This article is concerned with the history of vibration theory as it relates to music. For further information see Acoustics and Sound.
2. From Huygens to Sauveur and Newton.
SIGALIA DOSTROVSKY/MURRAY CAMPBELL (1–2), JAMES F. BELL, C. TRUESDELL/MURRAY CAMPBELL (3–6)
The basic ideas of the physics of music were first obtained in the 17th century. Acoustic science then consisted mainly in the study of musical sounds; in fact, music provided both questions and techniques for the study of vibration. Music gave experience in comparing the pitch and timbre of tones, and so the means for careful experiment on sound; musical instruments offered empirical information on the nature of vibration; and, rather remarkably, the Pythagorean ratios of traditional music theory provided frequency ratios.
Early in the 17th century it was realized that the sensation of pitch is appropriately quantified by vibrational frequency – that is, pitch ‘corresponds’ to frequency. This realization came as part of a preliminary understanding of consonance and dissonance. Once the correspondence had been made, it was possible to determine the relative vibrational frequencies of tones from the musical intervals they produced. When relative frequencies were known, there was the challenge of determining frequencies absolutely; and the first measurements were made during the century. The idea that pitch corresponds to frequency motivated efforts to understand overtones, since, during most of the 17th century, it seemed paradoxical that a single object could vibrate simultaneously at different frequencies. This paradox was resolved by the end of the century through an initial understanding of the ‘principle of superposition’. Also by this time the connection between overtones and timbre was noticed, and beats were explained quantitatively. During most of the century, sound was described as a succession of pulses, its wave nature being understood qualitatively. But late in the century the first mathematical analysis of the propagation of sound waves was made.
At least since the time of the Pythagoreans, musical intervals had been characterized by the length ratios of similar and equally tense strings – 2:1 for the octave, and so on. In music theory these ratios, although based on string lengths, were usually understood in purely arithmetical terms. As Palisca (1961) has shown, scepticism about arithmetical dogmatism in music led to an interest in the physical determinants of pitch. Around 1590 Vincenzo Galilei showed that various ratios could be associated with an interval. For example, if the strings’ tensions rather than their lengths were considered, the ratio for the octave would be 1:4 rather than 2:1. Also, Francis Bacon (1627) was dissatisfied with arithmetical analysis of musical sound and recommended that empirical information be obtained from instrument makers. Even Kepler (1619), who was inspired by the Pythagorean idea of a celestial harmony, criticized the Pythagoreans for overemphasizing arithmetical considerations when judging the consonance of musical intervals. But the validity of the traditional ratios remained apparent, so the emphasis shifted and there was interest in finding physical (rather than numerological) reasons for using them. Descartes (1650; written in 1618) suggested that the octave was the first consonance because it was the interval obtained most easily by overblowing flutes.
The ambiguity originally demonstrated by Vincenzo Galilei was resolved most clearly by his son Galileo (1564–1642). By considering the sound that reaches the ear rather than the vibrating object that produces it, Galileo came to realize that pitch corresponds to frequency and showed that musical intervals could be uniquely characterized by frequency ratios. To explain why some intervals are more consonant than others, one of the ‘musical problems’ that he proposed to solve, Galileo wrote, in 1638, that ‘the length of strings is not the direct and immediate reason behind the forms [ratios] of musical intervals, nor is their tension, nor their thickness, but rather, the ratios of the numbers of vibrations and impacts of air waves that go to strike our eardrum’ (see Drake, 1974). He asserted that the degree of consonance produced by a pair of tones is determined by the proportion of impacts from the higher tone that coincide with impacts from the lower. (Benedetti had realized by 1563 that pitch corresponded to frequency, but his idea had not become known. In the early 17th century Beeckman, Descartes and Mersenne made statements about this correspondence, but Galileo’s presentation was the clearest.)
Mersenne, who specialized in the physics of music, experimented extensively to relate vibrational frequencies to other properties of sources of sound. According to Mersenne’s Law, the (fundamental) frequency of a string of given material is proportional to √T/l√m, where l is the length, T the tension and m the mass per unit length. In his experiments to establish this, which he described (1636–7), Mersenne tuned pairs of strings that differed in one or two properties. The musical interval indicated the strings’ relative frequencies, and the dependence of frequency on each variable was found separately. To a certain extent the law was known by others; for example, the explicit statement that the frequency is inversely proportional to the square root of the density is due to Galileo. Mersenne tried to find for the vibrating air column a relation similar to the one he had found for the string. Although there is no simple precise relation between the frequency of a pipe and its dimensions, Mersenne did observe various effects of length, width and blowing pressure on the pitches of organ pipes, and he noted the octave difference between open and closed pipes.
Assuming that Mersenne’s Law holds even for a string so long that its vibrations can be counted visually, Mersenne estimated the frequency of a note, that is, he found experimentally the constant of proportionality in the law. Some decades later Robert Hooke may have made a direct measurement of frequency; in 1665 he wrote about estimating the frequency of a fly’s wings’ vibration from its buzz. In 1681 he demonstrated before the Royal Society a way of making musical sounds ‘by the striking of the teeth of several Brass Wheels, proportionally cut as to their numbers, and turned very fast round, in which it was observable that the equal or proportional stroaks of the Teeth, that is 2 to 1, 4 to 3, etc., made the musical [intervals]’. Hooke’s wheel (usually known as Savart’s wheel), which demonstrates explicitly that pitch corresponds to frequency, could have been used to make a rough measurement of frequencies.
Mersenne realized the importance of overtones, and urged his numerous correspondents to seek an explanation of them. In the case of an open string he identified at least four harmonic overtones, typically finding it paradoxical that a string should vibrate at different frequencies simultaneously. He also raised the problem of understanding instruments, such as the trumpet marine and the wind instruments, in which harmonics are produced separately.
Like Galileo, Christiaan Huygens was the son of a musician, played a number of instruments and was interested in consonance and dissonance. Around 1673, influenced by Mersenne, he gave a derivation (under simplifying assumptions) of Mersenne’s Law for the vibrating string. Like Mersenne, he was interested in overtones. He estimated absolute frequency, and understood the relationship between wavelength and pipe length.
In 1677 the mathematician John Wallis published a report on experiments showing that the overtones of a vibrating string, which are harmonic, are associated with the existence of nodal points on the string; according to him the phenomenon had been known to the Oxford musicians for a number of years. In his experiments a string was made to vibrate in a higher mode by resonance with a string tuned so that either its fundamental or one of its harmonics was at the pitch of the mode. Paper riders showed the nodal points. Wallis suggested that a similar situation must exist for wind instruments. He observed that the tone of a string was ‘rough’ if the string was excited at a potential node. In 1692 Francis Robartes published a similar description of nodes and harmonics in connection with his study of the scales of the trumpet and the trumpet marine.
At the beginning of the 18th century Joseph Sauveur proposed the development of a science of sound, which would be called ‘acoustique’. His own studies, which grew out of an interest in music and a background in mathematics, dealt with the physics of musical sound. He seems to have been the first to recognize that the frequency of the beats produced by a pair of notes is equal to their frequency difference, and was able to determine frequencies absolutely, probably to within a few per cent, by counting the beat frequency for two low-pitched organ pipes tuned a small semitone apart in just intonation (frequency ratio 25:24). Like Wallis and Robartes, but independently of them, he explained that nodes are present when a string vibrates in a higher mode. He introduced the terms ‘son harmonique’, ‘noeud’ and ‘ventre’.
Sauveur noted that organ builders had intuitively discovered the harmonic pitches, which they mixed by means of stops to obtain various timbres. As Fontenelle, secrétaire perpétuel of the Paris Académie, reported: ‘Nature [had] the strength to make musicians fall into the system of harmonic sounds, but they fell into it without knowing it, led only by their ear and their experience. Sauveur has given a very remarkable example of this’. Sauveur remarked that the organist mixes the stops ‘almost the way the painters mix colours’, and that ‘by the mixture of its stops, the organ is only imitating the harmony that nature observes in sonorous objects’. Fontenelle referred essentially to the principle of superposition in his explanation of harmonics: ‘each half, each third, each quarter of the string of an instrument makes its partial vibrations while the total vibration of the entire string continues’. In 1713 Sauveur ingeniously derived Mersenne’s Law with a constant of proportionality for the ideal string that was correct but for a factor of √10/π. He did this by considering a horizontal string, hanging in a curve because of the gravitational field, which he treated as a compound pendulum. (In the same year Brook Taylor also gave a derivation. His style of analysis belongs to the 18th century, Sauveur’s to the 17th.) The most important of Sauveur’s ideas for later developments was the implicit theme of superposition, which appeared in his studies of beats, harmonics and timbre.
Newton first commented on musical sound in some of his early papers on optics. In a major achievement of the Principia (1687–1726) he analysed the propagation of sound mathematically. (His analysis was correct except for his use of Boyle’s Law for the relation between pressure and volume rather than the adiabatic law, which was not known until the beginning of the 19th century.) Recognizing that the velocity of sound equals the product of wavelength and frequency, and using Mersenne’s and Sauveur’s determinations of the frequencies of organ pipes, Newton was able to conjecture ‘that the wavelengths, in the sounds of all open pipes, are equal to twice the lengths of the pipes’.
The audible overtones of taut strings and of other bodies used to make music are harmonious to the ear. In the 18th century it was popularly believed that the overtones of all natural bodies were harmonious (see for example BurneyH, i, 164), and some readers claimed that Rameau had founded his system of harmony upon this idea; in fact Rameau’s extensive passages on acoustics are confused and often misrepresented the knowledge in his time.
The dominant acoustic problem of the 18th century was to calculate the fundamental and the overtones of a given sonorous body. In 1713 Taylor attempted to determine the motion of the monochord on the basis of Newton’s rational mechanics. His pioneer approach was fruitful for later work, but he himself could not carry it through without restricting the shape of the string to be a single sine wave. He confirmed Mersenne’s Law and showed that its constant of proportionality is ½. In 1727 Johann Bernoulli proposed a model of the string as a set of n little balls connected by massless cords and calculated the fundamental frequency for a few small values of n. Both Taylor and Bernoulli had equations from which they could have calculated the frequencies and nodes of the overtones, but neither of them did so. Bernoulli’s discrete model, which had been proposed in the previous century, was to be studied in increasing detail, but it contributed nothing to acoustics.
In 1727, too, Euler published his Dissertatio physica de sono, a clear, short pamphlet that at once became a classic and guided acoustical research for about 75 years. Euler divided sounds into three kinds: the tremblings of solid bodies, such as the reeds of wind instruments, strings, chimes and drumskins; the sudden release of compressed or rarefied air, as by clapping the hands; and the oscillations of air, either free or confined by a chamber, such as the tube of a flute, an organ pipe or a trumpet. The first two kinds refer to the production of sound, the third also to its transmission to the ear.
Euler recalled Newton’s determination of the speed of sound in air. Although the ideas upon which that determination rested seemed to be correct, the numerical result was far too small to agree with measured values. The problem of correcting Newton’s analysis was to dog theorists for the whole century and to remain unsolved until 1868; however, the early theorists gradually came to see that mechanical principles could give correct ratios of frequencies, even if the pitch of the fundamental was incorrect, and they often reported their results in terms of such ratios. Fortunately for the earlier theories of music, it is the ratios that determine musical intervals. Similarly, pitches of all instruments were only uniformly incorrect; two instruments predicted to be consonant would be so indeed. Thus musical acoustics could make spectacular advances in the 18th century despite the standstill on what would seem to be the basic and central problem of physical acoustics.
In his next significant work on acoustics, Tentamen novae theoriae musicae (1739), Euler presented a developed theory of consonance, based upon an explicit, mathematical rule for determining the ‘simplicity’ of a set of frequencies such as those making up a chord. He derived his rule from ideas of the ancients, Ptolemy in particular. It could not take account of difference tones and summation tones, for they had not yet been reported, but it permitted Euler to determine by routine calculations the most complete systems of scales or modes ever published. The last chapter sketches a theory of modulation. Euler thus began to construct a mathematical theory of the consonance of a progression of chords.
The 7th and the combination of the 6th and the 5th have high measures of dissonance yet were often used by musicians of the 18th century. To explain this fact, Euler many years later suggested that ‘we must distinguish carefully the ratios that our ears really perceive from those that the sounds expressed as numbers include’ (‘Conjecture’, 1764). In the equally tempered scale there are no exact consonances, yet the ear seems to hear the ratio 2:3 when its irrational, equal-tempered substitute 1:20.583 is sounded. The ear tends to simplify the ratio perceived, especially if the dissonant tones follow after a harmonious progression; for example, 36–45–54–64 is indistinguishable from 36–45–54–63, which is the same set of ratios as 4–5–6–7. The paper closes with the suggestion that the music of the day had already replaced Leibniz’s basic numbers 2, 3 and 5, beyond which ‘music had not yet learned to count’, by 2, 3, 5 and 7. Apparently Euler was not familiar with Tartini’s beats, though by the time he wrote this work they had been demonstrated. In the succeeding paper on the ‘true character of modern music’ (1764), Euler reasserted the position he took in his treatise: consonances and dissonances are not essentially different but just sounds that are more ‘simple’ or less so, according to the value of his numerical measure. The ancients admitted a smaller range of consonances than do the moderns; both the ancient and the modern practices perfectly obey the principles of harmony, but the modern composers have achieved ‘a very considerable extension of the limits of ancient music’. Euler recommended that the musical scale of 12 notes based on the numbers 2, 3 and 5 be augmented by 12 more, based on the numbers 2, 3, 5 and 7; all of these ‘foreign notes’ are obtained by multiplying one of the usual notes by 7. A harpsichord with 24 keys per octave should be constructed so as to try out this extended scale. Presumably Euler did not know of Vito Trasuntino’s celebrated arcicembalo of 1606 (now in the Civico Museo of Bologna), which has 31 keys per octave. These works on the theory of harmony are exceptional in Euler’s research on acoustics, the rest of which concern strictly physical problems.
It is now known that a body in undergoing a free vibration at a single frequency must assume a shape proportional to some particular one. The amplitude of vibration is arbitrary, but both the proper frequency of the vibration and the generating shape, which is the normal mode that corresponds to it, are determined uniquely. The several normal modes correspond to the several pure tones that a free sonorous body may emit simultaneously; the sound of such a body is thus a mixture of its normal mode frequencies. What this mixture is depends upon the amplitudes given to the several normal modes, and therefore on the way in which the body is set into vibration. This idea, of central and indispensable importance in acoustics, is due to Daniel Bernoulli. One of the few scientists of the 18th century who produced important experiments as well as important theory, he formed it gradually upon the basis of accumulated musical experience, simple theoretical assumptions and calculations, and experimental checks. He was not only the first to conceive of vibratory motions in this way but also the first to calculate the complete set of proper frequencies and normal modes for a particular vibrating body. He chose to consider first a heavy cord hung up from one end, but he saw and stated that his ideas were general and would apply to musical bodies (see his papers of 1732–3 and 1734–5). His results showed that the partials of the hanging cord were incommensurable and not harmonious; he showed also that a normal mode of higher frequency had a greater number of nodes than did one of lower frequency.
Euler and Bernoulli in friendly competition poured out a torrent of research on the small vibrations of bodies. Of course the long-awaited explanation of the tones and frequencies of the monochord fell into their hands at once. All the modal shapes were found to be sine waves, as Taylor could have shown but did not; and their frequencies follow the series 1–2–3–4 …. The transverse vibrations of elastic bars or chimes, variously supported at their ends, confirmed abundantly the facts suggested by Bernoulli’s work on heavy cords (see Bernoulli, 1741–3, and Euler, 1734–5, 1744 and 1772). Only for exceptional bodies or conditions are the modal shapes sinusoidal; most of the partials are strikingly dissonant. The theory showed Bernoulli where to expect the nodes; by supporting a chime ‘with the tips of two fingers’ at the predicted nodes and then striking it he could easily induce a vibration in the corresponding pure mode. He recorded the tones ‘by observing, as best as I was able, the consonant sound on my harpsichord’. The idea that harmony could be founded upon the ‘naturalness’ of the progression of partials was destroyed. Rather, as it appeared, bodies fit for making music are most unusual ones, having been selected precisely because their series of partials are harmonious to the ear. Of course the traditional idea that ‘simplicity’ made a chord harmonious, which had been rendered precise by Euler, was not affected. Nature had been shown not to be simple in this regard. The idea, then commonly attributed to Rameau, that simplicity was harmonious because favoured by nature, was thereby destroyed. Euler (1739) also explained the long-known phenomenon of resonance by showing that if an (undamped) harmonic oscillator is driven at its natural frequency the amplitude of its vibration increases without limit.
The methods used to calculate the modal forms and frequencies of bodies were special; the motions considered, likewise, were special. Daniel Bernoulli had claimed that any free vibration of a sounding body could be regarded as a superposition of its simple modes with various amplitudes, but there were no general principles upon which proof of such a statement could be attempted. At the mid-century, the Newtonian framework of mechanics was vastly expanded by Euler and Jean Lerond d’Alembert to make possible a concise mathematical description, in principle, of all motions of a body. These statements came later to be called ‘differential equations of motion’; for bodies having infinitely many degrees of freedom, such as a bar or drumhead or chamber full of air, they are ‘field equations’, for they govern the local motion at each place in the body. On the basis of these equations, supplemented by suitable additional conditions on the boundary of the body, it is possible to ask whether Bernoulli’s principle of superposed simple oscillations is valid. The mathematical tools developed in the 18th century were insufficient to answer the question, but in the 19th century mathematicians were to justify application of Bernoulli’s idea to a multitude of vibrating bodies, and it forms an indispensable part of acoustics today.
A great controversy on this and related matters began in 1749, when d’Alembert published his discovery of the field equation for a taut cord; the other disputants were Euler, Bernoulli and, later, Joseph Louis Lagrange. From the very beginning the solutions of d’Alembert’s equation indicated that every possible form of string could be generated by suitable waves travelling both to the right and to the left with the same speed, namely, √T/m. Euler based all of his discussion of the matter on this fact alone; d’Alembert contended that only particular ‘equations’ were amenable to mathematics; Bernoulli claimed that use of a sufficient number of simple modes would explain everything with any accuracy desired. The dispute was of a technical nature, unresolvable with the mathematics then known, and in the 18th century it bore no direct fruit for acoustics. However, both Bernoulli and Euler pursued the subject in the constructive spirit that characterized all their work; each showed that his approach could produce new and valuable information about sounding bodies.
Bernoulli (1753 and 1762), proceeding by analogy to his concept of transverse vibrations in strings and bars, formed a special theory of longitudinal vibrations of air in wind instruments. He found that the partials of a closed pipe followed the progression 1–3–5–7 …. He analysed the tones of an organ pipe à cheminée and concluded that his results were in accord with his formula for determining the length of a consonant uniform pipe. He also calculated the series of partials of a conical horn.
The year 1759 is decisive for acoustics, for in that year Euler derived the general field equations for vibrations of air in one, two or three dimensions. The modern theory of aerial acoustics rests upon these ‘partial differential’ equations or upon modifications of them so as to take account of internal friction and the conduction of heat. Euler’s own research on them (1759) entered into only the simplest cases. He introduced the method of ‘separation of variables’, which is still the starting-point for solving many problems of aerial acoustics. He also determined the laws of propagation of cylindrical and spherical waves, calculating the diminution of their amplitudes as they spread out from a source. Euler had communicated some of his results to Lagrange during the weeks in which he discovered them. Lagrange applied them at once (1760–61) and extended the analysis in various directions. He obtained the field equations for longitudinal vibration of air in a tube of general cross-section; this equation was rediscovered over a century later, called the ‘Webster horn equation’, and put to extensive use in the design of loudspeakers.
One simple phenomenon remained ill-comprehended. If, as the theory asserts, sonic motion consists of waves running both ways continually, how can part of a string remain long at rest, and how can an echo be heard successively? In 1765 Euler easily showed how the opposing waves may simply annul each other for periods of time. He did so by explaining the general solution of the one-dimensional equation in terms of pulses which obey definite rules of reflection upon reaching a terminus such as the end of a wire, the vent or stopper of a pipe, or the face of a cliff. For example, he exhibited (1772) the complete solution for motion of a monochord plucked to triangular form and then released, thus settling the old problem of Beeckman, Mersenne and Taylor.
On the basis of this formulation of the transverse vibrations of a straight elastic bar in terms of a single partial differential equation, Euler in 1772 and 1774 calculated the frequencies and nodal forms of the modes corresponding to all six possible kinds of support. In 1782 Riccati published still more accurate calculations concerning the first six modes of a bar free at both ends, along with a detailed verification by experiments on chimes of brass and steel. The theory has been universally accepted ever since and is usually called ‘the Bernoulli-Euler theory’.
In the second half of the 18th century theories for vibrating bodies of more complicated kinds were proposed, but the only success lay in Euler’s field equations (‘De motu vibratorio tympanorum’, 1764) for a perfectly flexible drumhead, discovered in 1759. Euler obtained some particular solutions, but mathematical analysis was too primitive then to do much more. An epoch of acoustics closed with Euler’s death in 1783. That the great achievements of the 18th century were only imperfectly recognized and in some cases had to be rediscovered in the 19th may be due partly to the lack of a textbook or even a treatise for specialists. The nearest to the latter is Euler’s ‘Sectio quarta de motu aëris in tubis’ (1771), which is the last section of his treatise on fluid mechanics; the final chapters are devoted to the hyperbolic horn and the conical flute.
The research of the 18th century produced mainly theory such as to consolidate and extend understanding already formed by musical experience and by known experiments. It was nearly always presumed tacitly that music was first produced and then transmitted to the ear, which registered exactly the sound that fell upon it. There were occasional remarks about the nature of the ear, but little more than that. Rameau (1737) wrote: ‘What has been said of sonorous bodies should be applied equally to the fibres which carpet the bottom of the ear; these fibres are so many sonorous bodies, to which the air transmits its vibration, and from which the perception of sounds and harmony is carried to the soul’. Riccati suggested (1767) that the auditory nerve was ‘a bundle of nerves which by the smallest degrees pass from the lowest tone to the highest, and the one of these that corresponds to unison with a sounding body is set a-trembling’. Such an ear, if its fibres were tuned at intervals of one tenth of a comma, would smooth over small differences of frequency but otherwise would be a perfect receiver.
In 1787 appeared a pamphlet by Ernst Chladni, Entdeckungen über die Theorie des Klanges, which opened a new period in physical acoustics. Mathematics had dominated the subject for a century; now, for the first time, a master experimentist who understood and knew how to use existing theory broke new ground with experiments which at once suggested and demanded a theory of a new kind. To Chladni the idea of partials with their nodes corresponding to various kinds of support was second nature. He chose to determine them by experiment for thin, springy plates, and did so by scattering fine sand over the surface, supporting the plate at points conjectured to be nodal, and stroking the free edge with a violin bow (fig.1). He used this technique first to confirm the Bernoulli-Euler theory of straight rods; then he turned his attention to thin elastic surfaces, mainly circular discs but also square plates. The enormous variety of nodal patterns he obtained may be illustrated by the samples in fig.2. He recorded the frequencies in terms of musical pitches. His results reveal the intricacy of response that must be expected in the most idealized sounding-board of a musical instrument. To explain Chladni’s figures by a theory that would correlate the nodal patterns with the frequencies has remained a major open problem of acoustic theory; it has been a continuing stimulus to search of principle, and cannot be regarded as solved.
The publication of Chladni’s classic work Die Akustik (1802) gave a clear indication of change. In the new century the physics of music was to emphasize observation. By the end of that century, observation was to reveal that some of the problems of musical acoustics were more complex than they had seemed. The timbre and acoustical nuances of interest to the performing artist sometimes arose from effects too complicated to be included in the developing physical theories. For example, while the wolf note of the cello may be readily understood as an unavoidable problem of a resonant chamber of specified dimensions, the degree and type of orthotropy or directionally orientated elasticity in the cane of an oboe reed – possibly related in some way to the silicon content which makes cane from one particular region preferable to that from another – is a complicated problem far beyond the mathematical simplifications necessary to describe the physics of music.
To explain Chladni’s sand figures in the vibrating plate challenged several generations of experimentists and theorists. After a few initial mistaken steps, a theory of the vibrating plate with appropriate boundary conditions evolved in the 19th century, but further experiments by Savart, Faraday, Lord Rayleigh (J.W. Strutt) and many others revealed phenomena beyond its range. The dilemma remained until 1931, when Andrade and Smith showed that the sand did not rest along the stationary nodal curves but rather moved on the surface until it reached lines at which the maximum acceleration of the plate equalled the gravitational constant g. For oscillations of large amplitude such lines approach the nodal lines on either side, but in moderate motion they may be far from them (see fig.3). Thus Chladni and others after him were not right in interpreting the lines of sand as nodal lines; perhaps the lines of sand as interpreted by Andrade could be determined by the theory, but so far they have not been. The status of the now standard theory of elastic plates remains neither supported nor controverted by Chladni’s famous results. Andrade and Smith found, as had Rayleigh, that better consistency was obtained if the violin bow used by Chladni and those who followed him was replaced at a fixed location by a mechanical driver of constant frequency.
Experiments on the physics of music require precise determination of the absolute frequency of a sustained tone. Without such a determination it is not possible to analyse the frequencies of the various components of a musical note. Cagniard de Latour (1819) made a contribution to the measurement of absolute pitch (see fig.4). He developed the siren in a form, later improved by Seebeck (1841) and Helmholtz (1863), and perfected by Koenig (1867, published 1881), which made absolute measurement possible. To a bellows chamber Cagniard affixed one plate with a series of holes; he rotated a second plate with identical holes so as to open and close the matching holes alternately. He thus produced a tone of the desired loudness at a frequency determined by the speed of rotation. In the early 19th century it was difficult to maintain that speed constant, and many people throughout the rest of the century described their efforts to do so.
The most important tool for investigators in the 19th century was provided by Scheibler, a silk merchant who lived near Düsseldorf. In a pamphlet of 1834, in which he humbly acknowledged his limitations as a writer on science, he described his ‘tonometer’, which for the first time made it possible to determine absolute pitch precisely. His instrument was copied, improved and widely used for fundamental studies during the rest of the century. In fact, one of Scheibler’s original instruments was still being used by Alexander J. Ellis, 50 years after its construction, to determine the pitches of 16th- to 19th-century organs and the tuning-forks of historically prominent musicians.
In his long summary of these measurements Ellis observed, for a', frequencies ranging from 374 to 567 Hz (cycles per second). Although the frequency generally increased from the 16th century to the 19th, there were wide fluctuations at any given time (see Pitch). In this context the study by van der Pol and Addink (1939) is interesting; their electronic arrangement permitted them to measure continuously, to an accuracy of 0·2 Hz, the pitch of an orchestra during a performance. In comparing 450 radio broadcasts of orchestras in England, France, Germany and the Netherlands, they found that, although the total average pitch was a' = 440 Hz, there were national differences. More important, there were variations of pitch during individual performances, depending upon the dominance of the various sections of the orchestra or upon the soloist.
The tonometer Scheibler described in 1834 was an array of 56 tuning-forks. One fork was tuned to a' = 440 Hz, the second to a, one octave below, and the rest at 4-Hz intervals in the octave. Through the careful counting of beats, the absolute frequency of an unknown tone in any 4-Hz interval could be established. At a congress of physicists in Stuttgart in the year he described his invention Scheibler introduced a' = 440 Hz and pressed for its selection; that value became known as the ‘Stuttgart pitch’. Scheibler had made this choice not for some fundamental reason relating to scales, consonance or intonation, but because he had observed that 440 Hz was the mean of the frequencies of the a' on pianos in Vienna as they varied with temperature.
Scheibler’s equally tempered scale, which he calculated to four decimal places, was a consequence of his choice of a' = 440 Hz rather than a preconceived notion. Among the many attempts to provide an absolute standard this ‘Stuttgart pitch’ was unique in that for the first time the proposed value had meaning; it could be measured with precision. Scheibler’s tonometer, developed at the same time, provided the necessary tool.
One type of experiment in the 19th century was designed less as a part of the developing new ideas than as a confirmation of the conjecture of an earlier century. In the 18th century it had been relatively easy to perceive the nodes of transverse vibration of a string or a bar, but it was not until 1820 that Jean-Baptiste Biot, using polarized light through a transparent doubly refracting solid, demonstrated the existence of nodal patterns in the interior of a solid in longitudinal oscillation. This technique of photoelasticity had been discovered in a different context by Brewster (1816).
Kundt (1866), with a variation on Chladni’s original experiment with the sand figures on plates, demonstrated the existence of nodal patterns in a column of air in a transparent glass tube by means of his ‘dust figures’ (fig.5). He observed the distribution into nodal patterns of a fine powder which he had sprinkled in the tube. His famous dust figures became the main method in the 19th century for determining the velocity of sound in various gases and, by means of an ingenious adaptation, the velocity of sound in solids.
Knowledge of the physics of music in the early 19th century had advanced sufficiently to provide an essential background for Helmholtz’s experiments between 1854 and 1862. His work culminated in the penetrating study described in his great classic of 1863 Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik.
The theorists of the 18th century had tried again and again to repair Newton’s failure to calculate a value for the speed of aerial sound that squared with experiment. They had eliminated all sources of possible error but one: Newton’s having taken the pressure in sonic motion to be proportional to the density, just as it is in air at rest. The mechanical theories had yielded correct ratios of frequencies and hence correct musical intervals, but they could account for the diminution of sound only through its spreading outwards as it progressed. According to them, air once set in motion to and fro between the plane walls of an elastic box would go on sounding for ever. Real sounds die out in time. Otherwise there could be no music. It is equally necessary that all audible sounds travel at sensibly the same speed. A complete theory of propagation must account for all these facts.
Pierre-Simon de Laplace suggested by degrees in the years 1816–22 that the sonic motion was too rapid to allow differences of temperature in the neighbouring parts of the air to equalize through conduction of heat. He showed that in a ‘sudden compression’, later called an ‘adiabatic motion’, the pressure of a gas was proportional not simply to the density, but to the density raised to the power of the ratio of specific heats. Substituting this relation into Euler’s formula for the speed of sound in general, Laplace at once replaced Newton’s numerical value by one that agreed well with measurements. It was still not clear just why the motion should be adiabatic. The matter was settled in a masterly research by Gustav Kirchhoff (1868). He began from the general field equations of motion which James Clerk Maxwell had obtained from his kinetic theory of gases just the year before; these, in turn, reflected the idea of internal friction due to George Gabriel Stokes, the conduction of heat as described by Fourier, the theory of interconvertibility of heat and work as elaborated by James Prescott Joule and Rudolf Julius Emmanuel Clausius, and the framework of continuum mechanics established by Augustin-Louis Cauchy. All the centuries of experience in rational mechanics were brought to bear on the problem of determining the behaviour of plane sound waves in a gas. Kirchhoff proved that for waves of low frequency the speed was given approximately by Laplace’s formula, and that all such waves travelled at the same speed. However, for waves in the audible range the motion is not quite adiabatic, although it is more nearly so the slower (not the faster) is the oscillation. All sound waves are damped through the combined effects of viscosity and the conduction of heat, and sounds of higher pitch are damped out more quickly than are graver tones. ‘Ultrasonic’ waves, whose frequencies are very high, exhibit ‘dispersion’: the speed of propagation increases with frequency.
The velocity of sound in a solid had been measured first by Biot in 1808 in his experiments on waves propagating through nearly a kilometre of the newly constructed iron water pipes in Paris. Ingredients for Helmholtz’s analysis were provided by the rapid, phenomenal development of the linear theory of elasticity in the 1820s by Claude-Louis Navier, Siméon-Denis Poisson and Cauchy.
Meanwhile, in his work on the conduction of heat in the first decades of the 19th century Fourier had shown how to represent any curve by superposing sine waves corresponding to the frequencies 1, 2, 3 …. This theorem and generalizations or analogues of it not only substantiated Daniel Bernoulli’s viewpoint on acoustics but also became the basis for countless analyses of the tones, overtones and dissonant beats of musical instruments as well as unmusical ones.
Finally, as a prelude to the contributions of Helmholtz, there was the statement of Georg Ohm, better known for his law in electricity, who in 1843 proposed what became known as Ohm’s Law of Acoustics. Ohm suggested that musical sounds depended only on the distribution of energies among the harmonics and had no dependence on differences of phase. The physical demonstration of Ohm’s Law of Acoustics was the major accomplishment of Helmholtz, and the theory remained without effective challenge until the middle of the 20th century.
Helmholtz provided a classic analysis of the role of overtones. To do so he needed an instrument sufficient to identify the existence and determine the strength of a suspected overtone. Furthermore, he was the first to understand fully that analysis of consonance and dissonance, let alone the existence of combination tones and the timbre of musical instruments, required more than the physics of the vibrating structure; analysis had to include the interaction of the sound from that source of vibration impinging upon another vibrating structure, the human ear. Helmholtz met the need for an instrument by developing his resonator. In its early crude form it was a spherical glass chamber with a hole at either end (fig.7), of size such that when exposed to a specific frequency the resonant cavity enabled him to identify that frequency when one aperture of the sphere was placed in the ear by means of a melted wax earpiece.
To achieve a proper analysis, Helmholtz applied to the detailed study of the human ear his great talents as one of the outstanding physiologists of the 19th century. Helmholtz, who is said to have been a competent pianist in addition to having exceptional talents as a universalist in physics and physiology, developed a two-manual harmonium to produce the overtones, difference tones and summation tones he wished to study. He systematically tested the applicability of Ohm’s Law of Acoustics by using his metallic reeded harmonium, improved versions of the Cagniard de Latour siren and the Scheibler tonometer, and a greatly improved model of his own resonator, which Rudolph Koenig had perfected.
It is not possible here to give much detail of Helmholtz’s discoveries beyond his demonstration of the general principles enunciated in Ohm’s Law. He observed that the range of 30–40 Hz in the beating of high overtones produced the most unpleasant sensation. Some may contest this on aesthetic grounds, but Helmholtz’s research shed light on the centuries of debate on the subject of consonance.
Of the many particulars of his work beyond the consideration of Ohm’s Law, one must be mentioned because it set straight a previous misconception concerning the mystery of ‘Tartini’s beats’: if two pure tones are sounded, and if the ear hears a tone the frequency of which is the difference of the two, the tone heard is called a ‘difference tone’. Their discovery generally is credited to Tartini, who wrote of them in 1754. If date of publication is the criterion, however, the discovery must be credited to the German organist Georg Andreas Sorge, who published a description of the same phenomenon in 1745. Before Helmholtz the usual explanation for the difference tones is that stated by Thomas Young, among others. Such a tone, according to Young, was the beating of upper partials at a frequency high enough to provide a sound an octave below the lower of the two sounded notes. That this was a conjecture which led to absurdity was demonstrated by Helmholtz’s discovery of higher combination tones, which he called ‘summation tones’. The measured difference tone had a frequency which was the difference between two sounded notes, whereas the summation tone lay above the highest tone, with a frequency equivalent to the sum of the two sounding frequencies. Unlike the difference tone, which requires no special aural acuity to perceive, the summation tone is extremely difficult to hear. The discovery of its existence not only eliminated the conjectures of Young and others but also provided evidence for Helmholtz’s theory of the non-linearity of the ear.
A receptor such as the ear, as would be expected from a knowledge of the mechanical behaviour of other portions of the human body, performs in a manner which is best represented by recourse to non-linear mechanics. The expected behaviour is entirely different from that in the world of linearized physics, with its focus restricted to the inaudible whisper and the invisible vibration, that is, to infinitesimal phenomena. In the linearly interpreted world, the sums of different solutions are still solutions, superposition prevails, and elementary harmonic analysis is possible. This corresponds to the general behaviour of the sources of sound in musical instruments. If non-linearity prevails, the sums of solutions are not generally solutions, superposition does not apply, and the resulting response contains elements produced by the non-linear receptor itself. Thus the human ear, first described in mathematically acceptable terms by Helmholtz, is capable of providing sounds not actually present in the musical instruments which supply the stimuli. The purely aural harmonics are in the form of sums and differences of the frequencies of the fundamental and the overtones of a given note, and in the amazing matrix of possible combinations of sums and differences when an interval or a chord is sounded. Hence the rudimentary physics of the vibration of musical instruments provides an indication of general behaviour but cannot explain the timbre.
Like many solemn pronouncements in science, Ohm’s Law of Acoustics was more a summation of the accumulated wisdom of a preceding century than the sudden declaration of a newly conceived discovery. However, Helmholtz’s years of experimentation in musical acoustics gave far more than mere substance to those conjectures; he provided in enormous detail and breadth the foundations for a century of further research.
The research of Helmholtz and Koenig, like that of other scholars of similar calibre, was not at once universally accepted. Mercadier (1872) criticized Helmholtz for having concentrated on the study of sustained tones rather than upon the melodious flow of the music. Koenig, too, was attacked. A sequence of correspondence in Nature indicates that although many of his contemporaries revered Koenig, as is affirmed by Thompson (1890–91) when describing Koenig’s research and his visit to Koenig’s remarkable workshop at the Quai d’Anjou on the Seine, he had some detractors, Ellis being one of them. It is difficult to see how Ellis, on the basis of what turned out to be his own flimsy measurements on a single, questionable instrument, could claim that the tuning-forks of all Koenig’s tonometers were seriously in error, particularly in view of the accuracy which Koenig had achieved with his ‘clock tuning-fork’, among other instruments (see fig.8).
In Koenig’s clock tuning-fork, the clock mechanism was driven by a 64-Hz tuning-fork which, with added weights, was adjustable between 62 and 68 Hz. It was possible to observe the motion of the opposite prong of the tuning-fork under the microscope in order to examine the Lissajous figures. Matching the clock tuning-fork to an unknown frequency, and observing the loss or gain of time of the fork-driven clock compared with a standard clock, provided the most precise measurement of frequency in the 19th century. Koenig had also developed the Scheibler tonometer to a remarkable instrument which would permit the determination of frequency over the entire audible range with what may be called 20th-century precision.
The accuracy of Koenig’s tuning-forks became legendary. Each one was stamped with his own initials, to attest to his personal inspection of the validity of the stated frequency. Ellis, in his translator’s notes accompanying Helmholtz’s tome, described in detail the range of the tuning-forks and their prices in 1885 when they still were available from Koenig’s own workshop. Nevertheless, Ellis was adversely critical. He used Appunn’s reed tonometer, borrowed from the South Kensington Museum, and too rapidly concluded that Koenig’s tuning-forks erred by 2 to 12 Hz from their stated frequencies. Koenig, questioning the accuracy of the instrument Ellis had used, stoutly defended the accuracy of his own tuning-forks and of his correction of the earlier measurements of Lissajous, who had been appointed by the Académie to set the standard for the French ‘diapason normal’ of a' = 435 Hz. (Lissajous had used a siren and had introduced the famous Lissajous figures in carrying out the commission.)
That was not the end of the discussion. Rayleigh entered the argument. He described a series of experiments which revealed that, when two reeds were vibrating in the immediate vicinity of one another, as in the instrument Ellis had used, the observed beats would be in error, for stated reasons, if they were compared with those for the same unknown frequency ascertained by means of separated tuning-forks, as in Koenig’s instrument. Rayleigh’s analysis and experiment supported Koenig’s claim of precision. One could wish that in the final letter in this series Ellis had been as gracious to the craftsmanship of Koenig as he was to the ingenuity of Rayleigh.
The 18th century saw the laying of a firm theoretical foundation for the physics of music, and the mechanical ingenuity of the great 19th-century experimentalists succeeded in providing empirical support for many of the theoretical predictions concerning the nature of sound and the operation of musical instruments. 20th-century acoustics has been dominated by the electronic revolution, which placed in the hands of scientists an immensely improved set of tools for measurement and computation. The resulting increase in the precision of both experiment and calculation has revealed that classical linear acoustics is incapable of describing many important aspects of the physics of music. In the last decades of the 20th century it increasingly became recognized that non-linear dynamics must be used to describe the behaviour of continuously excited instruments such as woodwinds and bowed strings (see Hirschberg, Kergomard and Weinreich, eds., 1995).
One of the most important practical problems in acoustics which was solved by electronic techniques was that of obtaining an accurate graphical record of the waveform of a sound. In the 17th century Galileo had described the possibilities of interpreting displacement against time, using traces of scratches from a vibrating stylus drawn across a metallic surface. In 1849 Guillaume Wertheim, a notable experimental physicist in 19th-century solid mechanics, while studying the ‘deep tone’ longitudinal vibration of rods (see Bell, 1973), had recorded the detail of vibration by a method he attributed to Duhamel. A tiny needle attached to a vibrating rod produced a trace on a transversely moving glass plate coated with carbon. Such mechanical methods of determining waveforms, with their limitations of frequency and other problems, were improved throughout the second half of the 19th century. Before the development of the vacuum tube revolutionized such experiments, the ‘phonodeik’, devised by Dayton C. Miller (1909), gained much attention. The sensitive receiver of Miller’s device was a diaphragm of thin glass at the end of a resonator. A tiny steel spindle mounted in jewel bearings attached to the diaphragm by a thin thread made possible the photographing of the trace of the sound produced by the many different musical instruments which Miller investigated. The device still had limitations of frequency response which had characterized the previous methods of mechanical measurement in this area. Miller’s photographs filled the literature and influenced attitudes on musical acoustics until the mid-20th century.
The essential problem of all mechanical devices for recording sound vibrations lay in the difficulty of achieving an adequately rapid response from the mechanically moving parts. In 1897, J.J. Thomson demonstrated that a beam of electrons (which he described as a ‘cathode ray’) could be accelerated across an evacuated glass tube to create a visible spot on a fluorescent screen. A deflection of the beam by an electronic voltage could be observed as a movement of the visible spot. Since the electron beam, the only moving part of this system, is almost without mass, it is in principle capable of extremely rapid response. By the middle of the 20th century the cathode ray oscilloscope had become the standard method of observing the waveforms of the electrical signals generated by microphones or other vibration transducers (see Sound, §3).
Access to electronic amplification and filtering techniques made it much easier to determine the frequency and spectral content of sound. Frequency measurement systems based on tuning forks were replaced by electromechanical meters like the Stroboconn, and later by meters which used the piezoelectric effect in a quartz crystal to provide a highly stable frequency standard. In the last 30 years of the century the phenomenal development of computers and other microprocessor-based devices led to another revolution, in which analogue electrical techniques have largely been supplanted by digital techniques. The first stage in the investigation of a sound signal consists of a digital sampling, with the signal being stored in the computer memory as a set of numbers. Subsequently a variety of sophisticated signal processing techniques can be applied, and any portion of the recorded signal can be readily reconstructed for detailed study, modification or reproduction (see Recorded sound, §II, and Signal (ii)).
The impact of the age of electronics has been so great in terms of the variety and magnitude of measurement in musical acoustics, as well as in some other areas, that it would be impossible in a short history to do more than emphasize a few important points. In one sense, the successes of Helmholtz and of Koenig had led to a hiatus in which few advances were made until well into the 20th century, despite the continued interest in acoustics and musical acoustics of one of the great late 19th-century scientists, Rayleigh, who continued to experiment and write on the subject until his death in 1919. The most influential book ever published on acoustics is Rayleigh’s The Theory of Sound (1877), which gives a fairly reliable account of the accepted theory and experiment of its period.
By the end of the 19th century a fair understanding of impulsively excited instruments, such as plucked strings or struck bars, had been reached. If the energy imparted to the instrument by the excitation was not too great, the problem could be treated as the free decay of the normal modes of the system; the nature of the excitation determined the initial mode amplitudes. The situation was less satsifactory for continuously excited instruments, such as wind and bowed string instruments. In the 1850s Helmholtz, among others, had performed experiments to study the phenomenon of bowing, and had identified the idealised string behaviour now known as ‘Helmholtz motion’. Further studies of this motion were reported by C.V. Raman in 1918. Helmholtz had also investigated the excitation mechanisms of reed woodwind and brass instruments. Several researchers, including Rayleigh and especially Strouhal (1878), had developed an understanding of the principle of edge tones, which were believed to be related to sound production in the flute. However, although theoretical models based on these researches appeared to encompass the basic physics of the instruments, such musically important issues as the variation of timbre with change of playing parameters were not explained.
The pioneering work of Helmholtz and Rayleigh did not take into account an aspect of the physics of continuously excited musical instruments which is now recognised to be of crucial importance. This is the strongly non-linear relationship which is usually at the heart of the excitation mechanism. In the case of a reed woodwind instrument, for example, the relationship between the pressure difference across the reed and the rate of air flow through it is much more complicated than a simple linear proportionality. Because of this non-linearity, a sinusoidal driving pressure in the air column of the instrument can generate an air flow into the air column with a rich mixture of frequency components. Henri Bouasse (1929–30) was the first to recognize the significance of the non-linear excitation mechanism in wind instruments; his work in this field was later taken up and greatly extended by Arthur Benade (1976). A non-linear dynamical treatment of the bowed string has also been presented by McIntyre, Schumacher and Woodhouse (1983).
The classical theory of sound wave propagation rests on an approximation proposed by Euler, which is appropriate to waves of small amplitude. The original, non-linear partial differential equations of acoustics are replaced by linear ‘wave equations’. However, non-linear waves of air do occur in nature: they are an everyday occurrence in the flight of aeroplanes and also are easy to produce in the laboratory. A non-linear wave of condensation concentrates energy. If heard as a sound, such a wave seems to be louder after a short time. The linear acoustics, culminating in the work of Kirchhoff, cannot account for this ‘reinforcement’, for according to it, all parts of a wave travel at the same speed. Important research by Riemann (1858–9) and Hugoniot (1887, 1889) showed that Euler’s original, non-linear partial differential equations of gas dynamics, if not simplified by assuming the motion to be small from the outset, did predict reinforcement. Unfortunately, however, Euler’s theory predicts also that all plane waves will reinforce: every small condensation, not only those deliberately produced as music, would become audible. That is not surprising in a theory that allows sounds to continue unabated for ever because it neglects all internal causes of damping. Thus in the late 19th century two different theories, partly contradictory, were needed in order to explain the propagation of one and the same sound. This awkward artificiality persisted for a century. Finally Coleman and Gurtin (1965), working within a conceptual framework due in part to Walter Noll, succeeded in subsuming the two older theories as approximations to a master theory which displays the behaviour of plane waves of arbitrary amplitude in a body of dissipative material. They calculated a ‘critical amplitude’ determined by the conditions of the medium. Sounds feebler than that amplitude are damped from the start; sounds stronger than that are reinforced and hence are perceived as louder at first; of course they, too, are finally damped out by internal friction and the conduction of heat, according to Kirchhoff’s theory.
The human ear is an extremely sensitive organ, and the pressure variation caused by even a painfully loud sound is typically less than 0·1% of the mean atmospheric pressure. The small-amplitude approximation proposed by Euler is thus more than adequate for the description of almost all audible sound waves. Inside the mouthpiece of a trombone, on the other hand, the pressure can vary by around 10% when a very loud note is being played. It has been shown by Hirschberg and others (1996) that in these circumstances non-linear wave propagation in the trombone air column is musically important. Shock waves reminiscent of those generated by the ‘sonic boom’ of a supersonic aircraft are generated in the air column, giving rise to the rasping blare which characterizes the timbre of a loudly blown brass instrument.
There was one part of musical acoustics in which dramatic advances were made in the early 20th century, namely room acoustics, of obvious importance not only for the audience but also for the performing musician (see Acoustics, §I). Wertheim (1851), in a memoir on the vibration of sounds in air, had emphasized that although we spend our daily lives responding to sounds produced in rooms, cupboards, glasses, bottles and other receptacles of air, physicists in the mid-19th century were concerned solely with behaviour in infinite space and in cylindrical tubes with open ends. With his characteristic thoroughness in experiment, Wertheim proceeded to provide measurements to broaden this focus. His basic criticism of the experiments of his contemporaries, however, remained essentially valid until the contributions, between 1898 and 1917, of Wallace Clement Sabine.
Sabine’s collected papers on room acoustics present the results of a nearly single-handed achievement of importance in the physics of music. He brought the problems of reverberations, reflections, resonance and absorption in an auditorium to such a state of understanding that for the first time it became possible to design such a structure successfully on rational grounds, in advance of its construction. An outstanding example of his work was the design of the Boston Music Hall, which from its formal opening on 15 October 1900 was acclaimed for its fine acoustics. Directly from Sabine’s original ideas and experiments enormous development took place in the engineering and the physics of room acoustics, during the 20th century; by its close, sophisticated computer modelling techniques had made it possible for the acoustical consequences of architectural design choices to be explored and understood before the commencement of building.
Another important area which developed greatly in the 20th century was the study of the human response to particular types of sound. This field, in which physics, psychology and physiology overlap, is often described as psychophysics or psychoacoustics. Many of the findings of 19th-century acousticians were reports of their own perceptions, and were criticized by Mercadier and others for lack of objectivity. In the 20th century a careful methodology was developed, involving suitably designed tests on large numbers of subjects. To a large extent these tests answered the criticisms by providing data on average perceptions which were in principle subject to verification.
Great care is necessary in designing the protocols and techniques used in psychoacoustic testing. In 1924 Wegel and Lane reported on tests carried out using a faint probe tone to investigate the generation of aural harmonics. These are distortion products introduced by the ear when presented with a purely sinusoidal sound wave. The finding of Wegel and Lane that many such aural harmonics were clearly audible was later shown to be due to a fault in the experimental technique (see Clack, Edreich and Knighton, 1972). It is now accepted that although such distortion products are generated by the inner ear, they are normally at such a low level as to be musically insignificant.
The ability to record, manipulate and synthesize musical sounds has made it possible to design and carry out systematic and reproducible psychoacoustic tests which have helped to answer some of the questions which intrigued the acousticians of previous centuries. Ohm’s Law, stating that the human ear is insensitive to the relative phases of the partials in a complex sound, was conclusively disproved (see Plomp, 1976). In 1980 Carl Stumpf had suggested that the initial transient in the sound of a musical instrument was a crucial clue in identifying the instrument; this was verified by psychoacoustical studies carried out by Berger (1964) and by Grey and Moorer (1977).
At the end of the 20th century, the speed and memory capacity of computers continued to grow at a remarkable rate. As a consequence, it became possible to program into even a modest desktop computer a set of equations representing the theoretical behaviour of an instrument, and to obtain rapid predictions of the behaviour of the model. This proved to be of great value in testing theoretical predictions, since the equations are usually too complicated to be solved analytically. On the experimental side, laser-based vibration measuring equipment of very high accuracy has been added to the already impressive array of techniques being used to study both the detailed motion of musical instruments and the physiological responses of human and animal hearing systems.
Looking back over more than three centuries of sustained work on the physics of music, an acoustician at the end of the 20th century could not but marvel at the achievements of the past. Nevertheless, we are still a long way from being able to explain many of the features of musical sound and musical instruments which are of fundamental importance to musicians. To develop theoretical and experimental techniques to a level of accuracy that matches the sophistication of the musician’s perception is the great challenge facing musical acoustics at the dawn of the 21st century.
MersenneHU
J. Kepler: Harmonice mundi libri V (Linz, 1619; Eng. trans., 1997)
F. Bacon: Sylva sylvarum (London, 1627, 10/1676)
R. Descartes: Musicae compendium (Utrecht, 1650/R, 4/1695; Eng. trans., MSD, viii, 1961)
R. Hooke: Micrographia (London, 1665/R)
J. Wallis: ‘On the Trembling of Consonant Strings: Dr Wallis’s Letter to the Publisher concerning a New Musical Discovery’, Philosophical Transactions of the Royal Society, xii (1677), 839–42
I. Newton: Philosophiae naturalis principia mathematica (London, 1687/R, 3/1726; Eng. trans., 1729/R), ii/2, propositions xlvii–l; ed. A. Koyré and I.B. Cohen (Cambridge, 1972)
F. Robartes: ‘A Discourse concerning the Musical Notes of the Trumpet, and the Trumpet Marine, and of Defects of the Same’, Philosophical Transactions of the Royal Society, xvii (1692), 559–63
[B. Le B. de Fontenelle]: ‘Sur un nouveau systême de musique’, Histoire de l’Académie royale des sciences [1701] (Paris, 1704), 123–39
J. Sauveur: ‘Systême general des intervalles des sons’, ibid. [1701] (Paris, 1704), Mémoires, 297–364
[B. Le B. de Fontenelle]: ‘Sur l’application des sons harmoniques aux jeux d’orgues’, ibid. [1702] (Paris, 1704), 90–92
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B. Taylor: ‘De motu nervi tensi’, Philosophical Transactions of the Royal Society, xxviii [1713] (1714), 26–32
[B. Le B. de Fontenelle]: ‘Sur les cordes sonores, et sur une nouvelle détermination du son fixe’, Histoire de l’Académie royale des sciences [1713] (Paris, 1716), 68–75
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D. Bernoulli: ‘Theoremata de oscillationibus corporum filo flexibili connexorum et catenae verticaliter suspensae’, Commentarii Academiae scientiarum imperialis petropolitanae, vi (1732–3), 108–22
D. Bernoulli: ‘Demonstrationes theorematum suorum de oscillationibus corporum filo flexibili connexorum et catenae verticaliter suspensae’, ibid., vii (1734–5), 162–73
L. Euler: ‘De minimis oscillationibus corporum tam rigidorum quam flexibilium methodus nova et facilis’, ibid., vii (1734–5), 99–122; repr. in Opera omnia, II/x, ed. F. Stüssi (Leipzig, 1947), 17–34
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L. Euler: Tentamen novae theoriae musicae, ex certissimis harmoniae principiis dilucide expositae (St Petersburg, 1739); repr. in Opera omnia, III/i (Leipzig, 1926), 197–427
D. Bernoulli: ‘Excerpta ex litteris a Daniele Bernoulli ad Leonhardum Eulerum’, Commentarii Academiae scientiarum imperialis petropolitanae, xiii (1741–3), 1–15
D. Bernoulli: ‘De vibrationibus et sono laminarum elasticarum commentationes physico-mathematicae’, ibid., xiii (1741–3), 105–20
L. Euler: ‘Additamentum I de curvis elasticis’, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes (Lausanne and Geneva, 1744), 63–97; repr. in Opera omnia, I/xxiv (Leipzig, 1952)
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J.P. Rameau: Démonstration du principe de l’harmonie servant de base à tout l’art musical théorique et pratique (Paris, 1750/R)
D. Bernoulli: ‘Réflexions et éclaircissements sur les nouvelles vibrations des cordes exposées dans les Mémoires de l’Académie de 1747 & 1748’, Mémoires de l’Académie royale des sciences et des belles lettres de Berlin, ix (1753), 147–95
L. Euler: ‘De la propagation du son’, ibid., xv (1759), 185–264; repr. in Opera omnia, III/i (Leipzig, 1926), 428–507
J.-L. Lagrange: ‘Nouvelles recherches sur la nature et la propagation du son’, Mélanges de philosophie et de mathématique de la Société royale de Turin, ii (1760–61), 11–172; repr. in Oeuvres de Lagrange, i, ed. J.-A. Serret (Paris, 1867), 151–316
D. Bernoulli: ‘Recherches physiques, mécaniques et analytiques, sur le son et sur les tons des tuyaux d’orgues différemment construits’, Histoire de l’Académie des sciences de Paris (1762), 431–85
L. Euler: ‘Conjecture sur la raison de quelques dissonances généralement reçues dans la musique’, Mémoires de l’Académie royale des sciences et des belles lettres de Berlin, xx (1764), 165–73; repr. in Opera omnia, III/i (Leipzig, 1926), 508–15
L. Euler: ‘De motu vibratorio tympanorum, Novi commentarii Academiae scientiarum imperialis petropolitanae, x (1764), 243–60; repr. in Opera omnia, II/x, ed. F. Stüssi (Leipzig, 1947), 344–59
L. Euler: ‘Du véritable caractère de la musique moderne’, Mémoires de l Académie royale des sciences et des belles lettres de Berlin, xx (1764), 174–99; repr. in Opera omnia, III/i (Leipzig, 1926), 516–39
L. Euler: Eclaircissements plus détaillés sur la génération et la propagation du son, et sur la formation de l’écho, ibid., xxi (1765), 335–63; repr. in Opera omnia, III/i (Leipzig, 1926), 540–67
L. Euler: Sur le mouvement d’une corde qui au commencement n’a été ébranlée, que dans une partie’, ibid., xxi (1765), 307–23; repr. in Opera omnia, II/x, ed. F. Stüssi (Leipzig, 1947), 426–50
G. Riccati: ‘Delle vibrazioni delle corde sonore’, Delle corde ovvero fibre elastiche schediasmifisico-matematici (Bologna, 1767), 65–104
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