212edo

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.

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