166edo

The 166 equal divisions of the octave (166edo), or the 166(-tone) equal temperament (166tet, 166et) when viewed from a regular temperament perspective, divides the octave into 166 equal steps of size 7.229 cents each.

Theory

The principle interest of 166edo lies in the usefulness of its approximations; it tempers out 1600000/1594323, 225/224, 385/384, 540/539, 4000/3993, 325/324 and 729/728. It is an excellent tuning for the rank-3 temperament marvel, in both the 11-limit and in the 13-limit extension hecate, and the rank-2 temperament wizard, which also tempers out 4000/3993, giving the optimal patent val for all of these. In the 13-limit it tempers out 325/324, leading to hecate, and 1573/1568, leading to marvell, and tempering out both gives gizzard, the 72&94 temperament, for which 166 is an excellent tuning through the 19-limit.

Its prime factorization is 166 = 2 × 83.

166edo (as 83edo) contains a very good approximation of the harmonic 7th. It is 0.15121 cent flat of the just interval 7:4.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-263 166⟩	[⟨166 263]]	+0.237	0.237	3.27
2.3.5	1600000/1594323, [-31 2 12⟩	[⟨166 263 385]]	+0.615	0.568	7.86
2.3.5.7	225/224, 118098/117649, 1250000/1240029	[⟨166 263 385 466]]	+0.474	0.549	7.59
2.3.5.7.11	225/224, 385/384, 4000/3993, 322102/321489	[⟨166 263 385 466 574]]	+0.490	0.492	6.80
2.3.5.7.11.13	225/224, 325/324, 385/384, 1575/1573, 2200/2197	[⟨166 263 385 466 574 614]]	+0.498	0.449	6.21

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	33\166	238.55	147/128	Tokko
1	47\166	339.76	243/200	Amity / houborizic
1	81\166	585.54	7/5	Merman
2	30\166	216.87	17/15	Wizard / gizzard

Scales

• prisun