Some types of timpani produce a less focused sound than others and may need smaller cores and/or less felt than other types to give good projection.
Natural skin timpani heads generally need a smaller core and slightly more felt than plastic heads.
Very slack heads and low notes are generally better with cork or hard-felt cores.
of sound and the acoustic qualities of your concert venue
Very resonant acoustics such as cathedrals can make the timpani very indistinct with a very heavy bass quality to the sound. This type of acoustic needs a more sharply focused, less full sound than usual. You can achieve this by using smaller cores and thinner felt than usual. The smaller cores will produce less fullness and depth from your timpani and the thinner felt will give a clearer and more articulate sound.
Concert halls with very dry acoustics need the opposite treatment and you may wish to select larger cores to maximise the depth of tone and resonance produced by the timpani and use thicker felt to reduce the articulation heard by the audience.
When playing in smaller halls or with a small group of players, you may wish to reduce the fullness of sound by choosing smaller than normal cores so that the timpani are not overpowering.
The perceived hardness of the sticks