559edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 558edo559edo560edo →
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