Gallery of Z-polygon transversals: Difference between revisions
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-09 11:44:07 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>252369444</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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Below is a listing of some Z-polygon transverals for various well-known scales. Reading these into Scala and using the indicated subgroup generators for the horizonal and vertical factors in the "Lattice and player" under the "Analyze" pull-down menu in Scala, lattice diagrams of the convex closure of the scales in various planar temperaments can be obtained. Tempering the transversal in whatever tuning you favor you can make use of these convex closures; in fact, for microtemperaments such as breedsmic or ragismic you can keep the just intonation tuning and consider it tempered. The list below therefore covers some of the same ground as [[Diaconv scales]], but without giving an explicit tempering, something which is easily accomplished inside of Scala. | Below is a listing of some Z-polygon transverals for various well-known scales. Reading these into Scala and using the indicated subgroup generators for the horizonal and vertical factors in the "Lattice and player" under the "Analyze" pull-down menu in Scala, lattice diagrams of the convex closure of the scales in various planar temperaments can be obtained. Tempering the transversal in whatever tuning you favor you can make use of these convex closures; in fact, for microtemperaments such as breedsmic or ragismic you can keep the just intonation tuning and consider it tempered. The list below therefore covers some of the same ground as [[Diaconv scales]], but without giving an explicit tempering, something which is easily accomplished inside of Scala. | ||
=Septimal hexany= | |||
15/14 5/4 10/7 3/2 12/7 2 | |||
[[hexany_875|keemic]] | |||
[[hexany_245|sensamagic] | |||
[[hexany_126|starling]] | |||
[[hexany_1728|orwellismic]] | |||
[[hexany_1029|gamelismic]] | |||
[[hexany_225|marvel]] | |||
=7-limit diamond= | =7-limit diamond= | ||
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[[deka4375|ragismic]]</pre></div> | [[deka4375|ragismic]]</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Gallery of Z-polygon transversals</title></head><body><!-- ws:start:WikiTextTocRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Gallery of Z-polygon transversals</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Z-polytopes and convex closures">Z-polytopes and convex closures</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Septimal hexany">Septimal hexany</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#x7-limit diamond">7-limit diamond</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#x9-limit diamond">9-limit diamond</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Dekatesserany (2x2x2 chord cube)">Dekatesserany (2x2x2 chord cube)</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:16 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Z-polytopes and convex closures"></a><!-- ws:end:WikiTextHeadingRule:0 -->Z-polytopes and convex closures</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Z-polytopes and convex closures"></a><!-- ws:end:WikiTextHeadingRule:0 -->Z-polytopes and convex closures</h1> | ||
In geometry, a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_set" rel="nofollow">convex set</a> is a set of points such that for any two points in the set, the line segment connecting the points is also in the set. The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_hull" rel="nofollow">convex hull</a> of a set of points is the minimal convex set containing the given set, or in other words the intersection of all convex sets containing the set. A <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_lattice_polytope" rel="nofollow">Z-polytope</a> is a set of points with integer coordinates, such that every point with integer coordinates in its convex hull is already contained in the Z-polytope. A Z-polygon is a two-dimensional Z-polytope.<br /> | In geometry, a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_set" rel="nofollow">convex set</a> is a set of points such that for any two points in the set, the line segment connecting the points is also in the set. The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_hull" rel="nofollow">convex hull</a> of a set of points is the minimal convex set containing the given set, or in other words the intersection of all convex sets containing the set. A <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_lattice_polytope" rel="nofollow">Z-polytope</a> is a set of points with integer coordinates, such that every point with integer coordinates in its convex hull is already contained in the Z-polytope. A Z-polygon is a two-dimensional Z-polytope.<br /> | ||
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Below is a listing of some Z-polygon transverals for various well-known scales. Reading these into Scala and using the indicated subgroup generators for the horizonal and vertical factors in the &quot;Lattice and player&quot; under the &quot;Analyze&quot; pull-down menu in Scala, lattice diagrams of the convex closure of the scales in various planar temperaments can be obtained. Tempering the transversal in whatever tuning you favor you can make use of these convex closures; in fact, for microtemperaments such as breedsmic or ragismic you can keep the just intonation tuning and consider it tempered. The list below therefore covers some of the same ground as <a class="wiki_link" href="/Diaconv%20scales">Diaconv scales</a>, but without giving an explicit tempering, something which is easily accomplished inside of Scala.<br /> | Below is a listing of some Z-polygon transverals for various well-known scales. Reading these into Scala and using the indicated subgroup generators for the horizonal and vertical factors in the &quot;Lattice and player&quot; under the &quot;Analyze&quot; pull-down menu in Scala, lattice diagrams of the convex closure of the scales in various planar temperaments can be obtained. Tempering the transversal in whatever tuning you favor you can make use of these convex closures; in fact, for microtemperaments such as breedsmic or ragismic you can keep the just intonation tuning and consider it tempered. The list below therefore covers some of the same ground as <a class="wiki_link" href="/Diaconv%20scales">Diaconv scales</a>, but without giving an explicit tempering, something which is easily accomplished inside of Scala.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x7-limit diamond"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Septimal hexany"></a><!-- ws:end:WikiTextHeadingRule:2 -->Septimal hexany</h1> | ||
15/14 5/4 10/7 3/2 12/7 2<br /> | |||
<a class="wiki_link" href="/hexany_875">keemic</a><br /> | |||
<a class="wiki_link" href="/hexany_245">sensamagic][[hexany_126|starling</a><br /> | |||
<a class="wiki_link" href="/hexany_1728">orwellismic</a><br /> | |||
<a class="wiki_link" href="/hexany_1029">gamelismic</a><br /> | |||
<a class="wiki_link" href="/hexany_225">marvel</a><br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x7-limit diamond"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit diamond</h1> | |||
8/7 7/6 6/5 5/4 4/3 7/5 10/7 3/2 8/5 5/3 12/7 7/4 2<br /> | 8/7 7/6 6/5 5/4 4/3 7/5 10/7 3/2 8/5 5/3 12/7 7/4 2<br /> | ||
<a class="wiki_link" href="/diamond7_875">keemic</a><br /> | <a class="wiki_link" href="/diamond7_875">keemic</a><br /> | ||
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<a class="wiki_link" href="/diamond7_4375">ragismic</a><br /> | <a class="wiki_link" href="/diamond7_4375">ragismic</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x9-limit diamond"></a><!-- ws:end:WikiTextHeadingRule:6 -->9-limit diamond</h1> | ||
10/9 9/8 8/7 7/6 6/5 5/4 9/7 4/3 7/5 10/7 3/2 14/9 8/5 5/3 12/7 7/4 16/9 9/5 2<br /> | 10/9 9/8 8/7 7/6 6/5 5/4 9/7 4/3 7/5 10/7 3/2 14/9 8/5 5/3 12/7 7/4 16/9 9/5 2<br /> | ||
<a class="wiki_link" href="/diamond9_875">keemic</a><br /> | <a class="wiki_link" href="/diamond9_875">keemic</a><br /> | ||
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<a class="wiki_link" href="/diamond9_4375">ragismic</a><br /> | <a class="wiki_link" href="/diamond9_4375">ragismic</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Dekatesserany (2x2x2 chord cube)"></a><!-- ws:end:WikiTextHeadingRule:8 -->Dekatesserany (2x2x2 chord cube)</h1> | ||
21/20 15/14 35/32 9/8 5/4 21/16 35/24 3/2 49/32 25/16 105/64 7/4 15/8 2<br /> | 21/20 15/14 35/32 9/8 5/4 21/16 35/24 3/2 49/32 25/16 105/64 7/4 15/8 2<br /> | ||
<a class="wiki_link" href="/deka875">keemic</a><br /> | <a class="wiki_link" href="/deka875">keemic</a><br /> | ||