Fractional sharp notation

VERY WIP (I'll move it to the main namespace if it's finished.)

The fractional sharp notation (FSN) is a notation developed by CompactStar that is largely an extension of circle-of-fifths notation, which can be used for EDOs, and secondarily for temperaments and just intonation subgroups of rank 3. It represents all intervals with conventional accidentals, but with sharps and flats extended to have an arbitrary rational amount, denoted by a superscript or caret (like #^1/2 for half-sharp) except for in the case of single and double accidentals. This means that there is support for not only half-sharps and half-flats, but third-sharps, third-flats and so on, supporting almost all EDO systems. Multiples of 7edo up to 35edo, excluding 7edo itself, are not supported, but these are possible to be notated as subsets of larger EDOs. In particular, 14edo and 21edo can be notated as subsets of 42edo, 28edo as a subset of 56edo, and 35edo as a subset of 70edo or using the alternative fifth from 5edo. 1edo to 4edo are obviously notated as subsets, and 8edo is also required to be notated as a subset of 16edo, 24edo, or 32edo with no fifth except for 750 cents. Antidiatonic fifths may be notated using both the "major wider than minor" and "minor wider than major" systems, depending on what is needed.

The sharp and flat accidentals are always taken to raise and lower by an augmented union or apotome. In a given EDO system, the step size is 1 over the sharpness of a sharp. In interval naming, x-augmented and x-diminished raise and lower by x times a chromatic semitone. These are used in the same way as augmented and diminished normally are, but additionally, "augmented minor" and "diminished major" are used for constructions like "1/3-augmented minor 3rd", meaning a minor 3rd raised by 1/3 of an apotome (formerly these were described as a "fraction of major" like 1/3-major, the updated version was suggested by Frostburn). Since 1/2-augmented minor and 1/2-diminished major mean the same thing, they are replaced with the more conventional term "neutral".

Here is an example for 22edo:

Degree	Cents	Notation
0	0.000	perfect unison	P1	D
1	54.545	minor 2nd	m2	Eb
2	109.091	1/3-aug minor 2nd	1/3-AM2	Eb^2/3
3	163.636	1/3-dim major 2nd	1/3-dM2	Eb^1/3
4	218.182	major 2nd	M2	E
5	272.727	minor 3rd	m3	F
6	327.273	1/3-aug minor 3rd	1/3-Am3	F#^1/3
7	381.818	1/3-aug dim 3rd	1/3-dM3	F#^2/3
8	436.364	major 3rd	M3	F#
9	490.909	perfect fourth	P4	G
10	545.455	1/3-aug 4th, dim 5th	1/3-A4, d5	G#^1/3, Ab
11	600.000	2/3-aug 4th, 2/3-dim 5th	2/3-A4, 2/3-d5	G#^2/3, Ab^2/3
12	654.545	aug 4th, 1/3-dim 5th	A4, 1/3-d5	G#, Ab^1/3
13	709.091	perfect 5th	P5	A
14	763.636	minor 6th	m6	Bb
15	818.182	1/3-aug minor 6th	1/3-Am6	Bb^2/3
16	872.727	1/3-dim major 6th	1/3-dM6	Bb^1/3
17	927.273	major 6th	M6	B
18	981.818	minor 7th	m7	C
19	1036.364	1/3-aug minor 7th	1/3-Am7	C#^1/3
20	1090.909	1/3-dim major 7th	1/3-dM7	C#^2/3
21	1145.455	major 7th	M7	C#
22	1200.000	perfect octave	P8	D

Just intonation

This segment will probably removed or changed

Accidentals

Prime limit	Comma	Sharps
5	81/80	1/5
7	64/63	1/4
11	33/32	1/2
13	1053/1024	1/2
17	4131/4096	1/8
19	513/512	1/34
23	736/729	1/7