40edo: Difference between revisions

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== Theory ==

Up to this point, all the multiples of 5 have had the 720 cent [[blackwood]] 5th as their best approximation of [[3/2]]. [[35edo]] combined the small circles of blackwood and whitewood 5ths, almost equally far from just, requiring you to use both to reach all keys. 40edo adds a diatonic 5th that's closer to just. However, it is still the second flattest diatonic 5th, only exceeded by [[47edo]] in error, which results in it being inconsistent in the 5-limit - combining the best major and minor third will result in the blackwood 5th instead. As such, calling it a perfect 5th seems very much a misnomer. This fifth qualifies for [[flattone]], a variant of meantone with flat fifths, although 40edo's fifth is a bit extreme even for flattone. 40edo's fifth is flat enough that the meantone major third falls into submajor or even sharp neutral third territory at 360 cents, while the minor third is supraminor although not quite high enough to be considered neutral at 330 cents. The resulting [[5L 2s]] scale has large steps of 6 intervals and small ones of 5, putting sharps and flats right next to letters without any ups or downs in between and requiring up to 3 of them to notate more distant keys. It [[tempering_out|tempers out]] 648/625 in the [[5-limit]]; 225/224 and in the [[7-limit]]; 99/98, 121/120 and 176/175 in the [[11-limit]]; and 66/65 in the [[13-limit]].

Up to this point, all the multiples of 5 have had the 720 cent [[blackwood]] 5th as their best approximation of [[3/2]]. [[35edo]] combined the small circles of blackwood and whitewood 5ths, almost equally far from just, requiring you to use both to reach all keys. 40edo adds a diatonic 5th that's closer to just. However, it is still the second flattest diatonic 5th, only exceeded by [[47edo]] in error, which results in it being inconsistent in the 5-limit - combining the best major and minor third will result in the blackwood 5th instead. As such, calling it a perfect 5th seems very much a misnomer. This fifth qualifies for [[flattone]], a variant of meantone with flat fifths, although 40edo's fifth is a bit extreme even for flattone, with slightly sharper fifths like those of [[26edo]] or [[45edo]] being considered optimal for flattone. 40edo's fifth is flat enough that the meantone major third falls into submajor or even sharp neutral third territory at 360 cents, while the minor third is supraminor although not quite high enough to be considered neutral at 330 cents. The resulting [[5L 2s]] scale has large steps of 6 intervals and small ones of 5, putting sharps and flats right next to letters without any ups or downs in between and requiring up to 3 of them to notate more distant keys. It [[tempering_out|tempers out]] 648/625 in the [[5-limit]]; 225/224 and in the [[7-limit]]; 99/98, 121/120 and 176/175 in the [[11-limit]]; and 66/65 in the [[13-limit]].

40edo is more accurate on the 2.9.5.21.33.13.51.19 [[k*N_subgroups| 2*40 subgroup]], where it offers the same tuning as [[80edo|80edo]], and tempers out the same commas. It is also the first equal temperament to approximate both the 23rd and 19th harmonic, by tempering out the 9 cent comma to 4-edo, with 10 divisions therein.

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 {{EDO intro|40}}
 == Theory ==
-Up to this point, all the multiples of 5 have had the 720 cent [[blackwood]] 5th as their best approximation of [[3/2]]. [[35edo]] combined the small circles of blackwood and whitewood 5ths, almost equally far from just, requiring you to use both to reach all keys. 40edo adds a diatonic 5th that's closer to just. However, it is still the second flattest diatonic 5th, only exceeded by [[47edo]] in error, which results in it being inconsistent in the 5-limit - combining the best major and minor third will result in the blackwood 5th instead. As such, calling it a perfect 5th seems very much a misnomer. This fifth qualifies for [[flattone]], a variant of meantone with flat fifths, although 40edo's fifth is a bit extreme even for flattone. 40edo's fifth is flat enough that the meantone major third falls into submajor or even sharp neutral third territory at 360 cents, while the minor third is supraminor although not quite high enough to be considered neutral at 330 cents. The resulting [[5L 2s]] scale has large steps of 6 intervals and small ones of 5, putting sharps and flats right next to letters without any ups or downs in between and requiring up to 3 of them to notate more distant keys. It [[tempering_out|tempers out]] 648/625 in the [[5-limit]]; 225/224 and in the [[7-limit]]; 99/98, 121/120 and 176/175 in the [[11-limit]]; and 66/65 in the [[13-limit]].
+Up to this point, all the multiples of 5 have had the 720 cent [[blackwood]] 5th as their best approximation of [[3/2]]. [[35edo]] combined the small circles of blackwood and whitewood 5ths, almost equally far from just, requiring you to use both to reach all keys. 40edo adds a diatonic 5th that's closer to just. However, it is still the second flattest diatonic 5th, only exceeded by [[47edo]] in error, which results in it being inconsistent in the 5-limit - combining the best major and minor third will result in the blackwood 5th instead. As such, calling it a perfect 5th seems very much a misnomer. This fifth qualifies for [[flattone]], a variant of meantone with flat fifths, although 40edo's fifth is a bit extreme even for flattone, with slightly sharper fifths like those of [[26edo]] or [[45edo]] being considered optimal for flattone. 40edo's fifth is flat enough that the meantone major third falls into submajor or even sharp neutral third territory at 360 cents, while the minor third is supraminor although not quite high enough to be considered neutral at 330 cents. The resulting [[5L 2s]] scale has large steps of 6 intervals and small ones of 5, putting sharps and flats right next to letters without any ups or downs in between and requiring up to 3 of them to notate more distant keys. It [[tempering_out|tempers out]] 648/625 in the [[5-limit]]; 225/224 and in the [[7-limit]]; 99/98, 121/120 and 176/175 in the [[11-limit]]; and 66/65 in the [[13-limit]].
 edo is more accurate on the 2.9.5.21.33.13.51.19 [[k*N_subgroups| 2*40 subgroup]], where it offers the same tuning as [[80edo|80edo]], and tempers out the same commas. It is also the first equal temperament to approximate both the 23rd and 19th harmonic, by tempering out the 9 cent comma to 4-edo, with 10 divisions therein.