# 559edo

 ← 558edo 559edo 560edo →
Prime factorization 13 × 43
Step size 2.14669¢
Fifth 327\559 (701.968¢)
Semitones (A1:m2) 53:42 (113.8¢ : 90.16¢)
Consistency limit 11
Distinct consistency limit 11

559 equal divisions of the octave (abbreviated 559edo or 559ed2), also called 559-tone equal temperament (559tet) or 559 equal temperament (559et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 559 equal parts of about 2.15 ¢ each. Each step represents a frequency ratio of 21/559, or the 559th root of 2.

## Theory

559edo is a very strong 5-limit system. The equal temperament tempers out the luna comma, [38 -2 -15 and the minortone comma, [-16 35 -17 in the 5-limit, as well as the monzisma, [54 -37 2; 4375/4374, 2100875/2097152, and 282475249/281250000 in the 7-limit; 12005/11979, 41503/41472, 160083/160000, and 172032/171875 in the 11-limit. Rank-2 temperaments it supports include mitonic, lunatic, acrokleismic, monzism, and meridic.

### Prime harmonics

Approximation of prime harmonics in 559edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.013 +0.091 -0.668 +0.382 +0.975 +0.232 +0.877 +0.706 +0.834 -0.850
Relative (%) +0.0 +0.6 +4.2 -31.1 +17.8 +45.4 +10.8 +40.9 +32.9 +38.9 -39.6
Steps
(reduced)
559
(0)
886
(327)
1298
(180)
1569
(451)
1934
(257)
2069
(392)
2285
(49)
2375
(139)
2529
(293)
2716
(480)
2769
(533)

### Subsets and supersets

Since 559 factors into 13 × 43, 559edo contains 13edo and 43edo as subsets.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [886 -559 [559 886]] -0.0040 0.0040 0.19
2.3.5 [38 -2 -15, [-16 35 -17 [559 886 1298]] -0.0157 0.0168 0.78
2.3.5.7 4375/4374, 2100875/2097152, [-4 -2 -9 10 [559 886 1298 1569]] +0.0478 0.1109 5.16
2.3.5.7.11 4375/4374, 12005/11979, 41503/41472, 172032/171875 [559 886 1298 1569 1934]] 0.0161 0.1175 5.48
• 559et has a lower relative error than any previous equal temperaments in the 5-limit, past 441 and before 612.

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 90\559 182.47 10/9 Mitonic
1 90\559 193.20 352/315 Lunatic
1 116\559 249.02 [-27 11 3 1 Monzismic
1 147\559 315.56 6/5 Acrokleismic
13 232\559
(17\559)
498.03
(36.494)
4/3
(?)
Aluminium
43 232\559
(2\559)
498.03
(4.29)
4/3
(385/384)
Meridic

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