Air Loading
I refer to these air loaded modes as the harmonic preferred modes because the ratios set up a harmonic series minus the fundamental (a.k.a. missing fundamental) starting at the principal tone mode (1,1) up to what would be the seventh partial. This in itself is an ideal set of preferred modes because timpani cannot reach true harmonicity, only quasi or near- harmonicity. However, it is a standard for which one should strive when tempering heads. One may be able to reach harmonicity on a single note when tempering a head, but harmonicity will change when the pitch of the drum is changed due to factors such as the different resonant air modes in the bowl and the elasticity of the head as well as poor mechanical tolerances of the instrument or changes in barometric pressure and air density.
Figure 3f is a comparison chart showing the impact of air loading on an ideal membrane if the preferred modes were to be brought into a pure harmonic relationship. With a fundamental of A2 110 hz, when placed under the influence of air loading, the principal tone mode (1,1) would drop by 518 cents or the interval of a + perfect fourth. Mode (2,1) would be lowered by 336 cents or the interval of a + minor third. Modes (3,1) and (4,1) would be lowered about 200 cents or the interval of a major second and mode (5,1) about a half of a semitone. In the case of timpani, other factors such as the bowl have an influence as well. N.B. The actual frequency of mode (0,1) would be influenced as well. The stiffness of the air contained inside the bowl raises the frequencies somewhat of the lower concentric modes (0,1), (0,2), (0,3) etc., but since these modes do not contribute to the desired sound spectrum, they will not be displayed other than that of the approximate fundamental.
Fig. 3f
It seems quite amazing that the properties of air could shift the modes that much. Undoubtedly physicists agree because the effects of air loading on timpani has been the subject of numerous scientific studies.