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VERY WIP (I'll move it to the main namespace if it's finished.)
The '''fractional sharp notation''' (FSN) is a notation developed by [[User:CompactStar|CompactStar]] that is an extension of [[chain-of-fifths notation]], supporting almost all [[EDO]]s 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 #<sup>1/2</sup> 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.


This notation is mainly intended for equal temperaments
<nowiki>#</nowiki><sup>a/b</sup> (can be spoken as "a over b sharp") is always taken to raise by a/b chromatic semitones, and b<sup>a/b</sup> 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).


== Accidentals ==
== For EDOs ==
{|class="wikitable"
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]]. [[2L 5s|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 temperament]]s 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 [[2187/2048|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]]:
 
{| class="wikitable"
|-
! Degree
! Cents
! colspan="3"|Notation
|-
| 0
| 0.000
| perfect unison
| P1
| D
|-
| 1
| 70.588
| 1/2-aug unison, minor 2nd
| 1/2-A1, m2
| D#<sup>1/2</sup>, Eb
|-
| 2
| 141.176
| aug unison, neutral 2nd
| A1, n2
| D#, Eb<sup>1/2</sup>
|-
| 3
| 211.765
| major 2nd
| M2
| E
|-
| 4
| 282.353
| minor 3rd
| m3
| F
|-
| 5
| 352.941
| neutral 3rd
| n3
| F#<sup>1/2</sup>
|-
| 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#<sup>1/2</sup>, Ab
|-
|-
!Prime limit
| 9
!Comma
| 635.294
!Sharps
| aug 4th, 1/2-dim 5th
| A4, 1/2-d5
| G#, Ab<sup>1/2</sup>
|-
|-
|5
| 10
|[[81/80]]
| 705.882
|1/5
| perfect 5th
| P5
| A
|-
|-
|7
| 11
|[[64/63]]
| 776.471
|1/4
| minor 6th
| m6
| Bb
|-
|-
|11
| 12
|[[33/32]]
| 847.059
|1/2
| neutral 6th
| n6
| Bb<sup>1/2</sup>
|-
|-
|13
| 13
|[[1053/1024]]
| 917.647
|1/2
| major 6th
| M6
| B
|-
|-
|17
| 14
|[[4131/4096]]
| 988.235
|1/8
| minor 7th
| m7
| C
|-
|-
|19
| 15
|[[513/512]]
| 1058.824
|1/34
| neutral 7th, dim octave
| n7, d8
| C#<sup>1/2</sup>, Db
|-
|-
|23
| 16
|[[736/729]]
| 1129.412
| major 7th, 1/2-dim octave
| M7, 1/2-d8
| C#, Db<sup>1/2</sup>
|-
| 17
| 1200.00
| perfect octave
| P8
| D
|}


[[22edo]]:
{| class="wikitable"
|-
! Degree
! Cents
! colspan="3" |Notation
|-
|-
| 0
| 0.000
| perfect unison
| P1
| D
|-
| 1
| 54.545
| 1/3-aug unison, minor 2nd
| 1/3-A1, m2
| D#<sup>1/3</sup>, Eb
|-
| 2
| 109.091
| 2/3-aug unison, 1/3-aug minor 2nd
| 2/3-A1, 1/3-AM2
| D#<sup>2/3</sup>, Eb<sup>2/3</sup>
|-
| 3
| 163.636
| aug unison, 1/3-dim major 2nd
| A1, 1/3-dM2
| D#, Eb<sup>1/3</sup>
|-
| 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#<sup>1/3</sup>
|-
| 7
| 381.818
| 1/3-dim major 3rd
| 1/3-dM3
| F#<sup>2/3</sup>
|-
| 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#<sup>1/3</sup>, Ab
|-
| 11
| 600.000
| 2/3-aug 4th, 2/3-dim 5th
| 2/3-A4, 2/3-d5
| G#<sup>2/3</sup>, Ab<sup>2/3</sup>
|-
| 12
| 654.545
| aug 4th, 1/3-dim 5th
| A4, 1/3-d5
| G#, Ab<sup>1/3</sup>
|-
| 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<sup>2/3</sup>
|-
| 16
| 872.727
| 1/3-dim major 6th
| 1/3-dM6
| Bb<sup>1/3</sup>
|-
| 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#<sup>1/3</sup>
|-
| 20
| 1090.909
| 1/3-dim major 7th
| 1/3-dM7
| C#<sup>2/3</sup>
|-
| 21
| 1145.455
| major 7th
| M7
| C#
|-
| 22
| 1200.000
| perfect octave
| P8
| D
|}
|}
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[[Category:Notation]]
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