I took a side trip to find more triads in the 31-limit tonality diamond. With the diamond to the 11-limit, using either the otonality or utonality only, there are really only four valid triads, where valid is defined as having a pure major fifth, and four thirds: 7:6, 6:5, 5:4, 9:7. I systematically explored the 31-limit diamond and found many more. Here is a summary table of the thirds I found inside pure 3:2 fifths, in the otonality and utonality only.
Third name
third ratio
septimal
7:6
septimal
7:6
otonal minor
19:16
minor
6:5
supraminor
17:14
undecimal neutral
11:9
rastimic neutral
27:22
submajor
21:17
major
5:4
smaller undevicesimal major
24:19
septimal major
9:7
sensi supermajor
31:24
Each of these can be transposed to any key in the utonality or otonality, depending on from which they were derived. That’s a total of 180 distinct triads across the entire diamond, after some duplicated ones were eliminated. I played each one with a keyboard to find a reasonable diatonic scale for each, plus two 7th’s, major and minor. Now I plan to use these to pick more sonorous keys for the Allemande dance by Bach. But that’s not for today.
What we have today is just all 144 unique triads, sorted by the 1st degree, then the 3rd degree, starting with the most subminor going to the highest bearable super major. In order. Straight through. This is not music, but it is interesting to listen to.
