5L 2s/Temperaments: Difference between revisions

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Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[5L 2s]]) range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.

Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[5L 2s]]) range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.

== Meantone ==

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Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]

<div style="line-height:1.6;">Interval table (7-note MOS, 2.3.5.7 POTE tuning)</div>

<div style="line-height: 1.6;">Interval table (7-note MOS, 2.3.5.7 POTE tuning)</div>

{| class="wikitable right-1 right-2 sortable"

|-

! #

! Cents*

! class="unsortable"| Approximate ratios**

! class="unsortable" | Approximate ratios**

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Mapping generators: ~2, ~3

~~[[Wedgie]]: {{wedgie| 1 4 4 }}~~

[[Tuning ranges]]:

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Scales (Scala files): [[Flattone12]]

<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7.13 POTE tuning)</div>

<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.5.7.13 POTE tuning)</div>

{| class="wikitable right-1 right-2 sortable"

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! #Gens up

! Cents*

! class="unsortable"| Approximate ratios**

! class="unsortable" | Approximate ratios**

|-

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* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]

Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 ~~cents~~, equal to (1 + sqrt (19))/4.

Algebraic generator: Squarto, the positive root of {{nowrap|8''x''<sup>2</sup> − 4''x'' − 9}}, at 506.3239{{c}}, equal to <math>\frac{1+\sqrt{19}}{4}</math>.

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Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]

<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>

<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>

{| class="wikitable right-1 right-2 sortable"

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! #Gens up

! Cents*

! class="unsortable"| Approximate ratios**

! class="unsortable" | Approximate ratios**

|-

| 0

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* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]

Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 ~~cents~~. The recurrence converges quickly.

Algebraic generator: Cybozem, the real root of {{nowrap|15''x''<sup>3</sup> − 10''x''<sup>2</sup> − 18}}, which comes to 503.4257{{c}}. The recurrence converges quickly.

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* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

Algebraic generator: Cybozem; or else Radieubiz, the real root of {{nowrap|3''x''<sup>3</sup> + 6''x'' − 19}}. Unlike Cybozem, the recurrence for Radieubiz does not converge.

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* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 700.000]

[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 ~~cents~~.

[[Algebraic generator]]: Traverse, the positive real root of {{nowrap|''x''<sup>4</sup> + 2''x'' − 13}}, or 696.9529{{c}}.

{{Optimal ET sequence|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}

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Scales: [[Garibaldi12]], [[Garibaldi17]]

<div style="line-height:1.6;">Interval table (29-note MOS, 2.3.5.7.11.13.19 POTE tuning)</div>

<div style="line-height: 1.6;">Interval table (29-note MOS, 2.3.5.7.11.13.19 POTE tuning)</div>

{| class="wikitable right-1 right-2 sortable"

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! #Gens up

! Cents*

! class="unsortable"| Approximate ratios**

! class="unsortable" | Approximate ratios**

|-

| 0

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EDO generators: [[17edo|10\17]], [[29edo|17\29]], [[46edo|27\46]]

<div style="line-height:1.6;">Interval table (17-note MOS, 2.3.7.11.13 POTE tuning)</div>

<div style="line-height: 1.6;">Interval table (17-note MOS, 2.3.7.11.13 POTE tuning)</div>

{| class="wikitable right-1 right-2 sortable"

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! #Gens up

! Cents*

! class="unsortable"| Approximate ratios**

! class="unsortable" | Approximate ratios**

|-

| 0

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Scales: [[Archy5]], [[Archy7]], [[Archy12]]

<div style="line-height:1.6;">Interval table (7-note MOS, 2.3.7 POTE tuning)</div>

<div style="line-height: 1.6;">Interval table (7-note MOS, 2.3.7 POTE tuning)</div>

{| class="wikitable right-1 right-2 sortable"

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! #Gens up

! Cents*

! class="unsortable"| Approximate ratios**

! class="unsortable" | Approximate ratios**

|-

| 0

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Scales: [[Supra7]], [[Supra12]]

<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.7.11 POTE tuning)</div>

<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.7.11 POTE tuning)</div>

{| class="wikitable right-1 right-2 sortable"

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! #Gens up

! Cents*

! class="unsortable"| Approximate ratios**

! class="unsortable" | Approximate ratios**

|-

| 0

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EDO generators: [[22edo|13\22]], [[27edo|16\27]], [[49edo|29\49]]

<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>

<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>

{| class="wikitable right-1 right-2 sortable"

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! #Gens up

! Cents*

! class="unsortable"| Approximate ratios**

! class="unsortable" | Approximate ratios**

|-

| 0

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[[Mapping]]: [{{val|1 1 -3 4}}, {{val|0 1 9 -2}}]

~~[[Wedgie]]: {{wedgie|1 9 -2 12 -6 -30}}~~

[[Val]]s: {{Optimal ET sequence| 5, 17, 22, 27, 49 }}

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EDO generators: [[32edo|19\32]], [[37edo|22\37]]

<div style="line-height:1.6;">Interval table (22-note MOS, 2.3.5.7.13 POTE tuning)</div>

<div style="line-height: 1.6;">Interval table (22-note MOS, 2.3.5.7.13 POTE tuning)</div>

{| class="wikitable right-1 right-2 sortable"

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! #Gens up

! Cents*

! class="unsortable"| Approximate ratios**

! class="unsortable" | Approximate ratios**

|-

| 0

@@ Line 1: / Line 1: @@
 {{breadcrumb}}
-Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[5L 2s]]) range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.
+Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[5L&nbsp;2s]]) range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L&nbsp;2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.
 == Meantone ==
@@ Line 14: / Line 14: @@
 Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
-<div style="line-height:1.6;">Interval table (7-note MOS, 2.3.5.7 POTE tuning)</div>
+<div style="line-height: 1.6;">Interval table (7-note MOS, 2.3.5.7 POTE tuning)</div>
 <div class="mw-collapsible-content">
 {| class="wikitable right-1 right-2 sortable"
 |-
 ! &#35;
 ! Cents*
-! class="unsortable"| Approximate ratios**
+! class="unsortable" | Approximate ratios**
 |-
 | 0
@@ Line 61: / Line 60: @@
 Mapping generators: ~2, ~3
-[[Wedgie]]: {{wedgie| 1 4 4 }}
 [[Tuning ranges]]:
@@ Line 84: / Line 81: @@
 Scales (Scala files): [[Flattone12]]
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
-<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7.13 POTE tuning)</div>
+<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.5.7.13 POTE tuning)</div>
 <div class="mw-collapsible-content">
 {| class="wikitable right-1 right-2 sortable"
@@ Line 91: / Line 88: @@
 ! #Gens up
 ! Cents*
-! class="unsortable"| Approximate ratios**
+! class="unsortable" | Approximate ratios**
 |-
 | 0
@@ Line 164: / Line 161: @@
 * 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]
-Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
+Algebraic generator: Squarto, the positive root of {{nowrap|8''x''<sup>2</sup> − 4''x'' − 9}}, at 506.3239{{c}}, equal to <math>\frac{1+\sqrt{19}}{4}</math>.
 {{Optimal ET sequence|legend=1| 7, 19, 26, 45f }}
@@ Line 181: / Line 178: @@
 Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
-<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
+<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
 <div class="mw-collapsible-content">
 {| class="wikitable right-1 right-2 sortable"
@@ Line 188: / Line 185: @@
 ! #Gens up
 ! Cents*
-! class="unsortable"| Approximate ratios**
+! class="unsortable" | Approximate ratios**
 |-
 | 0
@@ Line 258: / Line 255: @@
 * 9-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]
-Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
+Algebraic generator: Cybozem, the real root of {{nowrap|15''x''<sup>3</sup> − 10''x''<sup>2</sup> − 18}}, which comes to 503.4257{{c}}. The recurrence converges quickly.
 {{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}
@@ Line 290: / Line 287: @@
 * 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]
-Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
+Algebraic generator: Cybozem; or else Radieubiz, the real root of {{nowrap|3''x''<sup>3</sup> + 6''x'' − 19}}. Unlike Cybozem, the recurrence for Radieubiz does not converge.
 {{Optimal ET sequence|legend=1| 12e, 19, 31, 81 }}
@@ Line 322: / Line 319: @@
 * 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 700.000]
-[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
+[[Algebraic generator]]: Traverse, the positive real root of {{nowrap|''x''<sup>4</sup> + 2''x'' − 13}}, or 696.9529{{c}}.
 {{Optimal ET sequence|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}
@@ Line 339: / Line 336: @@
 Scales: [[Garibaldi12]], [[Garibaldi17]]
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
-<div style="line-height:1.6;">Interval table (29-note MOS, 2.3.5.7.11.13.19 POTE tuning)</div>
+<div style="line-height: 1.6;">Interval table (29-note MOS, 2.3.5.7.11.13.19 POTE tuning)</div>
 <div class="mw-collapsible-content">
 {| class="wikitable right-1 right-2 sortable"
@@ Line 346: / Line 343: @@
 ! #Gens up
 ! Cents*
-! class="unsortable"| Approximate ratios**
+! class="unsortable" | Approximate ratios**
 |-
 | 0
@@ Line 486: / Line 483: @@
 EDO generators: [[17edo|10\17]], [[29edo|17\29]], [[46edo|27\46]]
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
-<div style="line-height:1.6;">Interval table (17-note MOS, 2.3.7.11.13 POTE tuning)</div>
+<div style="line-height: 1.6;">Interval table (17-note MOS, 2.3.7.11.13 POTE tuning)</div>
 <div class="mw-collapsible-content">
 {| class="wikitable right-1 right-2 sortable"
@@ Line 493: / Line 490: @@
 ! #Gens up
 ! Cents*
-! class="unsortable"| Approximate ratios**
+! class="unsortable" | Approximate ratios**
 |-
 | 0
@@ Line 587: / Line 584: @@
 Scales: [[Archy5]], [[Archy7]], [[Archy12]]
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
-<div style="line-height:1.6;">Interval table (7-note MOS, 2.3.7 POTE tuning)</div>
+<div style="line-height: 1.6;">Interval table (7-note MOS, 2.3.7 POTE tuning)</div>
 <div class="mw-collapsible-content">
 {| class="wikitable right-1 right-2 sortable"
@@ Line 594: / Line 591: @@
 ! #Gens up
 ! Cents*
-! class="unsortable"| Approximate ratios**
+! class="unsortable" | Approximate ratios**
 |-
 | 0
@@ Line 647: / Line 644: @@
 Scales: [[Supra7]], [[Supra12]]
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
-<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.7.11 POTE tuning)</div>
+<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.7.11 POTE tuning)</div>
 <div class="mw-collapsible-content">
 {| class="wikitable right-1 right-2 sortable"
@@ Line 654: / Line 651: @@
 ! #Gens up
 ! Cents*
-! class="unsortable"| Approximate ratios**
+! class="unsortable" | Approximate ratios**
 |-
 | 0
@@ Line 726: / Line 723: @@
 EDO generators: [[22edo|13\22]], [[27edo|16\27]], [[49edo|29\49]]
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow :auto;">
-<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
+<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
 <div class="mw-collapsible-content">
 {| class="wikitable right-1 right-2 sortable"
@@ Line 733: / Line 730: @@
 ! #Gens up
 ! Cents*
-! class="unsortable"| Approximate ratios**
+! class="unsortable" | Approximate ratios**
 |-
 | 0
@@ Line 789: / Line 786: @@
 [[Mapping]]: [{{val|1 1 -3 4}}, {{val|0 1 9 -2}}]
-[[Wedgie]]: {{wedgie|1 9 -2 12 -6 -30}}
 [[Val]]s: {{Optimal ET sequence| 5, 17, 22, 27, 49 }}
@@ Line 805: / Line 800: @@
 EDO generators: [[32edo|19\32]], [[37edo|22\37]]
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
-<div style="line-height:1.6;">Interval table (22-note MOS, 2.3.5.7.13 POTE tuning)</div>
+<div style="line-height: 1.6;">Interval table (22-note MOS, 2.3.5.7.13 POTE tuning)</div>
 <div class="mw-collapsible-content">
 {| class="wikitable right-1 right-2 sortable"
@@ Line 812: / Line 807: @@
 ! #Gens up
 ! Cents*
-! class="unsortable"| Approximate ratios**
+! class="unsortable" | Approximate ratios**
 |-
 | 0