What is the four tonic system? What's the theory behind it? How is it used in improvisation, harmonization, reharmonization?
4-tonic system means that a piece of music emphasizes four notes, usually equal intervals apart (in other words a diminished 7th), as its tonic centers. The chords and melodies generally related to those four tonal centers.
Coltrane's Giant Steps is a popular example of 3-tonic, leaping amid G, B, and Eb.
These don't have to be equal divisions of the chromatic scale, but that's pretty commonly what these terms are used to refer to in Jazz.
The best example of 4-tonic in a diminished scale which comes to mind to link to is Egg's Contrasong, which jumps around on these centers. Also, King Crimson's Larks' Tongues in Aspic Part One has some of this beginning at around the 2:50 point.