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 Hermann von Helmholtz (1821-1894). Their research in this area revolved around the investigation of a complex tone, a sound composed of several sinusoidal components, 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. His key finding was that the upper harmonics contribute to a sensation of a fundamental pitch, not because the fundamental is physically present, but because the pattern of harmonics repeats at the fundamental’s rate. The pitch is determined by this periodicity, not by any single spectral component.1 Figure 4a demonstrates Seebeck’s idea of periodicity pitch.
Fig. 4a
Periodicity Pitch Demonstration
courtesy of theflyingrobby
In 1843, Ohm formulated his law of hearing, which held that the human ear can be modeled as performing a spectral analysis, decomposing a complex tone into its sinusoidal components, where the frequency of the lowest component determines the pitch while the upper harmonics determine the timbre. In modern terms, Ohm’s law of hearing states 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. This means timbre is determined by which harmonics are present and how loud each one is, not by when each harmonic starts. 2 Figure 4b is an introduction to Fourier series and additive synthesis, techniques for building up complex tones from simple sinusoidal components (see Chapter 2: Harmonic Partials).
Fig. 4b
Intro to Fourier Series
courtesy of tummysak
In 1843, Ohm entered a heated debate with Seebeck that would continue for two decades in the scientific journal Annalen der Physik und Chemie. In 1863, Helmholtz revisited the debate and sided largely with Ohm, adopting a Fourier (spectral) analysis view of pitch perception, musical sound is determined primarially by its partials. Helmholtz also argued that 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 This effectively sided with Ohm in the historical debate, though the question would remain unsettled in certain respects for nearly a century.
Advances in laboratory equipment enabled further investigation into periodicity pitch. In 1938, J. F. Schouten, using optical equipment, generated periodic complex tones that lacked a fundamental frequency.14 His demonstrations showed that Seebeck’s conclusion was essentially correct, and that the missing fundamental effect could not be explained by peripheral difference tones alone (Helmholtz’s proposed mechanism). In 1940, Schouten published his theory, now known as the residue theory of pitch. 15 The missing fundamental effect was again acknowledged as a genuine phenomenon.
Schouten’s residue theory was scrutinized, and in 1962, R. J. Ritsma discovered that it 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 corresponding to harmonics above a certain rank did not create residue pitch. Ritsma called this the existence region of the tonal residue and found that it extends to only a certain range of harmonics which are able to produce residue pitch. Other harmonics outside of this existence region, may still be audible as spectral components, yet do not yield a strong residue pitch, an outcome that complicates the very premise of the residue theory.17
Roughly a century after Seebeck introduced periodicity pitch, technical developments and the ability to generate complex harmonic tones with a clearly defined number of harmonics enabled systematic investigation 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).21 Studies have shown that the missing fundamental phenomenon can be perceived with a harmonic spectrum of as few as five16, three18, and even only two19 consecutive harmonics. Other research has also shown that the perception of a missing fundamental can occur when sine tones of different frequencies are introduced separately to each ear via headphones.20 The data from these studies indicate that the perception of the missing fundamental is not caused by distortions introduced by the physics of the ear 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 perception, based on the theories of Seebeck, Ohm, and Helmholtz. The first school follows in the tradition of Seebeck and proposes a holistic, synthetic perception of sound, a synthetic mode (a term Helmholtz coined), where harmonics are synthesized or fuse into a single sound. The second school follows Ohm’s idea of real-time spectral analysis, an analytical mode (another term Helmholtz coined) where the sound is broken down and the individual harmonic components are detectable, even as a single pitch is perceived. These two concepts are also known as virtual pitch and spectral pitch, respectively, terms coined by Ernst Terhardt.21
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.28
- Spectral pitch is conceived as immediately corresponding to a directly resolved spectral component, a specific harmonic that the ear can isolate.
- Virtual pitch is modeled as being deduced from a set of spectral pitches at a higher stage of auditory processing, the brain infers a fundamental pitch from the pattern of resolved partials.
At its core, the Seebeck–Ohm debate is not merely historical—it points directly to one of the most important pitch phenomena in all of music: a listener can hear a pitch that is not present as a physical spectral component. This is the crux of Seebeck’s periodicity pitch and the reason Helmholtz felt compelled to explain it. The next page focuses on this phenomenon directly through listening demonstrations and examples, introducing what is now commonly called the missing fundamental effect, and why it is so relevant to timpani pitch.