Talk:Meantone: Difference between revisions

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::::: Your mappings of 17 and 19 are much better choices than the ones I used on [[User:Recentlymaterialized/Miscellaneous_rank-2_temperaments#Worsetone_.2812f_.26_67.29|worsetone]], thank you! I disagree, however, that prime 13 is badly represented by 12edo and 55edo. Mapped at +39 fifths, it is supported by [https://x31eq.com/pyscript/rt.html?ets=12f&limit=13 12f] and [https://x31eq.com/pyscript/rt.html?ets=55f&limit=13 55f] vals (and patent 67edo, of course), both of which have lower 13-limit error than their patent counterparts [https://x31eq.com/pyscript/rt.html?ets=12p&limit=13 12p] and [https://x31eq.com/pyscript/rt.html?ets=55p&limit=13 55p]. Because this 13 only needs another +9 fifths (and suffers less than a cent of error in the WE optimization), I don't see much reason to go no-13s in this case.

::::: Regardless, if you're searching for higher-limit extensions, it may be helpful to remove the 5 so that the optimizer doesn't obsess over the 5/1 (this is 1/6 comma meantone, we've accepted a sharp 5/4!). With this in mind, I suggest two possible 23-limit extensions using your mappings of 17 and 19: [https://sintel.pythonanywhere.com/result?edos=55f+%26+67&submit_edo=submit&subgroup=2.3.7.11.13.17.19.23 55f & 67] and [https://sintel.pythonanywhere.com/result?edos=55fi+67&submit_edo=submit&subgroup=2.3.7.11.13.17.19.23 55fi & 67]. The former is slightly more complex as it needs 13 extra fifths while the second only needs 10, but the two merge in 67edo anyways so you could probably use either one depending on the situation (if you're using 67edo, that is). [[User:Recentlymaterialized|Recentlymaterialized]] ([[User talk:Recentlymaterialized|talk]]) 04:31, 9 January 2025 (UTC)

:::::: I agree that using 12f instead of 12 and 55f instead of 55 improves the consistency, but either way, the 13th harmonic is very far off in those in relative error, which translates to huge absolute error for 12EDO and still enough error in 55EDO to stick out like a sore thumb even when sticking out in the same direction as nearby harmonics. (At least in 122cEDO, the absolute error finally gets small enough that you cloud plausibly gloss over it as long as you don't use it too much.) And either way, the huge fifthspan makes the representation of the 13th harmonic very brittle to small changes within the region around 1/6-comma meantone, and potentially inconsistent with the better higher (as well as lower) harmonics, so even if not dumping it permanently, I am still inclined to exclude it at least temporarily while exploring the higher harmonics, and then maybe back-extend to it later (maybe as a dual-13 system).

:::::: Although I can see why you might have also wanted different mappings for the 17th and 19th harmonics, since in 55EDO the error in these is in the opposite direction of that of most lower harmonics and/or each other (and in 122cEDO, it's a mixed bag). The problem is that the flat approximation to the 17th harmonic is pretty bad, and the 17th harmonic isn't far off enough in any of these to be a good candidate for a split harmonic.

:::::: I get fifthspan -18 for the 23rd harmonic for patent vals (except for the c wart on 122) for 12, 55, 67, and 122c; other than 12EDO, these are not too far off from just (well, 12EDO is, but that's as good as you're going to get for most harmonics with a small EDO like 12). If I instead use fifthspan +49, I get 12i, 55i, 67, and 122ci; only 67 is close to just. And fifthspan +49 is awfully brittle against a slight shift in tuning within the region around 1/6-comma meantone. So I would go with fifthspan -18 even though it is pointing the other way from the fifthspans for most of these harmonics.

:::::: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:51, 9 January 2025 (UTC)

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 ::::: Your mappings of 17 and 19 are much better choices than the ones I used on [[User:Recentlymaterialized/Miscellaneous_rank-2_temperaments#Worsetone_.2812f_.26_67.29|worsetone]], thank you! I disagree, however, that prime 13 is badly represented by 12edo and 55edo. Mapped at +39 fifths, it is supported by [https://x31eq.com/pyscript/rt.html?ets=12f&limit=13 12f] and [https://x31eq.com/pyscript/rt.html?ets=55f&limit=13 55f] vals (and patent 67edo, of course), both of which have lower 13-limit error than their patent counterparts [https://x31eq.com/pyscript/rt.html?ets=12p&limit=13 12p] and [https://x31eq.com/pyscript/rt.html?ets=55p&limit=13 55p]. Because this 13 only needs another +9 fifths (and suffers less than a cent of error in the WE optimization), I don't see much reason to go no-13s in this case.
 ::::: Regardless, if you're searching for higher-limit extensions, it may be helpful to remove the 5 so that the optimizer doesn't obsess over the 5/1 (this is 1/6 comma meantone, we've accepted a sharp 5/4!). With this in mind, I suggest two possible 23-limit extensions using your mappings of 17 and 19: [https://sintel.pythonanywhere.com/result?edos=55f+%26+67&submit_edo=submit&subgroup=2.3.7.11.13.17.19.23 55f & 67] and [https://sintel.pythonanywhere.com/result?edos=55fi+67&submit_edo=submit&subgroup=2.3.7.11.13.17.19.23 55fi & 67]. The former is slightly more complex as it needs 13 extra fifths while the second only needs 10, but the two merge in 67edo anyways so you could probably use either one depending on the situation (if you're using 67edo, that is). [[User:Recentlymaterialized|Recentlymaterialized]] ([[User talk:Recentlymaterialized|talk]]) 04:31, 9 January 2025 (UTC)
+:::::: I agree that using 12f instead of 12 and 55f instead of 55 improves the consistency, but either way, the 13th harmonic is very far off in those in relative error, which translates to huge absolute error for 12EDO and still enough error in 55EDO to stick out like a sore thumb even when sticking out in the same direction as nearby harmonics.  (At least in 122cEDO, the absolute error finally gets small enough that you cloud plausibly gloss over it as long as you don't use it too much.)  And either way, the huge fifthspan makes the representation of the 13th harmonic very brittle to small changes within the region around 1/6-comma meantone, and potentially inconsistent with the better higher (as well as lower) harmonics, so even if not dumping it permanently, I am still inclined to exclude it at least temporarily while exploring the higher harmonics, and then maybe back-extend to it later (maybe as a dual-13 system).
+:::::: Although I can see why you might have also wanted different mappings for the 17th and 19th harmonics, since in 55EDO the error in these is in the opposite direction of that of most lower harmonics and/or each other (and in 122cEDO, it's a mixed bag).  The problem is that the flat approximation to the 17th harmonic is pretty bad, and the 17th harmonic isn't far off enough in any of these to be a good candidate for a split harmonic.
+:::::: I get fifthspan -18 for the 23rd harmonic for patent vals (except for the c wart on 122) for 12, 55, 67, and 122c; other than 12EDO, these are not too far off from just (well, 12EDO is, but that's as good as you're going to get for most harmonics with a small EDO like 12).  If I instead use fifthspan +49, I get 12i, 55i, 67, and 122ci; only 67 is close to just.  And fifthspan +49 is awfully brittle against a slight shift in tuning within the region around 1/6-comma meantone.  So I would go with fifthspan -18 even though it is pointing the other way from the fifthspans for most of these harmonics.
+:::::: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:51, 9 January 2025 (UTC)