331edo

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← 330edo331edo332edo →
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 331-tone equal temperament (331tet), 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 of 331edo 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 (%) +38 +44 -23 -25 -7 +15 -18 +5 -6 +14 -30
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

Scales

Music

Francium