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5L 2s contains the pentatonic MOS [[2L_3s|2L 3s]] and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either [[7L_5s|7L 5s]] or [[5L_7s|5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (700c).
5L 2s contains the pentatonic MOS [[2L_3s|2L 3s]] and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either [[7L_5s|7L 5s]] or [[5L_7s|5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (700c).
== Rank-2 temperaments ==
Below are some important [[rank]]-2 [[temperaments]] with optimal generator size in the 5L 2s range. The temperaments are listed roughly in order of increasing generator size and child temperaments are extensions of low-[[complexity]] parent temperaments.
=== Meantone (12&19, 2.3.5) ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 696.239
EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
<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 class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #Gens up
! Cents <ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5, odd limit ≤ 27</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 696.2
| 3/2
|-
| 2
| 192.5
| 9/8, 10/9
|-
| 3
| 888.7
| 5/3
|-
| 4
| 385.0
| 5/4
|-
| 5
| 1081.2
| 15/8
|-
| 6
| 577.434
| 25/18
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [{{val|1 0 -4}}, {{val|0 1 4}}]
Comma: 81/80
Mapping generator: ~3
[[Tuning Ranges of Regular Temperaments|valid range]]: [685.714, 720.000] (7 to 5)
[[Tuning Ranges of Regular Temperaments|nice range]]: [694.786, 701.955] (1/3 comma to Pythagorean)
[[Tuning Ranges of Regular Temperaments|strict range]]: [694.786, 701.955]
{{EDOs|legend=1| 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb }}
[[Wedgie]]: {{wedgie|1 4 4}}
[[Badness]]: 0.00736
</div></div>
==== Flattone (19&26, 2.3.5.7.13) ====
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 693.7498
EDO generators: [[19edo|11\19]], [[26edo|15\26]], [[45edo|26\45]], [[64edo|37\64]]
Scales (Scala files): [[Flattone12]]
<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 class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #Gens up
! Cents <ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7.13, odd limit ≤ 27</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 693.7
| [[3/2]]
|-
| 2
| 187.5
| [[9/8]], [[10/9]]
|-
| 3
| 881.2
| [[5/3]]
|-
| 4
| 375.0
| [[5/4]], [[16/13]]
|-
| 5
| 1068.7
| 15/8, [[24/13]]
|-
| 6
| 562.5
| [[18/13]]
|-
| 7
| 56.2
|
|-
| 8
| 750.0
| [[20/13]]
|-
| 9
| 243.7
| [[8/7]]
|-
| 10
| 937.5
| [[12/7]]
|-
| 11
| 431.2
| [[9/7]]
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<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
[{{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 }}<nowiki>]</nowiki>
[[Eigenmonzo]]s: 2, 7/5
[[9-odd-limit|9-limit]] minimax
[{{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 }}<nowiki>]</nowiki>
Eigenmonzos: 2, 9/7
valid range: [692.308, 694.737] (26 to 19)
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.
[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}
[[Generator]]s: 2, 3
{{EDOs|legend=1| 7, 19, 26, 45 }}
[[Badness]]: 0.0386
</div></div>
==== Septimal meantone (19&12, 2.3.5.7) ====
Period: 1\1
Optimal ([[POTE]]) generator: 696.495
EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
<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 class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #Gens up
! Cents <ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 696.5
| 3/2
|-
| 2
| 193.0
| 9/8, 10/9
|-
| 3
| 889.5
| 5/3
|-
| 4
| 386.0
| 5/4
|-
| 5
| 1082.5
| 15/8, 28/15
|-
| 6
| 579.0
| 7/5
|-
| 7
| 75.5
| 21/20, 25/24, 28/27
|-
| 8
| 772.0
| 14/9, 25/16
|-
| 9
| 268.5
| 7/6
|-
| 10
| 965.0
| 7/4
|-
| 11
| 461.4
| 21/16
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<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
7 and [[9-odd-limit|9-limit]] minimax
[{{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
[[Tuning Ranges of Regular Temperaments|valid range]]: [694.737, 700.000] (19 to 12)
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.
[[Wedgie]]: {{wedgie|1 4 10 4 13 12}}
[[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
</div></div>
===== Meanpop (31&50, 2.3.5.7.11) =====
*-13 gens = 11/8
*Good EDO gens: 29\50
===== Huygens (31&43, 2.3.5.7.11) =====
*18 gens = 11/8
*Good EDO gens: 25\43
=== Schismic/Garibaldi (12&29, 2.3.5.7) ===
Commas: 225/224, 3125/3087
7-limit minimax tuning:
7-limit: [|1 0 0 0&gt;, |5/3 1/15 0 -1/15&gt;,
|5/3 -8/15 0 8/15&gt;, |5/3 -14/15 0 14/15&gt;<nowiki>]</nowiki>
[[Eigenmonzo|Eigenmonzo]]s: 2, 7/6
9-limit: [|1 0 0 0&gt;, |25/16 1/8 0 -1/16&gt;,
|5/2 -1 0 1/2&gt;, |25/8 -7/4 0 7/8&gt;<nowiki>]</nowiki>
Eigenmonzos: 2, 9/7
[[POTE generator]]: ~3/2 = 702.085
Mapping generator: ~3
Map: [&lt;1 0 15 25|, &lt;0 1 -8 -14|]
Wedgie: &lt;&lt;1 -8 -14 -15 -25 -10||
{{EDOs|legend=1| 12, 29, 41, 53, 94, 241c, 335cd, 576cd }}
Badness: 0.0216
=== 41&53, 2.3.5.7.11.13.19 ===
*-14 gens = 7/4
*23 gens = 11/8
*20 gens = 13/8
=== Parapyth (29&17, 2.3.7.11.13) ===
Commas: 169/168, 352/351, 364/363
[[Lp_tuning|POL2 generator]]: ~3/2 = 704.745 cents
Gencom: [2 3/2; 169/169 352/351 364/363]
Gencom mapping: [&lt;1 1 0 -6 -3 -1|, &lt;0 1 0 15 11 8|]
Sval map: [&lt;1 0 -21 -14 -9|, &lt;0 1 15 11 8|]
EDOs: 17, 46, 63
[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents
=== Archy (17&5, 2.3.7) ===
Comma: 64/63
[[Lp_tuning|POL2 generator]]: ~3/2 = 709.321
Gencom: [2 3/2; 64/63]
Gencom mapping: [&lt;1 1 0 4|, &lt;0 1 0 -2|]
Map: [&lt;1 2 2|, &lt;0 -1 2|]
EDOs: 5, 12, 17, 22, 27, 137bc
[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents
==== Supra (17&22, 2.3.7.11) ====
Commas: 64/63, 99/98
[[POTE_tuning|POTE generator]]: ~3/2 = 707.192
Gencom: [2 3/2; 64/63 99/98]
Gencom mapping: [&lt;1 1 0 4 7|, &lt;0 1 0 -2 -6|]
Sval map: [&lt;1 0 6 13|, &lt;0 1 -2 -6|]
EDOs: 5, 12, 17, 39c, 56c
[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
==== Superpyth (22&27, 2.3.5.7) ====
Commas: 64/63, 245/243
POTE generator: ~3/2 = 710.291
Map: [&lt;1 0 -12 6|, &lt;0 1 9 -2|]
Wedgie: &lt;&lt;1 9 -2 12 -6 -30||
EDOs: 5, 17, 22, 27, 49
Badness: 0.0323
==== Ultrapyth (27&37, 2.3.7.13/10) ====
*4 gens = 13/10


[[Category:Scales]]
[[Category:Scales]]
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