# 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

331 equal divisions of the octave (abbreviated 331edo or 331ed2), also called 331-tone equal temperament (331tet) or 331 equal temperament (331et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 331 equal parts of about 3.63 ¢ each. Each step represents a frequency ratio of 21/331, or the 331st root of 2.

## Theory

331edo is only consistent to the 5-odd-limit and the errors of both harmonics 3 and 5 are quite large, commending itself as a temperament of the 2.9.15.7.11.13.17.19 subgroup.

Using the patent val nonetheless, the equal temperament tempers out 5120/5103, 1959552/1953125 and 78125000/78121827 in the 7-limit; 3025/3024, 12005/11979, 16384/16335, 42875/42768, 43923/43750, 78408/78125, and 180224/180075 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
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
Absolute (¢) Relative (%)
2.9 [-1049 331 [331 1049]] 0.1402 0.1402 3.87
2.9.15 [-7 17 -12, [-81 12 11 [331 1049 1293]] +0.1238 0.1168 3.22
2.9.15.7 65625/65536, 420175/419904, 80387359983/80000000000 [331 1049 1293 929]] +0.1685 0.1275 3.52
2.9.15.7.11 9801/9800, 41503/41472, 137781/137500, 759375/758912 [331 1049 1293 929 1145]] +0.1499 0.1200 3.31
2.9.15.7.11.13 729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875 [331 1049 1293 929 1145 1225]] +0.0997 0.1568 4.33
2.9.15.7.11.13.17 729/728, 833/832, 1089/1088, 2025/2023, 10648/10647, 18816/18785 [331 1049 1293 929 1145 1225 1353]] +0.0791 0.1537 4.24

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 107\331 387.92 5/4 Würschmidt (331, 5-limit)

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

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