581edo: Difference between revisions

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


== Theory ==
== Theory ==
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.
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 ===
=== 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 ==
== 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|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 20: Line 29:
|-
|-
| 2.3
| 2.3
| {{monzo| 921 -581 }}
| {{Monzo| 921 -581 }}
| [{{val| 581 921 }}]
| {{Mapping| 581 921 }}
| -0.0891
| −0.0891
| 0.0891
| 0.0891
| 4.32
| 4.32
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| -29 -11 20 }}, {{monzo| 33 -34 9 }}
| {{Monzo| -29 -11 20 }}, {{monzo| 33 -34 9 }}
| [{{val| 581 921 1349 }}]
| {{Mapping| 581 921 1349 }}
| -0.0475
| −0.0475
| 0.0936
| 0.0936
| 4.53
| 4.53
Line 35: Line 44:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 33554432/33480783, 48828125/48771072
| 2401/2400, 33554432/33480783, 48828125/48771072
| [{{val| 581 921 1349 1631 }}]
| {{Mapping| 581 921 1349 1631 }}
| -0.0222
| −0.0222
| 0.0922
| 0.0922
| 4.46
| 4.46
Line 42: Line 51:
| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 19712/19683, 234375/234256
| 2401/2400, 3025/3024, 19712/19683, 234375/234256
| [{{val| 581 921 1349 1631 2010 }}]
| {{Mapping| 581 921 1349 1631 2010 }}
| -0.0261
| −0.0261
| 0.0828
| 0.0828
| 4.01
| 4.01
Line 49: Line 58:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078
| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078
| [{{val| 581 921 1349 1631 2010 2150 }}]
| {{Mapping| 581 921 1349 1631 2010 2150 }}
| -0.0259
| −0.0259
| 0.0756
| 0.0756
| 3.66
| 3.66
|-
|-
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 13013/13005
| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913
| [{{val| 581 921 1349 1631 2010 2150 2375 }}]
| {{Mapping| 581 921 1349 1631 2010 2150 2375 }}
| -0.0355
| −0.0355
| 0.0738
| 0.0738
| 3.58
| 3.58
|-
|-
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 10985/10982
| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913
| [{{val| 581 921 1349 1631 2010 2150 2375 2468 }}]
| {{Mapping| 581 921 1349 1631 2010 2150 2375 2468 }}
| -0.0283
| −0.0283
| 0.0717
| 0.0717
| 3.47
| 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 ===
=== 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 Octave
|-
! Generator<br>(Reduced)
! Periods<br>per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 64\581
| 64\581
| 132.19
| 132.19
| {{monzo| -38 5 13 }}
| {{Monzo| -38 5 13 }}
| [[Astro]]
| [[Astro]]
|-
|-
Line 95: Line 114:
| 49/40
| 49/40
| [[Newt]]
| [[Newt]]
|-
| 1
| 241\581
| 497.76
| 4/3
| [[Gary]]
|-
|-
| 1
| 1
Line 100: Line 125:
| 582.44
| 582.44
| 7/5
| 7/5
| [[Neptune]]
| [[Neptune]] (7-limit)
|-
|-
| 1
| 1
| 241\581
| 285\581
| 497.76
| 588.64
| 4/3
| 351/250
| [[Gary]]
| [[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]]

Latest revision as of 11:46, 20 May 2026

← 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

581 equal divisions of the octave (abbreviated 581edo or 581ed2), also called 581-tone equal temperament (581tet) or 581 equal temperament (581et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 581 equal parts of about 2.07 ¢ each. Each step represents a frequency ratio of 21/581, or the 581st root of 2.

Theory

581edo is a very strong 17-, 19- and 23-limit system, 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 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 supports and gives a good tuning for newt, the 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

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)
Approximation of prime harmonics in 581edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +0.635 +0.542 +0.703 -0.446 +0.162 +0.381 +0.499 -0.822 -0.006 -0.595 -1.026
Relative (%) +30.8 +26.2 +34.0 -21.6 +7.8 +18.4 +24.2 -39.8 -0.3 -28.8 -49.7
Steps
(reduced) 		3027
(122) 		3113
(208) 		3153
(248) 		3227
(322) 		3328
(423) 		3418
(513) 		3446
(541) 		3524
(38) 		3573
(87) 		3596
(110) 		3662
(176)

Subsets and supersets

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

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
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
  • 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

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
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\581 588.64 351/250 Garitritonic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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