volleo6144 wrote: ↑Sun Jun 28, 2020 5:08 am

I ... forgot to ask: What exactly makes an apotome-fraction or limma-fraction notation not work for specific colors? (I assume this is what you mean by "attempting to devise [a] notation for each color"?)

Sorry for the delay in responding. I'll talk in terms of apotome-fraction notations, but the same goes if you substitute "limma" for "apotome". In any case, we already have the only limma-fraction notation we need, so any future notation of this type will be apotome-fraction.

By the way, thanks for correcting my pronunciation of "a-POT-o-me". I had been pronouncing it "AP-o-toam" like "microtome" all these years!

An apotome-fraction notation that "works" over a given range of fifth-sizes and up to some maximum number of steps per apotome, is a sequence of symbols without accents, beginning with the natural and continuing with distinct upward single-shaft symbols, with boundaries between them defined as some (possibly irrational) positive fraction of the tempered apotome, covering up to at least the half-apotome, such that every EDO in that range has a distinct symbol for each number of steps up to the half-apotome, and every symbol represents a comma whose tempered size rounds to the number of steps it represents in every such EDO, where that comma is preferably the primary comma for that symbol, but may be a secondary comma which is the primary comma for an accented version of that symbol.

As I think

@cmloegcmluin once said, but I failed to appreciate at the time: Trojan (orange) is the epitome of an apotome-fraction notation. Gilbert and Sullivan? "He is the very model of a modern major general".

Finding that one cannot get such a notation to work, may just mean that we need to change the range of fifth sizes for a colour, from where my brief preliminary investigations suggested they should be. It may also mean that we're trying too hard in terms of steps per apotome and may just need to abandon some of the higher-numbered EDOs in the range (when we're pushing beyond 72-edo).

I note that an ideal situation would be to have notations overlap slightly, so that for EDOs near the boundary between colours, it wouldn't matter which of the two apotome-fraction notations you applied to it, they would both give the same notation for the EDO. The notation for 59edo is an example of this. Notice that although 59edo uses the gold notation, it has exactly the same notation as 66-edo, which is green and has the same number of steps to the apotome "#=9".

What's so special about a 692.2¢ fifth that makes it the boundary between purple and rose?

Nothing. Any value between 15\26 ≈ 690.91 ¢ and 19\33 ≈ 692.3 ¢ would do just as well. But if you ask, "Would 59b (34\59) be purple or rose?" then I'd say: Try both notations and see which ones work (if any). When you look at the limma-fraction (rose) notation, as set out a few posts back, you'll see it doesn't have enough symbols to do 7 steps to the limma. But I haven't checked whether /|) as 35M (purple) is valid as (i.e. has tempered value that rounds to) 1 step of 59b.

Is the boundary between black and gold just 800.0001¢ (4\6 is gold, but anything more is black)?

Is the boundary between rose and white just 654.5454¢ (6\11 is rose, but anything less is white)?

Probably. But again, the question is: Can you make the gold notation or rose notation "work" for something outside those bounds.

Also I just ... noticed: because of the linearity of the ♯=X lines on the 7xx¢ side and the EF=Y lines on the 6xx¢ side of the periodic table, the smaller EDOs for the integral group numbers are really just a distraction from the underlying pattern:

...

Cool.