User:Dave Keenan/sandbox: Difference between revisions

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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]] ~~when the EDO has no obvious~~ mapping from ~~just intonation~~, 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 "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.

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.

== History ==

[[Stein-Zimmermann notation]]

[[Chain-of-fifths_notation#Accidentals|Stein-Zimmermann notation]] can be viewed as a very simple apotome-fraction notation, only capable of notating half-apotomes.

On 24-Sep-2016 in the Facebook Group: Microtonal Music and Tuning Theory, [[User:Cryptic Ruse|Cryptic Ruse]] introduced [[https://www.facebook.com/groups/497105067092502/permalink/840445019425170/|a combination of apotome-fraction and limma-fraction notations]] to cover all EDOs up to 72.

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. When the EDO has fifths so narrow that the apotome becomes very small or negative (e.g. 33-EDO), limma-fraction notations must be used.

Although CRyptic later abandoned these notations, the principel was adopted by George Secor and Dave Keenan for the notation of EDOs with bad fifths, where a bad fidth is defined as one with an error of more tha 10 cents.

== Apotome-fraction notations ==

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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]] when the EDO has no obvious mapping from just intonation, 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 "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.
+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.
 == History ==
-[[Stein-Zimmermann notation]]
+[[Chain-of-fifths_notation#Accidentals|Stein-Zimmermann notation]] can be viewed as a very simple apotome-fraction notation, only capable of notating half-apotomes.
+On 24-Sep-2016 in the Facebook Group: Microtonal Music and Tuning Theory, [[User:Cryptic Ruse|Cryptic Ruse]] introduced [[https://www.facebook.com/groups/497105067092502/permalink/840445019425170/|a combination of apotome-fraction and limma-fraction notations]] to cover all EDOs up to 72.
+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. When the EDO has fifths so narrow that the apotome becomes very small or negative (e.g. 33-EDO), limma-fraction notations must be used.
+Although CRyptic later abandoned these notations, the principel was adopted by George Secor and Dave Keenan for the notation of EDOs with bad fifths, where a bad fidth is defined as one with an error of more tha 10 cents.
 == Apotome-fraction notations ==