The subjective tones (combination and difference tones) which can be produced by the interaction of the various harmonics of the sound of a musical instrument can, in some cases, add low-frequency “distortion products” that may reinforce the sense of pitch of the fundamental frequency. Most musical instruments (timpani are not one) produce a fundamental frequency plus several higher tones which are whole-number multiples of the fundamental. The beat frequencies between successive harmonics can produce difference-tone components at frequencies related to the fundamental, depending on the degree of nonlinearity in the ear (and/or playback chain). However, and this is crucial, the missing fundamental (virtual pitch) effect does not require subjective tones: listeners can still perceive a pitch at the fundamental even when no physical energy at that frequency is present and even when distortion products are negligible. This is called the missing fundamental effect. It plays an important role in sound by preserving the sense of pitch (including the perception of melody) when sound loses some of its lower frequencies.
This demonstration explores the relationship between the frequency content of a musical note and the pitch perceived by listeners. Musical notes are complex tones consisting of a fundamental frequency and higher harmonics (also known as partials) that are integral multiples of the fundamental frequency. The particular mix of partials is part (but only part!) of what gives different musical instruments their individual character. The pitch of the note is related to the fundamental frequency of the complex tone. However, the pitch of the note remains unchanged even if this fundamental frequency is removed.
Courtesy of Dr. William Robertson
The above sound file consists of a complex tone made up of a fundamental and nine higher harmonics. The first tone heard has all the frequencies; the second tone has the fundamental removed but maintains all of the higher harmonics. Each successive tone sequentially removes the lowest harmonic. Notice that although the character of each note changes, the pitch remains the same.
The pitch of a note is often determined by the higher quasi-harmonically related partials rather than the lowest partial present. In the example below, all frequency components below 250 Hz of the second timpani stroke have been removed. The perceived pitch is unchanged, though there is a considerable loss in depth or body of the resulting sound.29
The concept of the missing fundamental is important in the understanding of timpani sound production because the pitch portion of the spectrum created by timpani is comprised of a very narrow series of quasi-harmonic partials with a missing fundamental. These five or six partials of an inharmonic overtone series can correspond approximately to partials 2 (two) through 7 (seven) of the harmonic series. (see Preferred Modes & Timpani Sound Spectra)
The Six Quasi-Harmonic Overtones of Timpani Pitch
Preferred Modes 1,1 through 6,1
Timpani often produce a strong/clear sense of pitch when this series of quasi-harmonic partials is coaxed into a more nearly harmonic sequence by air loading (the surrounding air mass), in conjunction with the careful adjustment (tempering) of the head, and the volume of air contained within the bowl. When the accurate tensioning of the head is balanced with the current air loading of the head (which varies with air density), the missing fundamental effect can be produced giving a strong sense of pitch, and body to the sound of the instrument.
The missing fundamental effect has spawned many theories, but perhaps the most prominent is that of Virtual Pitch. The theory of Virtual Pitch was developed in 1969-1970 by Prof. Ernst Terhardt (Technische Universität München).22 Terhardt’s theory extends Giuseppe Tartini’s terzo suono (third sound), J. P. Rameau’s theory of fundamental bass, August Seebeck’s theory of periodicity pitch and J. F. Schouten Pitch of the Residue theory; all deal with the phenomenon of the missing fundamental and the way pitch can be inferred from partial patterns.
The missing fundamental effect explains what can happen perceptually: a listener can hear a stable pitch even when the spectrum contains little or no physical energy at that fundamental frequency. But it does not, by itself, explain how the auditory system arrives at that pitch, why the percept is strong in some situations and weak in others, or why different listeners sometimes report different “pitch centers” from the same complex tone.
To address those questions, psychoacousticians developed more complete models of pitch extraction. One of the most influential is Ernst Terhardt’s theory of Virtual Pitch, which formalizes how the brain can infer a fundamental from a pattern of partials—even when that fundamental is missing. This is especially relevant for timpani, where the perceived pitch often depends on a narrow set of quasi-harmonic partials and where the “missing fundamental” can become a practical tool for evaluating head balance. The next page applies Virtual Pitch directly to timpani sound, including how it relates to the principal tone, the preferred modes, and what players can measure with tuners and spectrograms.