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)

While the sharp 3 and 5 could have use in such chords, intervals involving them (such as ~7/6 with the sharp fifth) aren't really used, and intervals of 9 use the best 9. Dual-prime systems are very different from each other, and messy in general. For example, it is best for chords in 35edo to use the best 3, 5, 7, 9, and 11, leaving only "9/6" inconsistent in 4:5:6:7:9:11. The 8:10:12:15 chord doesn't fare well in 35edo, as there must be an interval with about 75% relative error no matter how it is tuned. It's so much more difficult when there are inconsistencies all over the place.

As for the table, I think there should be a column for ratios of the higher primes, ratios of flat 3 and 5, sharp 3 and 5, and best 9, 15, and 25. Inconsistent intervals should be italicized, and the reader will quickly see that the "sharp 3 and 5" column is almost entirely italic, while the other columns have much less inconsistencies. Also, the interval table template likes to exclude ratios if they have even slightly over 25% relative error, such as it excluding 7/5 in 224edo, while it includes ratios of higher primes that are best omitted from the ET's subgroup, so the table needs cleanup. This also shows that the template really needs rework, and explanation on the doc page on how to substitute it.--Overthink (talk) 04:28, 3 January 2026 (UTC)

The thing about using the sharp harmonic 3 or 5 is that they don't have other sharp harmonics to balance against (except each other), unlike (for instance) 55edo, and the sharp approximations are roughly 2X as far off from just as as the flat approximations. This means that, as you say, the sharp 3 and 5 column would be full of inconsistencies, which means it would not be very useful; however, the best 9, 15, and 25 column (come to think of it, even best 27) would be useful.

Related to this, we ought to have some formalized way of specifying a temperament that has both best 3 and best 9/best 5 and best 15 and 25/etc. As far as I know, the currently available temperament tools explicitly DON'T allow you to specify something like 2.3.5.9.15.25 — the logical use for that would be to use the best 3 or 5 first, and then for a ratio that has additional powers of either one, you combine it with the 3 or 5 that does the best job of correcting the error. With the current example of 57edo, you would thus default to 3♭ or 5♭, but then if it was part of a 9, 15, or 25, you would next use a 3♯ or 5♯, and if it was part of a 27 then you would go back to 3♭, etc. This could be formalized as 2.3.5.9.15.25.etc.

Lucius Chiaraviglio (talk) 08:06, 3 January 2026 (UTC)

. . . Which the current version of Just intonation subgroup also explicitly disallows. Lucius Chiaraviglio (talk) 11:44, 3 January 2026 (UTC)