# 557edo

 ← 556edo 557edo 558edo →
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 (%) +0.0 +17.6 -31.4 +30.3 +9.7 -14.5 +28.3 -9.6 +37.6 +10.5 -48.7
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

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