221edo

← 220edo

221edo

222edo →

Prime factorization	13 × 17
Step size	5.42986 ¢
Fifth	129\221 (700.452 ¢)
Semitones (A1:m2)	19:18 (103.2 ¢ : 97.74 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

221et tempers out 2109375/2097152 (semicomma) and 2541865828329/2500000000000 in the 5-limit; 1029/1024, 19683/19600, and 235298/234375 in the 7-limit, so that it provides the optimal patent val for the 7-limit hemiseven temperament.

Using the patent val, it tempers out 540/539, 2835/2816, 4375/4356, and 33614/33275 in the 11-limit; 364/363, 625/624, 1701/1690, and 2200/2197 in the 13-limit.

Using the 221ef val, it tempers out 385/384, 441/440, 24057/24010, and 43923/43750 in the 11-limit; 351/350, 676/675, 1287/1280, 1573/1568, and 14641/14625 in the 13-limit; 273/272, 561/560, 715/714, 833/832, 2187/2176, and 10648/10625 in the 17-limit, supporting the 17-limit hemiseven and the 11-limit triwell.

Odd harmonics

Approximation of odd harmonics in 221edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-1.50	-0.79	-2.31	+2.42	+2.53	+1.10	-2.30	-1.79	+1.13	+1.62	+1.59
	Relative (%)	-27.7	-14.6	-42.5	+44.7	+46.6	+20.3	-42.3	-32.9	+20.8	+29.8	+29.3
Steps (reduced)		350 (129)	513 (71)	620 (178)	701 (38)	765 (102)	818 (155)	863 (200)	903 (19)	939 (55)	971 (87)	1000 (116)

Subsets and supersets

221 factors into 13 × 17, with its subset edos 13edo and 17edo.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[-350 221⟩	⟨221 350]	0.4740	0.4742	8.73
2.3.5	[-21 3 7⟩, [-11 26 -13⟩	⟨221 350 513]	0.4299	0.3921	7.22
2.3.5.7	1029/1024, 19683/19600, 235298/234375	⟨221 350 513 620]	0.5282	0.3799	7.00

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	50\221	271.49	75/64	Orson
1	84\221	456.11	125/96	Qak
1	89\221	483.26	320/243	Hemiseven
1	93\221	504.98	104976/78125	Countermeantone
1	103\221	559.28	864/625	Tritriple