Big moves in roots can correspond to small moves in chords.
Imagine the 12 notes of a chromatic scale arranged around the hours of a clock: C at 12:00, C♯ at 1:00, D at 2:00, etc. The furthest apart two notes can be is 6 half steps, just as the furthest apart two times can be is 6 hours.
An interval of 6 half steps is called a tritone. That’s a common term in jazz. In classical music you’d likely say augmented fourth or diminished fifth. Same thing.
The largest possible movement in roots corresponds to almost the smallest possible movement between chords. Specifically, to go from a dominant seventh chord to another dominant seventh chord whose roots are a tritone apart only requires moving two notes of the chord a half step each.
For example, C and F♯ are a tritone apart, but a C7 chord and a F♯7 chord are very close together. To move from the former to the latter you only need to move two notes a half step.
Replacing a dominant seventh chord with one a tritone away is called a tritone substitution, or just tritone sub. It’s called this for two reasons. The root moves a tritone, but also the tritone inside the chord does not move. In the example above, the third and the seventh of the C7 chord become the seventh and third of the F♯7 chord. On the diagram, the dots at 4:00 and 10:00 don’t move.
Tritone substitutions are a common technique for making basic chord progressions more sophisticated. A common tritone sub is to replace the V of a ii-V-I chord progression, giving a nice chromatic progression in the bass line. For example, in the key of C, a D min – G7– C progression becomes D min – D♭7 – C.
I appreciate your series of posts on music and tonality. Seeing the overlap with math is wonderful.
I would make one small point about classical music. I have a degree in composition and piano. In music school we also called it a tritone. If it is augmented 4th or diminished 5th, that is a functional element that depends on context. I don’t know jazz, but I suspect there is a similar consideration such as what I’ve seen of jazz chord notation.