212edo
| ← 211edo | 212edo | 213edo → |
212 equal divisions of the octave (abbreviated 212edo or 212ed2), also called 212-tone equal temperament (212tet) or 212 equal temperament (212et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 212 equal parts of about 5.66 ¢ each. Each step represents a frequency ratio of 21/212, or the 212th root of 2.
Theory
212edo is distinctly consistent in the 15-odd-limit with harmonics of 3 through 13 all tuned flat. 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, 2080/2079, and 10648/10647.
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. It enables marveltwin chords, keenanismic chords, sinbadmic chords, and lambeth chords in the 13-odd-limit in addition to island chords in the 15-odd-limit.
The 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. This is related to the fact that 212edo splits steps of 53edo, which are mapped to a syntonic comma, in four.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.07 | -1.41 | -0.90 | -2.26 | -2.79 | +2.59 | +2.49 | +0.03 | +0.61 | -1.64 |
| Relative (%) | +0.0 | -1.2 | -24.9 | -15.9 | -40.0 | -49.3 | +45.8 | +43.9 | +0.5 | +10.8 | -29.0 | |
| Steps (reduced) |
212 (0) |
336 (124) |
492 (68) |
595 (171) |
733 (97) |
784 (148) |
867 (19) |
901 (53) |
959 (111) |
1030 (182) |
1050 (202) | |
Subsets and supersets
Since 212 factors into 22 × 53, 212edo has subset edos 2, 4, 53, and 106. As such, it can be used to tune the period-53 cartography temperament and the period-106 boiler temperment.
A step of 212edo is exactly 50 türk sents.
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 |
- 212et (212gh val) has a lower absolute error in the 19-limit than any previous equal temperaments, past 193 and followed by 217.
Rank-2 temperaments
Note: temperaments supported by 53et are not included.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 15\212 | 84.91 | 21/20 | Amicable / amorous / pseudoamical |
| 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 | 15\212 | 84.91 | 21/20 | Floral |
| 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\212) |
232.08 (5.66) |
8/7 (225/224) |
Schismerc / cartography |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct