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, and does not have compounding accidentals for large EDOs, although it will feature complex rational numbers as the amount of sharps. Excluding the no-fifth systems of 1edo-4edo and 8edo, there are 4 EDOs (all multiples of 7edo) that cannot be notated using the native fifth: . However, it is still possible to notate them as subsets of 42edo (for 14edo and 21edo), 56edo (for 28edo), and 70edo (for 35edo). 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".

Examples

17edo:

Degree	Cents	Notation
0	0.000	perfect unison	P1	D
1	70.588	minor 2nd	m2	Eb
2	141.176	neutral 2nd	n2	Eb^1/2
3	211.765	major 2nd	M2	E
4	282.353	minor 3rd	m3	F
5	352.941	neutral 3rd	n3	F#^1/2
6	423.529	major 3rd	M3	F#
7	494.118	perfect 4th	P4	G
8	564.706	1/2-aug 4th, dim 5th	1/2-A4, d5	G#^1/2, Ab
9	635.294	aug 4th, 1/2-dim 5th	A4, 1/2-d5	G#, Ab^1/2
10	705.882	perfect 5th	P5	A
11	776.471	minor 6th	m6	Bb
12	847.059	neutral 6th	n6	Bb^1/2
13	917.647	major 6th	M6	B
14	988.235	minor 7th	m7	C
15	1058.824	neutral 7th	n7	C#^1/2
16	1129.412	major 7th	M7	C#
17	1200.00	perfect octave	P8	D

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-dim major 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