589edo

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← 588edo589edo590edo →
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

Scales