Two Tetrad Chords, Polychords, Bitonal Chords

Here are some two triad chords. Most of these are only true enharmonically in 12-TET.

: .m7b5

C.m11b5b9 = C.dim + Bb.m
C.m11b5 = C.dim + Bb.maj
C.m7b5b9 = C.dim + Gb.maj

:: .m7

C.m11b9 = C.m + Bb.m
C.m11 = C.m + Bb.maj
C.m7#11 = C.m + Eb.m
C.m7b9 = C.m + G.dim
C.m9 = C.m + G.m
C.m7b9#11 = C.m + Gb.maj

:: .7

C.7b9 = C.maj + Bb.dim
C.11b9 = C.maj + Bb.m
C.11 = C.maj + Bb.maj
C.7#9#11 = C.maj + Eb.m
C.7#9 = C.maj + Eb.maj
C.7b9 = C.maj + G.dim
C.9 = C.maj + G.m
C.7b9#11 = C.maj + Gb.maj

: .maj7

C.maj7#9b13 = C.maj + Ab.m
C.maj11 = C.maj + B.dim
C.maj7#9#11 = C.maj + B.maj
C.maj7b13 = C.maj + E.maj

Here's a are some chords made of a (tetrad) and another (triad or tetrad):

:: .m7b5:

C.m7b5b13 = C.m7b5 + (Ab.maj or Ab.7)

C.m11b5b13 = C.m7b5 + (D.dim or D.m7b5)

C.m13b5 = C.m7b5 + (D.m or D.m7)

C.m13b5#11 = C.m7b5 + (D.maj or D.7)

C.m11b5b9b13 = C.m7b5 + (Db.maj or Db.maj7)

:: .m7:

C.m7b13 = C.m7 + (Ab.maj or Ab.maj7)

C.m11b9b13 = C.m7 + (Db.maj or Db.maj7)

C.m11b13 = C.m7 + (D.dim or D.m7b5)

C.m13 = C.m7 + (D.m or D.m7)

C.m13#11 = C.m7 + (D.maj or D.7)

:: .7:

C.11b9b13 = C.7 + (Db.maj or Db.maj7)

C.11b13 = C.7 + (D.dim or D.m7b5)

C.13 = C.7 + (D.m or D.m7)

C.13#11 = C.7 + (D.maj or D.7)

:: maj7:

C.maj11b9b13 = C.maj7 + (Db.maj or Db.maj7)

C.maj11b13 = C.maj7 + (D.dim or D.m7b5)

C.maj13 = C.maj7 + (D.m or D.m7)

C.maj13#11 = C.maj7 + (D.maj or D.7)

I like those a lot. I think they're all true exactly, instead of just enharmonically, at least in rank-2 interval space.

Here's an even larger set. Probably too long to be useful. I should compress it or filter or something. But maybe someone will like it.

  : .m7b5

C.m11b5 = C.m7b5 + Bb.maj

C.m11b5b9 = C.m7b5 + Bb.m

C.m11b5b9 = C.m7b5 + Gb.maj7

C.m13b5 = C.m7b5 + Bb.maj7

C.m7b5b9 = C.m7b5 + Eb.m7

C.m7b5b9 = C.m7b5 + Gb.maj

:: .m7

C.m11 = C.m7 + Bb.maj

C.m11 = C.m7 + G.m7

C.m11b9 = C.m7 + Bb.m

C.m11b9 = C.m7 + G.m7b5

C.m13 = C.m7 + Bb.maj7

C.m13#11 = C.m7 + D.7

C.m13#11 = C.m7 + D.maj

C.m13#9#11 = C.m7 + C.dim7

C.m13#9#11 = C.m7 + Eb.dim

C.m13#9#11 = C.m7 + Eb.dim7

C.m13#9#11 = C.m7 + Gb.dim

C.m13b9#11 = C.m7 + Eb.m7b5

C.m13b9#11 = C.m7 + Gb.m

C.m7#11 = C.m7 + Eb.m

C.m7#11b13 = C.m7 + Ab.7

C.m7b13 = C.m7 + Ab.maj

C.m7b13 = C.m7 + Ab.maj7

C.m7b9 = C.m7 + Eb.7

C.m7b9 = C.m7 + G.dim

C.m7b9#11 = C.m7 + Eb.m7

C.m7b9#11 = C.m7 + Gb.maj

C.m9 = C.m7 + Eb.maj7

C.m9 = C.m7 + G.m

:: .7

C.11 = C.7 + Bb.maj

C.11 = C.7 + G.m7

C.11#9b13 = C.7 + F.m7

C.11b13 = C.7 + Bb.7

C.11b13 = C.7 + D.dim

C.11b13 = C.7 + D.m7b5

C.11b9 = C.7 + Bb.m

C.11b9 = C.7 + G.m7b5

C.11b9b13 = C.7 + Bb.m7

C.13 = C.7 + Bb.maj7

C.13 = C.7 + D.m

C.13 = C.7 + D.m7

C.13#11 = C.7 + D.7

C.13#11 = C.7 + D.maj

C.13#9 = C.7 + F.7

C.13#9#11 = C.7 + C.dim7

C.13#9#11 = C.7 + Eb.dim

C.13#9#11 = C.7 + Eb.dim7

C.13b9#11 = C.7 + Gb.m

C.13b9#11 = C.7 + Gb.m7

C.7#9 = C.7 + Eb.maj

C.7#9#11 = C.7 + Eb.m

C.7#9#11b13 = C.7 + Ab.7

C.7#9b13 = C.7 + Ab.maj

C.7#9b13 = C.7 + Ab.maj7

C.7b9 = C.7 + Bb.dim

C.7b9 = C.7 + E.dim7

C.7b9 = C.7 + G.dim

C.7b9 = C.7 + G.dim7

C.7b9#11 = C.7 + Gb.7

C.7b9#11 = C.7 + Gb.maj

C.7b9b13 = C.7 + Bb.m7b5

C.9 = C.7 + E.m7b5

C.9 = C.7 + G.m

: .maj7

C.maj11 = C.maj7 + B.dim

C.maj11 = C.maj7 + G.7

C.maj11#9b13 = C.maj7 + F.m7

C.maj11#9b13 = C.maj7 + F.m7b5

C.maj13#9 = C.maj7 + F.7

C.maj13b9#11 = C.maj7 + Gb.m

C.maj13b9#11 = C.maj7 + Gb.m7

C.maj7#9#11 = C.maj7 + B.maj

C.maj7#9#11b13 = C.maj7 + Ab.7

C.maj7#9#11b13 = C.maj7 + Ab.m7

C.maj7#9b13 = C.maj7 + Ab.m

C.maj7#9b13 = C.maj7 + Ab.maj

C.maj7#9b13 = C.maj7 + Ab.maj7

C.maj7#9b13 = C.maj7 + E.maj7

C.maj7b13 = C.maj7 + E.maj

Ooh, if we include augmented triads, we get these polychords as well:

:: .m7b5

C.m13b5b9 = C.m7b5 + A.aug

C.m13b5b9 = C.m7b5 + F.aug

C.m9b5 = C.dim + D.aug

C.m9b5 = C.m7b5 + D.aug

:: .m7

C.m13b9 = C.m7 + F.aug

C.m9#11 = C.m + D.aug

C.m9#11 = C.m7 + D.aug

C.m13b9 = C.m7 + A.aug

:: .7

C.13b9 = C.7 + A.aug

C.13b9 = C.7 + F.aug

C.7b13 = C.7 + C.aug

C.7b13 = C.7 + E.aug

C.9#11 = C.7 + D.aug

C.9#11 = C.maj + D.aug

:: .maj7

C.maj7#9 = C.maj + B.aug

C.maj7#9 = C.maj + G.aug

C.maj7#9 = C.maj7 + B.aug

C.maj7#9 = C.maj7 + G.aug

C.maj7b13 = C.maj7 + C.aug

C.maj7b13 = C.maj7 + E.aug

Let me see if I can summarize the dominant ones from across all these categories.

Suppose you play a C.7 in your left hand, and some kind of Db, D natural, or D# chord in your right hand to cover ^9, ^11, ^13. Db for b9, D natural for 9, and D# for #9. For any of those roots in the right hand, you could play a diminished, minor, or major triad.

:: Db roots:

C.7b9 = C.7 + (Db.dim or Db.dim7)

C.7b9b13 = C.7 + Db.m

C.11b9b13 = C.7 + (Db.maj or Db.maj7)

C.13b9 = C.7 + Db.aug

:: D natural roots:

C.11b13 = C.7 + (D.dim or D.m7b5)

C.13 = C.7 + (D.m or D.m7)

C.13#11 = C.7 + (D.maj or D.7)

C.9#11 = C.7 + D.aug

# D sharp roots:

C.13#9#11 = C.7 + (D#.dim or D#.dim7)

C.7#9#11 = C.7 + D#.m

C.7#9 = C.7 + D#.maj

 Sometimes you can add a seventh, but .maj might go to .maj7 or .7, and .dim might go to .m7b5 or .dim7, and sometimes it just doesn't work to add a seventh at all. You can also play Db.aug or D.aug, but D#.aug is dumb. It has F## and A##, which are ##11 and ##13. The F## is enharmonic with G in 12-TET, so that disappears, but ##13 is a B natural, which doesn't belong in a C.7 chord.

Maybe that's still too much to memorzie. How about we look at right hand chords rooted on Bb.

C.7b9 = C.7 + (Bb.dim or Bb.dim7)

C.7b9b13 = C.7 + Bb.m7b5

C.11b9 = C.7 + Bb.m

C.11b9b13 = C.7 + Bb.m7

C.11b13 = C.7 + Bb.7

C.9#11 = C.7 + Bb.aug

This doesn't give us any #9 variants, and only 2 with natural 9. Also, C.11b9 is basically the same sonority as C.7b9, and C.11b9b13 is basically the same sonority as C.7b9b13, so this only gives us about four chord types. Not so good. Maybe the best thing is to think in terms of (Db, D, D#) with (.dim, .m, .maj, .aug) qualities, forget the 7ths in the right hand. That's only 11 options:

:: Db roots:

C.7b9 = C.7 + Db.dim

C.7b9b13 = C.7 + Db.m

C.11b9b13 = C.7 + Db.maj

C.13b9 = C.7 + Db.aug

:: D natural roots:

C.11b13 = C.7 + D.dim

C.13 = C.7 + D.m

C.13#11 = C.7 + D.maj

C.9#11 = C.7 + D.aug

# D sharp roots:

C.13#9#11 = C.7 + D#.dim

C.7#9#11 = C.7 + D#.m

C.7#9 = C.7 + D#.maj

And a few of them have the same sonority / the same accidentals relative to C.13, like (C.7b9 with C.13b9) and (C.7b9b13 with C.11b9b13) and (C.13#11 with C.9#11) and (C.13#9#11 with C.7#9#11) so you might skip learning one or the other of those. Not too hard. And that gives us just 7 dominant chords to learn the sound of and get comfortable with. Actually, when you're using weird dominant chords, it's not at all uncommon to also have b5, so there are many more options, but this is still a good start.