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 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 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. Multiples of [[7edo]] up to [[35edo]], 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 required to be notated as a subset of [[16edo]], [[24edo]], or [[32edo]] 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.
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 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, and does not have compounding accidentals for large EDOs, although it will feature complex rational numbers as the amount of sharps. Excluding the no-fifth systems of [[1edo]]-[[4edo]] and [[8edo]], there are 4 [[EDOs]] (all multiples of [[7edo]]) that cannot be notated using the native fifth: . 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.


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".  


Here is an example for [[22edo]]:
== 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
| minor 2nd
| m2
| Eb
|-
| 2
| 141.176
| neutral 2nd
| n2
| 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
| n7
| C#<sup>1/2</sup>
|-
| 16
| 1129.412
| major 7th
| M7
| C#
|-
| 17
| 1200.00
| perfect octave
| P8
| D
|}
 
[[22edo]]:
{| class="wikitable"
|-
|-
! Degree
! Degree
Line 152: Line 269:
| D
| D
|}
|}


== Just intonation ==
== Just intonation ==

Revision as of 08:22, 6 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 largely an extension of circle-of-fifths notation, which can be used for EDOs, and secondarily for temperaments 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 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 systems, and does not have compounding accidentals for large EDOs, although it will feature complex rational numbers as the amount of sharps. Excluding the no-fifth systems of 1edo-4edo and 8edo, there are 4 EDOs (all multiples of 7edo) that cannot be notated using the native fifth: . However, it is still possible to notate them as subsets of 42edo (for 14edo and 21edo), 56edo (for 28edo), and 70edo (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.

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 minor 2nd m2 Eb
2 141.176 neutral 2nd n2 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 n7 C#1/2
16 1129.412 major 7th M7 C#
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


Just intonation

This segment will probably removed or changed

Accidentals

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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