Gallery of Z-polygon transversals: Difference between revisions
Wikispaces>genewardsmith **Imported revision 347001212 - Original comment: Reverted to Oct 18, 2011 9:25 am: Spam** |
Wikispaces>genewardsmith **Imported revision 514035652 - Original comment: ** |
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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> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-06-16 00:26:28 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>514035652</tt>.<br> | ||
: The revision comment was: <tt> | : The revision comment was: <tt></tt><br> | ||
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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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Sometimes a normal sort of transversal using only two odd primes cannot be found for a given planar temperament, in which case the transversal will not work with Scala's lattice drawing command. In these cases an //oblique// transversal can be given; these have completely incorrect values considered as approximations, but with a correctly chosen val or tuning map, will give correct results for the temperament tuning. Since there is no reason not to, we use 2.3.5 5-limit vals for oblique vals, and give a corresponding //valid val// which will give correct N-edo tunings for the given oblique transversal. In Scala, under the pull-down menu for Modify, scroll down to Project, and check the Project the val option. Put the valid val into the box, apply it, and you transform the oblique transversal into the correct tuning. Oblique transversals are listed with an asterisk, like Roger Maris supposedly was. | Sometimes a normal sort of transversal using only two odd primes cannot be found for a given planar temperament, in which case the transversal will not work with Scala's lattice drawing command. In these cases an //oblique// transversal can be given; these have completely incorrect values considered as approximations, but with a correctly chosen val or tuning map, will give correct results for the temperament tuning. Since there is no reason not to, we use 2.3.5 5-limit vals for oblique vals, and give a corresponding //valid val// which will give correct N-edo tunings for the given oblique transversal. In Scala, under the pull-down menu for Modify, scroll down to Project, and check the Project the val option. Put the valid val into the box, apply it, and you transform the oblique transversal into the correct tuning. Oblique transversals are listed with an asterisk, like Roger Maris supposedly was. | ||
= | =Transversal listings= | ||
Below is a listing of some Z-polygon transverals and oblique transversals 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 Scala Temper command gives a number of options, and another tempering possibility is to use the edo with the optimal patent val. 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 and oblique transversals 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 Scala Temper command gives a number of options, and another tempering possibility is to use the edo with the optimal patent val. 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. | ||
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</pre></div> | </pre></div> | ||
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<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:16:&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:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#Z-polytopes and convex closures">Z-polytopes and convex closures</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Oblique transversals">Oblique transversals</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="# | <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:16:&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:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#Z-polytopes and convex closures">Z-polytopes and convex closures</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Oblique transversals">Oblique transversals</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Transversal listings">Transversal listings</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Septimal hexany">Septimal hexany</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#x7-limit diamond [-1, 0]^3 chord cube">7-limit diamond [-1, 0]^3 chord cube</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#x9-limit diamond">9-limit diamond</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Dekatesserany ([0,1]^3 chord cube)">Dekatesserany ([0,1]^3 chord cube)</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#x2)5 Dekany 1.3.5.7.11 (1.3 tonic)">2)5 Dekany 1.3.5.7.11 (1.3 tonic)</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | ||
<!-- ws:end:WikiTextTocRule:25 --><br /> | <!-- ws:end:WikiTextTocRule:25 --><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> | ||
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Sometimes a normal sort of transversal using only two odd primes cannot be found for a given planar temperament, in which case the transversal will not work with Scala's lattice drawing command. In these cases an <em>oblique</em> transversal can be given; these have completely incorrect values considered as approximations, but with a correctly chosen val or tuning map, will give correct results for the temperament tuning. Since there is no reason not to, we use 2.3.5 5-limit vals for oblique vals, and give a corresponding <em>valid val</em> which will give correct N-edo tunings for the given oblique transversal. In Scala, under the pull-down menu for Modify, scroll down to Project, and check the Project the val option. Put the valid val into the box, apply it, and you transform the oblique transversal into the correct tuning. Oblique transversals are listed with an asterisk, like Roger Maris supposedly was.<br /> | Sometimes a normal sort of transversal using only two odd primes cannot be found for a given planar temperament, in which case the transversal will not work with Scala's lattice drawing command. In these cases an <em>oblique</em> transversal can be given; these have completely incorrect values considered as approximations, but with a correctly chosen val or tuning map, will give correct results for the temperament tuning. Since there is no reason not to, we use 2.3.5 5-limit vals for oblique vals, and give a corresponding <em>valid val</em> which will give correct N-edo tunings for the given oblique transversal. In Scala, under the pull-down menu for Modify, scroll down to Project, and check the Project the val option. Put the valid val into the box, apply it, and you transform the oblique transversal into the correct tuning. Oblique transversals are listed with an asterisk, like Roger Maris supposedly was.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name=" | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Transversal listings"></a><!-- ws:end:WikiTextHeadingRule:4 -->Transversal listings</h1> | ||
Below is a listing of some Z-polygon transverals and oblique transversals 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 Scala Temper command gives a number of options, and another tempering possibility is to use the edo with the optimal patent val. 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 and oblique transversals 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 Scala Temper command gives a number of options, and another tempering possibility is to use the edo with the optimal patent val. 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 /> | ||