The density/viscothermal characteristics (viscosity, thermal conduction) of the volume of air enclosed within the bowl plays a critical role in the fine-tuning of the spectrum of each note on a timpano. The density/viscothermal characteristics determine how easily this internal air can be displaced by the motion of the vibrating head affecting how it interacts with the various modes of vibration of the head. The density/viscothermal characteristics of the internal air also affects how quickly the bowl volume can be normalized (i.e. equal to that of the external air mass above the head) via expulsion/induction of air through the vent at the bottom of the drum. Bowl material is a factor here as well. See Why Copper?
This volume of air also has its own modes of vibration, which can interact with the modes of vibration of the head, reinforcing and canceling the various modes. It is in fact the volume of air in the bowl that helps damp the actual fundamental of the drum so that pitch can be discerned by the listener (see Fleisher & Fastl). Too little air volume and it is easy to overplay the drum creating destructive hyper-resonances within the frequency range of the preferred modes.
The Rogers Accusonic timpani (1970s) were notorious for having this trait. The shallow bowl limited the volume of air that the bowl could contain for the extended range (1 octave) of the drum, limiting the usable dynamic range to somewhere between pianissimo and poco forte. Anything above that and the sound became distorted. In defense of the Rogers Accusonic, they were designed for use in the American elementary and middle school market, not for use with a full sized symphony orchestra.
Too much air and certain notes within the normal range of the drum suffer from a degraded clarity or pitch focus due to the lack of damping of the inharmonic fundamental and the secondary concentric modes, as well as the potential for exacerbated inharmonicity of the higher preferred modes (4,1 5,1, 6,1).
The four main areas of influence of the volume of air contained within the bowl are:
1) Muting of the inharmonic fundamental and its associated modes so that the partials of the more harmonic modes can predominate.
2) Reinforcing the partials of the head’s near-harmonic modes with similar resonant air modes inside the bowl.
3) Excessive motion at higher db levels causing the generation of hyper-resonances modifying the natural frequencies of the head adding coloration/distortion to the sound; this brings the stiffness to shear of the membrane into play , which when the drum is played at higher dynamic levels, raises the frequencies of the upper preferred modes.
4) Limiting or enhancing the motion of the concentric air modes, which acts as a restoring force (like a spring) to return the head to a neutral position. This affects head sustain and resonance of the diametric modes.
Unfortunately, when the first two influences are strong, so is influence No. 3, hence the reason that no uniform bowl shape has ever come into vogue.
If a bowl has too much air volume, there is:
1) not enough muting of the inharmonic partials for certain notes
2) the potential for causing exacerbated inharmonicity of the higher preferred modes (4,1 5,1, 6,1) at soft dynamic levels (unaffected by the stiffness to shear of the membrane)
3) a weakened reinforcement of the near harmonic partials by the air modes, which result in less clarity, immediacy and harmonicity of the pitch for certain notes
The drum may have considerable sustain and resonance, is easy to play, and can be played loudly without significant distortion, but for certain notes within the playing range of the drum, the response and projection of the pitch (i.e., clarity, immediacy and harmonicity) is noticeably diminished at softer dynamics.
If a bowl has too little air volume:
1) excessive air movement causes hyper-resonances (the drum is easy to overplay)
2) the clarity of the preferred modes is compromised at loud dynamic levels (affected by the stiffness to shear of the membrane)
3) The drum can have a smaller dynamic voice with limited sustain/resonance
The inharmonic fundamental may be significantly damped and the resonant air modes may interact with those of the head producing better immediacy and harmonicity of the pitch, but at louder dynamic levels (usually anything above forte), the excessive air movement interferes with the natural vibrations of the head adding hyper-resonances (unwanted coloration/distortion) to the sound waves emitted from the top of the head; the result is a drum with less power and a limited dynamic voice.
One might expect that venting a bowl with a smaller volume would alleviate the excessive air movement, but the vent-hole affects mostly the concentric modes, which have little influence on the pitch producing modes. Too much damping of the concentric modes in turn brings influence No. 4 into play, which can affect the sustain, resonance and feel/response of the head.
Adding to the mix of variables is the fact that ideally, this finite volume of air in the bowl should react the same for each pitch associated with the size and range of the drum, which will also be affected by the size, mass and tension of the head as well as the density/viscothermal characteristics of the surrounding air. Such a Goldilocks Zone cannot not exist with the design of modern variable pitched timpani.