Circulating and Traversing

Here is a new piece in 42edo. My main idea here was to use a relatively small scale. The scale here uses 9 notes per octave, out of the total of 42 per octave the tuning makes available. I like how four steps of the interval 35:24 comes very close to 32:7. This is rather like how four steps of the interval 3:2 comes very close to the interval 5:1. This second closeness is the syntonic comma, fundamental to conventional music. 35:24 is quite close to 3:2. Maybe the similarity of these two closenesses will let me make some music that doesn't sound too too unconventional?

The interval 48:25 (the inverse of 35:24) corresponds to 19 steps of 42edo. This table shows, in the second column, the sequence of pitch classes visited when moving by repeated intervals of 19 steps. The third and fourth columns show when this sequence comes close to the starting pitch (0). When the sequence comes closer than it has before, that marks a scale size that has the circulation property. Circulation means the scale can be shifted by just tweaking one of the included pitch classes. Scales of size 3, 5, 7, 9, 11, 20, and 31 circulate here. The scale used in this new piece is the scale of size 9, including the pitch classes 0, 7, 11, 15, 19, 26, 30, 34, and 38. The scale can be shifted in one direction by replacing 0 with 3, or in the other direction by replacing 26 with 23. These replacements correspond to the conventional sharps and flats of changing keys.

The other desired property of a scale is that it supports comma traversal. In this diagram, moving one cell to the right is moving a minor third, 6:5. Moving up a cell is moving by the interval 8:7. The nine pitch classes of this scale form a path that connects repeated appearances of pitch classes. This path corresponds to a comma traversal.

My algorithmic composition software is based on a consonance score for intervals. 3:2 gets a very good score; 35:24 does not get a very good score! 6:5 and 8:7 get reasonably good scores. This scale being based on a rather dissonant generator gave my software a bit of a challenge! I can tweak my consonance score calculator to some extent, but there are practical limits! So trying to build music using 35:24 as a parallel to 3:2 could only go so far! I tried to use a scale with only 7 notes per octave, the same as a conventional diatonic scale. But the resulting path really could not be traversed with reasonably consonant intervals! So I moved up to 9 notes per octave... it came out a lot better!