581edo: Difference between revisions

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Cleanup; +prime error table; +categories
+infobox; +RTT table and rank-2 temperaments
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{{Infobox ET
| Prime factorization = 7 × 73
| Step size = 2.06540¢
| Fifth = 340\581 (702.24¢)
| Semitones = 56:43 (115.66¢ : 88.81¢)
| Consistency = 25
}}
{{EDO intro|581}}
{{EDO intro|581}}


== 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 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.


=== Prime harmonics ===
{{Harmonics in equal|581|columns=11}}
{{Harmonics in equal|581|columns=11}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! 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
| {{monzo| 921 -581 }}
| [{{val| 581 921 }}]
| -0.0891
| 0.0891
| 4.32
|-
| 2.3.5
| {{monzo| -29 -11 20 }}, {{monzo| 33 -34 9 }}
| [{{val| 581 921 1349 }}]
| -0.0475
| 0.0936
| 4.53
|-
| 2.3.5.7
| 2401/2400, 33554432/33480783, 48828125/48771072
| [{{val| 581 921 1349 1631 }}]
| -0.0222
| 0.0922
| 4.46
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 19712/19683, 234375/234256
| [{{val| 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
| [{{val| 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, 13013/13005
| [{{val| 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, 10985/10982
| [{{val| 581 921 1349 1631 2010 2150 2375 2468 }}]
| -0.0283
| 0.0717
| 3.47
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per Octave
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! 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
| 282\581
| 582.44
| 7/5
| [[Neptune]]
|}


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

Revision as of 00:58, 30 August 2022

← 580edo 581edo 582edo →
Prime factorization 7 × 73
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 41 & 229 microtemperament, which has a neutral thirds generator. It is the first division after 270 with a lower 19-limit relative error, and the first past 311 with a lower 23-limit relative error, and not until 1578 do we reach a lower 23-limit relative error.

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, 13013/13005 [⟨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, 10985/10982 [⟨581 921 1349 1631 2010 2150 2375 2468]] -0.0283 0.0717 3.47

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per Octave 		Generator
(Reduced) 		Cents
(Reduced) 		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
1 282\581 582.44 7/5 Neptune
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