31edo: Difference between revisions

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== Theory ==

31edo's [[3/2|perfect fifth]] is flat of just by 5.2{{c}}, as befits a tuning of [[meantone]]. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31edo's approximation of [[7/4]], a cent flat, is also very close to just. Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).

31edo's [[3/2|perfect fifth]] is flat of just by 5.2{{c}}, as befits a tuning of [[meantone]]. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. To get a better approximation to quarter-comma meantone requires going to much higher edo sizes: [[112edo]] (as 112b) to get a major third closer to just in absolute error, or [[174edo]] (as 174b) to get a major third closer to just in relative error. 31edo's approximation of [[7/4]], a cent flat, is also very close to just. Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).

Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents. However, intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. This makes 31edo a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.

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 == Theory ==
-edo's [[3/2|perfect fifth]] is flat of just by 5.2{{c}}, as befits a tuning of [[meantone]]. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31edo's approximation of [[7/4]], a cent flat, is also very close to just. Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).
+edo's [[3/2|perfect fifth]] is flat of just by 5.2{{c}}, as befits a tuning of [[meantone]]. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. To get a better approximation to quarter-comma meantone requires going to much higher edo sizes: [[112edo]] (as 112b) to get a major third closer to just in absolute error, or [[174edo]] (as 174b) to get a major third closer to just in relative error. 31edo's approximation of [[7/4]], a cent flat, is also very close to just. Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).
 Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents. However, intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. This makes 31edo a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.