659edo
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Prime factorization
659 (prime)
Step size
1.82094¢
Fifth
385\659 (701.062¢)
Semitones (A1:m2)
59:52 (107.4¢ : 94.69¢)
Dual sharp fifth
386\659 (702.883¢)
Dual flat fifth
385\659 (701.062¢)
Dual major 2nd
112\659 (203.945¢)
Consistency limit
7
Distinct consistency limit
7
| ← 658edo | 659edo | 660edo → |
659 equal divisions of the octave (abbreviated 659edo or 659ed2), also called 659-tone equal temperament (659tet) or 659 equal temperament (659et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 659 equal parts of about 1.82 ¢ each. Each step represents a frequency ratio of 21/659, or the 659th root of 2.
Theory
659edo is consistent to the 7-odd-limit and its harmonic 3 is about halfway its steps. Using the 2.9.5.7.11.17.23.31 subgroup, it tempers out 1225/1224, 2025/2024, 5832/5831, 3520/3519, 3969/3968, 790625/790272 and 1740800/1740123. It supports counterultrakleismic.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.893 | -0.274 | -0.085 | +0.035 | +0.427 | +0.747 | +0.654 | +0.659 | -0.700 | +0.843 | -0.050 |
| Relative (%) | -49.0 | -15.1 | -4.7 | +1.9 | +23.5 | +41.0 | +35.9 | +36.2 | -38.4 | +46.3 | -2.7 | |
| Steps (reduced) |
1044 (385) |
1530 (212) |
1850 (532) |
2089 (112) |
2280 (303) |
2439 (462) |
2575 (598) |
2694 (58) |
2799 (163) |
2895 (259) |
2981 (345) | |
Subsets and supersets
659edo is the 120th prime EDO. 1318edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [2089 -659⟩ | [⟨659 2089]] | -0.0056 | 0.0056 | 1.02 |
| 2.9.5 | [10 -20 23⟩, [-83 13 18⟩ | [⟨659 2089 1530]] | +0.0357 | 0.0585 | 10.65 |
| 2.9.5.7 | 420175/419904, 703125/702464, [44 -14 5 -4⟩ | [⟨659 2089 1530 1850]] | +0.0343 | 0.0507 | 9.23 |
| 2.9.5.7.11 | 496125/495616, 420175/419904, 151263/151250, 2097152/2096325 | [⟨659 2089 1530 1850 2280]] | +0.0028 | 0.0777 | 14.15 |
| 2.9.5.7.11.13 | 1575/1573, 4096/4095, 86625/86528, 31250/31213, 650000/649539 | [⟨659 2089 1530 1850 2280 2439]] | -0.0313 | 0.1042 | 18.97 |