Talk:Meantone: Difference between revisions

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– [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:03, 11 November 2024 (UTC)

: Yes, this is the meantone extension supported by [[67edo]], mapping [[7/4]] to two [[128/125|dieses]] below [[9/5]] instead of the usual 1 diesis. I have [[User:Recentlymaterialized/Miscellaneous_rank-2_temperaments#~~Worsetone~~|the 13-limit extension]] documented on my user page (currently named "worsetone", though the name isn't final by any means), but it is not really an efficient temperament until you go to very high prime limits. [[User:Recentlymaterialized|Recentlymaterialized]] ([[User talk:Recentlymaterialized|talk]]) 19:12, 30 December 2024 (UTC)

: Yes, this is the meantone extension supported by [[67edo]], mapping [[7/4]] to two [[128/125|dieses]] below [[9/5]] instead of the usual 1 diesis. I have [[User:Recentlymaterialized/Miscellaneous_rank-2_temperaments#Worsetone_.2812f_.26_67.29|the 13-limit extension]] documented on my user page (currently named "worsetone", though the name isn't final by any means), but it is not really an efficient temperament until you go to very high prime limits. [[User:Recentlymaterialized|Recentlymaterialized]] ([[User talk:Recentlymaterialized|talk]]) 19:12, 30 December 2024 (UTC)

:: Starting to do the work for higher limits -- 11-limit for now. Computing manually, I find that assigning a fifthspan of +30 gets the right 11/8 (without warts) for 55EDO and 67EDO, but not 79EDO, and also gets the right 11/8 (with a c wart but no others) for 122EDO and 134EDO but not 110EDO or 146EDO. Amazingly, it even works for 12EDO without warts. I also tried fifthspan -25, but found that worked only for 55EDO, so fifthspan +30 it is. Fifthspan +30 coresponds to C-D𝄪𝄪, which corresponds to ((3/2)/(81/80)^(1/6))^30 / 2^17 in 1/6-comma meantone (which is more or less in the middle of the target region of meantone), which works out to |-27 10 5⟩, which yields a value of 551.119¢, which is just 0.119¢ flat of just 11/8 (551.318¢).

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:: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 10:19, 2 January 2025 (UTC) edited [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 10:56, 2 January 2025 (UTC)

::: Yes, excluding compton extensions, fifthspan +30 is the 11-limit extension with the lowest logflat badness (it's 0.0649609 gene smith tenney-euclidean badness, or 2.1475667 dirichlet badness). Similarly, my 13-limit extension with 3/1 at +39 generators is the 13-limit extension with the lowest logflat badness (0.0494043 smith TE, 2.0414469 dirichlet), excluding compton extensions. This 13-limit extension is supported by [[67edo]] unwarted; however, if you would like to get rid of the warts on 12edo and 55edo, [https://sintel.pythonanywhere.com/result?edos=12+%26+55&submit_edo=submit&subgroup=2.3.5.7.11.13 12 & 55] is an option, though it has much higher [[complexity]] and [[logflat badness]] because the 13/1 is reached with -28 fifths rather than a positive number of generators (and thus it cannot be played in otonal chords involving 7 and 11 without the use of the 67-note and larger MOS scales). [[User:Recentlymaterialized|Recentlymaterialized]] ([[User talk:Recentlymaterialized|talk]]) 16:03, 6 January 2025 (UTC)

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 – [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:03, 11 November 2024 (UTC)
-: Yes, this is the meantone extension supported by [[67edo]], mapping [[7/4]] to two [[128/125|dieses]] below [[9/5]] instead of the usual 1 diesis. I have [[User:Recentlymaterialized/Miscellaneous_rank-2_temperaments#Worsetone|the 13-limit extension]] documented on my user page (currently named "worsetone", though the name isn't final by any means), but it is not really an efficient temperament until you go to very high prime limits. [[User:Recentlymaterialized|Recentlymaterialized]] ([[User talk:Recentlymaterialized|talk]]) 19:12, 30 December 2024 (UTC)
+: Yes, this is the meantone extension supported by [[67edo]], mapping [[7/4]] to two [[128/125|dieses]] below [[9/5]] instead of the usual 1 diesis. I have [[User:Recentlymaterialized/Miscellaneous_rank-2_temperaments#Worsetone_.2812f_.26_67.29|the 13-limit extension]] documented on my user page (currently named "worsetone", though the name isn't final by any means), but it is not really an efficient temperament until you go to very high prime limits. [[User:Recentlymaterialized|Recentlymaterialized]] ([[User talk:Recentlymaterialized|talk]]) 19:12, 30 December 2024 (UTC)
 :: Starting to do the work for higher limits -- 11-limit for now.  Computing manually, I find that assigning a fifthspan of +30 gets the right 11/8 (without warts) for 55EDO and 67EDO, but not 79EDO, and also gets the right 11/8 (with a c wart but no others) for 122EDO and 134EDO but not 110EDO or 146EDO.  Amazingly, it even works for 12EDO without warts.  I also tried fifthspan -25, but found that worked only for 55EDO, so fifthspan +30 it is.  Fifthspan +30 coresponds to C-D𝄪𝄪, which corresponds to ((3/2)/(81/80)^(1/6))^30 / 2^17 in 1/6-comma meantone (which is more or less in the middle of the target region of meantone), which works out to |-27 10 5⟩, which yields a value of 551.119¢, which is just 0.119¢ flat of just 11/8 (551.318¢).
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 :: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 10:19, 2 January 2025 (UTC) edited [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 10:56, 2 January 2025 (UTC)
+::: Yes, excluding compton extensions, fifthspan +30 is the 11-limit extension with the lowest logflat badness (it's 0.0649609 gene smith tenney-euclidean badness, or 2.1475667 dirichlet badness). Similarly, my 13-limit extension with 3/1 at +39 generators is the 13-limit extension with the lowest logflat badness (0.0494043 smith TE, 2.0414469 dirichlet), excluding compton extensions. This 13-limit extension is supported by [[67edo]] unwarted; however, if you would like to get rid of the warts on 12edo and 55edo, [https://sintel.pythonanywhere.com/result?edos=12+%26+55&submit_edo=submit&subgroup=2.3.5.7.11.13 12 & 55] is an option, though it has much higher [[complexity]] and [[logflat badness]] because the 13/1 is reached with -28 fifths rather than a positive number of generators (and thus it cannot be played in otonal chords involving 7 and 11 without the use of the 67-note and larger MOS scales). [[User:Recentlymaterialized|Recentlymaterialized]] ([[User talk:Recentlymaterialized|talk]]) 16:03, 6 January 2025 (UTC)