581edo: Difference between revisions

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== 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 ===

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| 2.3

| {{monzo| 921 -581 }}

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

| {{mapping| 581 921 }}

| -0.0891

| 0.0891

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| 2.3.5

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

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

| {{mapping| 581 921 1349 }}

| -0.0475

| 0.0936

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| 2.3.5.7

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

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

| {{mapping| 581 921 1349 1631 }}

| -0.0222

| 0.0922

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| 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

| 0.0828

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| 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

| 0.0756

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| 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

| 0.0738

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| 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

| 0.0717

Line 70:

| 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

| 0.0800

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

Line 123:

| [[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 3: / Line 3: @@
 == 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 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 &amp; 311 microtemperament, which features a neutral-third generator.
 === Prime harmonics ===
@@ Line 21: / Line 21: @@
 | 2.3
 | {{monzo| 921 -581 }}
-| [{{val| 581 921 }}]
+| {{mapping| 581 921 }}
 | -0.0891
 | 0.0891
@@ Line 28: / Line 28: @@
 | 2.3.5
 | {{monzo| -29 -11 20 }}, {{monzo| 33 -34 9 }}
-| [{{val| 581 921 1349 }}]
+| {{mapping| 581 921 1349 }}
 | -0.0475
 | 0.0936
@@ Line 35: / Line 35: @@
 | 2.3.5.7
 | 2401/2400, 33554432/33480783, 48828125/48771072
-| [{{val| 581 921 1349 1631 }}]
+| {{mapping| 581 921 1349 1631 }}
 | -0.0222
 | 0.0922
@@ Line 42: / Line 42: @@
 | 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
 | 0.0828
@@ Line 49: / Line 49: @@
 | 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
 | 0.0756
@@ Line 56: / Line 56: @@
 | 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
 | 0.0738
@@ Line 63: / Line 63: @@
 | 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
 | 0.0717
@@ Line 70: / Line 70: @@
 | 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
 | 0.0800
@@ Line 82: / Line 82: @@
 |+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 123: / Line 123: @@
 | [[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]]