40edo: Difference between revisions

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'''40edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 40 parts of exactly 30 [[cent]]s each. 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. Despite all keys being reachable by stacking this 5th, it does not qualify as [[meantone]] either, as stacking 4 of them results in a near perfect tridecimal neutral third rather than a major one. 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 a lot of them to notate more distant keys. It [[tempering_out|tempers out]] 648/625 in the [[5-limit|5-limit]]; 225/224 and in the [[7-limit|7-limit]]; 99/98, 121/120 and 176/175 in the [[11-limit|11-limit]]; and 66/65 in the [[13-limit|13-limit]].

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			{{Infobox ET}}
	'''40edo''' is the [[Equal_division_of_the_octave\|equal division of the octave]] into 40 parts of exactly 30 [[cent]]s each. 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. Despite all keys being reachable by stacking this 5th, it does not qualify as [[meantone]] either, as stacking 4 of them results in a near perfect tridecimal neutral third rather than a major one. 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 a lot of them to notate more distant keys. It [[tempering_out\|tempers out]] 648/625 in the [[5-limit\|5-limit]]; 225/224 and in the [[7-limit\|7-limit]]; 99/98, 121/120 and 176/175 in the [[11-limit\|11-limit]]; and 66/65 in the [[13-limit\|13-limit]].		'''40edo''' is the [[Equal_division_of_the_octave\|equal division of the octave]] into 40 parts of exactly 30 [[cent]]s each. 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. Despite all keys being reachable by stacking this 5th, it does not qualify as [[meantone]] either, as stacking 4 of them results in a near perfect tridecimal neutral third rather than a major one. 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 a lot of them to notate more distant keys. It [[tempering_out\|tempers out]] 648/625 in the [[5-limit\|5-limit]]; 225/224 and in the [[7-limit\|7-limit]]; 99/98, 121/120 and 176/175 in the [[11-limit\|11-limit]]; and 66/65 in the [[13-limit\|13-limit]].