# 660edo

 ← 659edo 660edo 661edo →
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 660ed2), also called 660-tone equal temperament (660tet) or 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 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 (%) -7.5 -47.3 +14.6 -15.1 -22.5 -29.0 +45.2 +27.5 +36.8 +7.1 +44.9
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.