660edo

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← 659edo660edo661edo →
Prime factorization 22 × 3 × 5 × 11
Step size 1.81818¢
Fifth 386\660 (701.818¢) (→193\330)
Semitones (A1:m2) 62:50 (112.7¢ : 90.91¢)
Consistency limit 5
Distinct consistency limit 5

660 equal divisions of the octave (abbreviated 660edo), or 660-tone equal temperament (660tet), 660 equal temperament (660et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 660 equal parts of about 1.82 ¢ each. Each step of 660edo represents a frequency ratio of 21/660, or the 660th root of 2.

660edo is enfactored in the 5-limit, with the same tuning as 330edo, tempering out the schisma and the 22nd-octave major arcana comma, [-193 154 -22.

However, it is far better viewed as a no-5 system. It does tune well the 2.3.7.11.13 subgroup, with errors less than 50% on all pairs of intervals in that subgroup that also belong to the 15-odd-limit. In addition, in the no-5s 17-odd-limit, it only misses the pair {17/13, 26/17}.

Nonetheless, patent val does have some use. It tunes the undecimal dimcomp temperament and also provides the optimal patent val for the quadrant temperament in the 11-limit as well as the 13-limit. Furthermore, in 2.3.7 it is a septiruthenian system, and the patent val mapping for 5 allows the tuning of ruthenium temperament.

Other mappings can be considered. Taking a different mapping for 5, the 660c val tunes qintosec and atomic. In the 7-limit, it tunes decoid. In addition, one can combine the mappings for five to produce a 2.3.25 subgroup interpretation. There, 660edo has less error than on either vals and it tempers out the kwazy comma, as well as the landscape comma in the 2.3.25.7 subgroup.

Odd harmonics

Approximation of odd harmonics in 660edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) -0.137 -0.859 +0.265 -0.274 -0.409 -0.528 +0.822 +0.499 +0.669 +0.128 +0.817
relative (%) -8 -47 +15 -15 -22 -29 +45 +27 +37 +7 +45
Steps
(reduced)
1046
(386)
1532
(212)
1853
(533)
2092
(112)
2283
(303)
2442
(462)
2579
(599)
2698
(58)
2804
(164)
2899
(259)
2986
(346)

Subsets and supersets

Since 660 factors as 22 × 3 × 5 × 11, it has subset edos 1, 2, 3, 4, 5, 6, 10, 11, 12, 15, 20, 22, 30, 33, 44, 55, 60, 66, 110, 132, 165, 220, 330.