331edo

← 330edo 331edo 332edo →
Prime factorization	331 (prime)
Step size	3.62538 ¢
Fifth	194\331 (703.323 ¢)
Semitones (A1:m2)	34:23 (123.3 ¢ : 83.38 ¢)
Dual sharp fifth	194\331 (703.323 ¢)
Dual flat fifth	193\331 (699.698 ¢)
Dual major 2nd	56\331 (203.021 ¢)
Consistency limit	5
Distinct consistency limit	5

Template:EDO intro

Theory

331et tempers out 78125000/78121827, 5120/5103 and 1959552/1953125 in the 7-limit; 806736/805255, 1835008/1830125, 1019215872/1019046875, 12005/11979, 16384/16335, 2359296/2358125, 42875/42768, 180224/180075, 1684375/1679616, 968000/964467, 3025/3024, 78408/78125, 1362944/1361367, 4108797/4096000 and 43923/43750 in the 11-limit.

Odd harmonics

Approximation of odd harmonics in 331edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+1.37	+1.60	-0.85	-0.89	-0.26	+0.56	-0.66	+0.18	-0.23	+0.52	-1.08
Error	Relative (%)	+37.7	+44.2	-23.4	-24.5	-7.2	+15.4	-18.1	+5.0	-6.4	+14.3	-29.9
Steps (reduced)		525 (194)	769 (107)	929 (267)	1049 (56)	1145 (152)	1225 (232)	1293 (300)	1353 (29)	1406 (82)	1454 (130)	1497 (173)

Subsets and supersets

331edo is the 67th prime edo. 662edo, which doubles it, gives a good correction to the harmonics 3 and 5.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.9	[-1049 331⟩	⟨331 1049]	0.1402	0.1402	3.87
2.9.15	[-7 17 -12⟩, [-74 -5 23⟩	⟨331 1049 1293]	0.1494	0.1152	3.18
2.9.15.7	65625/65536, 420175/419904, 80387359983/80000000000	⟨331 1049 1293 929]	0.1878	0.1199	3.31
2.9.15.7.11	9801/9800, 41503/41472, 137781/137500, 759375/758912	⟨331 1049 1293 929 1145]	0.1653	0.1163	3.21
2.9.15.7.11.13	729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875, 53361/53248, 20336647/2028000	⟨331 1049 1293 929 1145 1225]	0.1125	0.1587	4.38
2.9.15.7.11.13.17	729/728, 833/832, 1089/1088, 2025/2023, 10648/10647, 14161/14157, 14175/14144, 43904/43875, 18816/18785, 92823/95744	⟨331 1049 1293 929 1145 1225 1353]	0.0901	0.1568	4.33

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	89\331	322.66	6/5	Magicaltet
1	137\331	496.68	5457/4096	Edson

Scales

Music

Silent Silence by Francium