Some microtonal composers with perfect pitch will just tune chords to minimize beating and then know the just intervals names for the frequency ratios they've tuned by hand.
I can't do that, but I'd still like to compose microtonal music with beautiful harmony. I probably should have tried using a computer model of acoustic beating, and I might do that next, but first I just listened to a bunch of tertian seventh chords and rated them "good", "okay", "bad". I recognize that all sorts of chords which sound bad in isolation can sound good through voice leading - there's definitely technical machinery for preparation and resolution of dissonance - but for I'm going to start by looking at consonance in isolation. Here are the first set of ratings I came up with:
[P1, Grm3, P5, Sbm7], # 1. Good
[P1, m3, P5, Sbm7], # 2. Good, but worse than previous
[P1, M3, P5, Sbm7], # 3. Good
[P1, AcM3, P5, Sbm7], # 4. Good, but a little worse than previous
[P1, Sbm3, P5, Grm7], # 5. Bad
[P1, Grm3, P5, Grm7], # 6. Okay
[P1, m3, P5, Grm7], # 7. Okay
[P1, M3, P5, Grm7], # 8. Good
[P1, AcM3, P5, Grm7], # 9. Good, but worse than previous
[P1, Sbm3, P5, m7], # 10. Bad
[P1, Grm3, P5, m7], # 11. Bad
[P1, m3, P5, m7], # 12. Good
[P1, M3, P5, m7], # 13. Okay
[P1, AcM3, P5, m7], # 14. Okay, but worse than previous
[P1, Sbm3, P5, Acm7], # 15. Bad
[P1, Grm3, P5, Acm7], # 16. Bad
[P1, m3, P5, Acm7], # 17. Okay
[P1, M3, P5, Acm7], # 18. Okay, but better than previous
[P1, AcM3, P5, Acm7], # 19. Bad or Okay?
[P1, Sbm3, P5, GrM7], # 20. Bad
[P1, Grm3, P5, GrM7], # 21. Bad
[P1, m3, P5, GrM7], # 22. Bad
[P1, M3, P5, GrM7], # 23. Good // Really
[P1, AcM3, P5, GrM7], # 24. Okay
[P1, Sbm3, P5, M7], # 25. Okay
[P1, Grm3, P5, M7], # 26. Bad
[P1, m3, P5, M7], # 27. Bad
[P1, M3, P5, M7], # 28. Okay
[P1, AcM3, P5, M7], # 29. Okay, but a little worse than previous
[P1, Sbm3, P5, AcM7], # 30. Bad
[P1, Grm3, P5, AcM7], # 31. Bad
[P1, m3, P5, AcM7], # 32. Bad
[P1, M3, P5, AcM7], # 33. Okay
[P1, AcM3, P5, AcM7], # 34. Okay
.
Next, I tried looking for patterns to explain/predict/compress my judgements. A few of the rating might have changed up or down a point on subsequent re-listenings as I investigated trends, but mostly those ratings were pretty stable for me. Here's what I came up with for explanatory trends:
First I looked at the relative-intervals between chord tones, instead of the absolute intervals in relation to the tonic, and rounded the just tunings to the nearest 20 cents. When I did that, I saw that chords with 400c or 440c as stepwise intervals almost always sound bad. I think this fact mostly boils down to the principle that one should not use anything but 5-limit m3 and 5-limit M3 for the second chord tone.
These two chords
[P1, Sbm3, P5, Sbm7], # Good
[P1, Grm3, P5, Sbm7], # Good
are exceptions to that rule and I don't understand them. Maybe the Sbm7 just sounds so good that it masks the badness of the large gap between the flat third and the pure fifth. Or maybe the 3rd and the 7th both being septimal has some merit and the Grm3 is approximating a Sbm3.
Within chords that don't have relative step intervals larger >= 400 cents, there's a trend that chords sound better with flatter 7ths (closer to the harmonic Sbm7) and sound worse with increasing size of the 7th.
This trend holds absolutely for chords with m3.
[P1, m3, Grm7], # 7. Good
[P1, m3, m7], # 12. Okay
[P1, m3, Acm7], # 17. Bad
[P1, m3, GrM7], # 22. Bad
[P1, m3, M7], # 27. Bad
Looking at only chords with M3, the trends is more bimodal:
[P1, M3, Grm7], # 8. Okay
[P1, M3, m7], # 13. Bad
[P1, M3, Acm7], # 18. Okay
[P1, M3, GrM7], # 23. Good
[P1, M3, M7], # 28. Good
So maybe Sbm7 and M7 are both good-in-principle when used with M3, and the Grm7 and GrM7 are acceptable approximations to those.
Now let's look at trends within the chords which have relative-intervals tuned to 400c or larger, which were mostly bad sounding.
The only good chords with 400c or 440c as rounded-tuned-relative steps are:
[400, 300, 300] : [P1, AcM3, P5, Grm7], # 9. Good
[400, 300, 380] : [P1, AcM3, P5, M7], # 29. Good
[400, 300, 260] : [P1, AcM3, P5, Sbm7], # 4. Good
Which all have intervals of 400 and 300 in their first two members, because they all have the Pythagorean [P1, AcM3, P5] as their base triad.
In chords with a relative-step interval of 400 or 440, any chord with a M7 sounds good or okay. More likely just okay. The version of such a chord instead using AcM7 will be a little worse, but otherwise similarly good.
If we use Sbm3 or Grm3 for the second chord tone, the chord will generally sound bad, but they both sound okay when paired with M7 or Sbm7.
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I looked for correspondences between the otonal and utonal spellings of the chords against my ratings of their consonance, and didn't find anything interesting. Just documenting that I did the fairly obvious thing.
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