Physiological Acoustics and Combination Tones

Abstract

Apart from a number of annual reports on the state of theoretical acoustics submitted to the Berlin Physical Society, Helmholtz’s first substantial exploration into the field of musical acoustics dealt with combination tones. As Helmholtz discovered, this is a physiological phenomenon that turned out to be of central importance for studies on just intonation. According to the law of the independence of sound waves – a law that reduces the considerations of complex cases of sound production and reception to those of simple sound waves – each component sound wave of a vibrating object produces a sound wave that can be regarded as traveling independently of every other component of the composite sound of the vibrating object. Although the law holds rigorously only in cases where the vibrations of all the parts of the vibrating body are small in dimension, this is the case for most of the string and air-column systems utilized in musical instruments. It is not the case where the intonation hinges on combination tones and the fine-tuned analysis of consonance, dissonance, beats, or other “interruptions of harmony” as Helmholtz referred to them. Alert to the idea that combination tones are markedly sensitive to inaccuracies of intonation, Helmholtz was led to perceive that the vibration of small non-negligible dimensions represents the essential anchor point for research on the problem of intonation on fixed-tone keyboards.

Apart from a number of annual reports on the state of theoretical acoustics submitted to the Berlin Physical Society, Helmholtz’s first substantial exploration into the field of musical acoustics dealt with combination tones. As Helmholtz discovered, this is a physiological phenomenon that turned out to be of central importance for studies on just intonation. According to the law of the independence of sound waves – a law that reduces the considerations of complex cases of sound production and reception to those of simple sound waves – each component sound wave of a vibrating object produces a sound wave that can be regarded as traveling independently of every other component of the composite sound of the vibrating object. Although the law holds rigorously only in cases where the vibrations of all the parts of the vibrating body are small in dimension, this is the case for most of the string and air-column systems utilized in musical instruments. It is not the case where the intonation hinges on combination tones and the fine-tuned analysis of consonance, dissonance, beats, or other “interruptions of harmony” as Helmholtz referred to them. Alert to the idea that combination tones are markedly sensitive to inaccuracies of intonation, Helmholtz was led to perceive that the vibration of small non-negligible dimensions represents the essential anchor point for research on the problem of intonation on fixed-tone keyboards.

It was long known that a vibrating string or column of air produces not only a tone corresponding to its fundamental frequency f, but also to a series of harmonics of frequency 2f, 3f, 4f….; that two tones of slightly different frequencies, sounded together, produce beats resulting from interference. Around the middle of the eighteenth century it was discovered that when two tones are sounded together a third tone phenomenon (later referred to as a difference tone) could be detected by a keen ear under normal conditions of music-making. Early descriptions of the third tone phenomenon occur in a treatise on composition (1745–1747) by Georg Sorge (1703–1778) and in a treatise on the acoustical foundations of harmony (1754) by the violinist virtuoso Giuseppe Tartini (1692–1770) who claimed to have discovered third tones (terzo suono) as he referred to them, while experimenting with double stops on the violin. Scientists at the time, Lagrange and Thomas Young for instance, postulated a beat theory for these difference tones and suggested that the beat frequency is large enough to be recognized as a tone itself. What was certain was that two tones with frequencies f₁ and f_2, when sounded together, produce a barely distinguishable third tone with frequency f₁–f₂ that does not belong to the harmonic series.

By the beginning of the nineteenth century the phenomenon of combination tones began to attract the attention of physicists such as Wilhelm Weber (1804–1891).¹ His paper of 1829 on Tartini tones led to a wealth of empirical data and a number of attempts to construct a descriptive theory to account for them. Weber supported the beat theory of difference tone, as Young and Lagrange had done earlier, but suggested that more than one beat tone was audible. Weber maintained that “once the physical basis of the Tartini tone is known, it becomes easy to determine the pitch of the Tartini tone in advance for each case, no matter what the ratio of the two beating tones might be. If such tones are equally possible, it is easy to determine which of them will be heard.”²

After reading Weber’s paper, the Finnish physicist Gustav Gabriel Hällström (1775–1844) suggested in 1832 as a challenge to Weber’s beat theory that not only the fundamental frequencies of tones but their harmonic partials should produce difference tones such as 2f₁–f_2, f₁–2f₂, 3f₂–f₂, etc.³ The challenge of demonstrating that the mechanism of the human ear, with its intricately constructed elastic and solid components, is able to interpret the observations as different real tones became for Hällström a matter of consuming interest. He calculated that second order tones should result from a blending of the first order combination tones (the Tartini tones) and the original generating tones; that third order tones should result from the second order tones, and so on. The extensive tables that Hällström drew up in comparing the data on observed combination tones with the predictions of his theory and other data from the literature were impressive, but the tones he predicted could not be observed at the time.⁴

In 1843 the Bavarian physicist Georg Simon Ohm (1789–1854) conducted a series of experiments with the polyphonic siren that led him to announce what came to be known as Ohm’s acoustical law.⁵ Like Ohm’s famous electrical law that relates current, voltage, and resistance in an electrical circuit, Ohm’s acoustical law (formulated in 1826) was structured on the work of the mathematical physicist Joseph Fourier (1768–1830). Fourier, the undisputed leader of the nineteenth century French analytical school, had shown in connection with studies on the propagation of heat in 1822 that any function can be expressed as the sum of a series of sines and cosines.⁶ When applied to the phenomena of sound, as it was by Ohm in 1843, the Fourier theorem specifies that any motion of air that corresponds to a composite group of musical tones is capable of being analyzed as a sum of harmonic vibrations; that each single harmonic vibration corresponds to a simple tone sensible to the ear having a simple specific pitch. According to Ohm’s acoustical law, the ear recognizes only sinusoidal waves as pure tones and is able, automatically, to perform an analysis of periodic sound waves into its component parts. This means that however complicated the motion producing the sound may be, the total sound is to be regarded as consisting of many non-interfering sinusoidal waves that are acting as if each wave is produced by itself.

Ohm’s acoustical law received little attention until it was so to speak “rediscovered” by Helmholtz and applied to the characterization of tone color (Klangfarbe) or timbre. He demonstrated that it was the specific combination of sinusoidal components that gives a particular sound its auditory character. In reference to the notion of Klangfarbe Helmholtz stated the Fourier theorem as follows: “Any given regular periodic form of vibration can be produced by the addition of simple vibrations having pitch numbers which are once, twice, thrice, four times, etc., as great as the pitch numbers of the given motion.”⁷

In 1856 Helmholtz presented the Berlin Academy of Sciences with the results of his own experimental investigations and theoretical interpretations on combination tones.⁸ Although the 1856 papers provide ample evidence of Helmholtz’s competence in the physics of vibrating systems, the cogency of his approach to characterizing and clarifying the phenomena of combination tones demonstrates in the main how firmly coupled his scientific thinking at this time in his career was with his wider concerns in the physiology of sensation. He had just completed the first volume of his handbook on physiological optics. With the decision to enlarge on combination tones and his perspective on the physiological studies of sensation by moving from optics to acoustics, Helmholtz became totally engrossed in problems unique to music theory. These interests were to capture his attention for more than a decade.

In the combination tone paper – his first research undertaking in acoustics – he was able to demonstrate the existence of the higher order difference tones that Hällström had predicted in 1832. The demonstration that Hällström’s difference tones are produced by the harmonics of the fundamental was crucial in the history of music theory because it signified that Ohm’s acoustical law remained intact. At the same time the experiments that Helmholtz had designed to detect difference tones revealed the existence of another series of combination tones that he referred to as “summation tones.”

The investigations of Weber, Ohm, and Hällström that have been mentioned thus far provide the point of departure for pursuing the nature of the experimental investigations and theoretical deliberations of Helmholtz’s own work on combination tones. He appraised the situation as follows:

Given m and n, whole numbers having no common divisor, it long has been known that two tones with frequencies mλ and (m + 1)λ produce the combination tone λ. On the other hand, for two tones with frequencies mλ and nλ, W. Weber and M. Ohm put forward the idea in a general way that the combination tone, likewise, has a frequency λ, whereas Hällström designated (m − n)λ as the first combination. At the same time he postulated a number of higher-order combination tones with frequencies (2n − m)λ, (3m − 2n)λ, etc. The higher-order combination tones were said to form lower-order combination tones with the original tones…. Poggendorff (then) correctly raised the question whether the so-called higher-order combination tones might not be the higher overtones [Nebentöne] that are present in practically all musical instruments.⁹

At the level of theory, several authors had offered explanations of the complexity of combination tones phenomena by assuming that the tones received by the ear are made up of a series of discontinuous impulses, the interpretation of which requires the ear to be endowed with special non-physical properties. For Helmholtz the physiologist, who in all of his deliberations had the basic principles of the physical sciences close to hand, the invention of subjective properties to explain a natural phenomenon such as the production of combination tones in the human ear was unacceptable. The suggestion in fact occasioned a blunt response: “I therefore will allow myself to present the Academy with a new explanation that is based entirely on known mechanical laws [emphasis added] that make it unnecessary to endow the human ear with any special properties.”¹⁰ Based on the notion that the principle of undisturbed superposition of oscillating motions is valid only so long as the motions are small, Helmholtz was able to demonstrate that combination tones should occur and be detectable when the motions become large enough for the square of the displacements to influence the motion. He calculated that among the oscillating parts of the ear only the tympanum (the eardrum) is singularly asymmetric and, with its hammer drawn inward, is able by displacement to influence the motion. “I therefore believe,” he emphasized, “that I am able to put forward the conjecture that it is the characteristic form of the tympanum that determines the formation of combination tones.”¹¹

With the aid of the siren, and by employing tuning forks and resonators to isolate and reinforce frequencies, Helmholtz was able to demonstrate the existence of the higher-order difference tones that Hällström had predicted. At the same time he discovered that under optimal experimental conditions difference tones resulting from the sum of the frequencies of two primary (generating) tones could be detected, if but faintly.

Thereby I discovered an until now unknown class of combination tones that do not fit the existing theories. I also discovered that there are objective combination tones that arise independently of the human ear. Finally, then, one is able to offer a theory of combination tones, very different from the hitherto existing theory, in which no special properties of the hearing nerves [Hörnerven] need to be postulated and which more adequately embrace all of the now known facts.

I will designate these new tones with the name summation tone [Summationstöne] in contrast to the already mentioned and long known tones that we can call difference tones [Differenztöne] because their frequencies are equal to the difference in frequency of their primary or third lower-order combination tones. I first became aware that such tones might exist and tried to hear them with resonators activated by tuning forks. I succeeded in doing so but with great difficulty because the tones of the tuning forks have only moderate strength and the combination tones became distinct only with stronger primary tones. The summation tones are weaker than the first-order difference tones and it therefore takes much practice and attentiveness to hear them at the weaker strengths of the primary tones.¹²

Helmholtz’s search for a theory to account for the various types of combination tones – one of which, the difference tones, had been recognized in its simplest form in the middle of the nineteenth century, and the other of which, the summation tones, he had discovered – eventually led him to conclude that it was not necessary to invoke non-mechanical principles to explain their existence. The anchor points that he had invoked to reach that conclusion included Ohm’s acoustical law, the Fourier theorem, and the recognition that it was necessary to reckon with the fact that the motions of the various components of the inner ear lie outside the limits of validity of the principle of independence of motion for sound waves. He concluded:

From what has been said it follows that one need not necessarily look for the cause of combination tones in the modes of sensation of the hearing nerves, but that for two simultaneously sounded tones of audible strength, the combination tones correspond, in the usual (that is, mechanical) way, to real vibrations of the tympanum [Trommelfelles] and the ossicles [Gehörknöchelchen]. Accordingly, the combination tones do not possess a mere subjective existence but would be able to exist objectively if only, to begin with, in the vibrating parts of the ear.¹³

In recent times it has been shown that the observation and explanation of combination tones are more significant in the development of musical acoustics and also are a more complicated physiological problem to unravel than anyone in the nineteenth century was in a position to realize in 1856 when Helmholtz tackled the problem. Combination tones are associated with non-linear systems, i.e., systems that introduce distortion during the transmission of sound. This occurs when the cochlea of the inner ear is presented with two tones having sufficient and similar intensities, as well as a frequency difference to be audible. “Non-linear” signifies in this context that the combination of two sounds with intensities a and b is not a simple linear sum, a + b, but is given by a formula involving powers of a and b. The linear formula is valid only for small intensities of sound so that combination tones normally are heard only when the original tones are sufficiently loud. The frequency of a combination tone is either the difference (difference tones) or the sum (summation tones) of the frequencies of the two primary notes or their multiples. Difference tones are readily realized in normal conditions of music-making. Summation tones, by contrast, are heard only under special conditions.

In conclusion to this section on Helmholtz’s experimental investigations on combination tones, it is significant finally to evaluate his reasons for highlighting this singularly complex phenomenon as a subject worth pursuing in some detail in connection with music. Combination tones, as Helmholtz discovered, are uniquely sensitive to inaccuracies of pitch and are therefore directly connected with intonation. The critical point at stake here is that the false combination tones of tempered intonation that show up as dissonances are more readily identifiable than they are for just intonation. The differences in terms of “pleasantness,” as Helmholtz recognized, are greatest in the higher octave of the scale because there the false combinational tones of the tempered intonation are more readily observable, and the number of beats for equal differences of pitch, as Helmholtz remarked, become larger and more pronounced in their roughness (Rauhigkeit).