Categorizing modes of scales other than major scale

Question

Modes of major scale are often ordered such that each consecutive scale can be obtained by altering one step of the previous one:

lydian     1  2  3 #4  5  6  7
ionian     1  2  3  4  5  6  7
mixolydian 1  2  3  4  5  6 b7
dorian     1  2 b3  4  5  6 b7
aeolian    1  2 b3  4  5 b6 b7
phrygian   1 b2 b3  4  5 b6 b7
locrian    1 b2 b3  4 b5 b6 b7

This way one can easily see how they gradually change from "bright" and "open" (lydian), towards "dark" and "closed" (locrian).

Not all scales however have modes which follow such simple dependence. In particular not the commonly used melodic minor and harmonic minor. Modes of these scales are typically presented as modes of major scale with alterations (see e.g. Melodic Minor Modes) but this doesn't explain much about how these modes relate to each other.

It would seem natural that since modes of a scale consist of the same intervals, just ordered differently, they should share some sonic properties. Is there some way to categorize modes of melodic minor and harmonic minor (and perhaps some other scales) in a way that shows connection between them?

Answer 1

I've never encountered something like this, and my guess is that this is untenable for at least two reasons:

1. Since the major scale is a conjunct subset of the circle of fifths (C major is F–C–G–D–A–E–B rearranged), this allows the clean "add one accidental for the next mode" that we see (since adding an accidental is equivalent to moving around the circle of fifths). This feature is not true for the melodic or harmonic minor scales.
2. The augmented second of the harmonic minor really messes with the ensuing scale constructions.

But because I'm a theorist, I tried to come up with something like your system. In order to fill in the blanks, I decided to combine the modes of both harmonic and melodic minor. I've ordered it to maximize common tones from one scale to the next, but as you see, there's no way to do it as cleanly as we do it with the major scale. The parenthesized numbers at the end of each mode show the number of common tones with the mode above, and only three times does the next mode have six tones in common with the prior one (as they all do in the modes of the major scale).

Melodic  Mode 3     1   2   3  ♯4  ♯5   6   7   1
Harmonic Mode 6     1  ♯2   3  ♯4   5   6   7   1   (5 tones in common)
Melodic  Mode 4     1   2   3  ♯4   5   6  ♭7   1   (5)
Harmonic Mode 3     1   2   3   4  ♯5   6   7   1   (4)
Harmonic Mode 4     1   2  ♭3  ♯4   5   6  ♭7   1   (6)
Melodic  Mode 1     1   2  ♭3   4   5   6   7   1   (5)
Harmonic Mode 1     1   2  ♭3   4   5  ♭6   7   1   (6)
Melodic  Mode 5     1   2   3   4   5  ♭6  ♭7   1   (5)
Melodic  Mode 2     1  ♭2  ♭3   4   5   6  ♭7   1   (4)
Harmonic Mode 5     1  ♭2   3   4   5  ♭6  ♭7   1   (5)
Melodic  Mode 6     1   2  ♭3   4  ♭5  ♭6  ♭7   1   (4)
Harmonic Mode 2     1  ♭2  ♭3   4  ♭5   6  ♭7   1   (5)
Melodic  Mode 7     1  ♭2  ♭3  ♭4  ♭5  ♭6  ♭7   1   (5)
Harmonic Mode 7     1  ♭2  ♭3  ♭4  ♭5  ♭6 ♭♭7   1   (6)

(Note that, for the purposes of creating this chart, I didn't consider enharmonic equivalence. For instance, I treated "♯2" and "♭3" as distinct entities.)

I'm sure it's possible to flesh this chart out so that every succeeding mode only differs by one pitch, but it would necessitate many more scales/modes than just the rotations of these two forms of minor. As a sample of this idea, I add in the diatonic modes in the chart below; now look at how many scales have six tones in common with their predecessor! (I've asterisked the new sixes.)

Ionian              1   2   3   4   5   6   7   1
Lydian              1   2   3  ♯4   5   6   7   1   (6)*
Melodic  Mode 3     1   2   3  ♯4  ♯5   6   7   1   (6)*
Harmonic Mode 6     1  ♯2   3  ♯4   5   6   7   1   (5)
Melodic  Mode 4     1   2   3  ♯4   5   6  ♭7   1   (5)
Mixolydian          1   2   3   4   5   6  ♭7   1   (6)*
Harmonic Mode 3     1   2   3   4  ♯5   6   7   1   (5)
Harmonic Mode 4     1   2  ♭3  ♯4   5   6  ♭7   1   (6)
Dorian              1   2  ♭3   4   5   6  ♭7   1   (6)*
Melodic  Mode 1     1   2  ♭3   4   5   6   7   1   (6)*
Harmonic Mode 1     1   2  ♭3   4   5  ♭6   7   1   (6)
Aeolian             1   2  ♭3   4   5  ♭6  ♭7   1   (6)*
Melodic  Mode 5     1   2   3   4   5  ♭6  ♭7   1   (6)*
Melodic  Mode 2     1  ♭2  ♭3   4   5   6  ♭7   1   (4)
Phrygian            1  ♭2  ♭3   4   5  ♭6  ♭7   1   (6)*
Harmonic Mode 5     1  ♭2   3   4   5  ♭6  ♭7   1   (6)*
Melodic  Mode 6     1   2  ♭3   4  ♭5  ♭6  ♭7   1   (4)
Locrian             1  ♭2  ♭3   4  ♭5  ♭6  ♭7   1   (6)*
Harmonic Mode 2     1  ♭2  ♭3   4  ♭5   6  ♭7   1   (6)*
Melodic  Mode 7     1  ♭2  ♭3  ♭4  ♭5  ♭6  ♭7   1   (5)
Harmonic Mode 7     1  ♭2  ♭3  ♭4  ♭5  ♭6 ♭♭7   1   (6)