A large drum seems to invite a strike at its centre: it looks like the strongest and most accurate target. Timpani are normally played closer to the rim, roughly one-third of the radius in from the hoop. The point is not to produce a different pitch. It is to change which vibration patterns dominate the head, and therefore the attack, balance of overtones and length of the resonance.
A central strike can give a pronounced impact yet lose the full, sustained bass associated with timpani. At a suitable point nearer the edge, the head vibrates more coherently, the perceived fundamental is stronger, and the resonance created with the bowl and enclosed air can unfold more clearly. Yamaha’s playing guide notes that moving the strike does not simply change pitch; it changes harmonics and tone colour. There is no single measurement for every drum and dynamic. Drum size, head tension, mallet hardness and musical context all affect the best point.
Listen to a timpani stroke in two stages: the brief contact of mallet and head, then the pitched low sound opening out afterwards. Good playing does not merely maximise the first impact. It uses the attack to lead the ear into the resonance. That also offers pianists a useful comparison: the instant of touch begins the sound, but the music continues in its decay, connection and surrounding space.
https://www.yamaha.com/en/musical_instrument_guide/timpani/play/play003.html
https://www.yamaha.com/en/musical_instrument_guide/timpani/mechanism/mechanism002.html
https://philharmonia.co.uk/resources/instruments/timpani/
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