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581edo: Difference between revisions

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{{EDO intro|581}}
{{Infobox ET}}
{{ED intro}}


581edo is a very strong 19- and 23-limit system, distinctly [[consistent]] to the [[25-odd-limit]]. It 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 [[support]]s and gives a good tuning for [[newt]], the 41 & 229 microtemperament, which has a neutral thirds generator. It is the first division after 270 with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and the first past 311 with a lower 23-limit relative error, and not until [[1578edo|1578]] do we reach a lower 23-limit relative error.
== Theory ==
581edo is a very strong [[17-limit|17-]], [[19-limit|19-]] and [[23-limit]] system, [[consistency|distinctly consistent]] to the [[25-odd-limit]], and except for [[27/23]] and its [[octave complement]], it is consistent to the [[27-odd-limit]].  


As an equal temperament, it [[tempering out|tempers out]] [[2401/2400]], [[33554432/33480783]], and [[48828125/48771072]] in the 7-limit; [[3025/3024]], [[19712/19683]], [[234375/234256]] in the 11-limit; [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the {{nowrap| 270 & 311 }} microtemperament, which features a neutral-third generator.
It notably achieves [[diamond monotone]] in the 71-odd-limit with the 581jks [[val]] (s is the [[wart]] for prime 67), which is a large improvement from the previous record of the 59-odd-limit, held by [[571edo]] using its [[patent val]].
=== Prime harmonics ===
{{Harmonics in equal|581|columns=11}}
{{Harmonics in equal|581|columns=11}}
{{Harmonics in equal|581|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 581edo (continued)}}
=== Subsets and supersets ===
Since 581 factors into primes as {{nowrap| 7 × 83 }}, 581edo contains [[7edo]] and [[83edo]] as subsets.
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 921 -581 }}
| {{Mapping| 581 921 }}
| −0.0891
| 0.0891
| 4.32
|-
| 2.3.5
| {{Monzo| -29 -11 20 }}, {{monzo| 33 -34 9 }}
| {{Mapping| 581 921 1349 }}
| −0.0475
| 0.0936
| 4.53
|-
| 2.3.5.7
| 2401/2400, 33554432/33480783, 48828125/48771072
| {{Mapping| 581 921 1349 1631 }}
| −0.0222
| 0.0922
| 4.46
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 19712/19683, 234375/234256
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 581 921 1349 1631 2010 2150 2375 2468 2628 }}
| −0.0155
| 0.0800
| 3.87
|}
* 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.
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 1
| 17\581
| 35.11
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 64\581
| 132.19
| {{Monzo| -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
| 351/250
| [[Garitritonic]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Newt]]
[[Category:Newt]]
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