571edo
| ← 570edo | 571edo | 572edo → |
571 equal divisions of the octave (abbreviated 571edo or 571ed2), also called 571-tone equal temperament (571tet) or 571 equal temperament (571et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 571 equal parts of about 2.1 ¢ each. Each step represents a frequency ratio of 21/571, or the 571st root of 2.
Theory
The equal temperament tempers out the parakleisma, [8 14 -13⟩, and the counterschisma, [-69 45 -1⟩, in the 5-limit, as well as the lafa comma, [77 -31 -12⟩; 2401/2400, 14348907/14336000, and 29360128/29296875 in the 7-limit; 3025/3024, 5632/5625, 41503/41472, and 17537553/17500000 in the 11-limit; 1001/1000, 1716/1715, 4096/4095, 17303/17280, and 107811/107653 in the 13-limit, supporting the 13-limit quasiorwell temperament; 1089/1088, 1701/1700, 2431/2430, 2601/2600, 5832/5831 and 7744/7735 in the 17-limit. The 7th harmonic is only 0.0007135 cents sharp in 571edo, as the denominator of a convergent to log27, after 109 and before 2393.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.029 | +0.376 | +0.001 | -0.705 | +0.103 | +0.123 | +0.911 | +0.097 | +0.195 | +0.323 |
| Relative (%) | +0.0 | -1.4 | +17.9 | +0.0 | -33.5 | +4.9 | +5.9 | +43.3 | +4.6 | +9.3 | +15.4 | |
| Steps (reduced) |
571 (0) |
905 (334) |
1326 (184) |
1603 (461) |
1975 (262) |
2113 (400) |
2334 (50) |
2426 (142) |
2583 (299) |
2774 (490) |
2829 (545) | |
Subsets and supersets
571edo is the 105th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-905 571⟩ | [⟨571 905]] | +0.0090 | 0.0090 | 0.43 |
| 2.3.5 | [8 14 -13⟩, [-69 45 -1⟩ | [⟨571 905 1326]] | −0.0480 | 0.0810 | 3.85 |
| 2.3.5.7 | 2401/2400, 14348907/14336000, 29360128/29296875 | [⟨571 905 1326 1603]] | −0.0361 | 0.0731 | 3.48 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 5632/5625, 14348907/14336000 | [⟨571 905 1326 1603 1975]] | +0.0119 | 0.1161 | 5.53 |
| 2.3.5.7.11.13 | 1001/1000, 1716/1715, 3025/3024, 4096/4095, 107811/107653 | [⟨571 905 1326 1603 1975 2113]] | +0.0053 | 0.1070 | 5.09 |
| 2.3.5.7.11.13.17 | 1001/1000, 1089/1088, 1716/1715, 2601/2600, 3025/3024, 4096/4095 | [⟨571 905 1326 1603 1975 2113 2334]] | +0.0002 | 0.0999 | 4.75 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 123\571 | 258.49 | [-32 13 5⟩ | Lafa |
| 1 | 129\571 | 271.10 | 90/77 | Quasiorwell |
| 1 | 147\571 | 315.24 | 6/5 | Parakleismic (5-limit) |
| 1 | 237\571 | 498.07 | 4/3 | Counterschismic |
| 1 | 248\571 | 521.19 | 875/648 | Maviloid |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct