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A '''fractional-3-limit notation''' is a kind of [[musical notation]] built on a [[chain-of-fifths notation]], which is used for notating [[EDOs]] or [[EDOs to ETs|ETs]] in a way that avoids favoring any mapping from JI, while preserving the notation of subset EDOs. Fractional-3-limit notations may be contrasted with two other kinds of chain-of-fifths notation for EDOs: ''JI-based notations'', like the good-fifths [[Sagittal|Sagittal notations]], which assume specific JI mappings, and ''step-count notations'', like [[Ups and downs|Ups and Downs notations]], which do not preserve the notation of subset EDOs. Fractional-3-limit notations assign symbols to fractions of some tempered 3-limit comma. In practice, this 3-limit comma is either the [[2187/2048|apotome]] (chromatic semitone) as represented by a sharp or flat, or the [[256/243|limma]] (diatonic semitone) as represented by the intervals B-C and E-F.
Testing a Stein-Zimmermann-Gould notation template {⁠{szg|...}} similar to the Sagittal template.


== History ==
text{{../szg| vvv# }}text<br>text{{../szg| ^^^b }}text
[[Chain-of-fifths_notation#Accidentals|Stein-Zimmermann notation]] can be viewed as a very simple apotome-fraction notation, notating only half-apotomes.
 
In 2016,<ref>https://www.facebook.com/groups/497105067092502/permalink/840445019425170</ref> [[Cryptic Ruse]] introduced the idea of using a combination of apotome-fraction and limma-fraction notations to cover all EDOs up to 72. This may have been the first proposal of a limma fraction notation. Although Cryptic Ruse later abandoned these ideas, they were adopted by George Secor and Dave Keenan to simplify the notation of EDOs with bad fifths in the Sagittal notation system.
 
== Sagittal fractional 3-limit notations ==
The Sagittal system uses fractional 3-limit notations only for EDOs with bad fifths, defined as having errors of more than 10.5 cents from just. EDOs with good fifths have JI-based notations.
 
When the EDO has fifths so narrow that the apotome becomes very small or negative (e.g. 33-EDO), a limma-fraction notation must be used. When the EDO has fifths so wide that the limma becomes very small or negative (e.g. 32-EDO), an apotome-fraction notation must be used.
 
The symbols were chosen from the Sagittal repertoire so they progressively increase in width and don't violate flag arithmetic. Beyond those requirements, the choice might have been arbitrary, but it turned out to be possible to choose symbols whose tempered-JI meaning is valid in most of the EDOs they notate.
 
=== Bad-fifths apotome-fraction notation ===
This notation is used for EDOs with fifths of 712.5 cents or more. These are the gold EDOs on the [https://sagittal.org/#periodic-table Periodic Table], except that EDOs 6, 8, 13, and 18 are usually notated as subsets of larger EDOs.


{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Sagittal bad-fifths apotome-fraction notation
|-
! colspan="2" | Up symbol !! rowspan="2" | Pronunciation<br />(sagispeak) !! rowspan="2" | Apotome fractions<br />represented
|-
! [[Sagittal#Evo|Evo]] !! [[Sagittal#Revo|Revo]]
|-
| colspan="2" | {{sagittal| )|  }} || rai  || 1/10, 1/9, 1/8, 1/7, 1/6
|-
| colspan="2" | {{sagittal|)~|  }} || slai  || 1/5, 2/9, 1/4
|-
| colspan="2" | {{sagittal| )|( }} || ranai || 3/10
|-
|-
| colspan="2" | {{sagittal| /| }} || pai  || 2/7, 1/3, 3/8, 2/5
{{#ifeq: {{{1|}}} | 60
| ! Semitones
{{!}} '''0''' {{!!}} {{frac|5}} {{!!}} {{frac|2|5}} {{!!}} {{frac|3|5}} {{!!}} {{frac|4|5}} {{!!}} '''1''' {{!!}} {{frac|1|1|5}} {{!!}} {{frac|1|2|5}} {{!!}} {{frac|1|3|5}} {{!!}} {{frac|1|4|5}} {{!!}} '''2''' {{!!}} {{frac|2|1|5}} {{!!}} {{frac|2|2|5}} {{!!}} {{frac|2|3|5}}
| ! Step offset
{{!}} '''0''' {{!!}} 1 {{!!}} 2 {{!!}} 3 {{!!}} 4 {{!!}} '''5''' {{!!}} 6 {{!!}} 7 {{!!}} 8 {{!!}} 9 {{!!}} '''10''' {{!!}} 11 {{!!}} 12 {{!!}} 13
}}
|-
|-
| colspan="2" | {{sagittal| /|) }} || patai || 3/7, 4/9
! rowspan="2" | Sharp symbol
| rowspan="4" | {{../szg| n |size=400%}}
| rowspan="2" | {{../szg| ^n |size=400%}}
| {{../szg| ^^n |size=400%|height=68px}}
| {{../szg| ^^^n |size=400%|height=78px}}
| rowspan="2" | {{../szg| v# |size=400%}}
| rowspan="2" | {{../szg| # |size=400%}}
| rowspan="2" | {{../szg| ^# |size=400%}}
| {{../szg| ^^# |size=400%|height=68px}}
| {{../szg| ^^^# |size=400%|height=78px}}
| rowspan="2" | {{../szg| vx |size=400%}}
| rowspan="2" | {{../szg| x |size=400%}}
| rowspan="2" | {{../szg| ^x |size=400%}}
| rowspan="2" | {{../szg| ^^x |size=400%}}
| rowspan="2" | {{../szg| ^^^x |size=400%}}
|-
|-
| colspan="2" | {{sagittal| (|\ }} || jakai || 1/2, 5/9, 4/7
| {{../szg| vvv# |size=400%|height=68px}}
| {{../szg| vv# |size=400%|height=68px}}
| {{../szg| vvvx |size=400%|height=68px}}
| {{../szg| vvx |size=400%|height=68px}}
|-
|-
| {{sagittal| \!  }}{{sagittal| # }} || {{sagittal| ||\ }} || sharp pao  || 3/5, 5/8, 2/3, 5/7
! rowspan="2" | Flat symbol
| rowspan="2" | {{../szg| vn |size=400%}}
| {{../szg| vvn |size=400%|height=68px}}
| {{../szg| vvvn |size=400%|height=68px}}
| rowspan="2" | {{../szg| ^b |size=400%}}
| rowspan="2" | {{../szg| b |size=400%}}
| rowspan="2" | {{../szg| vb |size=400%}}
| {{../szg| vvb |size=400%|height=68px}}
| {{../szg| vvvb |size=400%|height=68px}}
| rowspan="2" | {{../szg| ^bb |size=400%}}
| rowspan="2" | {{../szg| bb |size=400%}}
| rowspan="2" | {{../szg| vbb |size=400%}}
| rowspan="2" | {{../szg| vvbb |size=400%}}
| rowspan="2" | {{../szg| vvvbb |size=400%}}
|-
|-
| {{sagittal| )!( }}{{sagittal| # }} || {{sagittal|//||  }} || sharp ranao || 7/10
| {{../szg| ^^^b |size=400%|height=86px}}
|-
| {{../szg| ^^b |size=400%|height=68px}}
| {{sagittal|)~!  }}{{sagittal| # }} || {{sagittal| ~||\ }} || sharp slao  || 3/4, 7/9, 4/5
| {{../szg| ^^^bb |size=400%|height=86px}}
|-
| {{../szg| ^^bb |size=400%|height=68px}}
| {{sagittal| )!  }}{{sagittal| # }} || {{sagittal| (||~ }} || sharp rao  || 5/6, 6/7, 7/8, 8/9, 9/10
|-
| {{sagittal| # }} || {{sagittal| /||\ }} || sharp      || 1
|}
|}
The corresponding down notations replace sharps with flats and mirror the sagittals vertically, while their pronunciations replace "sharp" with "flat" and swap "ai" (high) and "ao" (down) endings.
The following table shows all the gold EDOs/ETs and shows how the notation of subsets is preserved.
=== Bad-fifths limma-fraction notation ===
This notation is used for EDOs with fifths of 791.5 cents or less. These are the rose EDOs on the [https://sagittal.org/#periodic-table Periodic Table], except that 11-EDO is usually notated as a subset of 22.


{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Sagittal bad-fifths limma-fraction notation
|-
! colspan="2" | Up symbol !! rowspan="2" | Pronunciation<br />(sagispeak) !! rowspan="2" | Limma fractions<br />represented
|-
! [[Sagittal#Evo|Evo]] !! [[Sagittal#Revo|Revo]]
|-
| colspan="2" | {{sagittal|  |( }} || nai  || 1/6, 1/5
|-
| colspan="2" | {{sagittal|  |) }} || tai  || 1/4, 1/3, 2/5
|-
| colspan="2" | {{sagittal|  |\ }} || kai  || 1/2
|-
|-
| colspan="2" | {{sagittal| (|\ }} || jakai || 3/5, 2/3, 3/4
{{#ifeq: {{{1|}}} | 168
| ! Semitones
{{!}} '''0''' {{!!}} {{frac|1|14}} {{!!}} {{frac|2|14}} {{!!}} {{frac|3|14}} {{!!}} {{frac|4|14}} {{!!}} {{frac|5|14}} {{!!}} {{frac|6|14}} {{!!}} {{frac|7|14}} {{!!}} {{frac|8|14}} {{!!}} {{frac|9|14}} {{!!}} {{frac|10|14}} {{!!}} {{frac|11|14}} {{!!}} {{frac|12|14}} {{!!}} {{frac|13|14}} {{!!}} '''1''' {{!!}} {{frac|1|1|14}} {{!!}} {{frac|1|2|14}} {{!!}} {{frac|1|3|14}}
| ! Step offset
{{!}} '''0''' {{!!}} 1 {{!!}} 2 {{!!}} 3 {{!!}} 4 {{!!}} 5 {{!!}} 6 {{!!}} 7 {{!!}} 8 {{!!}} 9 {{!!}} 10 {{!!}} 11 {{!!}} 12 {{!!}} 13 {{!!}} '''14''' {{!!}} 15 {{!!}} 16 {{!!}} 17
}}
|-
|-
| E-F{{nbhsp}}{{sagittal| !( }} || {{sagittal| ||) }}|| eef nao || 4/5, 5/6
! Sharp symbol
| rowspan="2" | {{../szg| n |size=500%|line-height=90px}}
| {{../szg| ^n |size=500%|height=90px}}
| {{../szg| ^^n |size=500%|height=90px}}
| {{../szg| ^^^n |size=500%|height=90px}}
| {{../szg| vvvt |size=500%|height=90px}}
| {{../szg| vvt |size=500%|height=90px}}
| {{../szg| vt |size=500%|height=90px}}
| {{../szg| t |size=500%|height=90px}}
| {{../szg| ^t |size=500%|height=90px}}
| {{../szg| ^^t |size=500%|height=90px}}
| {{../szg| ^^^t |size=500%|height=90px}}
| {{../szg| vvv# |size=500%|height=90px}}
| {{../szg| vv# |size=500%|height=90px}}
| {{../szg| v# |size=500%|height=90px}}
| {{../szg| # |size=500%|height=90px}}
| {{../szg| ^# |size=500%|height=90px}}
| {{../szg| ^^# |size=500%|height=90px}}
| {{../szg| ^^^# |size=500%|height=90px}}
|-
|-
| E-F || {{sagittal| ||\ }}|| eef || 1
! Flat symbol
| {{../szg| vn |size=500%|height=90px}}
| {{../szg| vvn |size=500%|height=90px}}
| {{../szg| vvvn |size=500%|height=90px}}
| {{../szg| ^^^d |size=500%|height=90px}}
| {{../szg| ^^d |size=500%|height=90px}}
| {{../szg| ^d |size=500%|height=90px}}
| {{../szg| d |size=500%|height=90px}}
| {{../szg| vd |size=500%|height=90px}}
| {{../szg| vvd |size=500%|height=90px}}
| {{../szg| vvvd |size=500%|height=90px}}
| {{../szg| ^^^b |size=500%|height=90px}}
| {{../szg| ^^b |size=500%|height=90px}}
| {{../szg| ^b |size=500%|height=90px}}
| {{../szg| b |size=500%|height=90px}}
| {{../szg| vb |size=500%|height=90px}}
| {{../szg| vvb |size=500%|height=90px}}
| {{../szg| vvvb |size=500%|height=90px}}
|}
|}
The corresponding down notations replace E-F with F-E (or B-C with C-B) and mirror the sagittals vertically, while their pronunciations replace "eef" with "fee" and swap "ai" (high) and "ao" (down) endings. The names "eef" and "fee" come from the interval E-F being a limma.
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Sagittal bad-fifths limma-fraction notation
|-
! EDOs !!  0  !! 1/6 !! 1/5 !! 1/4 !! 1/3 !! 2/5 !! 1/2 !! 3/5 !! 2/3 !! 3/4 !! 4/5 !! 5/6 !!  1 
|-
|-
| 7 || || || || || || || || || || || || || {{sagittal| /||\ }}
{{#ifeq: {{{1|}}} | 168
| ! Semitones
{{!}} {{frac|1|4|14}} {{!!}} {{frac|1|5|14}} {{!!}} {{frac|1|6|14}} {{!!}} {{frac|1|7|14}} {{!!}} {{frac|1|8|14}} {{!!}} {{frac|1|9|14}} {{!!}} {{frac|1|10|14}} {{!!}} {{frac|1|11|14}} {{!!}} {{frac|1|12|14}} {{!!}} {{frac|1|13|14}} {{!!}} '''2''' {{!!}} {{frac|2|1|14}} {{!!}} {{frac|2|2|14}} {{!!}} {{frac|2|3|14}}
| ! Step Offset
{{!}} 18 {{!!}} 19 {{!!}} 20 {{!!}} 21 {{!!}} 22 {{!!}} 23 {{!!}} 24 {{!!}} 25 {{!!}} 26 {{!!}} 27 {{!!}} '''28''' {{!!}} 29 {{!!}} 30 {{!!}} 31
}}
|-
|-
| 9, 14 || || || || || || || {{sagittal| (|\ }} || || || || || || {{sagittal| /||\ }}
! Sharp Symbol
| {{../szg| vvvt# |size=500%|height=90px}}
| {{../szg| vvt# |size=500%|height=90px}}
| {{../szg| vt# |size=500%|height=90px}}
| {{../szg| t# |size=500%|height=90px}}
| {{../szg| ^t# |size=500%|height=90px}}
| {{../szg| ^^t# |size=500%|height=90px}}
| {{../szg| ^^^t# |size=500%|height=90px}}
| {{../szg| vvvx |size=500%|height=90px}}
| {{../szg| vvx |size=500%|height=90px}}
| {{../szg| vx |size=500%|height=90px}}
| {{../szg| x |size=500%|height=90px}}
| {{../szg| ^x |size=500%|height=90px}}
| {{../szg| ^^x |size=500%|height=90px}}
| {{../szg| ^^^x |size=500%|height=90px}}
|-
|-
| 16, 21 || || || || || {{sagittal| |) }} || || || || {{sagittal| /|) }} || || || || {{sagittal| /||\ }}
! Flat Symbol
|-
| {{../szg| ^^^db |size=500%|height=90px}}
|
| {{../szg| ^^db |size=500%|height=90px}}
|-
| {{../szg| ^db |size=500%|height=90px}}
|
| {{../szg| db |size=500%|height=90px}}
|-
| {{../szg| vdb |size=500%|height=90px}}
|
| {{../szg| vvdb |size=500%|height=90px}}
| {{../szg| vvvdb |size=500%|height=90px}}
| {{../szg| ^^^bb |size=500%|height=90px}}
| {{../szg| ^^bb |size=500%|height=90px}}
| {{../szg| ^bb |size=500%|height=90px}}
| {{../szg| bb |size=500%|height=90px}}
| {{../szg| vbb |size=500%|height=90px}}
| {{../szg| vvbb |size=500%|height=90px}}
| {{../szg| vvvbb |size=500%|height=90px}}
|}
|}


The following table shows all the rose EDOs/ETs and shows how the notation of subsets is preserved.
<span style="font-size: 800%; line-height: 300px;">
 
{{../szg| bb }}{{../szg| db }}
[[Category:Notation]]
{{../szg| b }}
[[Category:Sagittal notation]]
{{../szg| d }}
{{../szg| n }}
{{../szg| t }}
{{../szg| # }}
{{../szg| t# }}
{{../szg| x }}
{{../szg| ^ }}
{{../szg| ^| }}
{{../szg| ^bb }}
{{../szg| ^db }}
{{../szg| ^b }}
{{../szg| ^d }}
{{../szg| ^n }}
{{../szg| ^t }}
{{../szg| ^# }}
{{../szg| ^t# }}
{{../szg| ^x }}
{{../szg| v }}
{{../szg| v| }}
{{../szg| vbb }}
{{../szg| vdb }}
{{../szg| vb }}
{{../szg| vd }}
{{../szg| vn }}
{{../szg| vt }}
{{../szg| v# }}
{{../szg| vt# }}
{{../szg| vx }}
{{../szg| ^^| }}
{{../szg| ^^bb }}
{{../szg| ^^db }}
{{../szg| ^^b }}
{{../szg| ^^d }}
{{../szg| ^^n }}
{{../szg| ^^t }}
{{../szg| ^^# }}
{{../szg| ^^t# }}
{{../szg| ^^x }}
{{../szg| vv| }}
{{../szg| vvbb }}
{{../szg| vvdb }}
{{../szg| vvb }}
{{../szg| vvd }}
{{../szg| vvn }}
{{../szg| vvt }}
{{../szg| vv# }}
{{../szg| vvt# }}
{{../szg| vvx }}
{{../szg| ^^^| }}
{{../szg| ^^^bb }}
{{../szg| ^^^db }}
{{../szg| ^^^b }}
{{../szg| ^^^d }}
{{../szg| ^^^n }}
{{../szg| ^^^t }}
{{../szg| ^^^# }}
{{../szg| ^^^t# }}
{{../szg| ^^^x }}
{{../szg| vvv| }}
{{../szg| vvvbb }}
{{../szg| vvvdb }}
{{../szg| vvvb }}
{{../szg| vvvd }}
{{../szg| vvvn }}
{{../szg| vvvt }}
{{../szg| vvv# }}
{{../szg| vvvt# }}
{{../szg| vvvx }}
</span>
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