2190edo
| ← 2189edo | 2190edo | 2191edo → |
2190 equal divisions of the octave (abbreviated 2190edo or 2190ed2), also called 2190-tone equal temperament (2190tet) or 2190 equal temperament (2190et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2190 equal parts of about 0.548 ¢ each. Each step represents a frequency ratio of 21/2190, or the 2190th root of 2.
2190edo is a very strong 13-limit system; no smaller division has a smaller 13-limit relative error, and nothing beats it until 2684. A basis for the 13-limit commas is {9801/9800, 10648/10647, 105644/105625, 140625/140608, 196625/196608}; also tempered out are 123201/123200 and 151263/151250. It is not as impressive beyond the 13-limit, though it does well in the 2.3.5.7.11.13.19.29 subgroup, holding the record of relative error until 14618.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.037 | -0.012 | -0.059 | -0.085 | +0.020 | +0.250 | +0.021 | +0.219 | +0.012 | +0.170 |
| Relative (%) | +0.0 | -6.8 | -2.3 | -10.7 | -15.5 | +3.7 | +45.6 | +3.9 | +39.9 | +2.2 | +31.0 | |
| Steps (reduced) |
2190 (0) |
3471 (1281) |
5085 (705) |
6148 (1768) |
7576 (1006) |
8104 (1534) |
8952 (192) |
9303 (543) |
9907 (1147) |
10639 (1879) |
10850 (2090) | |
Subsets and supersets
2190 factors into 2 × 3 × 5 × 73; among its divisors is the Woolhouse unit system, 730.
4380edo, which doubles 2190edo, provides a good correction to the harmonics 17 and 23.