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.
VERY WIP (I'll move it to the main namespace if it's finished.)


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 .
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 #<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. For interval naming, augmented and diminished are expanded to fractional values: a/b-augmented (abbreviated as a/b-A, as in [[User:M-yac/Neutral_Intervals_and_the_FJS#Generalizing|semi-neutral FJS]]) is a/b of a chromatic semitone above a major or perfect interval, and a/b-diminished (abbreviated as a/b-d) is a/b of a chromatic semitone below a minor or perfect interval. Additionally, for intervals between minor and major, a/b-augmented minor is used for a/b chromatic semitones above a minor interval, and a/b-diminished major is used for a/b chromatic semitones below a major interval (this was suggested by [[User:Frostburn|Frostburn]]). 1/2-augmented minor and 1/2-diminished major are identical, corresponding to halfway between minor and major, so both are replaced with the more conventional term "neutral" (abbreviated as lowercase n).


== For EDOs ==
== For EDOs ==
Line 7: Line 9:


== 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]]:
Line 27: Line 29:
| 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
Line 33: Line 35:
| 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
Line 69: Line 71:
| 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
Line 75: Line 77:
| 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
Line 111: Line 113:
| 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
Line 117: Line 119:
| 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
Line 156: Line 158:
| 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
Line 186: Line 188:
| 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
Line 258: Line 260:
| 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
Line 264: Line 266:
| major 7th
| major 7th
| M7
| M7
| C{{sharp}}
| C#
|-
|-
| 22
| 22

Revision as of 04:14, 26 December 2024

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, such as #1/2 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. For interval naming, augmented and diminished are expanded to fractional values: a/b-augmented (abbreviated as a/b-A, as in semi-neutral FJS) is a/b of a chromatic semitone above a major or perfect interval, and a/b-diminished (abbreviated as a/b-d) is a/b of a chromatic semitone below a minor or perfect interval. Additionally, for intervals between minor and major, a/b-augmented minor is used for a/b chromatic semitones above a minor interval, and a/b-diminished major is used for a/b chromatic semitones below a major interval (this was suggested by Frostburn). 1/2-augmented minor and 1/2-diminished major are identical, corresponding to halfway between minor and major, so both are replaced with the more conventional term "neutral" (abbreviated as lowercase n).

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. 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 temperaments 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 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:

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

22edo:

Degree Cents Notation
0 0.000 perfect unison P1 D
1 54.545 1/3-aug unison, minor 2nd 1/3-A1, m2 D#1/3, Eb
2 109.091 2/3-aug unison, 1/3-aug minor 2nd 2/3-A1, 1/3-AM2 D#2/3, Eb2/3
3 163.636 aug unison, 1/3-dim major 2nd A1, 1/3-dM2 D#, 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
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