The tritone is between ^2 and ^6 in minor but from what I understand, if resolved inward to scale degreees ^3 and ^5 it will sound like the mediant chord is now the new tonic. But what about if you resolve it outward. Is this some kind of modulation to A major since neither of the notes would be part of that scale? In Cm for example the D and Ab could resolve to Db/C# and to A natural.
^2 and ^6 in minor [...] from what I understand, if resolved inward to scale degreees ^3 and ^5 it will sound like the mediant chord is now the new tonic.
When played in isolation, perhaps, but in the context of a full realization of the harmony, I have serious misgivings about that hypothesis. For a counterexample, what about when ♮vii°7 resolves to i? I think despite the tritone notes moving towards the mediant chord's root and third, the minor tonic note's presence should easily prevail over the parallel major sound.
Is this some kind of modulation to A major since neither of the notes would be part of that scale?
Well, it could be. Those two notes at face value are pretty convincing for an A major change-up, but composers have the ability to use those notes to set up a lot of different ideas, not necessarily only modulations to A major.
Yes, it is possible to resolve the 2nd and 6th in a minor key to the flat supertonic and raised submediant, although perhaps a bit contrived of a solution to the ostensibly ambiguous 2-3, 6-5 resolution pattern. This resolves the tritone in the opposite direction; if previously the tritone in the minor key was going to an interval of a major third, now it will end up a sixth instead. Neither note is in the former key, so a modulation or temporary tonicization is one possible use for this idea.
Note that A major is the relative major of F# minor, which is the key exactly a tritone away from C minor. Any tritone can be reinterpreted as the same tritone relative to the key a tritone away. In this case, the interval between D and Ab (diminished fifth) was reinterpreted as being between D and G# (augmented fourth). This is an artifact of the tritone splitting the octave exactly in half (thus being its own inversion), and it is also the driving principle behind the tritone substitution in jazz theory.
Keep in mind that the scale degree functions change when that tritone is in the major or minor position.
In major the degrees are
^4 associated with the dominant chord, but in minor they are
^6 associated with the subdominant.
In major the
^7 would move up to
^1 like in
V6 I, but that
^2 in minor and it would be held when a subdominant moves to the dominant like in
I would say that major and minor are not "reverse" of each other and so when you switch between one key and its major/minor relative the scale degree function, or tendency tone directions, are also not "reserved."
When you "reverse" the movement, what you essentially are doing is inverting the scale degree and voice leading roles. With some enharmonic re-spelling
D A♭ (interval
E♭ G is
^7 ^4 to
^1 ^3 in
E♭ major, "reversing" it as
D G♯ (interval
C♯ A inverts the scales degrees to
^4 ^7 to
^3 ^1 in
A major. Both motions are dominant function but the keys are a tritone apart. This is the basic of the tritone substitution in jazz.
It's easier to see the relationships when comparing the two major keys
A rather than comparing
A major, because the converging motion of
D A♭ to
E♭ G is really characteristic of
Yes, you can use the tritone between 2 and 6 of the natural minor scale to lead all sorts of places! As you can with any tritone.
If we're in the world of functional harmony we must also consider the harmonic minor scale, which contains the same 4-7 tritone as the major scale. And (as mentioned in your other thread about the picardy 3rd) it can take the classic semitone resolution to a major tonic chord or the in-key resolution to the minor tonic. Same with the 2-6 tritone in the natural minor scale. Probably best to let one of the notes act as a leading-note and resolve a semitone up. But the other can resolve down by either a semitone or a full tone.
Tritones and diminished 7th chords have this in common-- they divide the octave equally. Composers VERY often use this fact to do just what you're talking about-- for example modulating to A major instead of resolving to E-flat or c minor.
A nice way to move forward might be something like this: C minor--> d + a-flat (=g#) --> A major --> d minor --> d-dim7 --> G7 -- c minor
This is one of the main ways that Romantic composers get around to so many keys. When in doubt, diminished 7th your way out! ;D