Marvel family
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The head of the marvel family is marvel, which tempers out 225/224, the septimal kleisma or marvel comma. Marvel has a [[Normal lists|normal list basis]] of [2, 3, 5]; hence a [[Harmonic Limit|5-limit]] scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by [[197edo]]. Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[Kleismic family|catakleismic temperament]]. Another temperament which does little damage to tuning accuracy is [[Pythagorean family|compton temperament]], for which [[240edo]] may be used. ===Vital statistics=== [[Comma]] c = 225/224 7-limit minimax: 3 and 5 1/4c flat, 7 just [|1 0 0 0>, |5/4 1/2 -1/2 1/4>, |5/4 -1/2 1/2 1/4>, |0 0 0 1>] [[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 6/5 9-limit minimax: 3 1/6c flat, 5 1/3c flat, 7 just [|1 0 0 0>, |5/6 2/3 -1/3 1/6>, |5/3 -2/3 1/3 1/3>, |0 0 0 1>] [[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 10/9 [[Minkowski lattice]] basis: secor length 1.256, 3/2 length 1.369 Angle(secor, 3/2) = 106.958 cents Map to lattice: [<0 0 -1 -2|, <0 1 -1 0|] [[edo|EDOs]]: [[197edo|197]] ==Eleven limit children== The second comma of the [[Normal lists|normal comma list]] defines which 11-limit family member we are looking at. Adding 4125/4096 gives unidecimal marvel, 91125/90112 gives prodigy, 5632/5625 minerva and 243/242 spectacle. ===Unidecimal marvel=== [[Comma|Commas]]: 225/224, 385/384 [[Minimax tuning]]: [|1 0 0 0 0>, |4/3 8/9 -1/3 0 -1/9>, |8/3 -2/9 1/3 0 -2/9>, |3 4/3 0 0 -2/3>, |8/3 -2/9 -2/3 0 7/9>] [[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 11/9 Lattice basis: secor length 1.0364 5/4 length 1.0759 Angle(secor, 5/4) = 104.028 degrees Map to lattice: [<0 -1 0 -2 1|, <0 -1 1 0 -2|] Map: [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|] [[Generator|Generators]]: 2, 3, 5 [[edo|Edos]]: [[19edo|19]], [[22edo|22]], [[31edo|31]], [[41edo|41]], [[50edo|50]], [[53edo|53]], [[72edo|72]], [[166edo|166]], [[197edo|197]] ===Prodigy=== [[Comma|Commas]]: 225/224, 441/440 [[Minimax tuning]]: [|1 0 0 0 0>, |13/12 1/2 -1/4 0 1/12>, |13/6 -1 1/2 0 1/6>, |3/2 -1 1/2 0 1/2>, |0 0 0 0 1>] [[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 11/8 Lattice basis: secor length 0.9111, 3/2 length 0.9477 Angle(secor, 3/2) = 65.933 Map to lattice: [<0 0 -1 -2 -3|, <0 1 -1 0 3|] Map: [<1 0 0 -5 -13|, <0 1 0 2 6|, <0 0 1 2 3|] [[Generator|Generators]]: 2, 3, 5 [[edo|EDOs]]: [[10edo|10]], [[12edo|12]], [[29edo|29]], [[31edo|31]], [[41edo|41]], [[72edo|72]] ===Minerva=== [[Comma|Commas]]: 99/98, 225/224 [[Minimax tuning]]: [|1 0 0 0 0>, |0 1 0 0 0>, |9/4 -1/2 0 0 1/4>, |-1/2 1 0 0 1/2>, |0 0 0 0 1>] [[Eigenmonzo|Eigenmonzos]]: 2, 3/2, 11/8 Map: [<1 0 0 -5 -9|, <0 1 0 2 2|, <0 0 1 2 4|] [[Generator|Generators]]: 2, 3, 5 [[edo|EDOs]]: [[9edo|9]], [[12edo|12]], [[21edo|21]], [[22edo|22]], [[31edo|31]], [[43edo|43]], [[53edo|53]], [[74edo|74]], [[75edo|75]], [[96edo|96]], [[127edo|127]] ===Spectacle=== [[Comma|Commas]]: 225/224, 243/242 [[Minimax tuning]]: [|1 0 0 0 0>, |1/5 0 0 0 2/5>, |2/5 -2 1 0 4/5>, |-19/5 -4 2 0 12/5>, |0 0 0 0 1>] [[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 11/8 Map: [<1 1 0 -3 2|, <0 2 0 4 5|, <0 0 1 2 0|] [[Generator|Generators]]: 2, 11/9, 5 [[edo|EDOs]]: [[10edo|10]], [[31edo|31]], [[41edo|41]], [[72edo|72]], [[240edo|240]], [[250edo|250]]
Original HTML content:
<html><head><title>Marvel family</title></head><body>The head of the marvel family is marvel, which tempers out 225/224, the septimal kleisma or marvel comma. Marvel has a <a class="wiki_link" href="/Normal%20lists">normal list basis</a> of [2, 3, 5]; hence a <a class="wiki_link" href="/Harmonic%20Limit">5-limit</a> scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by <a class="wiki_link" href="/197edo">197edo</a>. <br /> <br /> Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to <a class="wiki_link" href="/Kleismic%20family">catakleismic temperament</a>. Another temperament which does little damage to tuning accuracy is <a class="wiki_link" href="/Pythagorean%20family">compton temperament</a>, for which <a class="wiki_link" href="/240edo">240edo</a> may be used.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h3> --><h3 id="toc0"><a name="x--Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:0 -->Vital statistics</h3> <a class="wiki_link" href="/Comma">Comma</a> c = 225/224<br /> <br /> 7-limit minimax: 3 and 5 1/4c flat, 7 just<br /> [|1 0 0 0>, |5/4 1/2 -1/2 1/4>, |5/4 -1/2 1/2 1/4>, |0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 5/4, 6/5<br /> <br /> 9-limit minimax: 3 1/6c flat, 5 1/3c flat, 7 just<br /> [|1 0 0 0>, |5/6 2/3 -1/3 1/6>, |5/3 -2/3 1/3 1/3>, |0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 8/7, 10/9<br /> <br /> <a class="wiki_link" href="/Minkowski%20lattice">Minkowski lattice</a> basis: secor length 1.256, 3/2 length 1.369<br /> Angle(secor, 3/2) = 106.958 cents <br /> Map to lattice: [<0 0 -1 -2|, <0 1 -1 0|]<br /> <a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/197edo">197</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Eleven limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->Eleven limit children</h2> The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 11-limit family member we are looking at. Adding 4125/4096 gives unidecimal marvel, 91125/90112 gives prodigy, 5632/5625 minerva and 243/242 spectacle.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x-Eleven limit children-Unidecimal marvel"></a><!-- ws:end:WikiTextHeadingRule:4 -->Unidecimal marvel</h3> <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 385/384<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>: <br /> [|1 0 0 0 0>, |4/3 8/9 -1/3 0 -1/9>, |8/3 -2/9 1/3 0 -2/9>,<br /> |3 4/3 0 0 -2/3>, |8/3 -2/9 -2/3 0 7/9>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 10/9, 11/9<br /> <br /> Lattice basis: secor length 1.0364 5/4 length 1.0759<br /> Angle(secor, 5/4) = 104.028 degrees<br /> Map to lattice: [<0 -1 0 -2 1|, <0 -1 1 0 -2|]<br /> <br /> Map: [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 5<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/166edo">166</a>, <a class="wiki_link" href="/197edo">197</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x-Eleven limit children-Prodigy"></a><!-- ws:end:WikiTextHeadingRule:6 -->Prodigy</h3> <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 441/440<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br /> [|1 0 0 0 0>, |13/12 1/2 -1/4 0 1/12>, <br /> |13/6 -1 1/2 0 1/6>,<br /> |3/2 -1 1/2 0 1/2>, |0 0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 10/9, 11/8<br /> <br /> Lattice basis: secor length 0.9111, 3/2 length 0.9477<br /> Angle(secor, 3/2) = 65.933<br /> Map to lattice: [<0 0 -1 -2 -3|, <0 1 -1 0 3|]<br /> <br /> Map: [<1 0 0 -5 -13|, <0 1 0 2 6|, <0 0 1 2 3|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 5<br /> <a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/10edo">10</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/29edo">29</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/72edo">72</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="x-Eleven limit children-Minerva"></a><!-- ws:end:WikiTextHeadingRule:8 -->Minerva</h3> <a class="wiki_link" href="/Comma">Commas</a>: 99/98, 225/224<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br /> [|1 0 0 0 0>, |0 1 0 0 0>, |9/4 -1/2 0 0 1/4>, |-1/2 1 0 0 1/2>, |0 0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 3/2, 11/8<br /> <br /> Map: [<1 0 0 -5 -9|, <0 1 0 2 2|, <0 0 1 2 4|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 5<br /> <a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/9edo">9</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/21edo">21</a>, <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/43edo">43</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/74edo">74</a>, <a class="wiki_link" href="/75edo">75</a>, <a class="wiki_link" href="/96edo">96</a>, <a class="wiki_link" href="/127edo">127</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="x-Eleven limit children-Spectacle"></a><!-- ws:end:WikiTextHeadingRule:10 -->Spectacle</h3> <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 243/242<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br /> [|1 0 0 0 0>, |1/5 0 0 0 2/5>, |2/5 -2 1 0 4/5>, |-19/5 -4 2 0 12/5>, |0 0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 10/9, 11/8<br /> <br /> Map: [<1 1 0 -3 2|, <0 2 0 4 5|, <0 0 1 2 0|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 11/9, 5<br /> <a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/10edo">10</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/240edo">240</a>, <a class="wiki_link" href="/250edo">250</a></body></html>