566edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
566edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]]. | 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 === | ||
Since 566 factors into | 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]] (¢) | ||
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=== 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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| [[Orga]] | | [[Orga]] | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
Latest revision as of 06:30, 21 February 2025
| ← 565edo | 566edo | 567edo → |
566 equal divisions of the octave (abbreviated 566edo or 566ed2), also called 566-tone equal temperament (566tet) or 566 equal temperament (566et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 566 equal parts of about 2.12 ¢ each. Each step represents a frequency ratio of 21/566, or the 566th root of 2.
Theory
566edo is distinctly consistent in the 15-odd-limit. As an equal temperament, it 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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.188 | -0.448 | +0.079 | -0.081 | -0.952 | +1.052 | -0.693 | -0.713 | +0.811 | -0.159 |
| Relative (%) | +0.0 | -8.9 | -21.1 | +3.7 | -3.8 | -44.9 | +49.6 | -32.7 | -33.6 | +38.3 | -7.5 | |
| Steps (reduced) |
566 (0) |
897 (331) |
1314 (182) |
1589 (457) |
1958 (260) |
2094 (396) |
2314 (50) |
2404 (140) |
2560 (296) |
2750 (486) |
2804 (540) | |
Subsets and supersets
Since 566 factors into 2 × 283, 566edo contains 2edo and 283edo as subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-897 566⟩ | [⟨566 897]] | +0.0594 | 0.0594 | 2.80 |
| 2.3.5 | 32805/32768, [-3 -86 60⟩ | [⟨566 897 1314]] | +0.1039 | 0.0795 | 3.75 |
| 2.3.5.7 | 4375/4374, 32805/32768, [10 5 8 -13⟩ | [⟨566 897 1314 1589]] | +0.0709 | 0.0894 | 4.22 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 32805/32768, 825000/823543 | [⟨566 897 1314 1589 1958]] | +0.0614 | 0.0822 | 3.88 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 3025/3024, 15379/15360, 31250/31213 | [⟨566 897 1314 1589 1958 2094]] | +0.0941 | 0.1047 | 4.94 |
- 566et (566g val) has a lower absolute error in the 23-limit than any previous equal temperaments, past 525 and followed by 581.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 235\566 | 498.23 | 4/3 | Pontiac |
| 2 | 109\566 | 231.10 | 8/7 | Orga |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct