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== Rank-2 temperaments ==
== Rank-2 temperaments ==
Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the 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 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 (12&19, 2.3.5) ===
=== Meantone (12&19, 2.3.5) ===
{{main| Meantone }}
Period: 1\1
Period: 1\1


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<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">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+  
|+  
|-
|-
! #Gens up
! #
! Cents <ref>octave-reduced</ref>
! Cents<ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5, odd limit ≤ 27</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5, odd limit ≤ 27</ref>
|-
|-
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|}
|}
<references/></div></div>
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [{{val|1 0 -4}}, {{val|0 1 4}}]


Comma: 81/80
Comma list: 81/80


Mapping generator: ~3
[[Mapping]]: [{{val|1 0 -4}}, {{val|0 1 4}}]


[[Tuning Ranges of Regular Temperaments|valid range]]: [685.714, 720.000] (7 to 5)
Mapping generators: ~2, ~3


[[Tuning Ranges of Regular Temperaments|nice range]]: [694.786, 701.955] (1/3 comma to Pythagorean)
[[Wedgie]]: {{wedgie| 1 4 4 }}


[[Tuning Ranges of Regular Temperaments|strict range]]: [694.786, 701.955]
[[Tuning ranges]]:  


{{EDOs|legend=1| 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb }}
* valid range: [685.714, 720.000] (7 to 5)
* nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
* strict range: [694.786, 701.955]


[[Wedgie]]: {{wedgie|1 4 4}}
{{Vals|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}


[[Badness]]: 0.00736
[[Badness]]: 0.00736
</div></div>
</div></div>
==== Flattone (19&26, 2.3.5.7.13) ====
==== Flattone (19&26, 2.3.5.7.13) ====
Period: 1\1
Period: 1\1
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|}
|}
<references/></div></div>
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [{{val|1 0 -4 17 10}}, {{val|0 1 4 -9 -4}}]


[[7-odd-limit|7-limit]] minimax
[[Comma]] list: 81/80, 525/512


[{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }},
[[Mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]
{{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}<nowiki>]</nowiki>


[[Eigenmonzo]]s: 2, 7/5
Mapping generators: ~2, ~3


[[9-odd-limit|9-limit]] minimax
[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}


[{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }},
[[Minimax tuning]]:
{{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}<nowiki>]</nowiki>


Eigenmonzos: 2, 9/7
* [[7-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
: [[Eigenmonzo]]s: 2, 7/5


valid range: [692.308, 694.737] (26 to 19)
* [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }}, {{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}]
: Eigenmonzos: 2, 9/7


nice range: [692.353, 701.955]
[[Tuning ranges]]:
 
* valid range: [692.308, 694.737] (26 to 19)
strict range: [692.353, 694.737]
* nice range: [692.353, 701.955]
 
* strict range: [692.353, 694.737]
Mapping generator: ~3


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 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.


[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}
{{Vals|legend=1| 7, 19, 26, 45 }}


[[Generator]]s: 2, 3
[[Badness]]: 0.0386


{{EDOs|legend=1| 7, 19, 26, 45 }}
[[Badness]]: 0.0386
</div></div>
</div></div>


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|}
<references/></div></div>
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]


[[Comma]]s: 81/80, 126/125
[[Comma]] list: 81/80, 126/125


7 and [[9-odd-limit|9-limit]] minimax
[[Mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]


[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
Mapping generators: ~2, ~3


[[Eigenmonzo]]s: 2, 5
[[Wedgie]]: {{wedgie| 1 4 10 4 13 12 }}


[[Tuning Ranges of Regular Temperaments|valid range]]: [694.737, 700.000] (19 to 12)
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
: [[Eigenmonzo]]s: 2, 5


nice range: [694.786, 701.955]
[[Tuning ranges]]:
 
* valid range: [694.737, 700.000] (19 to 12)
strict range: [694.786, 700.000]
* nice range: [694.786, 701.955]
 
* strict range: [694.786, 700.000]
Mapping generator: ~3


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 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.


[[Wedgie]]: {{wedgie|1 4 10 4 13 12}}
{{Vals|legend=1| 12, 19, 31, 81, 112b, 143b }}


[[Vals]]: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb
[[Badness]]: 0.0137


[[Badness]]: 0.0137
</div></div>
</div></div>
===== Meanpop (31&50, 2.3.5.7.11) =====
===== Meanpop (31&50, 2.3.5.7.11) =====
Period: 1\1
Period: 1\1


Optimal ([[POTE]]) generator: 696.434
Optimal ([[POTE]]) generator: ~3/2 = 696.434


EDO generators: [[50edo|29\50]], [[69edo|40\69]], [[81edo|47\81]]
EDO generators: [[50edo|29\50]], [[69edo|40\69]], [[81edo|47\81]]


Mapping: Same as septimal meantone, plus -13 gens = 11/8
Mapping: Same as septimal meantone, plus -13 gens = 11/8
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
[[Map|Period-generator mapping]]: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
Comma list: 81/80, 126/125, 385/384


Commas: 81/80, 126/125, 385/384
Mapping: [{{val| 1 0 -4 -13 24 }}, {{val| 0 1 4 10 -13 }}]


[[11-odd-limit|11-limit]] minimax 1/4 comma
Mapping generator: ~2, ~3


[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }},
Minimax tuning:
{{Monzo| -3 0 5/2 0 0 }}, {{Monzo| 11 0 -13/4 0 0 }}]
* [[11-odd-limit]]: 1/4 comma
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| -3 0 5/2 0 0 }}, {{Monzo| 11 0 -13/4 0 0 }}]
: [[Eigenmonzo]]s: 2, 5


[[Eigenmonzo]]s: 2, 5
Tuning ranges:
* valid range: [694.737, 696.774] (19 to 31)
* nice range: [691.202, 701.955]
* strict range: [694.737, 696.774]


valid range: [694.737, 696.774] (19 to 31)
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.


nice range: [691.202, 701.955]
{{Vals|legend=1| 12e, 19, 31, 81 }}


strict range: [694.737, 696.774]
[[Badness]]: 0.0215
 
Mapping generator: ~3
 
[[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.
 
[[Generator]]s: 2, 3


{{EDOs|legend=1| 12, 19, 31, 50, 81 }}
[[Badness]]: 0.0215
</div></div>
</div></div>


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Mapping: Same as septimal meantone, plus 18 gens = 11/8
Mapping: Same as septimal meantone, plus 18 gens = 11/8
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
Comma list: 81/80, 126/125, 99/98
 
[[Comma]]s: 81/80, 126/125, 99/98
 
[[11-odd-limit|11-limit]] minimax
 
[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 25/16 -1/8 0 0 1/16 }}, {{Monzo| 9/4 -1/2 0 0 1/4 }},
{{Monzo| 21/8 -5/4 0 0 5/8 }}, {{Monzo| 25/8 -9/4 0 0 9/8 }}<nowiki>]</nowiki>
 
[[Eigenmonzo]]s: 2, 11/9


valid range: [696.774, 700.000] (31 to 12)
Mapping: [{{val| 1 0 -4 -13 -25 }}, {{val| 0 1 4 10 18 }}]


nice range: [691.202, 701.955]
Mapping generators: ~2, ~3


strict range: [696.774, 700.000]
Minimax tuning:  
* [[11-odd-limit]]
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 25/16 -1/8 0 0 1/16 }}, {{Monzo| 9/4 -1/2 0 0 1/4 }}, {{Monzo| 21/8 -5/4 0 0 5/8 }}, {{Monzo| 25/8 -9/4 0 0 9/8 }}]
: [[Eigenmonzo]]s: 2, 11/9


Mapping generator: ~3
Tuning ranges:  
* valid range: [696.774, 700.000] (31 to 12)
* nice range: [691.202, 701.955]
* strict range: [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 ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.


[[Generator]]s: 2, 3
{{Vals|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}


{{EDOs|legend=1| 12, 31, 43, 74, 105, 198be }}
Badness: 0.0170


[[Badness]]: 0.0170
</div></div>
</div></div>


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