The orthodox meaning of tritone substitution is replacing a V7 chord with the dominant seventh chord half an octave away. This works because the two notes that imply a dominant seventh chord–third and seventh–match between these two chords. If you play B and F together, it’s a jarring interval and they want to resolve to C and E. In other words, those two notes imply a G7 chord, especially in a C major context. However, you could call them F and Cb and resolve them in the opposite direction, to Gb and Bb, implying a Db7 chord. Tritone substitution is using one in place of the other in a V7-I cadence. A variation is to use the V7 and go to the substitution for an intermediate measure before resolving to the I, especially on turnarounds. To help establish the tonality, I put a ii chord in these example cadences: Dmin7-G7-C; Dmin7-Db7-C; Dmin7-G7-Db7-C.
What’s interesting to me is that a typical chord resolves to something a half step away from a tritone. G resolves to C or D, either side of the tritone Db. Put the other way, if you want to get from Db to C, substituting a chord a tritone away gets you there. Of course, if you just pound three chords in a row, Db-G-C, it’s an awfully high contrast of outside and consonance. However, if you play Db and C chords but play a melody of G scale to C, it’s still outside but it flows. If you play a melody that starts in a Db scale and drifts through a G scale into C, you get a nice mix of savory chords and sweet resolution.
I do both in chromatic progressions. In something like “A Night in Tunisia” that goes back and forth between e minor and F, I’m willing to do a Bb arpeggio or scale over the e minor on one measure somewhere in the late part of the phrase, resolving to the F and into the cadence. However, in a modal piece like “So What” or “Little Sunflower”, that’s entirely too angular. Instead I’ll work in the long measures of e minor more and more flatted fifths and major sevenths, and coming up on the F chord, start resolving those to the four and seven instead of to the tonic chord. Over e minor, play patterns of notes like Bb-A-G-E or D#-E-F#-E; then put the D# and D together in some sort of E-D#-D descending line to start suggesting an Eb-D resolution; then alter the motifs to include an F natural, then start leaving out the G in the first one to highlight the Bb-A resolution; and I’m already comfortably in F before the chord changes.
It’s either that or play up the sudden chord change after sixteen bars of pure e minor. I figure the head has defined that well enough, so I’m trying to draw smooth lines that blur that boundary into one continuous musical thought. At any rate, that’s what I’m going for.