In Search of the Missing Fundamental: by Richard K. Jones
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Seebeck vs. Ohm

Germany in the middle of the 19th century was the breeding ground for the modern science of psychoacoustics. Many of the ideas in use today regarding the perception of pitch originated during this time because of research done by August Seebeck (1805-1849), Georg Simon Ohm (1789-1854) and Herman von Helmholtz (1821-1894). Their research in this area revolved around the investigation of complex tone; tone composed of several sinusoidal tones with the lowest being the fundamental and the frequencies being multiples of the fundamental (harmonics).

In 1841, Seebeck observed what he called periodicity pitch in a complex tone. He described the sensation of a pitch like sound being perceived without there being an actual physical component in the stimulus at the perceived frequency. This effect is also known as the missing fundamental or subjective fundamental. Seebeck worked with a mechanical siren and was able to manipulate the odd harmonics to the point where the sound perceived always seemed to include the fundamental even though the fundamental was missing or very weak. He concluded that not only does the fundamental determine the pitch, but also that the upper harmonics contribute to a sensation of a fundamental pitch.1 Figure 4a demonstrates Seebeck’s idea of periodicity pitch.

Fig. 4a

Periodicity Pitch Demonstration
courtesy of

Seebeck’s theory of periodicity pitch was not acknowledged by Ohm and in 1843 Ohm postulated his acoustic phase law which argued that similar to a Fourier series, the human ear does a real time spectral analysis where the frequency of the lowest component determines the pitch while the upper harmonics determine the timbre. Ohm’s law of hearing is a statement of the fact that the perception of the tone of a sound is a function of the amplitudes of the harmonics and not of the phase relationships between them. The perceived quality of the sound depends only on the power of its spectrum and was independent of the phase angles of its frequency components. 2 Figure 4b is an introduction to Fourier series and additive synthesis (see chapter 2 harmonic partials).

Fig. 4b

Intro to Fourier Series
courtesy of

Shortly after announcing his acoustic phase law hypothesis, Ohm found himself in a bitter dispute with Seebeck and the debate continued for twenty years in the scientific journal Annalen der Physik und Chemie. In 1863, Helmholtz virtually ended the debate by siding with Ohm when he adopted the Fourier (spectral) analysis view of pitch perception where musical sound is determined only by its partials. Helmholtz also argued that the periodicity pitch could be explained as a nonlinear difference tone generated at the auditory periphery, i.e., distortions introduced by the physics of the ear.3 The missing fundamental vs spectral debate was finally settled, well, for almost century anyway.

With the advent of modern laboratory, equipment, more investigation into periodicity pitch was conducted. In 1938, J. F. Schouten using optical equipment generated periodic complex tones that lacked a fundamental frequency.4 His demonstrations showed that Seebeck’s conclusion was essentially correct and that the missing fundamental effect or periodicity pitch could not be explained as a nonlinear difference tone as Helmholtz had argued. Schouten went on to publish his theory which has become to be known as the residue theory of pitch in 1940.5 The missing fundamental effect was again recognized.

As with any good science, Schouten’s pitch residue theory was scrutinized and in 1962, R. J. Ritsma discovered that the pitch residue theory had its limitations too. It worked fine for resolving harmonics in the lower limits of a complex tone but it could not account for why pitch consisting of only harmonics above a certain point did not create residue pitch. Ritsma called this the existence region of the tonal residue and found that this existence region extends to only a certain range of harmonics which are able produce residue pitch. Other harmonics outside of this existence region which the inner ear should be able to resolve but doesn’t, contradicts the very premise of pitch residue theory.6

Within about a hundred years or so since Seebeck had introduced his periodicity pitch theory, technical developments and the possibility to actually generate complex harmonic tones with a clearly defined number of harmonics allowed for the systematic investigations of the missing fundamental effect. The missing fundamental effect is also known as periodicity pitch (Seebeck 1841,), residue pitch (Schouten 1938), virtual pitch (Terhardt 1972-74)7. Studies have shown that the missing fundamental phenomenon can be perceived with a harmonic spectra of five8, three9, and even only two10 consecutive harmonics. Other research has also shown that the perception of a missing fundamental can be perceived when a sine tones of different frequencies are introduced separately to each ear via head phones.11 The data from these studies indicate the perception of the missing fundamental is not caused by distortions introduced by the physics of the ear as Helmholtz suggested, but rather it is a function of the central pitch processing system of the brain.

Today, there remain essentially two schools of thought regarding human pitch precepts and these philosophies are based on the theories of Seebeck, Ohm and Helmholtz. The first school follows along the lines of Seebeck and propose a holistic perception of sound or synthetic mode (a term coined by Helmholtz) where harmonics are synthesized or fuse into a single sound. The second school follows Ohm’s idea of a real-time Fourier spectral analysis or analytical mode (another term coined by Helmholtz) where the sound is broken down and perceived as a spectral singularity pitch, i.e. a single pitch is perceived but the individual harmonic components are detectable. These two concepts are also know as virtual pitch and spectral pitch respectively; terms coined by Ernst Terhardt.12

The conceptual distinction between spectral pitch and virtual pitch is that both spectral pitch and virtual pitch ultimately are dependent on aural Fourier analysis; however, while any spectral pitch is conceived as immediately corresponding to a spectral singularity, virtual pitch is modeled as being deduced from a set of spectral pitches on another stage of auditory processing.13

The following pages are but a brief journey into how the human auditory system functions, but they do set the stage for how timpani pitch may or may not be perceived by the player or listener.