581edo: 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
| {{monzo| 921 -581 }}
| {{monzo| 921 -581 }}
| {{mapping| 581 921 }}
| {{mapping| 581 921 }}
| -0.0891
| &minus;0.0891
| 0.0891
| 0.0891
| 4.32
| 4.32
Line 32: Line 24:
| {{monzo| -29 -11 20 }}, {{monzo| 33 -34 9 }}
| {{monzo| -29 -11 20 }}, {{monzo| 33 -34 9 }}
| {{mapping| 581 921 1349 }}
| {{mapping| 581 921 1349 }}
| -0.0475
| &minus;0.0475
| 0.0936
| 0.0936
| 4.53
| 4.53
Line 39: Line 31:
| 2401/2400, 33554432/33480783, 48828125/48771072
| 2401/2400, 33554432/33480783, 48828125/48771072
| {{mapping| 581 921 1349 1631 }}
| {{mapping| 581 921 1349 1631 }}
| -0.0222
| &minus;0.0222
| 0.0922
| 0.0922
| 4.46
| 4.46
Line 46: Line 38:
| 2401/2400, 3025/3024, 19712/19683, 234375/234256
| 2401/2400, 3025/3024, 19712/19683, 234375/234256
| {{mapping| 581 921 1349 1631 2010 }}
| {{mapping| 581 921 1349 1631 2010 }}
| -0.0261
| &minus;0.0261
| 0.0828
| 0.0828
| 4.01
| 4.01
Line 53: Line 45:
| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078
| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078
| {{mapping| 581 921 1349 1631 2010 2150 }}
| {{mapping| 581 921 1349 1631 2010 2150 }}
| -0.0259
| &minus;0.0259
| 0.0756
| 0.0756
| 3.66
| 3.66
Line 60: Line 52:
| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913
| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913
| {{mapping| 581 921 1349 1631 2010 2150 2375 }}
| {{mapping| 581 921 1349 1631 2010 2150 2375 }}
| -0.0355
| &minus;0.0355
| 0.0738
| 0.0738
| 3.58
| 3.58
Line 67: Line 59:
| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913
| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913
| {{mapping| 581 921 1349 1631 2010 2150 2375 2468 }}
| {{mapping| 581 921 1349 1631 2010 2150 2375 2468 }}
| -0.0283
| &minus;0.0283
| 0.0717
| 0.0717
| 3.47
| 3.47
Line 74: Line 66:
| 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079
| 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079
| {{mapping| 581 921 1349 1631 2010 2150 2375 2468 2628 }}
| {{mapping| 581 921 1349 1631 2010 2150 2375 2468 2628 }}
| -0.0155
| &minus;0.0155
| 0.0800
| 0.0800
| 3.87
| 3.87
|}
{{comma basis end}}
* 581et has lower [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any previous equal temperaments in the 19- and 23-limit. It is the first after [[270edo|270]] with a lower 19-limit relative error, and the first after [[311edo|311]] with a lower 23-limit relative error. It is only bettered by [[742edo|742]] in terms of either 19-limit absolute error or 19-limit relative error, by [[718edo|718]] in terms of 23-limit absolute error, and not until [[1578edo|1578]] do we reach a lower 23-limit relative error.  
* 581et has lower [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any previous equal temperaments in the 19- and 23-limit. It is the first after [[270edo|270]] with a lower 19-limit relative error, and the first after [[311edo|311]] with a lower 23-limit relative error. It is only bettered by [[742edo|742]] in terms of either 19-limit absolute error or 19-limit relative error, by [[718edo|718]] in terms of 23-limit absolute error, and not until [[1578edo|1578]] do we reach a lower 23-limit relative error.  
* 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past [[494edo|494]] and followed by 742.  
* 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past [[494edo|494]] and followed by 742.  


=== 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 125: Line 111:
| {{monzo| -14 15 -4 }}
| {{monzo| -14 15 -4 }}
| [[Countritonic]] (5-limit)
| [[Countritonic]] (5-limit)
|}
{{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}}


[[Category:Newt]]
[[Category:Newt]]

Revision as of 05:03, 16 November 2024

← 580edo 581edo 582edo →
Prime factorization 7 × 83
Step size 2.0654 ¢ 
Fifth 340\581 (702.238 ¢)
Semitones (A1:m2) 56:43 (115.7 ¢ : 88.81 ¢)
Consistency limit 25
Distinct consistency limit 25

Template:EDO intro

Theory

581edo is a very strong 19- and 23-limit system, distinctly consistent to the 25-odd-limit. The equal temperament tempers out 2401/2400 in the 7-limit, 3025/3024, 19712/19683, 151263/151250 in the 11-limit, and 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647 in the 13-limit. It supports and gives a good tuning for newt, the 270 & 311 microtemperament, which features a neutral-third generator.

Prime harmonics

Approximation of prime harmonics in 581edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.283 -0.083 -0.151 +0.145 +0.092 +0.380 -0.095 -0.391 -1.006 -0.801
Relative (%) +0.0 +13.7 -4.0 -7.3 +7.0 +4.5 +18.4 -4.6 -18.9 -48.7 -38.8
Steps
(reduced) 		581
(0) 		921
(340) 		1349
(187) 		1631
(469) 		2010
(267) 		2150
(407) 		2375
(51) 		2468
(144) 		2628
(304) 		2822
(498) 		2878
(554)

Subsets and supersets

Since 581 factors into 7 × 83, 581edo contains 7edo and 83edo as subsets.

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [921 -581⟩ | [⟨581 921]] | −0.0891 | 0.0891 | 4.32 |- | 2.3.5 | [-29 -11 20⟩, [33 -34 9⟩ | [⟨581 921 1349]] | −0.0475 | 0.0936 | 4.53 |- | 2.3.5.7 | 2401/2400, 33554432/33480783, 48828125/48771072 | [⟨581 921 1349 1631]] | −0.0222 | 0.0922 | 4.46 |- | 2.3.5.7.11 | 2401/2400, 3025/3024, 19712/19683, 234375/234256 | [⟨581 921 1349 1631 2010]] | −0.0261 | 0.0828 | 4.01 |- | 2.3.5.7.11.13 | 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078 | [⟨581 921 1349 1631 2010 2150]] | −0.0259 | 0.0756 | 3.66 |- | 2.3.5.7.11.13.17 | 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913 | [⟨581 921 1349 1631 2010 2150 2375]] | −0.0355 | 0.0738 | 3.58 |- | 2.3.5.7.11.13.17.19 | 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913 | [⟨581 921 1349 1631 2010 2150 2375 2468]] | −0.0283 | 0.0717 | 3.47 |- | 2.3.5.7.11.13.17.19.23 | 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079 | [⟨581 921 1349 1631 2010 2150 2375 2468 2628]] | −0.0155 | 0.0800 | 3.87 Template:Comma basis end

  • 581et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first after 270 with a lower 19-limit relative error, and the first after 311 with a lower 23-limit relative error. It is only bettered by 742 in terms of either 19-limit absolute error or 19-limit relative error, by 718 in terms of 23-limit absolute error, and not until 1578 do we reach a lower 23-limit relative error.
  • 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past 494 and followed by 742.

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 17\581 | 35.11 | 1990656/1953125 | Gammic (5-limit) |- | 1 | 64\581 | 132.19 | [-38 5 13⟩ | Astro |- | 1 | 170\581 | 351.12 | 49/40 | Newt |- | 1 | 241\581 | 497.76 | 4/3 | Gary |- | 1 | 282\581 | 582.44 | 7/5 | Neptune (7-limit) |- | 1 | 285\581 | 588.64 | [-14 15 -4⟩ | Countritonic (5-limit) Template:Rank-2 end Template:Orf

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