Didymus rank-3 family
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-12-30 09:53:15 UTC.
- The original revision id was 288824195.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
These are the rank three temperaments tempering out the didymus comma, 81/80. 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. [[toc|flat]] =Euterpe= Commas: 81/80, 99/98 Minimax tuning [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |0 0 0 1 0>, |-1 0 -1/2 2 0>] Eigenmonzos: 2, 5, 7 Map: [<1 0 -4 0 1|, <0 1 4 0 -2|, <0 0 0 1 2|] EDOs: 5, 12, 26, 31, 88 Badness: 0.000536 =Calliope= Commas: 81/80, 45/44 Minimax tuning [|1 0 0 0 0>, |1 0 0 0 1/6>, |0 0 0 0 2/3>, |1 -1 0 1 1/6>, |0 0 0 0 1>] Eigenmonzos: 2, 7/3, 11 Map: [<1 0 -4 0 -6|, <0 1 4 0 6|, <0 0 0 1 0|] EDOs: 7, 12, 19, 26, 45 Badness: 0.000530 =Erato= Commas: 81/80, 126/125 Map: [<1 0 -4 -13 0|, <0 1 4 10 0], <0 0 0 0 1|] EDOs: 12, 19, 31, 81 Badness: 0.000558 =Clio= Commas: 81/80, 176/175 Minimax tuning ||1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |0 0 0 1 0>, |-4 0 2 1 0>] Eigenmonzos: 2, 5, 7 Map: [<1 0 -4 0 -12|, <0 1 4 0 8|, <0 0 0 1 1|] EDOs: 7, 12, 24, 31, 105, 129 Badness: 0.000738 =Polyhymnia= Commas: 81/80, 385/384 =Thalia= Commas: 81/80, 55/54 =Melpomene= Commas: 81/80, 56/55 =Urania= Commas: 81/80, 121/120 Lattice basis: 11/9 length 0.2536 8/7 length 2.807 Angle(11/9, 8/7) = 90 degrees Map to lattice: [<0 2 8 0 5|, <0 0 0 -1 0|] Map: [<1 1 0 0 2|, <0 2 8 0 5|, <0 0 0 1 0|] EDOs: 7, 24, 31 Badness: 0.000842 [[Spectrum of a temperament|Spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5 Scale: [[urania24]] =Terpsichore= Commas: 81/80, 540/539 Map: [<1 0 -4 0 -2|, <0 1 4 0 7|, <0 0 0 1 -2|] EDOs: 19, 31, 81, 112b Badness: 0.000850 Spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11
Original HTML content:
<html><head><title>Didymus rank three family</title></head><body>These are the rank three temperaments tempering out the didymus comma, 81/80. 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 /> <!-- ws:start:WikiTextTocRule:18:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#Euterpe">Euterpe</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Calliope">Calliope</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Erato">Erato</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Clio">Clio</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Polyhymnia">Polyhymnia</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Thalia">Thalia</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Melpomene">Melpomene</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Urania">Urania</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Terpsichore">Terpsichore</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> <!-- ws:end:WikiTextTocRule:28 --><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Euterpe"></a><!-- ws:end:WikiTextHeadingRule:0 -->Euterpe</h1> Commas: 81/80, 99/98<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, <br /> |0 0 0 1 0>, |-1 0 -1/2 2 0>]<br /> Eigenmonzos: 2, 5, 7<br /> <br /> Map: [<1 0 -4 0 1|, <0 1 4 0 -2|, <0 0 0 1 2|]<br /> EDOs: 5, 12, 26, 31, 88<br /> Badness: 0.000536<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Calliope"></a><!-- ws:end:WikiTextHeadingRule:2 -->Calliope</h1> Commas: 81/80, 45/44<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0>, |1 0 0 0 1/6>, |0 0 0 0 2/3>, <br /> |1 -1 0 1 1/6>, |0 0 0 0 1>]<br /> Eigenmonzos: 2, 7/3, 11<br /> <br /> Map: [<1 0 -4 0 -6|, <0 1 4 0 6|, <0 0 0 1 0|]<br /> EDOs: 7, 12, 19, 26, 45<br /> Badness: 0.000530<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Erato"></a><!-- ws:end:WikiTextHeadingRule:4 -->Erato</h1> Commas: 81/80, 126/125<br /> <br /> Map: [<1 0 -4 -13 0|, <0 1 4 10 0], <0 0 0 0 1|]<br /> EDOs: 12, 19, 31, 81<br /> Badness: 0.000558<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Clio"></a><!-- ws:end:WikiTextHeadingRule:6 -->Clio</h1> Commas: 81/80, 176/175<br /> <br /> Minimax tuning<br /> ||1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, <br /> |0 0 0 1 0>, |-4 0 2 1 0>]<br /> Eigenmonzos: 2, 5, 7<br /> <br /> Map: [<1 0 -4 0 -12|, <0 1 4 0 8|, <0 0 0 1 1|]<br /> EDOs: 7, 12, 24, 31, 105, 129<br /> Badness: 0.000738<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Polyhymnia"></a><!-- ws:end:WikiTextHeadingRule:8 -->Polyhymnia</h1> Commas: 81/80, 385/384<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Thalia"></a><!-- ws:end:WikiTextHeadingRule:10 -->Thalia</h1> Commas: 81/80, 55/54<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Melpomene"></a><!-- ws:end:WikiTextHeadingRule:12 -->Melpomene</h1> Commas: 81/80, 56/55<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Urania"></a><!-- ws:end:WikiTextHeadingRule:14 -->Urania</h1> Commas: 81/80, 121/120<br /> <br /> Lattice basis: 11/9 length 0.2536 8/7 length 2.807<br /> Angle(11/9, 8/7) = 90 degrees<br /> Map to lattice: [<0 2 8 0 5|, <0 0 0 -1 0|]<br /> <br /> Map: [<1 1 0 0 2|, <0 2 8 0 5|, <0 0 0 1 0|]<br /> EDOs: 7, 24, 31<br /> Badness: 0.000842<br /> <a class="wiki_link" href="/Spectrum%20of%20a%20temperament">Spectrum</a>: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5<br /> Scale: <a class="wiki_link" href="/urania24">urania24</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc8"><a name="Terpsichore"></a><!-- ws:end:WikiTextHeadingRule:16 -->Terpsichore</h1> Commas: 81/80, 540/539<br /> <br /> Map: [<1 0 -4 0 -2|, <0 1 4 0 7|, <0 0 0 1 -2|]<br /> EDOs: 19, 31, 81, 112b<br /> Badness: 0.000850<br /> Spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11</body></html>