Mathematicians and Music 3

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

The previous part of Archibald's talk is here

Mathematicians and Music

by R C Archibald, Brown University.


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, a constant, equal to unity)
d2y/dt2 = a2 d2y/dx2 ,
where the origin of coordinates was at the end of the chord whose length is l, the axis of x in the direction of the chord, and y the displacement at any time t. Of this equation he found the solution
y = f(at + x) - f(at - x),
where f represents any function such that f(z) = f(z + 2l). 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?

JOC/EFR August 2006

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