566edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! 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
| 2.3
Line 58: Line 50:
| 0.1047
| 0.1047
| 4.94
| 4.94
|}
{{comma basis end}}
* 566et (566g val) has a lower absolute error in the 23-limit than any previous equal temperaments, past [[525edo|525]] and followed by [[581edo|581]].  
* 566et (566g val) has a lower absolute error in the 23-limit than any previous equal temperaments, past [[525edo|525]] and followed by [[581edo|581]].  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 81: Line 67:
| 8/7
| 8/7
| [[Orga]]
| [[Orga]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 03:46, 16 November 2024

← 565edo 566edo 567edo →
Prime factorization 2 × 283
Step size 2.12014 ¢ 
Fifth 331\566 (701.767 ¢)
Semitones (A1:m2) 53:43 (112.4 ¢ : 91.17 ¢)
Consistency limit 15
Distinct consistency limit 15

Template:EDO intro

Theory

566edo is distinctly consistent in the 15-odd-limit. The equal temperament 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

Approximation of prime harmonics in 566edo
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

Template:Comma basis begin |- | 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 Template:Comma basis end

  • 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

Template:Rank-2 begin |- | 1 | 235\566 | 498.23 | 4/3 | Pontiac |- | 2 | 109\566 | 231.10 | 8/7 | Orga Template:Rank-2 end Template:Orf

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