2 3 5 7 11 13

Here is a new piece of music: 270edo scale 23.

This music is in the 270edo tuning system which divides octaves into 270 equal parts. This provides very precise tuning for a wide range of intervals. The music uses a scale with just 23 of the possible 270 pitches in each octave. The diagram above shows the pitches of the scale and their harmonic relationships. Pitches are labeled by integers, from 0 to 269 in each octave, and then starting up with 0 again for the next octave. In ascending order, the pitches in the scale are: 0, 10, 24, 36, 45, 62, 70, 77, 87, 97, 114, 122, 136, 148, 158, 168, 182, 199, 209, 228, 235, 245, and 260. The intervals between the successive pitches vary between 7 and 19 microsteps of the tuning system, so the spacing is reasonably even.

The arrows between pitches show the harmonic relationships. A green arrow represents a 3:2 frequency ratio (the best approximation in the 270edo tuning system), a perfect fifth in conventional tuning. Blue arrows are 5:4 ratio, major thirds in conventional tuning. Red arrows are 7:4, purple are 11:8, and yellow are 13:8; none of these are conventional intervals.

This scale supports traversal of a family of commas. The diagram above shows a traversal of the comma 1001:1000.

This is a traversal of 4096:4095.

This is a traversal of 41503:41472.

It looks to me like these three commas form a basis for all the commas that can be traversed in this scale. With six prime numbers involved in the tuning, the total set of commas tempered out by 270edo should be a five dimensional space. This scale supports traversal of a three dimensional subspace of that full set.

This traversal of 123201:123200 is a combination of the basis set of traversals.

I don't have a systematic way to construct scales like this, where the scales steps are reasonably even, where the graph of harmonic relationships is reasonable dense, and where comma traversals are supported. The set of commas tempered out by a tuning system already tend to have an irregular structure, so these sorts of scales probably won't be very regular either!