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 (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 equal tunings. Multiples of 7edo up to 35edo, and 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 needed to be notated as a subset of 16edo or 24edo 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. In a given EDO system, the step size is 1 over the sharpness of a sharp. In interval naming:

• x-minor represents x of the way from major to minor (0-minor is major and 1-minor is major, x below 1/2 is replaced by (1-x)-minor).
• x-major represents x of the way from minor to major (0-major is minor and 1-major is major, x below 1/2 is replaced by (1-x)-major).
• 1/2-minor and 1/2-major are identical so are called by the more conventional "neutral".
• x-augmented represents x of the way from perfect/major to augmented.
• x-diminished represents x of the way from perfect/minor to diminished.
• Augmented and diminished can be stacked more than once if needed, such as doubly augmented, and with fractional amounts as well like 3/2-augmented.

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	2/3-minor 2nd	2/3-m2	Eb2/3
3	163.636	2/3-major 2nd	2/3-M2	Eb1/3
4	218.182	major 2nd	M2	E
5	272.727	minor 3rd	m3	F
6	327.273	2/3-minor 3rd	2/3-m3	F#1/3
7	381.818	2/3-major 3rd	2/3-M3	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, Ab2/3
12	654.545	aug 4th, 1/3-dim 5th	A4, 1/3-d5	G#, Ab1/3
13	709.091	perfect 5th	P5	A
14	763.636	minor 6th	m6	Bb
15	818.182	2/3-minor 6th	2/3-m6	Bb2/3
16	872.727	2/3-major 6th	2/3-M6	Bb1/3
17	927.273	major 6th	M6	B
18	981.818	minor 7th	m7	C
19	1036.364	2/3-minor 7th	2/3-m7	C#1/3
20	1090.909	2/3-major 7th	2/3-M7	C#2/3
21	1145.455	major 7th	M7	C#
22	1200.000	perfect octave	P8	D

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