1889edo
| ← 1888edo | 1889edo | 1890edo → |
1889 equal divisions of the octave (abbreviated 1889edo or 1889ed2), also called 1889-tone equal temperament (1889tet) or 1889 equal temperament (1889et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1889 equal parts of about 0.635 ¢ each. Each step represents a frequency ratio of 21/1889, or the 1889th root of 2.
Theory
1889edo is strong in the 23-limit, though 1578, which among other things has a lower 23-limit relative error, rather puts it in the shade. It is distinctly consistent through the 27-odd-limit, but not, unlike 1578, to the 29-odd-limit. Even so, it should be noted that it supplies the optimal patent val for the 7-limit monzismic temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | +0.004 | -0.078 | -0.059 | +0.085 | -0.083 | -0.138 | -0.213 | -0.005 | +0.174 | -0.303 |
| relative (%) | +0 | +1 | -12 | -9 | +13 | -13 | -22 | -34 | -1 | +27 | -48 | |
| Steps (reduced) |
1889 (0) |
2994 (1105) |
4386 (608) |
5303 (1525) |
6535 (868) |
6990 (1323) |
7721 (165) |
8024 (468) |
8545 (989) |
9177 (1621) |
9358 (1802) | |
Subsets and supersets
1889edo is the 290th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2994 -1889⟩ | [⟨1889 2994]] | -0.0012 | 0.0012 | 0.19 |
| 2.3.5 | [54 -37 2⟩, [-66 -36 53⟩ | [⟨1889 2994 4386]] | +0.0104 | 0.0163 | 2.57 |
| 2.3.5.7 | 4375/4374, 3955078125/3954653486, [-57 16 10 3⟩ | [⟨1889 2994 4386 5303]] | +0.0131 | 0.0149 | 2.35 |
| 2.3.5.7.11 | 4375/4374, 151263/151250, 820125/819896, 1879453125/1879048192 | [⟨1889 2994 4386 5303 6535]] | +0.0055 | 0.0201 | 3.16 |
| 2.3.5.7.11.13 | 4375/4374, 4225/4224, 6656/6655, 4100625/4100096, 151263/151250 | [⟨1889 2994 4386 5303 6535 6990]] | +0.0084 | 0.0194 | 3.05 |
| 2.3.5.7.11.13.17 | 4375/4374, 12376/12375, 4225/4224, 14400/14399, 14875/14872, 194481/194480 | [⟨1889 2994 4386 5303 6535 6990 7721]] | +0.0120 | 0.0200 | 3.15 |
| 2.3.5.7.11.13.17.19 | 4375/4374, 12376/12375, 5985/5984, 4225/4224, 14400/14399, 6175/6174, 61965/61952 | [⟨1889 2994 4386 5303 6535 6990 7721 8024]] | +0.0167 | 0.0226 | 3.56 |
Rank-2 temperaments
| Periods per 8ve |
Generator (reduced)* |
Cents (reduced)* |
Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 392\1889 | 249.021 | [-26 18 1⟩ | Monzismic |
| 1 | 707\1889 | 449.127 | 35/27 | Semidimi |