212edo: Difference between revisions
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+rank-2 temperaments |
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Note: temperaments supported by 53et are not included. | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 15\212 | |||
| 84.91 | |||
| 21/20 | |||
| [[Amicable]] | |||
|- | |||
| 1 | |||
| 31\212 | |||
| 175.47 | |||
| 448/405 | |||
| [[Sesquiquartififths]] | |||
|- | |||
| 1 | |||
| 41\212 | |||
| 232.08 | |||
| 8/7 | |||
| [[Quadrawell]] | |||
|- | |||
| 1 | |||
| 67\212 | |||
| 379.25 | |||
| 56/45 | |||
| [[Marthirds]] | |||
|- | |||
| 2 | |||
| 11\212 | |||
| 62.26 | |||
| 28/27 | |||
| [[Eagle]] | |||
|- | |||
| 2 | |||
| 31\212 | |||
| 175.47 | |||
| 448/405 | |||
| [[Bisesqui]] | |||
|- | |||
| 2 | |||
| 97\212<br>(9\212) | |||
| 549.06<br>(50.94) | |||
| 11/8<br>(36/35) | |||
| [[Kleischismic]] | |||
|- | |||
| 4 | |||
| 56\212<br>(3\212) | |||
| 316.98<br>(16.98) | |||
| 6/5<br>(126/125) | |||
| [[Quadritikleismic]] | |||
|- | |||
| 4 | |||
| 88\212<br>(18\212) | |||
| 498.11<br>(101.89) | |||
| 4/3<br>(35/33) | |||
| [[Quadrant]] | |||
|- | |||
| 53 | |||
| 41\212<br>(1\198) | |||
| 232.08<br>(5.66) | |||
| 8/7<br>(225/224) | |||
| [[Schismerc]] / [[cartography]] | |||
|} | |} | ||
Revision as of 10:27, 25 July 2021
The 212 equal divisions of the octave (212edo), or the 212(-tone) equal temperament (212tet, 212et) when viewed from a regular temperament perspective, 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 |
Note: temperaments supported by 53et are not included.
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 15\212 | 84.91 | 21/20 | Amicable |
| 1 | 31\212 | 175.47 | 448/405 | Sesquiquartififths |
| 1 | 41\212 | 232.08 | 8/7 | Quadrawell |
| 1 | 67\212 | 379.25 | 56/45 | Marthirds |
| 2 | 11\212 | 62.26 | 28/27 | Eagle |
| 2 | 31\212 | 175.47 | 448/405 | Bisesqui |
| 2 | 97\212 (9\212) |
549.06 (50.94) |
11/8 (36/35) |
Kleischismic |
| 4 | 56\212 (3\212) |
316.98 (16.98) |
6/5 (126/125) |
Quadritikleismic |
| 4 | 88\212 (18\212) |
498.11 (101.89) |
4/3 (35/33) |
Quadrant |
| 53 | 41\212 (1\198) |
232.08 (5.66) |
8/7 (225/224) |
Schismerc / cartography |