870edo
| ← 869edo | 870edo | 871edo → |
870 equal divisions of the octave (abbreviated 870edo or 870ed2), also called 870-tone equal temperament (870tet) or 870 equal temperament (870et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 870 equal parts of about 1.38 ¢ each. Each step represents a frequency ratio of 21/870, or the 870th root of 2.
870edo is notably strong in the subgroup of Fermat primes, 2.3.5.17.
As an equal temperament, 870et tempers out [-53 10 16⟩ (kwazy comma) in the 5-limit; 250047/250000 and 2100875/2097152 in the 7-limit, supporting pnict. In the 11-limit it tempers out 12005/11979 and provides the optimal patent val for the corresponding rank-4 temperament.
Odd harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.114 | -0.107 | -0.550 | +0.406 | -0.528 | -0.128 | +0.418 | -0.688 | -0.612 | -0.208 |
| Relative (%) | +0.0 | +8.3 | -7.7 | -39.9 | +29.4 | -38.3 | -9.3 | +30.3 | -49.9 | -44.3 | -15.1 | |
| Steps (reduced) |
870 (0) |
1379 (509) |
2020 (280) |
2442 (702) |
3010 (400) |
3219 (609) |
3556 (76) |
3696 (216) |
3935 (455) |
4226 (746) |
4310 (830) | |
Subsets and supersets
Since 870 factors into 2 × 3 × 5 × 29, 870edo has subset edos 2, 3, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, and 435.
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