Talk:Meantone: Difference between revisions

Line 294:

:::: Looked at the 17th harmonic -- that's easy, it's equated to the diatonic semitone, so fifthspan -5. And the 19th harmonic is almost as easy, equated to the minor third, so fifthspan -3, except that if I haven't botched something by being in a hurry and up way too late, it actually doesn't quite work in 122EDO, so we get a ch wart here (EDO list: 12, 55, 67, 122ch -- most are close, but 122ch has high relative error). [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 12:00, 8 January 2025 (UTC)

::::: 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)

@@ Line 294: / Line 294: @@
 :::: Looked at the 17th harmonic -- that's easy, it's equated to the diatonic semitone, so fifthspan -5.  And the 19th harmonic is almost as easy, equated to the minor third, so fifthspan -3, except that if I haven't botched something by being in a hurry and up way too late, it actually doesn't quite work in 122EDO, so we get a ch wart here (EDO list:  12, 55, 67, 122ch -- most are close, but 122ch has high relative error).  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 12:00, 8 January 2025 (UTC)
+::::: 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)