Talk:Meantone: Difference between revisions

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: 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)

:: 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-~~B𝄪𝄪~~, 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¢).

:: 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¢).

:: Have not yet tried to compute badness, since when I go to [[Tenney-Euclidean_temperament_measures#TE_logflat_badness|the page for that]], I run into 2 problems currently beyond my knowledge to solve: What the symbol "∧" means in this context, and how [[Wedgies and Multivals|wedgies]] work, for which the article itself says "This page may be difficult to understand to those unfamiliar with the mathematical concepts involved." So I will simply note for now that the 11th harmonic accuracy is extremely high (even in the actual non-warted EDOs other than 134EDO, where it is about to go overboard in relative error), but with very high complexity, and having overall 11-limit accuracy dragged down by drift in the 7th harmonic, the drift and sharpness in the 5th harmonic, and (of course) the classic ~1/6-comma meantone flat 3rd harmonic. Would be interesting to see what the badness comes out to without the 13th harmonic.

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:: Have not yet tried to see if I can get rid of the warts you found in the 13th harmonic -- will have to do this later. I see that the mapping I have not yet counting the 13th harmonic matches what you have for the second mapping you have under Worsetone, but maybe something different needs to be done with the 13th harmonic.

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

:: [[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)

@@ Line 279: / Line 279: @@
 : 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)
-:: 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-B𝄪𝄪, 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¢).
+:: 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¢).
 :: Have not yet tried to compute badness, since when I go to [[Tenney-Euclidean_temperament_measures#TE_logflat_badness|the page for that]], I run into 2 problems currently beyond my knowledge to solve:  What the symbol "∧" means in this context, and how [[Wedgies and Multivals|wedgies]] work, for which the article itself says "This page may be difficult to understand to those unfamiliar with the mathematical concepts involved."  So I will simply note for now that the 11th harmonic accuracy is extremely high (even in the actual non-warted EDOs other than 134EDO, where it is about to go overboard in relative error), but with very high complexity, and having overall 11-limit accuracy dragged down by drift in the 7th harmonic, the drift and sharpness in the 5th harmonic, and (of course) the classic ~1/6-comma meantone flat 3rd harmonic.  Would be interesting to see what the badness comes out to without the 13th harmonic.
@@ Line 285: / Line 285: @@
 :: Have not yet tried to see if I can get rid of the warts you found in the 13th harmonic -- will have to do this later.  I see that the mapping I have not yet counting the 13th harmonic matches what you have for the second mapping you have under Worsetone, but maybe something different needs to be done with the 13th harmonic.
-:: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 10:19, 2 January 2025 (UTC)
+:: [[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)