Gallery of Z-polygon transversals: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2011-08-31 13:12:44 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-31 13:15:27 UTC</tt>.<br>

: The original revision id was <tt>~~249738308~~</tt>.<br>

: The original revision id was <tt>249740068</tt>.<br>

: 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>

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=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/~~Integer_points_in_convex_polyhedra~~|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, or 2-polytope.

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, or 2-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-polygon transversal//, and in case of a [[planar temperament]], where the Z-polytope lies in a plane, a Z-polygon transversal.

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<br />

<h1 id="toc0"><a name="Z-polytopes and convex closures"></a>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/~~Integer_points_in_convex_polyhedra~~" 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, or 2-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, or 2-polytope.<br />

<br />

If a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a> of rank r+1 has no elements which are fractions of the octave, one of the generators can be taken as &quot;2&quot;, and the octave-equivalent pitch classes form a free abelian group of rank r, which can be written as an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tuple" rel="nofollow">r-tuple</a> of integers [a1 a2 ... ar]. A <a class="wiki_link" href="/periodic%20scale">periodic scale</a> 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 <a class="wiki_link" href="/transversal">transversal</a> 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 <em>Z-polygon transversal</em>, and in case of a <a class="wiki_link" href="/planar%20temperament">planar temperament</a>, where the Z-polytope lies in a plane, a Z-polygon transversal. <br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-08-31 13:12:44 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-31 13:15:27 UTC</tt>.<br>
-: The original revision id was <tt>249738308</tt>.<br>
+: The original revision id was <tt>249740068</tt>.<br>
 : 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>
@@ Line 9: / Line 9: @@
 =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/Integer_points_in_convex_polyhedra|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, or 2-polytope.
+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, or 2-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-polygon transversal//, and in case of a [[planar temperament]], where the Z-polytope lies in a plane, a Z-polygon transversal.
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 &lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;br /&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/Integer_points_in_convex_polyhedra" 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, or 2-polytope.&lt;br /&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, or 2-polytope.&lt;br /&gt;
 &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-polygon 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;