557edo

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← 556edo557edo558edo →
Prime factorization 557 (prime)
Step size 2.1544¢ 
Fifth 326\557 (702.334¢)
Semitones (A1:m2) 54:41 (116.3¢ : 88.33¢)
Consistency limit 5
Distinct consistency limit 5

557 equal divisions of the octave (abbreviated 557edo or 557ed2), also called 557-tone equal temperament (557tet) or 557 equal temperament (557et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 557 equal parts of about 2.15 ¢ each. Each step represents a frequency ratio of 21/557, or the 557th root of 2.

Theory

557edo is only consistent to the 5-odd-limit. Using the patent val, the equal temperament tempers out [3 -18 11 (quartonic comma) and [-74 13 23 (sesesix comma), as well as [77 -31 -12 (lafa comma) in the 5-limit; 65625/65536, 420175/419904 and 2460375/2458624 in the 7-limit; 1375/1372, 4000/3993, 19712/19683, 43923/43904, 180224/180075, and 322102/321489 in the 11-limit. It supports fifthplus, although 171edo is better suited for that purpose.

Prime harmonics

Approximation of prime harmonics in 557edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.000 +0.379 -0.676 +0.653 +0.208 -0.312 +0.610 -0.206 +0.810 +0.225 -1.050
relative (%) +0 +18 -31 +30 +10 -14 +28 -10 +38 +10 -49
Steps
(reduced)
557
(0)
883
(326)
1293
(179)
1564
(450)
1927
(256)
2061
(390)
2277
(49)
2366
(138)
2520
(292)
2706
(478)
2759
(531)

Subsets and supersets

557edo is the 102nd prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [883 -557

[557 883]]

-0.1195 0.1195 5.55
2.3.5 [3 -18 11, [-74 13 23

[557 883 1293]]

+0.0174 0.2169 10.07

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 228\557 491.203 3645/2744 Fifthplus

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

Scales