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. Note: "apotome" is pronounced "a-POT-o-me".
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 having errors of more than 10.5 cents from just. EDOs with good fifths have JI-based notations.
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 Periodic Table, except that EDOs 6, 8, 13 and 18 are usually notated as subsets of larger EDOs.
| Up symbol | Pronunciation (sagispeak) |
Apotome fractions represented | |
|---|---|---|---|
| Evo | Revo | ||
| | 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 | |
| | | sharp pao | 3/5, 5/8, 2/3, 5/7 |
| | | sharp ranao | 7/10 |
| | | sharp slao | 3/4, 7/9, 4/5 |
| | | sharp rao | 5/6, 6/7, 7/8, 8/9, 9/10 |
| | | 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 Periodic Table, except that 11-EDO is usually notated as a subset of 22.
| Up symbol | Pronunciation (sagispeak) |
Limma fractions represented | |
|---|---|---|---|
| Evo | Revo | ||
| | nai | 1/6, 1/5 | |
| | tai | 1/4, 1/3, 2/5 | |
| | kai | 1/2 | |
| | jakai | 3/5, 2/3, 3/4 | |
| E-F | | eef nao | 4/5, 5/6 |
| E-F | | 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 and shows how the notation of subsets is preserved.