581edo: Difference between revisions

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== 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 270 & 311 microtemperament, which features a neutral-third generator.  
581edo is a very strong 19- and 23-limit system, [[consistency|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 270 & 311 microtemperament, which features a neutral-third generator.  


=== Prime harmonics ===
=== Prime harmonics ===
Line 21: Line 21:
| 2.3
| 2.3
| {{monzo| 921 -581 }}
| {{monzo| 921 -581 }}
| [{{val| 581 921 }}]
| {{mapping| 581 921 }}
| -0.0891
| -0.0891
| 0.0891
| 0.0891
Line 28: Line 28:
| 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
Line 35: Line 35:
| 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
Line 42: Line 42:
| 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
Line 49: Line 49:
| 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
Line 56: Line 56:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913
| 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
Line 63: Line 63:
| 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, 4914/4913
| 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
Line 70: Line 70:
| 2.3.5.7.11.13.17.19.23
| 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
| 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079
| [{{val| 581 921 1349 1631 2010 2150 2375 2468 2628 }}]
| {{mapping| 581 921 1349 1631 2010 2150 2375 2468 2628 }}
| -0.0155
| -0.0155
| 0.0800
| 0.0800
Line 82: Line 82:
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
Line 123: Line 123:
| [[Countritonic]] (5-limit)
| [[Countritonic]] (5-limit)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


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

Revision as of 09:47, 25 October 2023

← 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. 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 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)

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 [-14 15 -4⟩ Countritonic (5-limit)

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

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