557edo: Difference between revisions
Jump to navigation
Jump to search
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..." |
Rework; cleanup; clarify the title row of the rank-2 temp table |
||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|557}} | {{EDO intro|557}} | ||
==Theory== | |||
== 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 | === 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" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|883 -557}} | | {{monzo| 883 -557 }} | ||
|{{ | | {{mapping| 557 883 }} | ||
| -0.1195 | | -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 }} | ||
|{{ | | {{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 | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|228\557 | | 228\557 | ||
|491.203 | | 491.203 | ||
| | | 3645/2744 | ||
|[[ | | [[Fifthplus]] | ||
|} | |} | ||
==Scales== | <nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | ||
== Scales == | |||
* [[Laz5]] | * [[Laz5]] | ||
* [[Laz9]] | * [[Laz9]] | ||
Revision as of 11:02, 29 October 2023
| ← 556edo | 557edo | 558edo → |
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
| 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
| 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