Kleismic rank three family
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These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. 7 and 9 limit minimax [|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>] [[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3 Map: [<1 0 1 0|, <0 6 5 0|, <0 0 0 1|] [[Generator|Generators]]: 2, 6/5, 7 [[edo|Edos]]: [[15edo|15]], [[19edo|19]], [[23edo|23]], [[34edo|34]], [[53edo|53]], [[68edo|68]], [[72edo|72]], [[87edo|87]], [[121edo|121]], [[125edo|125]], [[140edo|140]], [[212edo|212]], 333, 405 ==Enlil== [[Comma|Commas]]: 15625/15552, 385/384 [[Minimax tuning]] [|1 0 0 0 0>, |0 1 0 0 0>, |1 5/6 0 0 0>, |3 1/12 0 1/2 -1/2>, |3 1/12 0 -1/2 1/2>] [[Eigenmonzo|Eigenmonzos]]: 2, 4/3, 14/11 Map: [<1 0 1 0 6|, <0 6 5 0 1|, <0 0 0 1 -1|] [[Generator|Generators]]: 2, 6/5, 7 EDOs: 53, 68, 72, 140, 159, 212, 284 Badness: 0.000798 ===13-limit=== [[Comma|Commas]]: 325/324, 385/384, 625/624 13 and 15 limit minimax tuning [|1 0 0 0 0 0>, |4/3 0 0 -2/9 -2/9 4/9>, |19/9 0 0 -5/27 -5/27 10/27>, |28/9 0 0 13/27 -14/27 1/27>, |28/9 0 0 -14/27 13/27 1/27>, |28/9 0 0 -14/27 -14/27 28/27>] [[Eigenmonzo|Eigenmonzos]]: 2, 13/11, 14/11 Map: [<1 0 1 0 6 0|, <0 6 5 0 1 14|, <0 0 0 1 -1 0|] [[Generator|Generators]]: 2, 6/5, 7 [[edo|Edos]]: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[53edo|53]], [[57edo|57]], [[68edo|68]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[212edo|212]], 284, 299 Badness: 0.000680 Scales: [[enlil19_13]] ==Enki== [[Comma|Commas]]: 3025/3024, 4375/4356 [[Minimax tuning]] [|1 0 0 0 0>, |-18/13 0 6/13 -6/13 12/13>, |-2/13 0 5/13 -5/13 10/13>, |-2/13 0 -8/13 8/13 10/13>, |0 0 0 0 1>] [[Eigenmonzo|Eigenmonzos]]: 2, 11/8, 7/5 Map: [<1 0 1 0 1|, <0 6 5 0 4|, <0 0 0 2 1|] [[Generator|Generators]]: 2, 6/5, 66/25 [[edo|Edos]]: [[15edo|15]], [[49edo|49]], [[57edo|57]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], [[231|231]], 265, 280, 337, 352, 439 Badness: 0.00109 ===13-limit=== [[Comma|Commas]]: 325/324, 364/363, 625/624 13 and 15 limit [|1 0 0 0 0 0>, |2/3 0 -2/3 0 0 2/3>, |14/9 0 -5/9 0 0 5/9>, |0 0 0 1 0 0>, |13/9 0 -4/9 1/2 0 4/9>, |14/9 0 -14/9 0 0 14/9>] [[Eigenmonzo|Eigenmonzos]]: 2, 8/7 13/10 Map: [<1 0 1 0 1 0|, <0 6 5 0 4 14|, <0 0 0 2 1 0|] [[Generator|Generators]]: 2, 6/5, 66/25 [[edo|Edos]]: [[15edo|15]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], 280, 439 Badness: 0.000874
Original HTML content:
<html><head><title>Kleismic rank three family</title></head><body>These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.<br /> <br /> 7 and 9 limit minimax<br /> [|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7/6, 4/3<br /> <br /> Map: [<1 0 1 0|, <0 6 5 0|, <0 0 0 1|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 6/5, 7<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/23edo">23</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/121edo">121</a>, <a class="wiki_link" href="/125edo">125</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/212edo">212</a>, 333, 405<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Enlil"></a><!-- ws:end:WikiTextHeadingRule:0 -->Enlil</h2> <a class="wiki_link" href="/Comma">Commas</a>: 15625/15552, 385/384<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a><br /> [|1 0 0 0 0>, |0 1 0 0 0>, |1 5/6 0 0 0>, <br /> |3 1/12 0 1/2 -1/2>, |3 1/12 0 -1/2 1/2>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 4/3, 14/11<br /> <br /> Map: [<1 0 1 0 6|, <0 6 5 0 1|, <0 0 0 1 -1|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 6/5, 7<br /> EDOs: 53, 68, 72, 140, 159, 212, 284<br /> Badness: 0.000798<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Enlil-13-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->13-limit</h3> <a class="wiki_link" href="/Comma">Commas</a>: 325/324, 385/384, 625/624<br /> <br /> 13 and 15 limit minimax tuning<br /> [|1 0 0 0 0 0>, |4/3 0 0 -2/9 -2/9 4/9>, <br /> |19/9 0 0 -5/27 -5/27 10/27>, <br /> |28/9 0 0 13/27 -14/27 1/27>, <br /> |28/9 0 0 -14/27 13/27 1/27>, <br /> |28/9 0 0 -14/27 -14/27 28/27>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 13/11, 14/11<br /> <br /> Map: [<1 0 1 0 6 0|, <0 6 5 0 1 14|, <0 0 0 1 -1 0|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 6/5, 7<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/57edo">57</a>, <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/159edo">159</a>, <a class="wiki_link" href="/212edo">212</a>, 284, 299<br /> Badness: 0.000680<br /> <br /> Scales: <a class="wiki_link" href="/enlil19_13">enlil19_13</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-Enki"></a><!-- ws:end:WikiTextHeadingRule:4 -->Enki</h2> <a class="wiki_link" href="/Comma">Commas</a>: 3025/3024, 4375/4356<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a><br /> [|1 0 0 0 0>, |-18/13 0 6/13 -6/13 12/13>, <br /> |-2/13 0 5/13 -5/13 10/13>, <br /> |-2/13 0 -8/13 8/13 10/13>, |0 0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 11/8, 7/5<br /> <br /> Map: [<1 0 1 0 1|, <0 6 5 0 4|, <0 0 0 2 1|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 6/5, 66/25<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/49edo">49</a>, <a class="wiki_link" href="/57edo">57</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/159edo">159</a>, <a class="wiki_link" href="/193edo">193</a>, <a class="wiki_link" href="/231">231</a>, 265, 280, 337, 352, 439 <br /> Badness: 0.00109<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x-Enki-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h3> <a class="wiki_link" href="/Comma">Commas</a>: 325/324, 364/363, 625/624<br /> <br /> 13 and 15 limit<br /> [|1 0 0 0 0 0>, |2/3 0 -2/3 0 0 2/3>, <br /> |14/9 0 -5/9 0 0 5/9>, |0 0 0 1 0 0>, <br /> |13/9 0 -4/9 1/2 0 4/9>, <br /> |14/9 0 -14/9 0 0 14/9>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 8/7 13/10<br /> <br /> Map: [<1 0 1 0 1 0|, <0 6 5 0 4 14|, <0 0 0 2 1 0|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 6/5, 66/25<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/159edo">159</a>, <a class="wiki_link" href="/193edo">193</a>, 280, 439<br /> Badness: 0.000874</body></html>