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>2011-09-09 11:45:52 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-09 17:13:09 UTC</tt>.<br>

: The original revision id was <tt>~~252370104~~</tt>.<br>

: The original revision id was <tt>252479686</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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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.

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 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.

=Septimal hexany=

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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-polytope 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 />

<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 />

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 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 />

<h1 id="toc1"><a name="Septimal hexany"></a>Septimal hexany</h1>

@@ 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>2011-09-09 11:45:52 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-09 17:13:09 UTC</tt>.<br>
-: The original revision id was <tt>252370104</tt>.<br>
+: The original revision id was <tt>252479686</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 13: / Line 13: @@
 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.
+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 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.
 =Septimal hexany=
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 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;br /&gt;
-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 &amp;quot;Lattice and player&amp;quot; under the &amp;quot;Analyze&amp;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 &lt;a class="wiki_link" href="/Diaconv%20scales"&gt;Diaconv scales&lt;/a&gt;, but without giving an explicit tempering, something which is easily accomplished inside of Scala.&lt;br /&gt;
+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 &amp;quot;Lattice and player&amp;quot; under the &amp;quot;Analyze&amp;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 &lt;a class="wiki_link" href="/Diaconv%20scales"&gt;Diaconv scales&lt;/a&gt;, but without giving an explicit tempering, something which is easily accomplished inside of Scala.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Septimal hexany"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Septimal hexany&lt;/h1&gt;