I would use "tonal center" as fairly synonymous with "tonic". The reference pitch, around which others being used "revolve" or are heard in relation to.

If I were talking about a performance in "raag" (melodic mode in North Indian classical music), I would not talk about the "key" it was in, rather I'd say it has a tonal center. I would say an Arab 'maqaam" has a tonal center...and that one might at some point play certain pitches and pitch patterns to effect another tonal center.

Though this is more or less (i.e., without being pedantic) the same as tonic, the word tonic is (subjectively?) suggestive to me of a certain slice of Western tonal music (i.e. that world of thinking in harmonic relationships), so I would not tend to say it.

"Tonal center" — which I'm not making any particular argument for, just noting how I might use it — has less "baggage" as the concept of "key." You could still say "key," but that may have some connotations that you wish to avoid.