I agree that 1:3:9 inconsistency (or equivalently: more than 25% relative error in the fifth) is another kind of "bad fifth", and such divisions do not require a native fifth notation. They can make do with a subset notation. But we would like to give, at least the low numbered ones, a native fifth notation of some kind (whether JI-based, ap-frac or lim-frac), when this can be made simple enough. Where "simple enough" means: using only Spartan symbols, plus (in order of decreasing simplicity/increasing prime-limit) kai
I suggest we tackle this in two stages. The first stage is nearly complete and consists of agreeing, for each division from 5 to 72, what is its best-if-any JI-based notation, its best-if-any apotome-fraction notation, its best-if-any limma-fraction notation and its best-if-any subset notation. I feel we should include all such notations in sagittal.pdf. These four notation types could be abbreviated JB AF LF and SS. And I note that in many cases JB = AF.
The second stage would be to decide which single notation to call the "preferred" or "default" notation for each division. This is the one that we would want to be selected in Scala when the user types SET NOTATION SA<n> where <n> is the number of the division.
I think it is very desirable that the boundaries between the regions where the four different notation types are preferred, should consist of straight lines on the above diagram, and with a good deal of reflective symmetry about the line of pythagoreans (just fifths). The divisions for which JB = AF make this easier since the boundary between these types then becomes somewhat arbitrary. Examples of straight lines are: steps per apotome, steps per limma, steps per octave, absolute fifth-error, relative fifth error.
I have responded to your recent proposals for new JI-based notations for some divisions—in most cases accepting them. I'd appreciate if you would address those I have not (yet) accepted, and address my recent JI notation proposals, including changing
everywhere that it is used as a half-apotome with a 13-limit meaning, including in the AF notation for wide fifths.
I note that 44-edo is 1:3:5:11:13-consistent, and the following 1:3:5:11:13 JI-based notation is valid:
It also happens to be the same as the apotome-fraction notation used for other divisions having 6 steps to the apotome, 30 and 37. Its other neighbouring 6-step-to-the-apotome division is 51-edo. 51's standard notation differs in using the 7-comma symbol
for 1 step. This would not be valid in 44-edo as the 7-comma vanishes there.
I will return to 61 and 66 later, but for now I want to flag two possible native-fifth notations for 61 that need to be investigated for validity.
possible JB same as 68-edo
possible AF or JB