In Search of the Missing Fundamental: by Richard K. Jones
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Acoustic Properties of Timpani

The English physicist John William Strutt, 3rd Baron Rayleigh (1842-1919) also known simply as Lord Rayleigh was one of the first to document studies on the acoustics of timpani in the English language. In his seminal work The Theory of Sound (1877)/(2d ed. 1894), he wrote the following about kettle-drums.

In the case of the kettle-drums the matter is further complicated by the action of the shell, which limits the motion of air upon one side of the membrane. From the fact that kettle-drums are struck not in the center, but a point about midway between the center and the edge, we may infer that the vibrations which it is desired to excite are not of the symmetrical class. The sound is indeed but little affected when the central point is touched with the finger. Under these circumstances the principal vibration (1) is that with one nodal diameter and no nodal circle, and to this correspond the greater part of the sound obtained in the normal use of the instrument. Other tones, however, are audible, which correspond with vibrations characterized (2) by two modal diameters and no nodal circle, (3) by three nodal diameters and no nodal circles, (4) by one nodal diameter and one nodal circle. By observation with resonators upon a large kettle-drum of 25 inches diameter the pitch of (2) was found to be about a fifth above (1), that of (3) about a major seventh above (1), and that of (4) a little higher again, forming an imperfect octave with the principal tone. For the corresponding modes of a uniform perfectly flexible membrane vibrating in vacuo, the thoretical interavls are those represented by the ratios 1:34, 1:66, 1:83 respectively. 1

Taking into consideration that Lord Rayleigh did not have modern laboratory equipment with which to work and that the “English” timpani of his day were considered to be less than desirable than those made in Germany at the time, it is remarkable that his results were as accurate as they were.2 In fact, they may have been different if he had used timpani of a higher quality. Arthur H. Benade in his book Fundamentals of Musical Acoustics relates that in personal correspondence with noted English scholar P. R. Kirby, Kirby stated that the drums used in Lord Rayleigh’s experiments were “second hand and not properly tuned”. Perhaps this accounts for the interval of a major seventh in his analysis. 3

Since Lord Rayleigh’s time, the physics behind how timpani produce near harmonic spectra, ones that can be perceived as musical pitch, has been the subject of numerous scientific investigations and the topic of writings by many professional timpanists.

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