569edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 568edo569edo570edo →
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

Music

Francium