220edo

← 219edo

220edo

221edo →

Prime factorization	22 × 5 × 11
Step size	5.45455¢
Fifth	129\220 (703.636¢)
Semitones (A1:m2)	23:15 (125.5¢ : 81.82¢)
Consistency limit	7
Distinct consistency limit	7

220 equal divisions of the octave (abbreviated 220edo or 220ed2), also called 220-tone equal temperament (220tet) or 220 equal temperament (220et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 220 equal parts of about 5.45 ¢ each. Each step represents a frequency ratio of 2^1/220, or the 220th root of 2.

Theory

Using the patent val, it tempers out [20 -17 3⟩ (rodan comma) and [23 6 -14⟩ (vishnuzma) in the 5-limit; 6144/6125, 10976/10935, and 390625/388962 in the 7-limit; 1331/1323, 1375/1372, 2200/2187, and 16384/16335 in the 11-limit; 325/324, 352/351, 1001/1000, 1573/1568, and 2704/2695 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 220edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+1.68	+0.96	+2.08	-2.09	-0.41	-0.53	+2.64	-1.32	+2.49	-1.69	-1.00
	Relative (%)	+30.8	+17.6	+38.2	-38.4	-7.5	-9.7	+48.4	-24.2	+45.6	-31.0	-18.4
Steps (reduced)		349 (129)	511 (71)	618 (178)	697 (37)	761 (101)	814 (154)	860 (200)	899 (19)	935 (55)	966 (86)	995 (115)

Subsets and supersets

Since 220 factors into 2² × 5 × 11, 220edo has subset edos 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[349 -220⟩	[⟨220 349]]	−0.5304	0.5302	9.72
2.3.5	[20 -17 3⟩, [23 6 -14⟩	[⟨220 349 511]]	−0.4912	0.4364	8.00
2.3.5.7	6144/6125, 10976/10935, 390625/388962	[⟨220 349 511 618]]	−0.5538	0.3932	7.21

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	43\220	234.55	729/640	Rodan (5-limit)
1	83\220	452.73	125/81	Maja (5-limit)
2	13\220	70.91	25/24	Vishnu (5-limit)
11	91\220 (9\220)	496.36 (49.09)	4/3 (36/35)	Hendecatonic
20	91\220 (3\220)	496.36 (16.36)	4/3 (126/125)	Degrees

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct