581edo
| ← 580edo | 581edo | 582edo → |
581 equal divisions of the octave (abbreviated 581edo or 581ed2), also called 581-tone equal temperament (581tet) or 581 equal temperament (581et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 581 equal parts of about 2.07 ¢ each. Each step represents a frequency ratio of 21/581, or the 581st root of 2.
Theory
581edo is a very strong 19- and 23-limit system, distinctly consistent to the 25-odd-limit. As an equal temperament, it tempers out 2401/2400 in the 7-limit, 3025/3024, 19712/19683, 151263/151250 in the 11-limit, and 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647 in the 13-limit. It supports and gives a good tuning for newt, the 270 & 311 microtemperament, which features a neutral-third generator.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.283 | -0.083 | -0.151 | +0.145 | +0.092 | +0.380 | -0.095 | -0.391 | -1.006 | -0.801 |
| Relative (%) | +0.0 | +13.7 | -4.0 | -7.3 | +7.0 | +4.5 | +18.4 | -4.6 | -18.9 | -48.7 | -38.8 | |
| Steps (reduced) |
581 (0) |
921 (340) |
1349 (187) |
1631 (469) |
2010 (267) |
2150 (407) |
2375 (51) |
2468 (144) |
2628 (304) |
2822 (498) |
2878 (554) | |
Subsets and supersets
Since 581 factors into 7 × 83, 581edo contains 7edo and 83edo as subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [921 -581⟩ | [⟨581 921]] | −0.0891 | 0.0891 | 4.32 |
| 2.3.5 | [-29 -11 20⟩, [33 -34 9⟩ | [⟨581 921 1349]] | −0.0475 | 0.0936 | 4.53 |
| 2.3.5.7 | 2401/2400, 33554432/33480783, 48828125/48771072 | [⟨581 921 1349 1631]] | −0.0222 | 0.0922 | 4.46 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 19712/19683, 234375/234256 | [⟨581 921 1349 1631 2010]] | −0.0261 | 0.0828 | 4.01 |
| 2.3.5.7.11.13 | 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078 | [⟨581 921 1349 1631 2010 2150]] | −0.0259 | 0.0756 | 3.66 |
| 2.3.5.7.11.13.17 | 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913 | [⟨581 921 1349 1631 2010 2150 2375]] | −0.0355 | 0.0738 | 3.58 |
| 2.3.5.7.11.13.17.19 | 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913 | [⟨581 921 1349 1631 2010 2150 2375 2468]] | −0.0283 | 0.0717 | 3.47 |
| 2.3.5.7.11.13.17.19.23 | 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079 | [⟨581 921 1349 1631 2010 2150 2375 2468 2628]] | −0.0155 | 0.0800 | 3.87 |
- 581et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first after 270 with a lower 19-limit relative error, and the first after 311 with a lower 23-limit relative error. It is only bettered by 742 in terms of either 19-limit absolute error or 19-limit relative error, by 718 in terms of 23-limit absolute error, and not until 1578 do we reach a lower 23-limit relative error.
- 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past 494 and followed by 742.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 17\581 | 35.11 | 1990656/1953125 | Gammic (5-limit) |
| 1 | 64\581 | 132.19 | [-38 5 13⟩ | Astro |
| 1 | 170\581 | 351.12 | 49/40 | Newt |
| 1 | 241\581 | 497.76 | 4/3 | Gary |
| 1 | 282\581 | 582.44 | 7/5 | Neptune (7-limit) |
| 1 | 285\581 | 588.64 | [-14 15 -4⟩ | Countritonic (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct