User:CompactStar/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 an extension of chain-of-fifths notation, supporting a wide range of EDO and 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.
As in the original chain-of-fifths notation, the sharp and flat accidentals are always taken to raise and lower by an augmented union or chromatic semitone. For interval naming, augmented and diminished are expanded to fractional values: a/b-augmented (abbreviated as a/b-A, as in semi-neutral FJS) is a/b of a chromatic semitone above a major or perfect interval, and a/b-diminished (abbreviated as a/b-d) is a/b of a chromatic semitone below a minor or perfect interval. Additionally, for intervals between minor and major, a/b-augmented minor is used for a/b chromatic semitones above a minor interval, and a/b-diminished major is used for a/b chromatic semitones below a major interval (this was suggested by Frostburn). 1/2-augmented minor and 1/2-diminished major are identical, corresponding to halfway between minor and major, so both are replaced with the more conventional term "neutral" (abbreviated as lowercase 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. 36However, 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
| 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 |
| 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 |