3889edo
| ← 3888edo | 3889edo | 3890edo → |
3889 equal divisions of the octave (abbreviated 3889edo or 3889ed2), also called 3889-tone equal temperament (3889tet) or 3889 equal temperament (3889et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3889 equal parts of about 0.309 ¢ each. Each step represents a frequency ratio of 21/3889, or the 3889th root of 2.
Theory
3889edo is consistent to the 27-odd-limit, and except for 29/19 and its octave complement, it is consistent to the 31-odd-limit.
As an equal temperament, it tempers out the pirate comma, the starscape comma, and the quartisma. Some of the simpler commas it tempers out in the higher limits include 6656/6655, 123201/123200 and 1990656/1990625 in the 13-limit; 12376/12375, 14400/14399 in the 17-limit; 5776/5775, 6175/6174, 14365/14364, 23409/23408, 28900/28899, 43681/43680 in the 19-limit; 5083/5082, 7866/7865, and 8625/8624 in the 23-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.025 | +0.007 | +0.061 | +0.083 | -0.003 | -0.044 | -0.059 | -0.041 | +0.096 | +0.040 |
| Relative (%) | +0.0 | +8.1 | +2.2 | +19.7 | +27.0 | -1.0 | -14.3 | -19.0 | -13.2 | +31.2 | +13.1 | |
| Steps (reduced) |
3889 (0) |
6164 (2275) |
9030 (1252) |
10918 (3140) |
13454 (1787) |
14391 (2724) |
15896 (340) |
16520 (964) |
17592 (2036) |
18893 (3337) |
19267 (3711) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.134 | +0.148 | +0.079 | +0.063 | +0.036 | +0.124 | +0.112 | -0.006 | -0.113 | -0.070 | -0.124 |
| Relative (%) | +43.6 | +48.0 | +25.6 | +20.4 | +11.7 | +40.1 | +36.2 | -2.1 | -36.7 | -22.8 | -40.3 | |
| Steps (reduced) |
20260 (815) |
20836 (1391) |
21103 (1658) |
21602 (2157) |
22276 (2831) |
22878 (3433) |
23065 (3620) |
23591 (257) |
23916 (582) |
24072 (738) |
24515 (1181) | |
Subsets and supersets
3889edo is the 539th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [6164 -3889⟩ | [⟨3889 6164]] | −0.0079 | 0.0079 | 2.56 |
| 2.3.5 | [-90 -15 49⟩, [56 -91 38⟩ | [⟨3889 6164 9030]] | −0.0062 | 0.0068 | 2.20 |
| 2.3.5.7 | 645700815/645657712, [2 -20 14 -1⟩, [46 -14 -3 -6⟩ | [⟨3889 6164 9030 10918]] | −0.0101 | 0.0089 | 2.88 |
| 2.3.5.7.11 | 1771875/1771561, 3294225/3294172, 14348907/14348180, 104857600/104825259 | [⟨3889 6164 9030 10918 13454]] | −0.0129 | 0.0098 | 3.18 |
| 2.3.5.7.11.13 | 6656/6655, 123201/123200, 492128/492075, 823680/823543, 1990656/1990625 | [⟨3889 6164 9030 10918 13454 14391]] | −0.0106 | 0.0103 | 3.34 |
| 2.3.5.7.11.13.17 | 6656/6655, 12376/12375, 28561/28560, 37180/37179, 361250/361179, 937125/937024 | [⟨3889 6164 9030 10918 13454 14391 15896]] | −0.0075 | 0.0121 | 3.92 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 602\3889 | 185.755 | [24 4 -13⟩ | Pirate |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct