Earlier in this topic, Dave proposed that to break a tie between two commas with the same absolute value of their 3-exponent, the negative one should be considered more complex. But then later the notion of directed commas arose, to handle a counterexample volleo6144 had found where two commas had the exact same 3-exponent and size category and absolute value of their exponents beyond 3. I expect that if someone with stronger math skills than I were to review the proof Dave gave here, they might find that it assumes directed comma names to work.
So: do I understand this new development correctly, that now commas with negative 3-exponents and those with positive 3-exponents are independent from each other with respect to the complexity level in their names?
Let me give a concrete example. Previously we would refer to
[-4 4 -1 ⟩ 81/80 21.506¢
as the 5C, or classic-Comma, and we would refer to
[-34 20 1⟩ 17433922005/17179869184 25.414¢
as the c5C, or complex-classic-Comma.
However, nowadays these would be the 1/5C and 5C, respectively. Is that correct? I suspect @Dave Keenan may balk at the prospect of 5C referring to anything other than the syntonic comma, so I thought I should illuminate this example to ensure this is really what we want. Or he might take umbrage with the fact that something with ATE of 20 isn't complex (or is 21 the cutoff?).
I also have a question about 3-limit comma names: should they be directed, e.g. should the Pythagorean large diesis [27 -17⟩ be the "1/3L" because its 3-exponent is negative? Certainly 3-limit comma names behave differently than others — for starters, we don't name by the power of 3; we just name with "3", i.e. the Pythagorean comma is the 3C, not the 531441C. So I wouldn't be surprised if they similarly simplify to being undirected. Also, I do not believe we could ever find a similar counterexample where two 3-limit commas had the same ATE and size category, because then one of them would be negative in cents and the other would be positive (the only thing you have left to change is the 2-exponent, and good luck changing that to get any other size category).
My code currently can do comma names either factorized or unfactorized, i.e. 1/455n or 1/5.7.13n. But over email some time ago Dave and I thought that there might be good reason to go for a hybridized factorization scheme with a threshold, specifically, that whenever any side of a comma's quotient has three or more prime factors, it should be factorized.
- 65/77n should not be factorized; it has 4 total prime factors, but there's only 2 on each side of the vinculum.
- The 1/18.104.22.168n should be factorized, because even though it only has two different prime factors, it has 3 or more on one side.
Should I build in the edo naming approach touched upon here?
Should I build in the ability to recognize and substitute (parse and format) words such as Pythagorean, classic, septimal, etc. for numbers 3-, 5-, 7- etc. where appropriate?