling of tones pleasing. In the whole discussion I have necessarily had a metaphysical basis, wherein the cause is contained why a piece of music can give one pleasure and the basis for it is to be located, and why a thing to us pleasing is to another displeasing.To this Bernoulli replied:-
I cannot readily divine wherein that principle should exist, however metaphysical it may be, whereby the reason could be given why one could take pleasure in a piece of music, and why a thing pleasant for us, may for another be unpleasant. One has indeed a general idea of harmony that it is charming if it is well arranged and the consonances are well managed; but, as it is well known, dissonances in music also have their use since by means of them the charm of the immediately following consonances is brought out the better, according to the common saying opposita juxta se posita magis elucescunt [opposites placed together shine brighter]; also in the art of painting, shadows must be relieved by light.Euler's treatise does not seem to have met with unqualified favour. Brewster reports Fuss to have said:-
... it had no great success as it contained too much geometry for musicians, and too much music for geometers.Helmholtz gives a good deal of space to setting forth the psychological considerations which Euler explains had influenced him to found his relations of consonances to whole numbers.
But here we must leave this "myriad-minded" eighteenth century genius.
And now there is time for but the briefest references to mathematics and music during the past one hundred years - the century in which niceties of mathematical calculation were surely contributory to the improvement of such instruments as the flute and organ, to the wonders of phonograph-record manufacture, of broadcasted concerts, and of sound-wave photography - the century in which Helmholtz and Rayleigh lived and worked.
Helmholtz's epoch-making work, Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik, first appeared in 1862. The great feature of this work is the formulation and proof of the laws by which the ear bears musical sounds from one or more distinct sources; how the theory of combined musical sounds is reduced to the theory of combined simple sounds. The starting point of these discoveries was the fact, recognized by Rameau just two hundred years ago, that upper partials were associated with fundamental tones. From these laws we learn the nature of consonance and dissonance, knowledge so necessary for building up a system of harmony; we learn the principles which determined those degrees of musical sound selected by various nations at various times; we understand the reasons for the simple ratios of the lengths of strings producing consonant tones and the limitation of the numbers of these ratios; and we appreciate the value of temperaments for different instruments.
In his Tonempfindung Helmholtz relegated to appendices the purely mathematical discussions. For example, the third appendix is On the motion of plucked strings; the fifth is On the vibrational forms of pianoforte strings; the sixth is an Analysis of the motion of violin strings; and the seventh is On the theory of pipes. He goes into such matters more extensively in the volume of his lectures on the mathematical, principles of acoustics.
Such subjects are also treated in masterly fashion by Rayleigh in his Theory of Sound and in his papers. Among other works will be mentioned only the mathematical elements of music, as presented some twenty-five years earlier by Airy, senior wrangler and astronomer.
In such works, in the comparatively recent notable paper in this country by Harvey Davis, on vibrations of a rubbed string, and, of course, in other mathematical treatments of similar material, Fourier series must enter in a fundamental manner. With specified conditions the series and its coefficients for a given tone or combination of tones may be determined. Or, if we have a graph of the vibrations corresponding to such tones, the series may also be calculated, various terms in the series corresponding to simple elements compounded in the tone or tones.
During the past twenty years photography has contributed in a remarkable manner to the analysis of musical sounds. In England, from 1905 to 1912, E H Barton and his associates published a series of papers illustrated by photographs of vibration curves particularly as issuing from the violin strings, bridge, and belly.
In India, five years ago, R C V Raman published an extensive bulletin On the mechanical theory of the vibration of bowed strings and of musical instruments of the violin family, with experimental verification of the results. It is illustrated by reproductions of many photographs, those of the wolf-notes, so well known to stringed-instrument players, having especial interest. The more recent publications of S Garten and F Kleinknecht contain a discussion of tones produced by the voice. And with us the work that D C Miller, of the Case School of Applied Science, has done in this connection is known to many, not only through his volume on The Science of Musical Sounds, but also through his remarkably interesting public lectures where his extraordinary instrument called the phonodeik, which photographically records sound waves, may also be used for projecting traces of the waves, as generated, on the screen of a lecture platform.
For the mathematician a great advantage of a photograph is that he can, after much labour, from it calculate the corresponding Fourier series. But in the laboratory, work of this kind is often saved by the employment of a machine called the harmonic analyzer. The first instrument of this kind was made by Lord Kelvin in 1878; two were put forth by Henrici in 1894, and among others is that of Michelson and Stratton, constructed in 1898. By means of a Henrici machine, when the stylus of the instrument is moved along the curve of the photograph the numerical values of the coefficients in the corresponding Fourier series may be read off. In 1910 Miller reconstructed a Henrici analyzer so as to care for thirty components with precision. That is, a tone made up of 30 simple tones can be analyzed and the coefficients of the corresponding number of terms in the Fourier series written down. Regarding this kind of work I must not pause to do more than suggest that it has applications of high importance for tone generation and for perfecting musical instruments.
In concluding references to activities of the past one hundred years, I should, however, take time to recall that when, in these latest days, there arose a question as to the manner in which our present musical notation for equal temperament scales could best be simplified, it was a former president of this Association who brought forward a scheme so beautifully simple that further advance in this regard cannot be imagined.
Speculation as to music of the future furnishes tempting themes for discussion. I shall merely mention some of these in conclusion.
The possibilities of melody and harmony in the trinity of musical fundamentals have, within the limits of our hampering scale systems, been largely explored. But what is to be the future of the almost untried vast rhythmic possibilities so intimately bound up with mathematical relations? Practically all of our music is modulo 2, 3, 4, 6, 8, 9, 12; but why not have modulo 5, 7, 10, 11, 13, for example, or combinations of these moduli in the same measures?
Again, is it not within the realms of possibility that some day the inadequacies of the present vehicle of musical expression may lead us to revive some of the ideals of Greek music during the golden period of Aristoxenus?
And yet again, when we recall the many results in connection with musical tones found empirically by makers of musical instruments but for which no satisfactory explanations have been furnished by the mathematician or physicist, may we not conclude that when such explanations are forthcoming, a new era shall have dawned in the evolution of musical instruments?
Last Updated August 2006