However, it turns out that there are two accidentals that when one of them is removed the promethean set does become consistent. One of them is , which if removed leaves a set that forces 400edo with a sharp 23rd harmonic instead of a flat one, and the other is , that if removed forces 971edo. That of course prompted me to try and make notations for these edos (which will probably be of interest to someone, since they both have excellent 5ths and 400edo is compatible with cents).
First, for 400edo I had a bit of a challenge. The beginning wasn't hard, but I'm not sure how to represent 14 and 17 degrees. That's the notation I have so far:
? ?? ?? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23For 14 degrees I have two competing commas:
- , because it is similar to the apotome complement of . The problem with it is that its primary value is 23-limit and isn't a valid comma here. This problem can be solved with a suitable secondary value, which may be 23-limit or not, but I have no idea how to find one.
- , because it's spartan and its primary comma fits the value perfectly. The problem is that it will lead to 24 degrees being notated as which is not consistent with flag arithmetic.
Second, for 971edo I had kind of an opposite problem, where all the accidentals are mapped to distinct degrees but there are not enough of them, so we have to use accent marks. The schisma maps to 2\971 and the mina to 0, but the JI value for the double mina maps to 1\971, so I devised this Olympian notation:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56And I guess the places where there is only a need for one accented accidental would have on the lower one in revo as well (which is not the apotome complement of the base accidental of the apotome complement interval). Also, 33 steps could be notated with if we can find a valid secondary comma for it.
Any thoughts?