Loop Search

Recently I was exploring the 270edo tuning system, and a 90 note scale. I wanted a symmetric scale, which this subset is. I also wanted a scale that would traverse the comma 2080:2079, which includes the primes 2, 3, 5, 7, 11, and 13. I was wondering whether that scale might support traversal of other commas, too. This is a lot of notes in a space of rather high dimensionality! So I wrote a program! I can switch the set of primes, the tuning system, and the scale, all very easily, to explore the possibilities for comma traversals.

The program looks at loops for each note in the scale: sometimes the loops that pass through one note are different than those that pass through a different note. Experimenting with this program, I see that happening sometimes. For this particular 90 note subset of 270edo, each note has the same set of loops:

• | 12 -2 -1 -1 0 -1 > = 4096:4095
• | -7 -1 2 0 -1 2 > = 4225:4224
• | -10 -1 -1 1 0 3 > = 15379:15360
• | 2 1 -1 -3 1 1 > = 1716:1715

These strings of numbers are the powers of the primes 2, 3, 5, 7, 11, and 13, so they give the prime factorization of the ratios. These commas can be combined arbitrarily to form a whole linear space of commas that can be traversed with this scale. My new program looks for a minimal basis for this space.

This 90 note subset is symmetric: the 270 notes of the tuning system are divided into 10 blocks of 27 notes each. The scale picks out 9 notes from each block, at the same positions in each block. I ran the program with only 8 notes per block: then the sets of commas traversable had a basis of three instead of four commas. When I increased the number of notes per block, some notes had a basis set of five commas. So 9 notes per block does seem like a nice threshold.

Another scale I have explored at some point - I don't remember when! - is a 21 note scale in 72edo, generated by the semitone 16:15. 72edo works well with primes 2, 3, 5, 7, and 11, but not with 13. So I ommitted 13 in this analysis. Again, all the notes in the scale had the same space of commas traversable:

• | -10 1 0 3 0 > = 1029:1024
• | -5 2 2 -1 0 > = 225:224
• | -7 -1 1 1 1 > = 385:384