581edo: Difference between revisions

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== 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 }}

| {{mapping| 581 921 }}

| -0.0891

| −0.0891

| 0.0891

| 4.32

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Line 24:

| {{monzo| -29 -11 20 }}, {{monzo| 33 -34 9 }}

| {{mapping| 581 921 1349 }}

| -0.0475

| −0.0475

| 0.0936

| 4.53

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Line 31:

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

| {{mapping| 581 921 1349 1631 }}

| -0.0222

| −0.0222

| 0.0922

| 4.46

Line 46:

Line 38:

| 2401/2400, 3025/3024, 19712/19683, 234375/234256

| {{mapping| 581 921 1349 1631 2010 }}

| -0.0261

| −0.0261

| 0.0828

| 4.01

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| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078

| {{mapping| 581 921 1349 1631 2010 2150 }}

| -0.0259

| −0.0259

| 0.0756

| 3.66

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Line 52:

| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913

| {{mapping| 581 921 1349 1631 2010 2150 2375 }}

| -0.0355

| −0.0355

| 0.0738

| 3.58

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| 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

| −0.0283

| 0.0717

| 3.47

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| 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

| −0.0155

| 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"~~

~~|+Table of~~ rank-2 ~~temperaments by generator~~

~~! Periods<br>per 8ve~~

! Generator*

! Cents*

! Associated<br>Ratio*

~~! Temperaments~~

|-

| 1

Line 125:

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| {{monzo| -14 15 -4 }}

| [[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]]

@@ Line 12: / Line 12: @@
 == Regular temperament properties ==
-{| class="wikitable center-4 center-5 center-6"
+{{comma basis begin}}
-! 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 }}
 | {{mapping| 581 921 }}
-| -0.0891
+| &minus;0.0891
 | 0.0891
 | 4.32
@@ Line 32: / Line 24: @@
 | {{monzo| -29 -11 20 }}, {{monzo| 33 -34 9 }}
 | {{mapping| 581 921 1349 }}
-| -0.0475
+| &minus;0.0475
 | 0.0936
 | 4.53
@@ Line 39: / Line 31: @@
 | 2401/2400, 33554432/33480783, 48828125/48771072
 | {{mapping| 581 921 1349 1631 }}
-| -0.0222
+| &minus;0.0222
 | 0.0922
 | 4.46
@@ Line 46: / Line 38: @@
 | 2401/2400, 3025/3024, 19712/19683, 234375/234256
 | {{mapping| 581 921 1349 1631 2010 }}
-| -0.0261
+| &minus;0.0261
 | 0.0828
 | 4.01
@@ Line 53: / Line 45: @@
 | 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078
 | {{mapping| 581 921 1349 1631 2010 2150 }}
-| -0.0259
+| &minus;0.0259
 | 0.0756
 | 3.66
@@ Line 60: / Line 52: @@
 | 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913
 | {{mapping| 581 921 1349 1631 2010 2150 2375 }}
-| -0.0355
+| &minus;0.0355
 | 0.0738
 | 3.58
@@ Line 67: / Line 59: @@
 | 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
+| &minus;0.0283
 | 0.0717
 | 3.47
@@ Line 74: / Line 66: @@
 | 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
+| &minus;0.0155
 | 0.0800
 | 3.87
-|}
+{{comma basis end}}
 * 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"
+{{rank-2 begin}}
-|+Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
-! Generator*
-! Cents*
-! Associated<br>Ratio*
-! Temperaments
 |-
 | 1
@@ Line 125: / Line 111: @@
 | {{monzo| -14 15 -4 }}
 | [[Countritonic]] (5-limit)
-|}
+{{rank-2 end}}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+{{orf}}
 [[Category:Newt]]