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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.


On 24-Sep-2016 in the Facebook Group: Microtonal Music and Tuning Theory, [[Cryptic Ruse]] introduced the idea of using [https://www.facebook.com/groups/497105067092502/permalink/840445019425170/ |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.
{| class="wikitable center-all"
|-
{{#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
}}
|-
! 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%}}
|-
| {{../szg| vvv# |size=400%|height=68px}}
| {{../szg| vv# |size=400%|height=68px}}
| {{../szg| vvvx |size=400%|height=68px}}
| {{../szg| vvx |size=400%|height=68px}}
|-
! 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%}}
|-
| {{../szg| ^^^b |size=400%|height=86px}}
| {{../szg| ^^b |size=400%|height=68px}}
| {{../szg| ^^^bb |size=400%|height=86px}}
| {{../szg| ^^bb |size=400%|height=68px}}
|}


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.  
{| class="wikitable center-all"
|-
{{#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
}}
|-
! 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}}
|-
! 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}}
|}
{| class="wikitable center-all"
|-
{{#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
}}
|-
! 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}}
|-
! 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}}
|}


Although Cryptic Ruse later abandoned these notations, the idea was adopted by George Secor and Dave Keenan to simplify the notation of EDOs with bad fifths in the Sagittal notation system. Sagittal defines a bad fifth as one with an error of more than 10.5 cents from just.
<span style="font-size: 800%; line-height: 300px;">
 
{{../szg| bb }}{{../szg| db }}
== Apotome-fraction notations ==
{{../szg| b }}
 
{{../szg| d }}
== Limma-fraction notations ==
{{../szg| n }}
 
{{../szg| t }}
[[Category:Notation]]
{{../szg| # }}
[[Category:Sagittal notation]]
{{../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>
Retrieved from "https://en.xen.wiki/w/User:Dave_Keenan/sandbox"