212edo: Difference between revisions

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== Theory ==

212 = 4 × 53, and it shares the 3rd, 5th, and 13th [[Overtone series|harmonics]] with [[53edo]], but the mapping differs for 7 and 11.

It tempers out the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]].

It tempers out the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]]. In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]). In the [[11-limit]], [[385/384]], [[1375/1372]], [[6250/6237]], [[9801/9800]] and [[14641/14580]]; in the [[13-limit]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[1716/1715]], and [[2080/2079]].

In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]).

In the [[11-limit]], 385/384 ~~([[keenanisma~~]]), 1375/1372 ~~([[moctdel comma~~]]), 6250/6237 ~~([[liganellus comma~~]]), 9801/9800 ~~([[kalisma~~]]) and 14641/14580 ~~([[semicanousma~~]]).

In the [[13-limit]], 325/324 ~~([[marveltwin comma~~]]), 625/624 ~~([[tunbarsma~~]]), 676/675 ~~([[island comma~~]]), 1001/1000 ~~([[sinbadma~~]]), 1716/1715 ~~([[lummic comma~~]]), 2080/2079 ~~([[ibnsinma~~]]).

It is distinctly [[consistent]] in the [[15-odd-limit]] with harmonics of 3 through 13 all tuned flat. It is the [[optimal patent val]] for 7- and 13-limit [[Kleismic family #Quadritikleismic|quadritikleismic temperament]], the 7-limit rank-3 [[Kleismic rank three family #Kleismic|kleismic temperament]], and the 13-limit rank-3 [[Breed family #Agni|agni temperament]]. 212gh val shows some potential beyond 15-odd-limit. Also, using 212bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning almost identical to the POTE tuning for 5-limit meantone.

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 == Theory ==
 = 4 × 53, and it shares the 3rd, 5th, and 13th [[Overtone series|harmonics]] with [[53edo]], but the mapping differs for 7 and 11.
-It tempers out the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]].
+It tempers out the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]]. In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]). In the [[11-limit]], [[385/384]], [[1375/1372]], [[6250/6237]], [[9801/9800]] and [[14641/14580]]; in the [[13-limit]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[1716/1715]], and [[2080/2079]].
-In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]).
-In the [[11-limit]], 385/384 ([[keenanisma]]), 1375/1372 ([[moctdel comma]]), 6250/6237 ([[liganellus comma]]), 9801/9800 ([[kalisma]]) and 14641/14580 ([[semicanousma]]).
-In the [[13-limit]], 325/324 ([[marveltwin comma]]), 625/624 ([[tunbarsma]]), 676/675 ([[island comma]]), 1001/1000 ([[sinbadma]]), 1716/1715 ([[lummic comma]]), 2080/2079 ([[ibnsinma]]).
 It is distinctly [[consistent]] in the [[15-odd-limit]] with harmonics of 3 through 13 all tuned flat. It is the [[optimal patent val]] for 7- and 13-limit [[Kleismic family #Quadritikleismic|quadritikleismic temperament]], the 7-limit rank-3 [[Kleismic rank three family #Kleismic|kleismic temperament]], and the 13-limit rank-3 [[Breed family #Agni|agni temperament]]. 212gh val shows some potential beyond 15-odd-limit. Also, using 212bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning almost identical to the POTE tuning for 5-limit meantone.