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I've been messing around a bit on guitar playing with some thoughts on dissonance and harmony. Basically, I'm trying to analyze chords and their resolutions in terms of their symmetry. The idea is that more symmetrical chords have more ambiguity since their tonal center isn't clear, which allows them to resolve to lots of different other chords. So, staying in the key of C major for simplicity: you can see Cmaj is a very asymmetrical chord.
The distance between C and E is a major third, the distance between E and G is a minor third, the distance between G and C is a perfect fourth. The character of the chord is dictated by these intervals, such that if, for example, you put the chord in first inversion, it doesn't become a Gmaj chord: C stays the center of gravity of the chord.
Compare this to a Csus4 chord, where the intervals are a perfect fourth, a whole step, and then another perfect fourth. This chord has more symmetry than a major chord because it has two perfect fourth intervals: when you put a Csus4 in second inversion, it's identical to an Fsus2. There's a nice balance of consonance and dissonance in suspended chords that gives them that spacey ambiguity.
But anyway, what I'm interested in is fully symmetrical chords. Just working in twelve-tone equal temperament, the candidates for symmetrical chords are derived from the divisors of 12, which are 1, 2, 3, 4, 6, and 12. 1, 2, 6 and 12 correspond to stacking half-steps, whole steps, tritones, and nothing/an octave. These aren't very good-sounding candidates, only 3 and 4 -- stacking minor thirds and major thirds -- lead to "canonical" chords. These correspond to diminished seventh chords and augmented chords; I've been focusing just on the diminished sevenths.
C
[F] G
Bb [D]
Eb A
[Ab] E
Db [B]
Gb
Sticking close to C major and looking at Bdim7, you can see the symmetry of the chord by looking at where all the notes are on the circle of fifths (Notably, Ab is not in the key of C). So my thought process is to try and explore ways to use the symmetry and dissonance of diminished chords to do interesting key changes. Generally speaking, "easier" key changes usually involve moving to nearby keys on the circle of fifths, so C to G is hardly a noticeable key change, but C to Gb is a world of difference.
But what I'm wondering is if a dim7 centric framework can give an alternative framework of what constitutes feasible key changes, where changing keys by thirds or even semitones makes harmonic sense. i.e., if I play in B Locrian with a heavy emphasis on the Idim, can I recontextualize the Idim as a IIIdim and jump to D Locrian, or a Vdim and jump to F Locrian?
As I mentioned, Ab is notably not in the scale, but I think used carefully, having one of the notes of the Idim7 be non-diatonic could be helpful. Borrowing non-diatonic notes can help set you up for a key change, so if I'm playing in B Locrian and I introduce an Ab into the scale, it could make the key change smoother. i.e., you might anticipate from the Ab that I'm actually moving into the key of Ab, or it might just help set me up to move to D Locrian, since Ab is the V of D Locrian. I think those are both probably stronger key changes than from B Locrian to F Locrian.
Since a diminished chord can resolve easily to either a major chord (root note down a half step) or to a minor chord (fifth up a half step), between the four notes in a given dim7 chord, you can resolve to like eight different keys, if we're taking the simplest assumption that you're resolving to the root note of either a major or a minor key. So there's a lot of potential there I think.
It's a whole lot of thinking and not much playing because I've been practicing guitar just trying to get any good at playing lead stuff, and I'm really not that familiar with playing in keys anywhere outside of like F through A lol... and even then it's a lot of C and G. But I thought this was interesting and it's motivating me to practice my scales all around the circle of fifths a little bit more, and to mess around in different Locrian modes to see if I can't figure out a way to make some sense of this... I'm becoming kind of confident in B Locrian but trying to make it sound less samey, and working on practicing more in the key of A major and Ab Locrian so I can have a starting point for playing around between some Locrian modes that are a minor third apart.
Hopefully something interesting comes of this...
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