Mothra[11] Harmony

Someone asked around generally for help harmonizing the mothra[11] scale, but didn't describe it. I couldn't find a description online at the time but still offered to help if they'd give me something to go on. But they didn't. Two months later I thought of it again and did another search. This time, I found this description of mothra[11] in relative steps of 31-EDO: 

    [5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 1].

I don't care much about xenharmonic scales like these, but it kind of stuck in my craw that the person wouldn't describe the scale they wanted help with, so now I'm going to analyze it really well out of spite.

If we accumulate the relative scale steps we get absolute steps: 

    [0, 5, 6, 11, 12, 17, 18, 23, 24, 29, 30, 31]

And harmonizing that is dirt simple; almost every scale degree has a root position septimal triad as a diatonic option:

^0: [P1, SpM3, P5]

^5: [P1, Sbm3, P5]

^6: [P1, SpM3, P5]

^11: [P1, Sbm3, P5]

^12: [P1, SpM3, P5]

^18: [P1, SpM3, P5]

^24: [P1, Sbm3, P5]

^30: [P1, Sbm3, P5]

We're just missing ^[17, 23, and 29], and even those notes are participating in the harmonies above, just not as root notes; you could readily put septimal [1, 3, 6] and [1, 4, 6] inversions of septimal chords on those scale degrees. Those three scale degrees all have Sbm3 and/or m3 as options above for harmonization, but they don't have very good options for a chordal fifth. If you really want a root position triad on those scale degrees, you're going to need to make use of some unusual chords, like perhaps

[0, 12, 19]\31 _ [13:17:20] ~ [P1, ExReA3, ReA5] # [1/1, 17/13, 20/13]

[0, 7, 14]\31 _ [36:42:49] ~ [P1, Sbm3, SbSbd5] # [1/1, 7/6, 49/36]

[0, 7, 19]\31 _ [30:35:46] ~ [P1, Sbm3, Nb5] # [1/1, 7/6, 23/15]

[0, 7, 19]\31 _ [36:42:55] ~ [P1, Sbm3, AsGr5] # [1/1, 7/6, 55/36]

[0, 7, 19]\31 _ [42:49:64] ~ [P1, Sbm3, SpGr5] # [1/1, 7/6, 32/21]

[0, 7, 20]\31 _ [33:38:51] ~ [P1, FaDem3, ExDeA5] # [1/1, 38/33, 17/11]

[0, 8, 19]\31 _ [15:18:23] ~ [P1, m3, Nb5] # [1/1, 6/5, 23/15]

[0, 8, 19]\31 _ [35:42:54] ~ [P1, m3, Sp5] # [1/1, 6/5, 54/35]

[0, 8, 20]\31 _ [11:13:17] ~ [P1, PrDem3, ExDeA5] # [1/1, 13/11, 17/11]

Each line has the tempered tuning of the chord in 31-EDO steps, then an otonal representation of the just tuning of the chord, then interval names, then the just tuning of the chord.

Those are options, and I don't hate the sound of them, but I also think septimal triads on 8 of the scale degrees should be plenty for music making.

If you want to extend the root position septimal triads to four note chords / tetrads, I think these are probably the best options:

[0, 7, 18, 24]\31 _ [42:49:63:72] ~ [P1, Sbm3, P5, SpM6] # [1/1, 7/6, 3/2, 12/7]

[0, 7, 18, 25]\31 _ [12:14:18:21] ~ [P1, Sbm3, P5, Sbm7] # [1/1, 7/6, 3/2, 7/4]

[0, 7, 18, 26]\31 _ [30:35:45:54] ~ [P1, Sbm3, P5, m7] # [1/1, 7/6, 3/2, 9/5]

[0, 11, 18, 25]\31 _ [28:36:42:49] ~ [P1, SpM3, P5, Sbm7] # [1/1, 9/7, 3/2, 7/4]

[0, 11, 18, 29]\31 _ [14:18:21:27] ~ [P1, SpM3, P5, SpM7] # [1/1, 9/7, 3/2, 27/14]

With those chords defined, here are the diatonic tetrad options:

^0: [P1, SpM3, P5] + [SpM7]

^5: [P1, Sbm3, P5] + [SpM6 or Sbm7 or m7]

^6: [P1, SpM3, P5] +  [Sbm7]

^11: [P1, Sbm3, P5] + [Sbm7 or m7]

^12: [P1, SpM3, P5] + [Sbm7]

^18: [P1, SpM3, P5] + [Sbm7]

^24: [P1, Sbm3, P5] + [SpM6 or Sbm7]

^30: [P1, Sbm3, P5] + [SpM6 or Sbm7]

Easy.