557edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|557}} ==Theory== 557et tempers out 645700815/645657712, 2460375/2458624, 65625/65536 and 420175/419904 in the 7-limit; 820125/819896, 209715..."
 
Francium (talk | contribs)
4,621 edits
m changed EDO intro to ED intro
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|557}}
{{ED intro}}
==Theory==
 
557et tempers out [[645700815/645657712]], 2460375/2458624, [[65625/65536]] and 420175/419904 in the 7-limit; 820125/819896, 2097152/2096325, [[4000/3993]], 78675968/78594219, [[536870912/535869675]], 180224/180075, 1375/1372, 184549376/184528125, [[19712/19683]], 43923/43904, 20614528/20588575 and 322102/321489 in the 11-limit.
== Theory ==
===Prime harmonics===
557edo is only [[consistent]] to the [[5-odd-limit]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] {{monzo| 3 -18 11 }} (quartonic comma) and {{monzo| -74 13 23 }} (sesesix comma), as well as {{monzo| 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 [[support]]s [[fifthplus]], although [[171edo]] is better suited for that purpose.
 
=== Prime harmonics ===
{{Harmonics in equal|557}}
{{Harmonics in equal|557}}
===Subsets and supersets===
 
557edo is the 102nd [[prime EDO]].
=== Subsets and supersets ===
==Regular temperament properties==
557edo is the 102nd [[prime edo]].
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! 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
! rowspan="2" | [[Subgroup]]
|{{monzo|883 -557}}
! rowspan="2" | [[Comma list]]
|{{val|557 883}}
! rowspan="2" | [[Mapping]]
| -0.1195
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 883 -557 }}
| {{mapping| 557 883 }}
| −0.1195
| 0.1195
| 0.1195
| 5.55
| 5.55
|-
|-
|2.3.5
| 2.3.5
|{{monzo|3 -18 11}}, {{monzo|-74 13 23}}
| {{monzo| 3 -18 11 }}, {{monzo| -74 13 23 }}
|{{val|557 883 1293}}
| {{mapping| 557 883 1293 }}
| +0.0174
| +0.0174
| 0.2169
| 0.2169
| 10.07
| 10.07
|}
|}
===Rank-2 temperaments===
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|228\557
| 228\557
|491.203
| 491.203
|4/3
| 3645/2744
|[[Sesesix]]
| [[Fifthplus]]
|}
|}
==Scales==
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Scales ==
* [[Laz5]]
* [[Laz5]]
* [[Laz9]]
* [[Laz9]]
== Music ==
; [[Francium]]
* "Are You A Youth Medium?" from ''Questions'' (2024) – [https://open.spotify.com/track/4S3oY6c2VRNHBrYD16owoF Spotify] | [https://francium223.bandcamp.com/track/are-you-a-youth-medium Bandcamp] | [https://www.youtube.com/watch?v=OAxdvEZJ-8M YouTube] – sidi in 557edo tuning

Latest revision as of 05:59, 21 February 2025

← 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 distinct

Scales

Music

Francium
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