User:Dave Keenan/sandbox
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 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 notations, which assume specific JI mappings, and step-count notations, like 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 apotome (chromatic semitone) as represented by a sharp or flat, or the limma (diatonic semitone) as represented by the intervals B-C and E-F.
History
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 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.
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.
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 fifths with errors of more than 10.5 cents from just. EDOs with good fifths have JI-based notations.
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 Periodic Table, although it is recommended that EDOs with extremely wide fifths, 6, 8, 13 and 18 are notated as subsets of larger EDOs with good fifths.
| Symbol | Pronunciation | Apotome fractions represented |
|---|---|---|
| | rai | 1/10, 1/9, 1/8, 1/7, 1/6 |
| | slai | 1/5, 2/9, 1/4 |
| | ranai | 3/10 |
| | pai | 2/7, 1/3, 3/8, 2/5 |
| | patai | 3/7, 4/9 |
| | jakai | 1/2, 4/7, 5/9 |
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 Periodic Table, although it is recommended that 11-EDO, which has an extremely narrow fifth, is notated as a subset of 22.
Symbol Limma fractions represented |( 1/6, 1/5 |) 1/4, 1/3, 2/5 |\ 1/2 (|\ 3/5, 2/3, 3/4 ||) 4/5, 5/6 ||\ 1 limma