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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.  


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{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #
! #
! Cents<ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5, odd limit ≤ 27</ref>
! class="unsortable"| Approximate ratios**
|-
|-
| 0
| 0
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| 25/18
| 25/18
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />* 2.3.5, odd limit &le; 27</div></div>


Comma list: 81/80
Comma list: 81/80
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<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5.7.13, odd limit ≤ 27</ref>
! class="unsortable"| Approximate ratios**
|-
|-
| 0
| 0
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| [[9/7]]
| [[9/7]]
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.5.7.13, odd limit &le; 27</div></div>


[[Comma list]]: 65/64, 81/80, 105/104
[[Comma list]]: 65/64, 81/80, 105/104
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<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref>
! class="unsortable"| Approximate ratios**
|-
|-
| 0
| 0
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| 21/16
| 21/16
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.5.7, odd limit &le; 27</div></div>
[[Comma]] list: 81/80, 126/125
[[Comma]] list: 81/80, 126/125


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<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5.7.11.13.19</ref>
! class="unsortable"| Approximate ratios**
|-
|-
| 0
| 0
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| 13/10
| 13/10
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** In the 2.3.5.7.11.13.19 subgroup</div></div>
[[Comma list]]: 190/189, 209/208, 225/224, 275/273, 325/324
[[Comma list]]: 190/189, 209/208, 225/224, 275/273, 325/324


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<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.7.11.13, odd limit ≤ 27</ref>
! class="unsortable"| Approximate ratios**
|-
|-
| 0
| 0
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| 21/16
| 21/16
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.7.11.13, odd limit &le; 27</div></div>
[[Comma list]]: 169/168, 352/351, 364/363
[[Comma list]]: 169/168, 352/351, 364/363


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<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.7, odd limit ≤ 27</ref>
! class="unsortable"| Approximate ratios**
|-
|-
| 0
| 0
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|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.7, odd limit &le; 27</div></div>
[[Comma]]: 64/63
[[Comma]]: 64/63


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<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.7.11, odd limit ≤ 27</ref>
! class="unsortable"| Approximate ratios**
|-
|-
| 0
| 0
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|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.7.11, odd limit &le; 27</div></div>
Comma list: 64/63, 99/98
Comma list: 64/63, 99/98


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<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref>
! class="unsortable"| Approximate ratios**
|-
|-
| 0
| 0
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| 10/7
| 10/7
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.5.7, odd limit &le; 27</div></div>
[[Comma list]]: 64/63, 245/243
[[Comma list]]: 64/63, 245/243


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<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5.7.13, odd limit ≤ 15</ref>
! class="unsortable"| Approximate ratios**
|-
|-
| 0
| 0
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|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.5.7.13, odd limit &le; 15</div></div>
Comma list: 64/63, 91/90, 4394/4375
Comma list: 64/63, 91/90, 4394/4375


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