Talk:57edo

Canonical subgroup for the interval table

Treat this edo as dual-3 dual-5 or plain-3 plain-5? —FloraC (talk) 13:22, 2 January 2026 (UTC)

I was trying to fill in the automatically generated output (from the reverted {{interval table}} temporary change) to get the full 31-limit (since the automatic output goes all the way to the 31st harmonic, although maybe it would be best to prune out the impractically high primes). We accidentally edit-clashed when I was still filling in missing pieces from the automatic output (I was going to fill in 3/2 and 4/3 but then the edit clash happened). I tried to put back your changes thereafter, except for the deletion of ratios of 9. I was motivated by the previous interval table not actually having any intervals in it at all (and wanted to look at it to see if 57edo could be used in some sort of relative of Orwell temperament); but doing it right takes time and much realizing after the fact of hitting Save that it still needs work (but probably a good idea to Save multiple times anyway, in case something fries). Maybe next time I should put some kind of "Under Construction" temporary heading on it? Lucius Chiaraviglio (talk) 13:45, 2 January 2026 (UTC)

. . . Although come to think of it, it might not be a bad idea to add an extra column eventually of additional ratios of 9, 15, or 25 tending sharp (seems to me that the 3rd and 5th harmonics aren't close enough to the midpoint to warrant sharp 3rd or 5th harmonics, when so many of the next higher harmonics are pretty close to just). Lucius Chiaraviglio (talk) 14:26, 2 January 2026 (UTC)

When you left out 3/2 and 4/3 I thought you intended it to be a dual-3 dual-5, so I edited to formalize it that way, tho I figured the 5-limit part from 19edo would also make sense. That's why I was asking. —FloraC (talk) 14:32, 2 January 2026 (UTC)

Understandable since you caught me in the middle of the process, and I was belated in realizing that the automated interval table generator didn't fill in 3/2 and 4/3 (who would have thought?). I think tripled 19edo + highly accurate higher primes component (as was already in the description) makes a lot of sense for 57edo, which deserves more exploration. Lucius Chiaraviglio (talk) 22:53, 2 January 2026 (UTC)

I think this edo should be dual-3 dual-5, as not only should the intervals themselves be considered, but rather chords they are in. For example, a 16:19:24:28:36 chord by direct approximation of each harmonic would have only one inconsistent interval, 36/24 (not the same number of steps as 24/16), with +65.7% relative error. In contrast, by patent val, all of 36/16, 36/19, and 36/28 are inconsistent, having -68.6%, -55.4%, and -66.6% error respectively. Harmonics 9 is very common in chords, so in most cases prime 3 should be dual even if it has just about 1/3 relative error (though 49edo is an exception, due to the sharpness of 5, 7, and 11). Prime 5 should also be dual, as harmonics 15 and 25 are relatively simple, and have about -70% relative error by patent val. In general, things get messy when there's inconsistencies, and EDOs don't work like JI.--Overthink (talk) 04:10, 3 January 2026 (UTC)