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Fractional sharp notation: Difference between revisions

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The '''fractional sharp notation''' (FSN) is a notation developed by [[User:CompactStar|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 {{sharp}}<sup>1/2</sup> for half-sharp. 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 [[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.


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. {{shar}}<sup>x</sup> and .
<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).


== For EDOs ==
== For EDOs ==
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== For rank-2 temperaments ==
== 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]]) 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.  
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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| 1/2-aug unison, minor 2nd
| 1/2-aug unison, minor 2nd
| 1/2-A1, m2
| 1/2-A1, m2
| D{{sharp}}<sup>1/2</sup>, Eb
| D#<sup>1/2</sup>, Eb
|-
|-
| 2
| 2
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| aug unison, neutral 2nd
| aug unison, neutral 2nd
| A1, n2
| A1, n2
| D{{sharp}}, Eb<sup>1/2</sup>
| D#, Eb<sup>1/2</sup>
|-
|-
| 3
| 3
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| neutral 3rd
| neutral 3rd
| n3
| n3
| F{{sharp}}<sup>1/2</sup>
| F#<sup>1/2</sup>
|-
|-
| 6
| 6
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| major 3rd
| major 3rd
| M3
| M3
| F{{sharp}}
| F#
|-
|-
| 7
| 7
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| 1/2-aug 4th, dim 5th
| 1/2-aug 4th, dim 5th
| 1/2-A4, d5
| 1/2-A4, d5
| G{{sharp}}<sup>1/2</sup>, Ab
| G#<sup>1/2</sup>, Ab
|-
|-
| 9
| 9
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| aug 4th, 1/2-dim 5th
| aug 4th, 1/2-dim 5th
| A4, 1/2-d5
| A4, 1/2-d5
| G{{sharp}}, Ab<sup>1/2</sup>  
| G#, Ab<sup>1/2</sup>  
|-
|-
| 10
| 10
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| neutral 7th, dim octave
| neutral 7th, dim octave
| n7, d8
| n7, d8
| C{{sharp}}<sup>1/2</sup>, Db
| C#<sup>1/2</sup>, Db
|-
|-
| 16
| 16
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| major 7th, 1/2-dim octave
| major 7th, 1/2-dim octave
| M7, 1/2-d8
| M7, 1/2-d8
| C{{sharp}}, Db<sup>1/2</sup>
| C#, Db<sup>1/2</sup>
|-
|-
| 17
| 17
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| 1/3-aug unison, minor 2nd
| 1/3-aug unison, minor 2nd
| 1/3-A1, m2
| 1/3-A1, m2
| D{{sharp}}<sup>1/3</sup>, Eb
| D#<sup>1/3</sup>, Eb
|-
|-
| 2
| 2
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| 2/3-aug unison, 1/3-aug minor 2nd
| 2/3-aug unison, 1/3-aug minor 2nd
| 2/3-A1, 1/3-AM2
| 2/3-A1, 1/3-AM2
| D{{sharp}}<sup>2/3</sup>, Eb<sup>2/3</sup>
| D#<sup>2/3</sup>, Eb<sup>2/3</sup>
|-
|-
| 3
| 3
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| aug unison, 1/3-dim major 2nd
| aug unison, 1/3-dim major 2nd
| A1, 1/3-dM2
| A1, 1/3-dM2
| D{{sharp}}, Eb<sup>1/3</sup>
| D#, Eb<sup>1/3</sup>
|-
|-
| 4
| 4
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| 1/3-aug minor 3rd
| 1/3-aug minor 3rd
| 1/3-Am3
| 1/3-Am3
| F{{sharp}}<sup>1/3</sup>
| F#<sup>1/3</sup>
|-
|-
| 7
| 7
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| 1/3-dim major 3rd
| 1/3-dim major 3rd
| 1/3-dM3
| 1/3-dM3
| F{{sharp}}<sup>2/3</sup>
| F#<sup>2/3</sup>
|-
|-
| 8
| 8
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| major 3rd
| major 3rd
| M3
| M3
| F{{sharp}}
| F#
|-
|-
| 9
| 9
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| 1/3-aug 4th, dim 5th
| 1/3-aug 4th, dim 5th
| 1/3-A4, d5
| 1/3-A4, d5
| G{{sharp}}<sup>1/3</sup>, Ab
| G#<sup>1/3</sup>, Ab
|-
|-
| 11
| 11
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| 2/3-aug 4th, 2/3-dim 5th
| 2/3-aug 4th, 2/3-dim 5th
| 2/3-A4, 2/3-d5
| 2/3-A4, 2/3-d5
| G{{sharp}}<sup>2/3</sup>, Ab<sup>2/3</sup>
| G#<sup>2/3</sup>, Ab<sup>2/3</sup>
|-
|-
| 12
| 12
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| aug 4th, 1/3-dim 5th
| aug 4th, 1/3-dim 5th
| A4, 1/3-d5
| A4, 1/3-d5
| G{{sharp}}, Ab<sup>1/3</sup>
| G#, Ab<sup>1/3</sup>
|-
|-
| 13
| 13
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| 1/3-aug minor 7th
| 1/3-aug minor 7th
| 1/3-Am7
| 1/3-Am7
| C{{sharp}}<sup>1/3</sup>
| C#<sup>1/3</sup>
|-
|-
| 20
| 20
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| 1/3-dim major 7th
| 1/3-dim major 7th
| 1/3-dM7
| 1/3-dM7
| C{{sharp}}<sup>2/3</sup>
| C#<sup>2/3</sup>
|-
|-
| 21
| 21
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| major 7th
| major 7th
| M7
| M7
| C{{sharp}}
| C#
|-
|-
| 22
| 22
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| D
| D
|}
|}
{{Navbox notation}}
[[Category:Notation]]
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