212edo: Difference between revisions
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== Theory == | == 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. | 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 | |||
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. | 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. | ||
Revision as of 09:54, 25 July 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, 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.
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".
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 2401/2400, 15625/15552, 32805/32768 | [⟨212 336 492 595]] | +0.243 | 0.244 | 4.30 |
| 2.3.5.7.11 | 385/384, 1375/1372, 6250/6237, 14641/14580 | [⟨212 336 492 595 733]] | +0.325 | 0.273 | 4.82 |
| 2.3.5.7.11.13 | 325/324, 385/384, 625/624, 1375/1372, 10648/10647 | [⟨212 336 492 595 733 784]] | +0.396 | 0.296 | 5.23 |
| 2.3.5.7.11.13.17 | 289/288, 325/324, 385/384, 442/441, 625/624, 10648/10647 | [⟨212 336 492 595 733 784 866]] (212g) | +0.447 | 0.301 | 5.32 |
| 2.3.5.7.11.13.17.19 | 289/288, 325/324, 361/360, 385/384, 442/441, 513/512, 625/624 | [⟨212 336 492 595 733 784 866 900]] (212gh) | +0.485 | 0.299 | 5.27 |