# 189edo

 ← 188edo 189edo 190edo →
Prime factorization 33 × 7
Step size 6.34921¢
Fifth 111\189 (704.762¢) (→37\63)
Semitones (A1:m2) 21:12 (133.3¢ : 76.19¢)
Dual sharp fifth 111\189 (704.762¢) (→37\63)
Dual flat fifth 110\189 (698.413¢)
Dual major 2nd 32\189 (203.175¢)
Consistency limit 7
Distinct consistency limit 7

189 equal divisions of the octave (abbreviated 189edo or 189ed2), also called 189-tone equal temperament (189tet) or 189 equal temperament (189et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 189 equal parts of about 6.35 ¢ each. Each step represents a frequency ratio of 21/189, or the 189th root of 2.

189edo is consistent to the 7-odd-limit, but harmonics 3 and 7 are about halfway between its steps. It has good approximations to 5, 9, 11, 19, and 21, making it suitable for a 2.9.5.21.11.19 subgroup interpretation.

Using the full 13-limit patent val nonetheless, the equal temperament tempers out 15625/15552 (kleisma) and [53 -29 -3 in the 5-limit; 4000/3969, 6144/6125, and 537824/531441 in the 7-limit, supporting the hemikleismic temperament. It tempers out 896/891, 1331/1323, 1375/1372, and 16896/16807 in the 11-limit; 169/168, 352/351, 364/363, and 1001/1000 in the 13-limit.

### Odd harmonics

Approximation of odd harmonics in 189edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +2.81 +0.99 +2.60 -0.74 +1.06 -2.43 -2.55 +2.98 +0.90 -0.94 +0.30
Relative (%) +44.2 +15.6 +41.0 -11.6 +16.7 -38.3 -40.2 +47.0 +14.2 -14.8 +4.7
Steps
(reduced)
300
(111)
439
(61)
531
(153)
599
(32)
654
(87)
699
(132)
738
(171)
773
(17)
803
(47)
830
(74)
855
(99)

### Subsets and supersets

Since 189 factors into 33 × 7, 189edo contains 3, 7, 9, 21, 27, and 63 as its subsets. 378edo, which doubles it, provides a good correction for the approximation of 3 and 7.