557edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 35: Line 27:
| 0.2169
| 0.2169
| 10.07
| 10.07
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
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| 3645/2744
| 3645/2744
| [[Fifthplus]]
| [[Fifthplus]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Scales ==
== Scales ==

Revision as of 01:31, 16 November 2024

← 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

Template:EDO intro

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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 228\557 | 491.203 | 3645/2744 | Fifthplus Template:Rank-2 end Template:Orf

Scales

Music

Francium
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