I've recently come across some interesting properties of the circle of fifths in relation to diatonic modes. I'm wondering whether this has been noted elsewhere and if so, in what capacity? A name to Google would help. So far, I've found nothing.
The general idea is that tritone can move chromatically and imply a modal shift in fifths. Each diatonic mode has one tritone in it and it's position relative to the root can be considered distinctive of that mode as no two modes share the same tritone position. Let's consider an exercise. Say a tritone is imperfect and needs to be either widened to a perfect fifth or narrowed to a perfect fourth. We'll start with a series of narrowings using D Dorian as a starting point. I chose it for its symmetry, but it's really arbitrary. It will work regardless.
D Dorian's tritone is from F to B, the minor third to the major sixth. We'll flatten the B to a Bb, making the interval a perfect fourth. Well now the the interval from E to Bb is a tritone. We flatten the E to Eb. Note that we don't further flatten the Bb. We're operating with fourths, so we must use the inversion spelled as an augmented fourth, not a diminished fifth. We'll continue this process. We have the mode, the major scale it is derived from, and the note that was just flattened from the last mode:
Mode Major Scale Flattened Note D Dorian C -- D Aeolian F Bb D Phrygian Bb Eb D Locrian Eb Ab Db Lydian Ab Db Db Ionian Db Gb Db Mixolydian Gb Cb Db Dorian Cb Fb
We can do the same for sharpening diminished fifths into perfect fifths:
Mode Major Scale Sharpened Note D Dorian C -- D Mixolydian G F# D Ionian D C# D Lydian A G# D# Locrian E D# D# Phrygian B A# D# Aeolian F# E# D# Dorian C# B#
Of course, we see that the corresponding series leads us the circle of fifths with sharps and the circle of fourths with flats. A circle of fourths moves the tritone down a halfstep for each pass until it is a perfect fifth away from its original position and the mode is now the same, just a chromatic half step down. The circle of fifths performs the complement. So again, does this have a name and who has written about it?
Edit: I'm specifically asking about a mode's tritone moving chromatically as the modes move through the circle of fifths.