Gallery of Z-polygon transversals: Difference between revisions

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<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]

=Z-polytopes and convex closures=

In geometry, a [[http://en.wikipedia.org/wiki/Convex_set|convex set]] 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 [[http://en.wikipedia.org/wiki/Convex_hull|convex hull]] 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 [[http://en.wikipedia.org/wiki/Convex_lattice_polytope|Z-polytope]] 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.

If a [[Regular Temperaments|regular temperament]] of rank r+1 has no elements which are fractions of the octave, one of the generators can be taken as "2", and the octave-equivalent pitch classes form a free abelian group of rank r, which can be written as an [[http://en.wikipedia.org/wiki/Tuple|r-tuple]] of integers [a1 a2 ... ar]. A [[periodic scale]] in this regular temperament is then a finite set of such r-tuples, and the minimal Z-polytope containing this set is the convex closure of the scale. If a just intonation tuning is chosen, one where every generator is tuned as a just interval belonging to the p-limit group or subgroup the temperament maps from, then a [[transversal]] is obtained; for any tuning of the temperament, the specific notes can be obtained by mapping from the notes of the transversal. Such a transversal may be called a //Z-polytope transversal//, and in case of a [[planar temperament]], where the Z-polytope lies in a plane, a Z-polygon transversal.  

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.

=9-limit diamond=

<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Gallery of Z-polygon transversals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Z-polytopes and convex closures"&gt;Z-polytopes and convex closures&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#x7-limit diamond"&gt;7-limit diamond&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt; | &lt;a href="#x9-limit diamond"&gt;9-limit diamond&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;

&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Z-polytopes and convex closures"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Z-polytopes and convex closures&lt;/h1&gt;

In geometry, a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_set" rel="nofollow"&gt;convex set&lt;/a&gt; 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_hull" rel="nofollow"&gt;convex hull&lt;/a&gt; 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_lattice_polytope" rel="nofollow"&gt;Z-polytope&lt;/a&gt; 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.&lt;br /&gt;

If a &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;regular temperament&lt;/a&gt; of rank r+1 has no elements which are fractions of the octave, one of the generators can be taken as &amp;quot;2&amp;quot;, and the octave-equivalent pitch classes form a free abelian group of rank r, which can be written as an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tuple" rel="nofollow"&gt;r-tuple&lt;/a&gt; of integers [a1 a2 ... ar]. A &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; in this regular temperament is then a finite set of such r-tuples, and the minimal Z-polytope containing this set is the convex closure of the scale. If a just intonation tuning is chosen, one where every generator is tuned as a just interval belonging to the p-limit group or subgroup the temperament maps from, then a &lt;a class="wiki_link" href="/transversal"&gt;transversal&lt;/a&gt; is obtained; for any tuning of the temperament, the specific notes can be obtained by mapping from the notes of the transversal. Such a transversal may be called a &lt;em&gt;Z-polytope transversal&lt;/em&gt;, and in case of a &lt;a class="wiki_link" href="/planar%20temperament"&gt;planar temperament&lt;/a&gt;, where the Z-polytope lies in a plane, a Z-polygon transversal. &lt;br /&gt;

&lt;a class="wiki_link" href="/diamond7_225"&gt;marvel&lt;/a&gt;&lt;br /&gt;

&lt;a class="wiki_link" href="/diamond9_245"&gt;sensamagic&lt;/a&gt;&lt;br /&gt;