The Missing Fundamental
The subjective tones (combination and difference tones) which are produced by the beating of the various harmonics of the sound of a musical instrument help to reinforce the 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 the successive harmonics constitute subjective tones which are at the same frequency as the fundamental and therefore reinforce the sense of pitch of the fundamental note being played. If the lower harmonics are not produced you still hear the tone as having the pitch of the nonexistent fundamental because of the presence of these beat frequencies. 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.
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.17
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 vary narrow series of quasi-harmonic partials with a missing fundamental. These five or six partials of an inharmonic overtone series amount 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 only produce a strong/clear sense of pitch when this series of quasi-harmonic partials is coaxed into a harmonic sequence by the weight of the earth’s atmosphere (air loading), and by the careful adjustment (tempering) of the head. 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 Universitat Munchen).15 Terhardt’s theory extends 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.
A study of Virtual Pitch and timpani can be found in the next section.