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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 largely an extension of [[circle-of-fifths notation]], which can be used for [[EDO]]s, and secondarily for [[temperament]]s and [[just intonation]] subgroups of [[rank]] 3. It represents all intervals with conventional accidentals, but with sharps and flats extended to have an arbitrary rational amount, denoted by a superscript (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 equal tunings. Multiples of [[7edo]] up to [[35edo]], and excluding [[7edo]] itself, are not supported, but these are possible to be notated as subsets of larger EDOs. In particular, [[14edo]] and [[21edo]] can be notated as subsets of  [[42edo]], [[28edo]] as a subset of [[56edo]], and [[35edo]] as a subset of [[70edo]] or using the alternative fifth from [[5edo]]. [[1edo]] to [[4edo]] are obviously notated as subsets, and [[8edo]] is also needed to be notated as a subset of [[16edo]] or [[24edo]] with no fifth except for 750 cents. [[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.
<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. In a given EDO system, the step size is 1 over the [[sharpness]] of a sharp. In interval naming:
== For EDOs ==
* x-minor represents x of the way from major to minor (0-minor is major and 1-minor is major, x below 1/2 is replaced by (1-x)-minor).
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.
* x-major represents x of the way from minor to major (0-major is minor and 1-major is major, x below 1/2 is replaced by (1-x)-major).
* 1/2-minor and 1/2-major are identical so are called by the more conventional "neutral".
* x-augmented represents x of the way from perfect/major to augmented.
* x-diminished represents x of the way from perfect/minor to diminished.
* Augmented and diminished can be stacked more than once if needed, such as doubly augmented, and with fractional amounts as well like 3/2-augmented.


Here is an example for [[22edo]]:
== 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 center-all right-2 left-3"
{| 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
|-
| 9
| 635.294
| aug 4th, 1/2-dim 5th
| A4, 1/2-d5
| G#, Ab<sup>1/2</sup>
|-
| 10
| 705.882
| perfect 5th
| P5
| A
|-
| 11
| 776.471
| minor 6th
| m6
| Bb
|-
| 12
| 847.059
| neutral 6th
| n6
| Bb<sup>1/2</sup>
|-
| 13
| 917.647
| major 6th
| M6
| B
|-
| 14
| 988.235
| minor 7th
| m7
| C
|-
| 15
| 1058.824
| neutral 7th, dim octave
| n7, d8
| C#<sup>1/2</sup>, Db
|-
| 16
| 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
! Degree
Line 28: Line 142:
| 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
| 2/3-minor 2nd
| 2/3-aug unison, 1/3-aug minor 2nd
| 2/3-m2
| 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
| 2/3-major 2nd
| aug unison, 1/3-dim major 2nd
| 2/3-M2
| A1, 1/3-dM2
| Eb<sup>1/3</sup>
| D#, Eb<sup>1/3</sup>
|-
|-
| 4
| 4
Line 58: Line 172:
| 6
| 6
| 327.273
| 327.273
| 2/3-minor 3rd
| 1/3-aug minor 3rd
| 2/3-m3
| 1/3-Am3
| F#<sup>1/3</sup>
| F#<sup>1/3</sup>
|-
|-
| 7
| 7
| 381.818
| 381.818
| 2/3-major 3rd
| 1/3-dim major 3rd
| 2/3-M3
| 1/3-dM3
| F#<sup>2/3</sup>
| F#<sup>2/3</sup>
|-
|-
Line 112: Line 226:
| 15
| 15
| 818.182
| 818.182
| 2/3-minor 6th
| 1/3-aug minor 6th
| 2/3-m6
| 1/3-Am6
| Bb<sup>2/3</sup>
| Bb<sup>2/3</sup>
|-
|-
| 16
| 16
| 872.727
| 872.727
| 2/3-major 6th
| 1/3-dim major 6th
| 2/3-M6
| 1/3-dM6
| Bb<sup>1/3</sup>
| Bb<sup>1/3</sup>
|-
|-
Line 136: Line 250:
| 19
| 19
| 1036.364
| 1036.364
| 2/3-minor 7th
| 1/3-aug minor 7th
| 2/3-m7
| 1/3-Am7
| C#<sup>1/3</sup>
| C#<sup>1/3</sup>
|-
|-
| 20
| 20
| 1090.909
| 1090.909
| 2/3-major 7th
| 1/3-dim major 7th
| 2/3-M7
| 1/3-dM7
| C#<sup>2/3</sup>
| C#<sup>2/3</sup>
|-
|-
Line 159: Line 273:
|}
|}


=== Accidentals ===
{{Navbox notation}}
{|class="wikitable"
 
|-
[[Category:Notation]]
!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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