581edo: Difference between revisions

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~~{{novelty}}{{stub}}~~{{Infobox ET}}

== 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 &~~amp;~~ 311 microtemperament, which features a neutral-third generator.

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|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~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 20:

Line 29:

|-

| 2.3

| {{~~monzo~~| 921 -581 }}

| {{Monzo| 921 -581 }}

| [{{~~val~~| 581 921 }}]

| {{Mapping| 581 921 }}

| -0.0891

| −0.0891

| 0.0891

| 4.32

|-

| 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

| 4.53

Line 35:

Line 44:

| 2.3.5.7

| 2401/2400, 33554432/33480783, 48828125/48771072

| [{{~~val~~| 581 921 1349 1631 }}]

| {{Mapping| 581 921 1349 1631 }}

| -0.0222

| −0.0222

| 0.0922

| 4.46

Line 42:

Line 51:

| 2.3.5.7.11

| 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

| 4.01

Line 49:

Line 58:

| 2.3.5.7.11.13

| 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

| 3.66

Line 56:

Line 65:

| 2.3.5.7.11.13.17

| 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

| 3.58

Line 63:

Line 72:

| 2.3.5.7.11.13.17.19

| 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

| 3.47

Line 70:

Line 79:

| 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

| [{{~~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

| 3.87

|}

* 581et has a lower [[Tenney-Euclidean temperament measures #TE simple badness|relative ~~error~~]] 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.

=== Rank-2 temperaments ===

{| 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 per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

Line 96:

Line 106:

| 64\581

| 132.19

| {{~~monzo~~| -38 5 13 }}

| {{Monzo| -38 5 13 }}

| [[Astro]]

|-

Line 104:

Line 114:

| 49/40

| [[Newt]]

|-

| 1

| 241\581

| 497.76

| 4/3

| [[Gary]]

|-

| 1

Line 112:

Line 128:

|-

| 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:Newt]]

@@ Line 1: / Line 1: @@
-{{novelty}}{{stub}}{{Infobox ET}}
+{{Infobox ET}}
-{{EDO intro|581}}
+{{ED intro}}
 == Theory ==
-edo 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 &amp; 311 microtemperament, which features a neutral-third generator.
+edo 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|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|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 20: / Line 29: @@
 |-
 | 2.3
-| {{monzo| 921 -581 }}
+| {{Monzo| 921 -581 }}
-| [{{val| 581 921 }}]
+| {{Mapping| 581 921 }}
-| -0.0891
+| −0.0891
 | 0.0891
 | 4.32
 |-
 | 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
 | 4.53
@@ Line 35: / Line 44: @@
 | 2.3.5.7
 | 2401/2400, 33554432/33480783, 48828125/48771072
-| [{{val| 581 921 1349 1631 }}]
+| {{Mapping| 581 921 1349 1631 }}
-| -0.0222
+| −0.0222
 | 0.0922
 | 4.46
@@ Line 42: / Line 51: @@
 | 2.3.5.7.11
 | 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
 | 4.01
@@ Line 49: / Line 58: @@
 | 2.3.5.7.11.13
 | 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
 | 3.66
@@ Line 56: / Line 65: @@
 | 2.3.5.7.11.13.17
 | 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
 | 3.58
@@ Line 63: / Line 72: @@
 | 2.3.5.7.11.13.17.19
 | 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
 | 3.47
@@ Line 70: / Line 79: @@
 | 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
-| [{{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
 | 3.87
 |}
-* 581et has a lower [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] 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.
 === Rank-2 temperaments ===
 {| 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 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 96: / Line 106: @@
 | 64\581
 | 132.19
-| {{monzo| -38 5 13 }}
+| {{Monzo| -38 5 13 }}
 | [[Astro]]
 |-
@@ Line 104: / Line 114: @@
 | 49/40
 | [[Newt]]
+|-
+| 1
+| 241\581
+| 497.76
+| 4/3
+| [[Gary]]
 |-
 | 1
@@ Line 112: / Line 128: @@
 |-
 | 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:Newt]]