114edo

I've been exploring diaschismic tunings, tunings that temper out the diaschisma comma, 2048:2025. 2025 = 25 * 81, so the comma gets tempered by making both perfect fifths and major thirds a bit sharp. A few days ago I looked at 90edo, which has a perfect fifth sharp enough to do all the work, leaving the major third very close to just. Today's exploration is 114edo, which has perfect fifths and major thirds equally sharp, sharing the workload.

Looking at the tuning errors of 114edo, in the upper left area one can see that 3:1 and 5:1, corresponding to perfect fifths and major thirds, are not so accurate, being sharp enough to temper out the diaschisma. Since they are equally sharp, minor thirds, 5:3, are quite precise. But what jumped out at me with this table of errors is that 114edo has a quite accurate approximation for 7:1. That should spice things up!

Staring at the tonnetz diagram for a while and contemplating 7:1 which corresponds to 92 steps of 114... well, two steps of 7:1 would be 70 steps, which comes back near the the starting point. This corresponds to the comma 245:243, which I see people call the minor Bohlen-Pierce diesis. I looked for a scale that would support traversals of this and the diaschisma. I came up with the 24 note scale highlighted above. This is the union of two scales generated by the interval of 62 steps of 114edo, the two scales offset by 67 steps. As I have thought more about this, probably offsetting the scales by 5 steps would have been better, but I got pretty far down the road with the 67 step offset, so that's what I have here.

The tonnetz diagram layout shows perfect fifths, moving from a cell to its neighbor on the right, and major thirds, moving from a cell to its neighbor above. I used colors to show movement by 7:1. Each blue cell can move to a purple cell by a 7:1 interval. So an example of a traversal of 245:243 would be to start at a (blue) 0 cell and leap to a (purple) 92 cell, moving with a 7:1 interval. Then move by two perfect fourths, stepping left two cells, to a (blue) 72 cell. Leap by 7:1 again to a (purple) 50 cell. From there, move left three cells, three perfect fourths, to a 77 cell, and then up one cell, a major third, which brings one back to a 0 cell where the traversal started.

Here's what the scale sounds like: 114edo scale 24.