390edo

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← 389edo390edo391edo →
Prime factorization 2 × 3 × 5 × 13
Step size 3.07692¢
Fifth 228\390 (701.538¢) (→38\65)
Semitones (A1:m2) 36:30 (110.8¢ : 92.31¢)
Consistency limit 3
Distinct consistency limit 3

390 equal divisions of the octave (abbreviated 390edo), or 390-tone equal temperament (390tet), 390 equal temperament (390et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 390 equal parts of about 3.08 ¢ each. Each step of 390edo represents a frequency ratio of 21/390, or the 390th root of 2.

Theory

390et is enfactored in the 5-limit, with the same tuning as 65edo. But its approximation to higher harmonics are improved, so that it is suitable for use in the 2.3.7.11.13.17.23.31.41 subgroup.

Using the patent val nonetheless, it tempers out 2401/2400 and 3136/3125 in the 7-limit, supporting hemiwürschmidt.

Prime harmonics

Approximation of prime harmonics in 390edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.00 -0.42 +1.38 +0.40 -0.55 -0.53 -0.34 +0.95 -0.58 +1.19 -0.42
relative (%) +0 -14 +45 +13 -18 -17 -11 +31 -19 +39 -14
Steps
(reduced)
390
(0)
618
(228)
906
(126)
1095
(315)
1349
(179)
1443
(273)
1594
(34)
1657
(97)
1764
(204)
1895
(335)
1932
(372)

Subsets and supersets

Since 390 factors into 2 × 3 × 5 × 13, 390edo has subset edos 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, and 195. 780edo, which doubles it, gives a good correction to the harmonic 5.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.7 118098/117649, 34451725707/34359738368 [390 618 1095]] 0.0395 0.1685 5.48
2.3.7.11 118098/117649, 1362944/1361367, 235782657/234881024 [390 618 1095 1349]] 0.0693 0.1548 5.03
2.3.7.11.13 729/728, 10648/10647, 16848/16807, 1574573/1572864 [390 618 1095 1349 1443]] 0.0839 0.1415 4.60
2.3.7.11.13.17 729/728, 1089/1088, 16848/16807, 65637/65536, 95823/95744 [390 618 1095 1349 1443 1594]] 0.0838 0.1292 4.20