Fractional 3-limit notation
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.
In 2016,[1] 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 have consistent 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 apotome-fractions and on the right for 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 Periodic Table.
| 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, 5/9, 4/7 | |
| | | 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. Notice how the notation of subsets is preserved; there are never two different symbols in the same column.
| EDOs b-ETs |
Sharpness ( = ) |
Apotome fraction | ||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1/10 | 1/9 | 1/8 | 1/7 | 1/6 | 1/5 | 2/9 | 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 | | ||||||||||||||||||||||||||||||||
| 10 | 2 | | | |||||||||||||||||||||||||||||||
| (8), 15 | 3 | | | | ||||||||||||||||||||||||||||||
| (6), (13), 20 | 4 | | | | | |||||||||||||||||||||||||||||
| (18), 25, 32 | 5 | | | | | | ||||||||||||||||||||||||||||
| 23b, 30, 37 | 6 | | | | | | | |||||||||||||||||||||||||||
| 35b, 42 | 7 | | | | | | | | ||||||||||||||||||||||||||
| 47b | 8 | | | | | | | | | |||||||||||||||||||||||||
| [59] | 9 | | | | | | | | | | ||||||||||||||||||||||||
| 64b | 10 | | | | | | | | | | | |||||||||||||||||||||||
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 best justified[2] 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.
With the gold EDOs that are multiples of 5, and with 23b, Sagittal recommends using a pentatonic notation that avoids the nominals B and F (the extremes on the chain of nominal fifths FCGDAEB), as the limmas B-C and E-F are of zero size or negative for them.
Bad-fifths limma-fraction notation
This notation is used for EDOs with fifths of 691.5 cents or less. These are the rose colored EDOs on the Periodic Table.
| 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 | | limmai nao | 4/5, 5/6 |
| E-F | | limmai | 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 "ai" (high) and "ao" (down) endings.
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.
| EDOs b-ETs |
Limmanosity (EF = ) |
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 | | ||||||||||||
| 9, 14 | 2 | | | |||||||||||
| (11), 16, 21 | 3 | | | | ||||||||||
| 23, 28, 33 | 4 | | | | | |||||||||
| 30b, 35, 40 | 5 | | | | | | ||||||||
| 42b, 47 | 6 | | | | | | | |||||||
(11)-EDO is in parenthesis because its fifth is so narrow it is usually notated as a subset of 22-EDO which has a good fifth.
To obtain the Evo notations, delete the multishaft symbols. There are no equivalent combinations with conventional sharps or flats, as limmas are conventionally notated as the intervals B-C and E-F. The multishaft symbols are not required, even in Revo notation; they merely allow for alternative spellings there. In theory, sharps and flats could be used with the notations for 33, 40, and 47 because the sharp is one step for them, while it is zero or negative for the other rose EDOs and Sagittal never uses symbols for commas that have been tempered out or tempered negative. But Sagittal recommends not using sharps or flats, conventional or sagittal, with any limma-fraction notation.
- ↑ https://www.facebook.com/groups/497105067092502/permalink/840445019425170
- ↑ In just intonation, doesn't notate any common ratio that would make it valid as 1\59. It does however notate the uncommon ratio 55/49 = 5·11/7² by representing, as a secondary comma, 441/440 which tempers to 1\59 if only prime 3 is tempered or using the 59d map (second-best approximation of prime 7). Thanks to Roee Sinai for this JI-based justification. 55/49 ranks 87th in popularity among 2,3-free equivalence classes of ratios, according to N2D3P9.