212edo: Difference between revisions
mNo edit summary |
Expansion |
||
| Line 5: | Line 5: | ||
212 = 4 × 53, and it shares the 3rd, 5th, and 13th [[Overtone series|harmonics]] with [[53edo]], but the mapping differs for 7 and 11. | 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]], | 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 ([[ | 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 [[11-limit]], 385/384 ([[keenanisma]]), 1375/1372 ([[moctdel comma]]), 6250/6237 ([[liganellus comma]]), 9801/9800 ([[kalisma]]) and 14641/14580 ([[semicanousma]]). | ||
| Line 13: | Line 13: | ||
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]]). | 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]], and the 13-limit rank | 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. | ||
=== Prime intervals === | === Prime intervals === | ||
| Line 20: | Line 20: | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:53edo]] | [[Category:53edo]] | ||
[[Category:Kleismic]] | |||
[[Category:Agni]] | [[Category:Agni]] | ||
[[Category:Quadritikleismic]] | [[Category:Quadritikleismic]] | ||
Revision as of 12:27, 23 April 2021
212 equal temperament divides the octave into 212 equal parts of 5.660 cents each.
Theory
212 = 4 × 53, and it shares the 3rd, 5th, and 13th harmonics with 53edo, but the mapping differs for 7 and 11.
It tempers out the same commas (15625/15552, 32805/32768, 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 (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 quadritikleismic temperament, the 7-limit rank-3 kleismic temperament, and the 13-limit rank-3 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.
Prime intervals
Script error: No such module "primes_in_edo".