# 581edo

 ← 580edo 581edo 582edo →
Prime factorization 7 × 83
Step size 2.0654¢
Fifth 340\581 (702.238¢)
Semitones (A1:m2) 56:43 (115.7¢ : 88.81¢)
Consistency limit 25
Distinct consistency limit 25

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. The equal temperament 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

Approximation of prime harmonics in 581edo
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

Table of rank-2 temperaments by generator
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 it is distinct