Shape or Volume?
In speaking of the bowl Benade states:
The varying degree in perfection in preserving the correct air-to-membrane relationship is what explains the observation by musicians that every drum plays best at one particular frequency in its range of usability. 20
In 1984, perhaps some hope was given to the timpanists’ belief of “the bowl shaping the sound” with the publication of the paper, Effects of air loading on timpani membrane vibrations by Richard S. Christian, Robert E. Davis, Arnold Tubis, Craig A. Anderson, Ronald I. Mills, and Thomas D. Rossing. This paper dealt with theoretical calculations (Green function method) of timpani modal frequencies and decay times of timpano bowl enclosures with varying volume. Their conclusion:
Although the timpani-membrane frequency shifts are more sensitive to total kettle volume then they are to the details of kettle geometry, the specifics of kettle shape may be important for the fine details of the timpani spectrum (e.g., the frequencies and relative strength of the partials other than the fm1), for the instrument response times, etc. This may well account for the various preferences timpani players have concerning kettle geometries. 21
In 1988, Robert Eugene Davis in his Ph.D. dissertation Mathematical modeling of the orchestral timpani, used Green function and boundary integral methods to investigate the effects of air loading and kettle shape on the vibrational spectrum. He mathematically calculated timpani modal frequencies for five different kettle shapes and concluded:
The timpani spectrum depends weakly on the detailed kettle shape as long as the correct volume of the kettle with regard to the size of the membrane has been met. 22
Rossing’s analysis is simple and to the point:
Is kettle shape important in determining timpani sound? No. Careful studies indicate that the shape of the timpani kettle is quite unimportant in determining timpani sound provided that the volume is kept in correct range. 23
One must bear in mind that Rossing’s definition of timpani sound is based on the harmonicity of the preferred modes and not the overall sound of the timpani, which may include inharmonic partials (collateral color) for the purpose of color and projection.
Fleischer and Fastl investigated the influence of bowl vibrations, and bowl shape by using a single 73cm timpano frame and interchanging three different bowl shapes with different finishes. Their study showed :
In comparing the determined oscillation frequencies with the results of harmonic frequencies in the audible range of the timpani sound, there is no indication that the bowl actively contributes to the sound radiation. There are no components with the same frequencies of sound as that of the bowl. A constructive interference of sound is therefore excluded.
The volume of the air enclosed influences the intervals of the partial tones, to an especially high extent the frequency of the 01 tone. To keep this inharmonic tone inaudible, the kettle should be small. With respect to the intervals of main tone, quint and octave, however, there is an optimum volume. 24
Why does bowl shape not affect the sound with respect to the near-harmonic partials of the preferred modes? Is there an explanation? Rossing explains.
These results may surprise some timpanists, since articles in the literature frequently refer to “shaping” the sound by using kettles with hemispherical, parabolic or other shapes. A simple fact that helps explain why kettle shape is unimportant is that the the sound wavelengths are so much larger than the kettle dimensions. (At 140 Hz, for example the wavelength is 2.5 meters, and even at 44o it is about 0.8 m). Thus the mode frequencies of the enclosed air, which depend on kettle volume are virtually unaffected by the shape of the kettle. 25
The volume of air inside of the bowl effectively fine tunes the partials created by the preferred modes helping to bring them into a near-harmonic relationship. Most modern timpani bowls seem to have the right volume to optimize the harmonicity of the principal partials, but Fleischer and Fastl believe that there is an optimum volume based on the diameter of the head.
To date, partials other than that of the preferred modes (collateral color) has not been a subject of scientific investigation with respect to exactly what is the overall sound of the timpani spectrum; the complete timpani sound that timpanists hear. More studies in this area are certainly needed.
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