Werckismic temperaments
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[[toc|flat]] =Rank four= These are rank three temperaments tempering out 441/440, but we can began by looking at the rank four temperament. Minimax tuning [|1 0 0 0 0>, |1/2 2/3 1/6 -1/3 1/6>, |0 0 1 0 0>, |1 -2/3 1/3 1/3 1/3>, |0 0 0 0 1>] Eigenmonzos: 2, 5, 11, 9/7 Map: [<1 0 0 0 -3|, <0 1 0 0 2|, <0 0 1 0 -1|, <0 0 0 1 2|] EDOs: 72, 118, 130, 159, 171, 190, 248, 289, 320 =History= Commas: 441/400, 4000/3993 Minimax tuning [|1 0 0 0 0>, |4/17 12/17 0 -6/17 6/17>, |-27/17 4/17 0 -2/17 19/17>, |8/17 -10/17 0 5/17 12/17>, |0 0 0 0 1>] Eigenmonzos: 2, 11, 9/7 Map: [<1 2 0 0 1|, <0 -6 0 7 2|, <0 0 1 1 1|] Generators: 2, 21/20, 5 EDOs: 58, 72, 130, 202, 217, 289, 491 Badness: 0.000604 ==13-limit== Commas: 364/363, 441/440, 1001/1000 Minimax tuning [|1 0 0 0 0 0>, |5/4 0 0 -3/8 0 3/8>, |0 0 1 0 0 0>, |7/8 0 1 7/16 0 -7/16>, |5/4 0 1 1/8 0 -1/8>, |7/8 0 1 -9/16 0 9/16>] Eigenmonzos: 2, 5, 13/7 Map: [<1 2 0 0 1 2|, <0 -6 0 7 2 -9|, <0 0 1 1 1 1|] Generators: 2, 21/20, 5 EDOs: 58, 72, 87, 130, 217, 289 Badness: 0.000540 [[Spectrum of a temperament|Spectrum]]: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7 =Varuna= Commas: 441/440, 8019/8000 Map: [<2 0 0 9 12|, <0 1 0 -4 -6|, <0 0 1 2 3|] EDOs: 12, 26, 46, 58, 72, 118, 130, 248, 320 Generators: 99/70, 3, 5 Badness: 0.000419
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<html><head><title>Werckismic temperaments</title></head><body><!-- ws:start:WikiTextTocRule:8:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#Rank four">Rank four</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#History">History</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Varuna">Varuna</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> <!-- ws:end:WikiTextTocRule:13 --><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Rank four"></a><!-- ws:end:WikiTextHeadingRule:0 -->Rank four</h1> These are rank three temperaments tempering out 441/440, but we can began by looking at the rank four temperament.<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0>, |1/2 2/3 1/6 -1/3 1/6>, |0 0 1 0 0>, <br /> |1 -2/3 1/3 1/3 1/3>, |0 0 0 0 1>]<br /> Eigenmonzos: 2, 5, 11, 9/7<br /> <br /> Map: [<1 0 0 0 -3|, <0 1 0 0 2|, <0 0 1 0 -1|, <0 0 0 1 2|]<br /> EDOs: 72, 118, 130, 159, 171, 190, 248, 289, 320<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="History"></a><!-- ws:end:WikiTextHeadingRule:2 -->History</h1> Commas: 441/400, 4000/3993<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0>, |4/17 12/17 0 -6/17 6/17>, <br /> |-27/17 4/17 0 -2/17 19/17>, <br /> |8/17 -10/17 0 5/17 12/17>, |0 0 0 0 1>]<br /> Eigenmonzos: 2, 11, 9/7<br /> <br /> Map: [<1 2 0 0 1|, <0 -6 0 7 2|, <0 0 1 1 1|]<br /> Generators: 2, 21/20, 5<br /> EDOs: 58, 72, 130, 202, 217, 289, 491<br /> Badness: 0.000604<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="History-13-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit</h2> Commas: 364/363, 441/440, 1001/1000<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0 0>, |5/4 0 0 -3/8 0 3/8>, |0 0 1 0 0 0>, <br /> |7/8 0 1 7/16 0 -7/16>, |5/4 0 1 1/8 0 -1/8>, <br /> |7/8 0 1 -9/16 0 9/16>]<br /> Eigenmonzos: 2, 5, 13/7<br /> <br /> Map: [<1 2 0 0 1 2|, <0 -6 0 7 2 -9|, <0 0 1 1 1 1|]<br /> Generators: 2, 21/20, 5<br /> EDOs: 58, 72, 87, 130, 217, 289<br /> Badness: 0.000540<br /> <br /> <a class="wiki_link" href="/Spectrum%20of%20a%20temperament">Spectrum</a>: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Varuna"></a><!-- ws:end:WikiTextHeadingRule:6 -->Varuna</h1> Commas: 441/440, 8019/8000<br /> <br /> Map: [<2 0 0 9 12|, <0 1 0 -4 -6|, <0 0 1 2 3|]<br /> EDOs: 12, 26, 46, 58, 72, 118, 130, 248, 320<br /> Generators: 99/70, 3, 5<br /> Badness: 0.000419</body></html>