Fractional sharp notation: Difference between revisions

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VERY WIP (I'll move it to the main namespace if it's finished.)
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 [[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. 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: [[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. [[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.  Accidentals do not stack for large EDOs because of the superscript notation, but the amount of sharps can often be a complicated rational number.
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 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. 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. [[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.  Accidentals do not stack for large EDOs because of the superscript notation, but the amount of sharps can often be a complicated rational number.


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".  
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".  
Line 22: Line 22:
| 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
Line 106: Line 106:
| 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
Line 268: Line 268:
| P8
| P8
| D
| D
|}
== 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
|}
|}

Revision as of 08:20, 7 March 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 or caret (like #1/2 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. 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. Antidiatonic fifths may be notated using both the "major wider than minor" and "minor wider than major" systems, depending on what is needed. Accidentals do not stack for large EDOs because of the superscript notation, but the amount of sharps can often be a complicated rational number.

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 Frostburn). Since 1/2-augmented minor and 1/2-diminished major mean the same thing, they are replaced with the more conventional term "neutral".

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 minor 2nd m2 Eb
2 109.091 1/3-aug minor 2nd 1/3-AM2 Eb2/3
3 163.636 1/3-dim major 2nd 1/3-dM2 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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