Coltrane changes, Slonimsky-Schillinger Symmetric System, three-tonic system, four-tonic system: what are they? How are they used in improvisation, composition, harmonization, and reharmonization? Are there more of them?
The central basis of a multi-tonic system is that the underlying scale or primary note collection is symmetrical, thus allowing several notes within the collection to behave as tonic since they are all approached and left in the same way. In other words: in a C major scale, the tonic has an entirely unique relationship to the other six notes of the scale—at least one of those intervals will be different than any other note of the scale—and this is actually true of every scale degree. Although they will all share some relationships with their other scale degree partners, no two scale degrees have the same battery of relationships. Thus, every note can have a unique role, including one note behaving as tonic (although we can shift which note gets treated as tonic modally, we will vastly change the feel of the music when we change it). An opposite extreme would be the whole-tone scale (one of the systems that Slonimsky and Schillinger discuss) in which every note has the same possible relationships with every other note, and thus any note could be tonic at any time without changing the feel at all. That would be a six-tonic system.
The three-tonic system is just another name for the Coltrane changes. If we divide the octave symmetrically such that there are only three different pitches, we get an augmented triad. Note that Coltrane uses plenty of other notes from the full chromatic, but the symmetrically distributed augmented triad notes all play a tonic (or quasi-tonic) role throughout the progression, especially since each one is preceded by a V7.
EDIT TO ADD EXAMPLE: The chord progression for the beginning of Giant Steps is (in an overarching B-tonic environment):
I've written it to show the (quasi) tonics on the beginning of each line. Each one is preceded by its own V7. Note that the "tonics" are B, G and Eb, which outline an augmented triad.
A four-tonic system was also explored by Coltrane, after the Giant Steps augmented explorations. Here we divide the octave into four symmetrically-divided notes, thus producing the notes of a fully-diminished seventh chord. If we treat each of those notes as a tonic or quasi-tonic then we're working in a 4-tonic system.
Slonimsky/Schillinger also talk about dividing the octave in two to get two tonics a tritone apart. I believe this could be seen as part of the basis for tritone substitution, except I'm not aware of the tritone substitute being treated like a tonic. It mostly seems to exist as a substitute for V.