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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.
 
And when a single-shaft symbol only has flags on one side of its shaft, they are always on the left for the apotome-fractions and on the right for the limma-fractions.
 
=== Bad-fifths apotome-fraction notation ===
This notation is used for EDOs with fifths of 712.5 cents or more. These are the gold colored EDOs on the [https://sagittal.org/#periodic-table Periodic Table].


{| 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
{{#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
}}
|-
|-
! [[Sagittal#Evo|Evo]] !! [[Sagittal#Revo|Revo]]
! 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| )| }} || rai  || 1/10, 1/9, 1/8, 1/7, 1/6
| {{../szg| vvv# |size=400%|height=68px}}
| {{../szg| vv# |size=400%|height=68px}}
| {{../szg| vvvx |size=400%|height=68px}}
| {{../szg| vvx |size=400%|height=68px}}
|-
|-
| colspan="2" | {{sagittal|)~| }} || slai  || 1/5, 2/9, 1/4
! 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%}}
|-
|-
| colspan="2" | {{sagittal| )|( }} || ranai || 3/10              
| {{../szg| ^^^b |size=400%|height=86px}}
|-
| {{../szg| ^^b |size=400%|height=68px}}
| colspan="2" | {{sagittal| /|  }} || pai  || 2/7, ↑ , 1/3, 3/8, 2/5
| {{../szg| ^^^bb |size=400%|height=86px}}
|-
| {{../szg| ^^bb |size=400%|height=68px}}
| colspan="2" | {{sagittal| /|) }} || patai || 3/7, 4/9
|-
| colspan="2" | {{sagittal| (|\ }} || jakai || 1/2, 5/9, 4/7
|-
| {{sagittal| \!  }}{{sagittal| # }} || {{sagittal|  ||\ }} || sharp pao  || 3/5, 5/8, 2/3, ↓ , 5/7
|-
| {{sagittal| )!( }}{{sagittal| # }} || {{sagittal|//||  }} || sharp ranao ||               7/10
|-
| {{sagittal|)~!  }}{{sagittal| # }} || {{sagittal| ~||\ }} || sharp slao  || 3/4, 7/9, 4/5
|-
| {{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. Notice how the notation of subsets is preserved; there are never two different symbols in the same column.


{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Revo sagittal bad-fifths apotome-fraction notations
|-
! rowspan="2" | EDOs<br />b-ETs !! rowspan="2" | sharpness !! colspan="33" | Apotome fraction 
|-
!  0  !! 1/10 !! 1/9 !! 1/8 !! 1/7 !! 1/6 !! 1/5 !! 2/5 !! 1/4 !! 2/7 !! 3/10 !! 1/3 !! 3/8 !! 2/5 !! 3/7 !! 4/9 !! 1/2 !! 5/9 !! 4/7 !! 3/5 !! 5/8 !! 2/3 !! 7/10 !! 5/7 !! 3/4 !! 7/9 !! 4/5 !! 5/6 !! 6/7 !! 7/8 !! 8/9 !! 9/10 !!  1 
|-
| 5 || 1 || || || || || || || || || || || || || || || || || || || || || || || || || || || || || || || || || {{sagittal| /||\ }}
|-
| 10 || 2 || || || || || || || || || || || || || || || || || {{sagittal| (|\ }} || || || || || || || || || || || || || || || || {{sagittal| /||\ }}
|-
| (8), 15 || 3 || || || || || || || || || || || || {{sagittal| /| }} || || || || || || || || || || {{sagittal| ||\ }} || || || || || || || || || || || {{sagittal| /||\ }}
|-
| (6), (13), 20 || 4 || || || || || || || || || {{sagittal| )~| }} || || || || || || || || {{sagittal| (|\ }} || || || || || || || || {{sagittal| ~||\ }} || || || || || || || || {{sagittal| /||\ }}
|-
| (18), 25, 32 || 5 || || || || || || || {{sagittal| )~| }} || || || || || || || {{sagittal| /| }} || || || || || || {{sagittal| ||\ }} || || || || || || || {{sagittal| ~||\ }} || || || || || || {{sagittal| /||\ }}
|-
|-
| 23b, 30, 37 || 6 || || || || || || {{sagittal| )~| }} || || || || || || {{sagittal| /| }} || || || || || {{sagittal| (|\ }} || || || || || {{sagittal| ||\ }} || || || || || || || || || || || {{sagittal| /||\ }}
{{#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
}}
|-
|-
| 35b, 42 || 7 || || || || || || || || || || || || || || || || || || || || || || || || || || || || || || || || || {{sagittal| /||\ }}
! 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}}
|-
|-
| 47b || 8 || || || || || || || || || {{sagittal| )~| }} || || || || || || || || {{sagittal| (|\ }} || || || || || || || || {{sagittal| ~||\ }} || || || || || || || || {{sagittal| /||\ }}
! Flat symbol
|-
| {{../szg| vn |size=500%|height=90px}}
| [59] || 9 || || || || || || || || || || || || {{sagittal| /| }} || || || || || || || || || || {{sagittal| ||\ }} || || || || || || || || || || || {{sagittal| /||\ }}
| {{../szg| vvn |size=500%|height=90px}}
|-
| {{../szg| vvvn |size=500%|height=90px}}
| 64b || 10 || || || || || || || {{sagittal| )~| }} || || || || {{sagittal| )|( }} || || || {{sagittal| /| }} || || || {{sagittal| (|\ }} || || || {{sagittal| ||\ }} || || || {{sagittal| //|| }} || || || || {{sagittal| ~||\ }} || || || || || || {{sagittal| /||\ }}
| {{../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}}
|}
|}
EDOs in parentheses (6), (8), (13), and (18) have fifths that are so wide they are usually notated as subsets of larger EDOs with good fifths. [59]-EDO is in square brackets because it is green not gold on the periodic table, and so it should have a JI-based notation. However, its fifth is very close to the bad-fifth threshold and its 1-step symbol is only justified by the apotome-fraction notation, so its notation is a hybrid.
To obtain the Evo notations, replace the multishaft symbols above with their Evo equivalents from the previous table.
=== Bad-fifths limma-fraction notation ===
This notation is used for EDOs with fifths of 791.5 cents or less. These are the rose colored EDOs on the [https://sagittal.org/#periodic-table Periodic Table].
{| class="wikitable center-all"
|+ Sagittal bad-fifths limma-fraction accidentals
|-
! 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
|-
| E-F{{nbhsp}}{{sagittal|  !( }} || {{sagittal| ||) }}|| eef nao || 4/5, 5/6
|-
| E-F || {{sagittal| ||\ }}|| eef || 1
|}
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.
The following table shows all the rose EDOs/ETs. Notice how the notation of subsets is preserved; there are never two different symbols in the same column.
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Revo sagittal bad-fifths limma-fraction notations
|-
|-
! rowspan="2" | EDOs<br />b-ETs !! rowspan="2" | eEFness !! colspan="13" | Limma fraction 
|-
! 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 || 1 || || || || || || || || || || || || || {{sagittal| ||\ }}
|-
| 9, 14 || 2 || || || || || || || {{sagittal| (|\ }} || || || || || || {{sagittal| ||\ }}
|-
|-
| (11), 16, 21 || 3 || || || || || {{sagittal| |) }} || || || || {{sagittal| /|) }} || || || || {{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
}}
|-
|-
| 23, 28, 33 || 4 || || || || {{sagittal| |) }} || || || {{sagittal| (|\ }} || || || {{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}}
|-
|-
| 30b, 35, 40 || 5 || || || {{sagittal| |( }} || || || {{sagittal| |) }} || || {{sagittal| /|) }} || || || {{sagittal| ||) }} || || {{sagittal| ||\ }}
! Flat Symbol
|-
| {{../szg| ^^^db |size=500%|height=90px}}
| 42b, 47 || 6 || || {{sagittal| |( }} || || || {{sagittal| |) }} || || || || {{sagittal| /|) }} || || || {{sagittal| ||) }} || {{sagittal| ||\ }}
| {{../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}}
|}
|}


(11)-EDO is in parenthesis because its fifth is so narrow that it is usually notated as a subset of 22 which has a good fifth.
<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>

Latest revision as of 06:51, 28 May 2026

Testing a Stein-Zimmermann-Gould notation template {⁠{szg|...}} similar to the Sagittal template.

texttext
texttext

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
Flat symbol
Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Sharp symbol
Flat symbol
Step Offset 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Sharp Symbol
Flat Symbol

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