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

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.

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.

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

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 (Reduced)	Cents (Reduced)	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

Music

Eliora

• Etude in Amicable, Bisesqui, and 53edo