Fractional sharp notation

The fractional sharp notation (FSN) is a notation developed by CompactStar that is an extension of chain-of-fifths notation, supporting almost all EDOs and several rank-2 temperament systems. It represents all intervals with conventional accidentals, but with sharps and flats extended to have an arbitrary rational amount, denoted by a superscript, such as #^1/2 for half-sharp, except for in the case of single and double accidentals. If ASCII compatibility is required, superscripts can be substituted for carets–in this case, #^(a/b) is preferred over #^a/b for clarity.

#^a/b (can be spoken as "a over b sharp") is always taken to raise by a/b chromatic semitones, and b^a/b is always taken to lower by a/b chromatic semitones. The "augmented" and "diminished" qualifiers for interval names are also extended to arbitrary rational amounts, where a/b-augmented (a/b-A) widens the interval by a/b chromatic semitones and a/b-diminished (a/b-d) narrows the interval by a/b chromatic semitones. Intervals between minor and major are expressed as a/b-augmented minor or a/b-diminished major (this was suggested by User:Frostburn). For example, 1/3 of the way from a minor third to a major third is a 1/3-augmented minor third, while 2/3 of the way from a minor third to a major third is a 1/3-diminished major third. Because 1/2-augmented minor and 1/2-diminished major are identical, they are instead referred to by the more conventional "neutral" (n).

For EDOs

By using a tempered fifth, almost all EDO tunings are supported, since there is support for not only half-sharps and half-flats, but third-sharps, third-flats and so on. Excluding 1edo-4edo and 8edo, there are four EDOs (all multiples of 7edo) that cannot be notated using the native fifth: 14edo, 21edo, 28edo and 35edo. However, it is still possible to notate them with subset notation, using 42edo's notation for 14edo and 21edo, 56edo's notation for 28edo, and 70edo's notation for 35edo. 35edo can additionally be notated using the b val sharp fifth from 5edo. Antidiatonic fifths may be notated using both the "major wider than minor" and "major narrower than minor" systems, with the former involving swapping sharps/flats, major/minor and augmented/diminished with each other. Accidentals do not stack for large EDOs because of the superscript notation, but the amount of sharps can often be a complicated rational number.

For rank-2 temperaments

A few rank-2 temperaments can be notated, but only ones which have a period of an unsplit octave, and in which the generator can be expressed as an FSN interval category. For example, neutral temperament can have the generator notated as n3, and porcupine temperament can have the generator notated as 1/3-dM2, because the difference between the generator and 9/8 (represented by 81/80, 45/44 and etc.) is equated to 1/3 of an apotome in porcupine. Semaphore is an example of a temperament which does not qualify, because there is no FSN category that implies a semifourth.

Examples

17edo:

Degree	Cents	Notation
0	0.000	perfect unison	P1	D
1	70.588	1/2-aug unison, minor 2nd	1/2-A1, m2	D#1/2, Eb
2	141.176	aug unison, neutral 2nd	A1, n2	D#, Eb1/2
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#, Ab1/2
10	705.882	perfect 5th	P5	A
11	776.471	minor 6th	m6	Bb
12	847.059	neutral 6th	n6	Bb1/2
13	917.647	major 6th	M6	B
14	988.235	minor 7th	m7	C
15	1058.824	neutral 7th, dim octave	n7, d8	C#1/2, Db
16	1129.412	major 7th, 1/2-dim octave	M7, 1/2-d8	C#, Db1/2
17	1200.00	perfect octave	P8	D

22edo:

Degree	Cents	Notation
0	0.000	perfect unison	P1	D
1	54.545	1/3-aug unison, minor 2nd	1/3-A1, m2	D#1/3, Eb
2	109.091	2/3-aug unison, 1/3-aug minor 2nd	2/3-A1, 1/3-AM2	D#2/3, Eb2/3
3	163.636	aug unison, 1/3-dim major 2nd	A1, 1/3-dM2	D#, Eb1/3
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, 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	1/3-aug minor 6th	1/3-Am6	Bb2/3
16	872.727	1/3-dim major 6th	1/3-dM6	Bb1/3
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