# 569edo

 ← 568edo 569edo 570edo →
Prime factorization 569 (prime)
Step size 2.10896¢
Fifth 333\569 (702.285¢)
Semitones (A1:m2) 55:42 (116¢ : 88.58¢)
Consistency limit 5
Distinct consistency limit 5

569 equal divisions of the octave (abbreviated 569edo or 569ed2), also called 569-tone equal temperament (569tet) or 569 equal temperament (569et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 569 equal parts of about 2.11 ¢ each. Each step represents a frequency ratio of 21/569, or the 569th root of 2.

## Theory

569edo is only consistent to the 5-odd-limit. The equal temperament tempers out 1600000/1594323 (amity comma) in the 5-limit. Using the patent val, it tempers out 65625/65536, 823543/820125, and 33554432/33480783 in the 7-limit; 42592/42525, 42875/42768, 43923/43904, and 151263/151250 in the 11-limit.

### Prime harmonics

Approximation of prime harmonics in 569edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.330 -0.373 -0.812 -0.879 +0.949 +0.493 -0.149 +0.197 -0.403 +0.131
Relative (%) +0.0 +15.6 -17.7 -38.5 -41.7 +45.0 +23.4 -7.1 +9.3 -19.1 +6.2
Steps
(reduced)
569
(0)
902
(333)
1321
(183)
1597
(459)
1968
(261)
2106
(399)
2326
(50)
2417
(141)
2574
(298)
2764
(488)
2819
(543)

### Subsets and supersets

569edo is the 104th prime edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [902 -569 [569 902]] -0.1040 0.1040 4.93
2.3.5 1600000/1594323, [-94 -11 48 [569 902 1321]] -0.0157 0.1510 7.16
2.3.5.7 65625/65536, 823543/820125, 1600000/1594323 [569 902 1321 1597]] +0.0605 0.1858 8.81
2.3.5.7.11 42592/42525, 42875/42768, 43923/43904, 151263/151250 [569 902 1321 1597 1968]] +0.0992 0.1834 8.70

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 161\569 339.54 243/200 Paramity

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Francium