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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.


The '''fractional sharp notation''' (FSN) is a notation developed by [[User:CompactStar|CompactStar]] that is an extension of [[circle-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 #<sup>1/2</sup> 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. Excluding [[1edo]], [[3edo]], [[4edo]] and [[8edo]], which do not have a diatonic or antidiatonic fifth, there are 4 [[EDOs]] (all multiples of [[7edo]]) that cannot be notated using the native fifth. These are [[14edo]], [[21edo]], [[28edo]] and [[35edo]]. However, it is still possible to notate them as subsets of [[42edo]] (for 14edo and 21edo), [[56edo]] (for 28edo), and [[70edo]] (for 35edo). [[2L 5s|Antidiatonic]] fifths may be notated using both the "major wider than minor" and "minor wider than major" systems, depending on what is needed. and does not have compounding accidentals for large EDOs, although it will feature complex rational numbers as the amount of sharps
<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).


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 [[User:Frostburn|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 ==
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 ==
== Examples ==
[[17edo]]:
[[17edo]]:
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| 1
| 1
| 70.588
| 70.588
| minor 2nd
| 1/2-aug unison, minor 2nd
| m2
| 1/2-A1, m2
| Eb
| D#<sup>1/2</sup>, Eb
|-
|-
| 2
| 2
| 141.176
| 141.176
| neutral 2nd
| aug unison, neutral 2nd
| n2
| A1, n2
| Eb<sup>1/2</sup>
| D#, Eb<sup>1/2</sup>
|-
|-
| 3
| 3
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| 15
| 15
| 1058.824
| 1058.824
| neutral 7th
| neutral 7th, dim octave
| n7
| n7, d8
| C#<sup>1/2</sup>
| C#<sup>1/2</sup>, Db
|-
|-
| 16
| 16
| 1129.412
| 1129.412
| major 7th
| major 7th, 1/2-dim octave
| M7
| M7, 1/2-d8
| C#
| C#, Db<sup>1/2</sup>
|-
|-
| 17
| 17
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| 1
| 1
| 54.545
| 54.545
| minor 2nd
| 1/3-aug unison, minor 2nd
| m2
| 1/3-A1, m2
| Eb
| D#<sup>1/3</sup>, Eb
|-
|-
| 2
| 2
| 109.091
| 109.091
| 1/3-aug minor 2nd
| 2/3-aug unison, 1/3-aug minor 2nd
| 1/3-AM2
| 2/3-A1, 1/3-AM2
| Eb<sup>2/3</sup>
| D#<sup>2/3</sup>, Eb<sup>2/3</sup>
|-
|-
| 3
| 3
| 163.636
| 163.636
| 1/3-dim major 2nd
| aug unison, 1/3-dim major 2nd
| 1/3-dM2
| A1, 1/3-dM2
| Eb<sup>1/3</sup>
| D#, Eb<sup>1/3</sup>
|-
|-
| 4
| 4
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|}
|}


{{Navbox notation}}


 
[[Category:Notation]]
== Just intonation ==
This segment will probably removed or changed
 
=== Accidentals ===
{|class="wikitable"
|-
!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
|}
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