589edo
| ← 588edo | 589edo | 590edo → |
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
| 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 |