# 212edo

 ← 211edo 212edo 213edo →
Prime factorization 22 × 53
Step size 5.66038¢
Fifth 124\212 (701.887¢) (→31\53)
Semitones (A1:m2) 20:16 (113.2¢ : 90.57¢)
Consistency limit 15
Distinct consistency limit 15

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

The equal temperament 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

Approximation of prime harmonics in 212edo
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.

Table of rank-2 temperaments by generator
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)
4 88\212
(18\212)
498.11
(101.89)
4/3
(35/33)