Somehow, though, that's not how the math works! This is a table of tuning errors for 13edo: how far off it is from just intonation for a variety of intervals. The number in the upper left, 0.395, is the most fundamental error, the error for perfect fifths. A just tuned perfect fifth lies 39.5% of the way between two of the notes of 13edo. It could hardly be worse!
So, maybe we can try to make some music using intervals that 13edo approximates well. Still focusing on the region of simpler intervals toward the upper left, 9:5 and 11:1 are approximated quite well. Let's try using those as building blocks!
So here is an unconventional tonnetz diagram based on these two intervals. 9:5 is equivalent to 10:9 which is one form of a conventional whole step. 2 steps of 13edo form a interval a bit flatter than 2 steps of 12edo; 10:9 is a bit flatter than 9:8. So far, so good! In the diagram, moving right or left from a cell is moving up or down by 10:9. Moving down is moving by 11:8; moving up is moving by 16:11. These are not conventional intervals at all! These are close to the dissonant tritone: 11:8 is a bit flatter; 16:11 is a bit sharper. Then again, 11:8 is a bit sharper than a perfect fourth, 4:3, and 16:11 is a bit flatter than a perfect fifth, 3:2. This is how 13edo works: it chops the pitch spectrum up in a very different way!
Cells labeled 0 occur in many places. A tonnetz diagram like this is really an unrolled torus. A path from one 0 cell to another 0 cell represents a loop on the torus, which is a traversal of a comma. Starting from one 0 cell, each other 0 cell represents a different comma, a just interval that is quite close to unison. There is a path 0, 2, 4, 6, 0 that corresponding to a traversal of the comma 8019:8000. Extending this path looks like a nice way to build a scale with six notes per octave.
Here's what it sounds like: 13edo steer
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