A comma is named as the ratio it notates with any factors of 2 and 3 removed (5103:5120 becomes 7:5, which becomes 5:7 because the lower number comes first), together with a size category (twice 5103:5120 is in between [317 -200> and [-19 12>, so 5103:5120 is a kleisma). These size categories are delineated by halves (or square roots?) of various 3-limit commas, which are mostly relevant to specific EDOs.Dave Keenan wrote: ↑Thu May 28, 2020 10:45 amI agree with your argument. But there's no need to check. That property was guaranteed by generating candidates with steadily increasing 3-exponents, starting from zero and working in both positive and negative directions.

Here's the Comma Namer spreadsheet. [attached: CommaNamer.xlsx]

These upper bounds are, in increasing order:

schismina (n): [-84 53> / 2, or about 1.81¢. Half of Mercator's comma (1s or 3s), or 26.5 times the interval by which each fifth is flattened in 53-EDO.

schisma (s): [317 -200> / 2, or about 4.50¢.

kleisma (k): [-19 12> / 2, or about 11.73¢. Half of the Pythagorean comma (1C or 3C), or six times the interval by which each fifth is flattened in 12-EDO.

comma (C): [27 -17> / 2, or about 33.38¢.

S-diesis (S): [8 -5> / 2, or about 45.11¢. Half of the Pythagorean limma, or m2.

M-diesis (M): [-11 7> / 2, or about 56.84¢. Half of the Pythagorean apotome, or sharp.

L-diesis (L): [-30 19> / 2, or about 68.57¢.

S-semitone (SS): [-49 31> / 2, or about 80.30¢.

M-semitone or limma (MS): [-3 2> / 2, or about 101.96¢. The 2:3 perfect fifth minus half an octave, or half an 8:9 "major" tone.

L-semitone (LS): [62 -39> / 2, or about 111.88¢.

Apotome (A): [-106 67> / 2, or about 115.49¢. "Apotome" is pronounced similarly to "epitome", and the epitome of mispronunciation is how long I thought it was pronounced with emphasis on the "a".

After these, the existing comma boundaries here are added to [-22 14> / 2 = [-11 7>, the Pythagorean apotome, to yield [295 -186> / 2 for s+A, etc. up to double-apotome (A+A) with an upper bound of [-128 81> / 2 ≈ 229.18¢. Limma-plus-apotome can be named as "whole-tone" instead, because a minor second (a limma) plus a sharp (an apotome) is a major second (a whole tone).

When there is a simpler ratio (by smaller 3-exponent) that fits the primes and the size category, the prefixes "complex", "supercomplex", "hypercomplex", "ultracomplex", "5-complex", "6-complex", etc. may be added to signify how many smaller 3-exponents "work" for it. (Strangely, the spreadsheet in its current form has a glitch where any comma that's supercomplex or beyond will still just be listed as "complex", and I'm not sure which of [50516 -31867, 0 0 -1, 0 0 0, -1> and [-50500 31867, 0 0 -1, 0 0 0, -1> is counted as less complex.)

users.bigpond.net.au isn't responding (it isn't showing me any kind of error; it just isn't responding), but archive.org exists, so I checked the other sources out. The first one refers to a particular way of assigning "wide" or "narrow" labels to the existing labels used in the second one (based on Miracle temperament, which is 72-EDO but with 7\72 instead of 1\72 as the generator), and the second one lays out a systematic naming system for 11-limit intervals based on 31-EDO.AP20 in the spreadsheet wrote: [The] "plus-apotome" names are only intended to be used if the interval is really being considered as a comma, otherwise a different naming system applies. See http://users.bigpond.net.au/d.keenan/Mu ... Naming.txt or http://users.bigpond.net.au/d.keenan/Mu ... Naming.htm

P1 = 0\31, M2 = 5\31, M3 = 10\31, P4 = 13\31, P5 = 18\31, M6 = 23\31, M7 = 28\31, P8 = 31\31.

For seconds, thirds, sixths, and sevenths, the modifiers for successive 31-EDO steps are: subdiminished, diminished, subminor, minor, neutral, major, supermajor, augmented, superaugmented.

For unisons, fourths, fifths, and octaves, the modifiers are: doubly diminished, subdiminished, diminished, sub, perfect (0), super, augmented, superaugmented, double augmented.

For both the comma system and the larger-interval system, the names "Pythagorean", "classic" (???), "septimal", "undecimal", and "tridecimal" can be used instead of "1" (or "3", which can replace "1" for 3-limit intervals), "5", "7", "11", and "13" when no other prime is present ("classic-to-septimal kleisma" is certainly not a correct full name for the 5:7k) in full names.

Obviously, these should usually accompany the interval's ratio itself, not replace it, but I didn't make the names; the other people here did.

Even with a post as extensive as this, I'm sure I missed something, and I'm sure I'd have missed something no matter how many times I revised this, so this should probably do for now.