# 589edo

 ← 588edo 589edo 590edo →
Prime factorization 19 × 31
Step size 2.03735¢
Fifth 345\589 (702.886¢)
Semitones (A1:m2) 59:42 (120.2¢ : 85.57¢)
Dual sharp fifth 345\589 (702.886¢)
Dual flat fifth 344\589 (700.849¢)
Dual major 2nd 100\589 (203.735¢)
Consistency limit 7
Distinct consistency limit 7

589 equal divisions of the octave (abbreviated 589edo or 589ed2), also called 589-tone equal temperament (589tet) or 589 equal temperament (589et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 589 equal parts of about 2.04 ¢ each. Each step represents a frequency ratio of 21/589, or the 589th root of 2.

## Theory

589edo is consistent to the 7-odd-limit, but harmonic 3 is about halfway between its steps. As every other step of 1178edo, the approximations to lower harmonics are not impressive, making it only suitable for a 2.9.15.21.19 subgroup interpretation, in which case it is identical to 1178edo. The full 17-limit patent val, however, is plausible since all the harmonics from 3 to 17 are tuned sharp. Using the patent val, the equal temperament tempers out 420175/419904 in the 7-limit; 3025/3024, 117649/117612, 422576/421875, 456533/455625, 644204/643125, 766656/765625, 1953125/1948617, 3294225/3294172, 4302592/4296875, 55296000/55240493, 85937500/85766121, 107495424/107421875 and 781258401/781250000 in the 11-limit.

### Odd harmonics

Approximation of odd harmonics in 589edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.931 +0.783 +0.953 -0.175 +0.804 +0.898 -0.323 +0.987 -0.060 -0.153 -0.770
Relative (%) +45.7 +38.4 +46.8 -8.6 +39.5 +44.1 -15.9 +48.4 -2.9 -7.5 -37.8
Steps
(reduced)
934
(345)
1368
(190)
1654
(476)
1867
(100)
2038
(271)
2180
(413)
2301
(534)
2408
(52)
2502
(146)
2587
(231)
2664
(308)

### Subsets and supersets

Since 589 factors into 19 × 31, 589edo contains 19edo and 31edo as subsets. 1178edo, which doubles it, gives good corrections to harmonics 3, 5, 7, 11, 13, and 17.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [-1867 589 [589 1867]] +0.0276 0.0276 1.35
2.9.5 [-37 19 -10, [72 0 -31 [589 1867 1368]] -0.0940 0.1734 8.51