I was playing with microtonality. Specifically, equal temperaments other than 12edo. Since the point of equal temperaments is versatility, I wanted them to admit a notion of key signatures similar to those of the standard 12edo. My requirements were:
The white keys shall represent the default scale.
The perfect fifth (C-to-G) shall approximate Pythagorean, and the white keys shall be along the generated circle of fifths.
No monotonic scales.
Sharps (resp. flats) shall be positive (resp. negative).
One unit of sharp (that is, for example, half-sharp if appropriate) on an appropriate white key shall transpose the scale (whatever the default scale was).
After some investigation, it turns out many of EDOs don't admit a heptatonic scale. For 22edo as an example, it doesn't admit a heptatonic scale, but a pentatonic scale. Picking the pentatonic major mode as an example, some examples of key signatures are (+/- for one step, #/b for two steps):
(White key position in steps: C=0, D=4, E=8, G=13, A=17) C pentatonic major: C, D, E, G, A C# pentatonic major: C#, D#, E#, G#, A# D pentatonic major: D, E, G-, A, C- E pentatonic major: E, G-, A-, C-, D- E+ pentatonic major: E+, G, A, C, D G pentatonic major: G, A, C-, D, E A pentatonic major: A, C-, D-, E, G-
For 41edo as another example, it doesn't admit a heptatonic scale either, but a dodecatonic scale. That is, we need to take extra characters for some white keys. Let me notate them by Cyrillic (+/- for one step):
(White key position in steps: C=0, Д=3, D=7, Э=10, E=14, F=17, Ф=21, G=24, Я=27, A=31, Б=34, B=38) C chromatic: C, Д, D, Э, E, F, Ф, G, Я, A, Б, B Ф chromatic: Ф, G, Я+, A, Б+, B, C+, Д+, D, Э+, E, F+ Ф+ chromatic: Ф+, G+, Я++, A+, Б++, B+, C++, Д++, D+, Э++, E+, F++ F chromatic: F, Ф-, G, Я, A, Б, B, C, Д, D, Э, E G chromatic: G, Я, A, Б, B, C, Д+, D, Э, E, F, Ф
So far, what I've defined are the white key names, the scale they represent, and how key signatures work for them. This arises some questions. The standard five-lines score is designed for the major scale, which is heptatonic. Would it be appropriate to modify the number of lines, so pentatonic/dodecatonic/whatever scale could be notated over? If so, how would clefs work?