135edo

The 135 equal divisions of the octave (135edo), or the 135(-tone) equal temperament (135tet, 135et) when viewed from a regular temperament perspective, is the equal division of the octave into 135 parts of about 8.89 cents each.

Theory

135edo is consistent to the 7-odd-limit, but there is a large relative delta for the 5th and the 13th harmonics.

Using the 135f val ⟨135 214 313 379 467 499], which tends flat, 135et tempers out 32805/32768 (schisma) and 30517578125/29386561536 (quintriyo comma) in the 5-limit; 225/224, 3125/3087, and 28824005/28697814 in the 7-limit, 385/384, 540/539, 2200/2187, 12005/11979 and the quartisma in the 11-limit; 169/168 and 364/363 in the 13-limit.

Using the 135c val ⟨135 214 314 379 467 500], which tends sharp, it tempers out 1594323/1562500 and 50331648/48828125 in the 5-limit; 126/125, 10976/10935, and 589824/588245 in the 7-limit; 176/175, 441/440, 14641/14580 and 16384/16335 in the 11-limit; 196/195, 351/350, 352/351, 676/675, and 6656/6655 in the 13-limit.

As every other step of the full 13-limit monster – 270et, 135et probably makes more sense as a 2.3.7.11 subgroup temperament, where it tempers out the garischisma and the symbiotic comma.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[214 -135⟩	[⟨135 214]]	-0.0843	0.0843	0.95
2.3.7	33554432/33480783, 40353607/40310784	[⟨135 214 379]]	-0.0637	0.0747	0.84
2.3.7.11	19712/19683, 41503/41472, 43923/43904	[⟨135 214 379 467]]	-0.0328	0.0840	0.94