557edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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== Regular temperament properties == | == 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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 25: | Line 26: | ||
| {{monzo| 883 -557 }} | | {{monzo| 883 -557 }} | ||
| {{mapping| 557 883 }} | | {{mapping| 557 883 }} | ||
| | | −0.1195 | ||
| 0.1195 | | 0.1195 | ||
| 5.55 | | 5.55 | ||
| Line 39: | Line 40: | ||
=== 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 | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Fifthplus]] | | [[Fifthplus]] | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
Latest revision as of 05:59, 21 February 2025
| ← 556edo | 557edo | 558edo → |
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
| 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 distinct