In Search of the Missing Fundamental: by Richard K. Jones
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Rossing Et al.

Rossing also observed that perhaps the most significant contribution of the bowl is to act as a baffle, which acoustically separates the top and bottom sides of the head. This increases the radiation efficiency, which decreases the decay times, especially the modes of the lower frequencies. This may seem counter intuitive to want to decrease the decay times of the lower modes (especially the preferred modes), but when some of those lower modes detract from the harmonicity, this can be a desirable trait.

Fig. 3l

Figure 3l charts the decay times of the lower mode frequencies of a timpano both with and without the bowl. Without the bowl, all of the modes are not very efficient at radiating their energy; therefore they have longer decay times, which seems desirable. However, this includes the inharmonic lower concentric modes as well as the more harmonic preferred modes. With a bowl, most of the lower modes radiate energy more efficiently and consequently they decay much faster. When a timpano is struck a quarter of the distance between the edge and the center, the inharmonic modes will radiate their energy much more efficiently and decay faster leaving the more harmonic preferred modes to dominate the sound spectrum.

Fig. 3m

Figure 3m is a sound spectrum graphing the amplitude, frequency and decay time of a 26 inch Ludwig Professional Series timpano (with a bowl). The more inharmonic concentric modes have less amplitude and faster decay time than do the diametric preferred modes (1,1), (2,1), (3,1), (4,1) and (5,1). Note the quick decay time of the principal tone mode (1,1) leaving modes (2,1), (3.1) and (4,1) to carry the pitch.8

Fig, 3n

Figure 3n (courtesy of Fleischer & Fastl) is a sound spectrum graphing the amplitude, frequency and decay time of a 73.66 cm Kolberg timpano. Again, the principal tone mode (1,1) decays much faster than the secondary modes (2,1), (3,1), (4,1).

Due to the effect of the baffle created by the bowl, not only are the lower concentric modes dampened, but so also is the principal tone somewhat. This can lead to what timpanists refer to as the overbearing fifth and pitch creep in the spectrum. When tempering timpani heads, it is imperative to adjust the frequency of the principal tone at each tuning lug so that it includes many strong  near-harmonic partials. When the frequency of the principal tone is not consistent from lug to lug, the overall strength of perceived pitch is severely diminished and permutations of the more audible mode 2,1 will tend to dominate the spectra. An overbearing fifth as well as a perceived pitch shift is likely to occur once the principal tone begins to decay. When the tension of the head is adjusted in such a manner that the secondary preferred modes are focused on creating  the virtual pitch of the principal tone (mode 1,1), mode 2,1 then becomes less pronounced/overbearing and the overall pitch will be perceived as being more harmonic in nature. On occasion, the missing fundamental can be perceived. See Pleading the Fifth.

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