An apotome is a chromatic semitone (Eb:E, Bb:B, F:F#, C:C#). A limma is a diatonic semitone (B:C, E:F). If you want to know how many steps there are in the major-whole-tone (A:B, C:D, D:E, F:G, G:A), just add together the steps per apotome and the steps per limma. The number of steps in the EDO's best fifth can be obtained by taking the EDO number itself and adding both its steps per apotome and its steps per limma and dividing by 2. e.g. (22+3+1)/2 = 13.

So this diagram tells us how the nominals, sharps and flats are spaced in a chain-of-fifths notation for a given EDO. If the steps-per-apotome are negative or zero, we don't use sharps or flats. If the steps-per-limma are negative or zero, we don't use the nominals B and F.

Of course this diagram alone, doesn't tell you what accidentals to use for each number of steps. And it only works for native-fifth notations, not subset notations (which are likely to be preferred for 6. 8, 13, 18 and 11 edos, as subsets of 12, 24, 26, 36 and 22 edos respectively).

The blue area shows EDOs whose best fifth has an error no greater than 7.5 cents. Meantone and superpythagorean temperaments are just inside those boundaries, on opposite sides. Pythagoreans, such as the bolded 41 and 53 are on a straight diagonal line that runs down the middle of the blue area, through 94.

Presently,

*all*Sagittal EDO notations are JI-based, in the sense that a symbol always represents the tempered size of its comma role in just intonation. But this results in an explosion of obscure symbols being used for the EDOs in the red and amber areas, many of which are very simple and so don't deserve to have obscure symbols. Cryptic Ruse pointed out this problem, and showed that it could be solved by using fractional-3-limit notations, using only a small number of accidental symbols. He gave the semantics (in bold below) but left the symbols unassigned.

Here's my latest version of this proposal. I've only shown the upward symbols, but of course there are matching downward symbols.

[Edit: You can skip to a later version, that George Secor and I agree on, here: http://forum.sagittal.org/viewtopic.php?p=729#p729.]

Symbol Pronunciation Apotome fractions represented Limma fractions represented

nai 1/10, 1/9, 1/8 apotome, 1/7 limma

pai 1/5, 1/6, 1/7, 2/9 apotome 1/6, 1/5,1/4, 2/7limma

tai1/4, 2/7, 3/10, 1/3apotome1/3, 2/5, 3/7 limma

phai 3/8, 2/5, 3/7, 4/9 apotome1/2 limma

vai1/2 apotome4/7, 3/5, 2/3 limma

= phao-sharp 5/9, 4/7, 3/5, 5/8 apotome 5/7, 3/4, 4/5, 5/6 limma

= tao-sharp 2/3, 7/10, 5/7, 3/4 apotome 6/7 limma

= pao-sharp 7/9, 6/7, 5/6, 4/5 apotome 1 limma

= nao-sharp 7/8, 8/9, 9/10 apotome

= sharp1 apotome

Notice how the width of the Sagittal symbols increases steadily with the size of their alteration. You will also notice that these 5 symbols and their apotome-complements are in the Spartan subset of Sagittal -- the simplest and most-commonly-used subset by far. My previous version of this proposal did not use Spartan symbols because it was thought that they should not be used when they do not represent the tempering of their defined 11-limit commas.

However, I think it is more important to limit the number of symbols that need to be learned. I think we should add to Sagittal a rule that says:

When the notational fifth is bad enough, the symbols cease to represent the tempered value of their higher-limit comma and revert to their untempered value, which is then treated as a fraction of an apotome or limma. If the fifth is narrower than that of 19-edo, they represent a fraction of a limma. If the fifth is wider than that of 22-edo, they represent a fraction of an apotome.