In Search of the Missing Fundamental: by Richard K. Jones
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Inharmonic Motion

The sound spectra produced by timpani are not generated by vibrating columns of air or vibrating strings, but rather from vibrating circular membranes. Air columns and strings vibrate with an overtone series that is harmonic (integer multiples of the fundamental frequency). Vibrating circular membranes do not vibrate with a harmonic series yet they do generate an overtone series; this series not harmonic. Consequently, the motion from a vibrating circular membrane is inharmonic. How then do timpani produce harmonic pitch? The following information, from the Georgia State University HyperPhysics website is an explanation of how timpani work.

The Timpani 4
Courtesy of Georgia State University HyperPhysics

    1. A timpano has a round head stretched over a sealed enclosure. The tension may be altered by means of a foot pedal which actuates tensioning elements tightening the head. The pedal connects to the lugs which control the tension in the membrane.
    2. The round head of a timpano (membrane) can vibrate in a large number of vibrational modes. The fundamental is not preferred; it is a damped, muffled sound and does not produce as pleasing a sound as when the head is struck a few inches from the rim. At this point, the timpano’s fundamental mode is not excited.
    3. The player chooses a striking point which will emphasize the preferred modes of the circular membrane. The resulting sounded frequencies are further influenced by the enclosed air cavity.

Most timpanists are taught to strike the drum in a manner which is said to allow the fundamental tone of the drum to ring as much as possible. Many instrument manufacturers speak of the rich fundamental tone that their drums produce. If the above statements hold true that a timpano’s fundamental is damped, from where then does the sound come? Why is the fundamental not preferred? What are these preferred modes?

These are all excellent questions. The answers can be found by first defining what an ideal circular membrane in fact is, and then by taking a close look at some of the vibrational modes of an ideal circular membrane.

The mid-19th century English physicist Lord Rayleigh (1842-1919) who was one of the first to research vibrating circular membranes as they pertain to timpani defined a theoretical membrane as:

a perfectly flexible and infinitely thin lamina of solid matter of uniform material and thickness which is stretched in all directions by a tension so great as to remain sensibly unaltered during the vibrations and displacements contemplated. 5

Dr. Dan Russell, The Pennsylvania State University has a more modern definition of what an ideal circular membrane is:

an absolutely round membrane, infinitely thin, perfectly flexible, completely homogeneous, evenly and uniformly tensioned where the outer circular edge of the membrane constitutes a fixed boundary condition in an “in vacuo” state (in a vacuum). This type of membrane exists in theory only. 6

For vibrating timpani heads, the conditions are somewhat different simply because one does not play timpani in a vacuum, but the mode shapes are almost the same as those of the ideal circular membrane. Investigating these mode shapes and then comparing them to the string and air column’s harmonic series will help shed some light on the aforementioned questions regarding the preferred modes and fundamental of the timpani sound spectra. It will also begin to bring to light how timpani are able to generate sound which the human ear perceives as having a sense of pitch even though they do not vibrate with a harmonic series.

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