2053edo
| ← 2052edo | 2053edo | 2054edo → |
2053 equal divisions of the octave (abbreviated 2053edo or 2053ed2), also called 2053-tone equal temperament (2053tet) or 2053 equal temperament (2053et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2053 equal parts of about 0.585 ¢ each. Each step represents a frequency ratio of 21/2053, or the 2053rd root of 2.
Theory
2053edo is consistent to the 5-odd-limit, its harmonic 7 is exactly between its steps. It is strong in the 2.3.5.13.19.23.31 subgroup, tempering out 359424/359375, 22816/22815, 96876/96875, 497705/497664, 4784000/4782969, and 369664/369603. Using the 2.3.5.13.19.37 subgroup, it tempers out 7696/7695.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.042 | +0.048 | -0.292 | -0.125 | -0.002 | +0.256 | +0.003 | +0.074 | -0.254 | +0.020 |
| Relative (%) | +0.0 | +7.2 | +8.2 | -50.0 | -21.3 | -0.3 | +43.9 | +0.5 | +12.7 | -43.5 | +3.5 | |
| Steps (reduced) |
2053 (0) |
3254 (1201) |
4767 (661) |
5763 (1657) |
7102 (943) |
7597 (1438) |
8392 (180) |
8721 (509) |
9287 (1075) |
9973 (1761) |
10171 (1959) | |
Subsets and supersets
2053edo is the 310th prime edo. 4106edo, which doubles it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [3254 -2053⟩ | [⟨2053 3254]] | −0.0133 | 0.0133 | 2.28 |
| 2.3.5 | [37 25 -33⟩, [93 -66 5⟩ | [⟨2053 3254 4767]] | −0.0157 | 0.0114 | 1.95 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 959\2053 | 560.546 | 864/625 | Whoosh |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct