I made this in response to a challenge on a tuning group on facebook. 23edo, the tuning system that divides octaves into 23 equal parts, is quite strange. Conventional tuning divides octaves into 12 equal parts. Dividing each of these parts in half, one obtains 24edo. The intervals in 23edo will be quite different than those in 24edo, or in 12edo.
This is a table of errors for 23edo. The numbers on the top and left side are the numerators and denominators of fractions for just intervals. The numbers in the matrix are the errors as a fraction of a step of 23edo. For example, the just interval 5:3, a major sixth, is 16.95 steps of 23edo, i.e. it is 0.05 away from 17 steps, the interval available in the tuning. So 0.05 appears at the column labeled 5 and the row labeled 3.
The errors of 23edo can be contrasted with the errors of 24edo, given by the table above. The most fundamental difference is that the error for 3:1 is very low for 24edo but very large for 23edo. 3:1 corresponds to a perfect fifth. A tuning like 23edo that doesn't work well with perfect fifths... is going to be strange!
The composition algorithm I use is based on scores for intervals. The scores have two components: a good score is given when an interval in the tuning is very close to a simple frequency ratio. Intervals that are far from all simple ratios and only close to complex ratios are given poor scores, i.e. they are suppressed in the random selection of pitches.
Outside of octaves, the intervals given the best scores, i.e. that will appear most often in this piece, are:
- 17 steps, corresponding to 5:3
- 9 steps, corresponding to 21:16
- 1 step, corresponding to 33:32 and 25:24
- 15 steps, corresponding to 11:7
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