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 or caret (like #1/2 for half-sharp) except for in the case of single and double accidentals.The sharp and flat accidentals are always taken to raise and lower by an augmented union or apotome. 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".
For EDOs
Fractional sharp notes by using a 3-limit mapping. 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. Antidiatonic fifths may be notated using both the "major wider than minor" and "minor wider than major" systems, depending on what is needed. Accidentals do not stack for large EDOs because of the superscript notation, but the amount of sharps can often be a complicated rational number.
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 | minor 2nd | m2 | Eb |
| 2 | 109.091 | 1/3-aug minor 2nd | 1/3-AM2 | Eb2/3 |
| 3 | 163.636 | 1/3-dim major 2nd | 1/3-dM2 | 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 |