566edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
566edo is | 566edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the [[schisma]] in the 5-limit; 4375/4374 ([[ragisma]]), 65625/65536 ([[horwell comma]]), and 14348907/14336000 ([[skeetsma]]) in the 7-limit; [[3025/3024]] in the 11-limit; [[1716/1715]] and [[2080/2079]] in the 13-limit. It notably supports [[pontiac]] and [[orga]]. | ||
The 566g val is interesting in the higher limits, and in the 23-limit in particular it has a great rating in terms of absolute error. It tempers out [[1156/1155]], 1275/1274, 2431/2430, [[2500/2499]] and [[2601/2600]] in the 17-limit; [[1445/1444]], [[1521/1520]] and [[1729/1728]] in the 19-limit; 1105/1104 and 2025/2024 in the 23-limit. | The 566g val is interesting in the higher limits, and in the 23-limit in particular it has a great rating in terms of absolute error. It tempers out [[1156/1155]], 1275/1274, 2431/2430, [[2500/2499]] and [[2601/2600]] in the 17-limit; [[1445/1444]], [[1521/1520]] and [[1729/1728]] in the 19-limit; 1105/1104 and 2025/2024 in the 23-limit. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
566edo | Since 566 factors into 2 × 283, 566edo contains [[2edo]] and [[283edo]] as subsets. | ||
== 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]] (¢) | ||
| Line 26: | Line 27: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -897 566 }} | | {{monzo| -897 566 }} | ||
| | | {{mapping| 566 897 }} | ||
| +0.0594 | | +0.0594 | ||
| 0.0594 | | 0.0594 | ||
| Line 33: | Line 34: | ||
| 2.3.5 | | 2.3.5 | ||
| 32805/32768, {{monzo| -3 -86 60 }} | | 32805/32768, {{monzo| -3 -86 60 }} | ||
| | | {{mapping| 566 897 1314 }} | ||
| +0.1039 | | +0.1039 | ||
| 0.0795 | | 0.0795 | ||
| Line 40: | Line 41: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 4375/4374, 32805/32768, {{monzo| 10 5 8 -13 }} | | 4375/4374, 32805/32768, {{monzo| 10 5 8 -13 }} | ||
| | | {{mapping| 566 897 1314 1589 }} | ||
| +0.0709 | | +0.0709 | ||
| 0.0894 | | 0.0894 | ||
| Line 47: | Line 48: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 3025/3024, 4375/4374, 32805/32768, 825000/823543 | | 3025/3024, 4375/4374, 32805/32768, 825000/823543 | ||
| | | {{mapping| 566 897 1314 1589 1958 }} | ||
| +0.0614 | | +0.0614 | ||
| 0.0822 | | 0.0822 | ||
| Line 54: | Line 55: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 1716/1715, 2080/2079, 3025/3024, 15379/15360, 31250/31213 | | 1716/1715, 2080/2079, 3025/3024, 15379/15360, 31250/31213 | ||
| | | {{mapping| 566 897 1314 1589 1958 2094 }} | ||
| +0.0941 | | +0.0941 | ||
| 0.1047 | | 0.1047 | ||
| Line 63: | Line 64: | ||
=== 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 | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br> | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 82: | Line 84: | ||
| [[Orga]] | | [[Orga]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||