Convex scale: Difference between revisions
Wikispaces>keenanpepper **Imported revision 265993642 - Original comment: ** |
Wikispaces>keenanpepper **Imported revision 266010180 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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:keenanpepper|keenanpepper]] and made on <tt>2011-10-18 | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-10-18 12:23:17 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>266010180</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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(Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module.) | (Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module.) | ||
===Convex set=== | ===Convex set=== | ||
A convex set is a set that includes all convex combinations of its elements.</pre></div> | A convex set is a set that includes all convex combinations of its elements. | ||
==Examples== | |||
* Every [[MOSScales|MOS]] is convex. | |||
* In fact, every [[distributionally even]] scale is convex. | |||
* Every [[Fokker blocks|Fokker block]] is convex. | |||
* Every untempered [[Tonality diamond|tonality diamond]] is convex. | |||
* [[Gallery of Z-polygon transversals]]</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>Convex scale</title></head><body>In a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>, a <strong>convex scale</strong> is a set of pitches that form a <strong>convex set</strong> in the interval lattice of the temperament. The &quot;regular temperament&quot; is often <a class="wiki_link" href="/Just%20intonation">JI</a>, in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.<br /> | <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>Convex scale</title></head><body>In a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>, a <strong>convex scale</strong> is a set of pitches that form a <strong>convex set</strong> in the interval lattice of the temperament. The &quot;regular temperament&quot; is often <a class="wiki_link" href="/Just%20intonation">JI</a>, in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.<br /> | ||
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(Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module.)<br /> | (Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module.)<br /> | ||
<!-- ws:start:WikiTextHeadingRule:5:&lt;h3&gt; --><h3 id="toc2"><a name="x-Formal definition-Convex set"></a><!-- ws:end:WikiTextHeadingRule:5 -->Convex set</h3> | <!-- ws:start:WikiTextHeadingRule:5:&lt;h3&gt; --><h3 id="toc2"><a name="x-Formal definition-Convex set"></a><!-- ws:end:WikiTextHeadingRule:5 -->Convex set</h3> | ||
A convex set is a set that includes all convex combinations of its elements.</body></html></pre></div> | A convex set is a set that includes all convex combinations of its elements.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc3"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:7 -->Examples</h2> | |||
<ul><li>Every <a class="wiki_link" href="/MOSScales">MOS</a> is convex.</li><li>In fact, every <a class="wiki_link" href="/distributionally%20even">distributionally even</a> scale is convex.</li><li>Every <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> is convex.</li><li>Every untempered <a class="wiki_link" href="/Tonality%20diamond">tonality diamond</a> is convex.</li><li><a class="wiki_link" href="/Gallery%20of%20Z-polygon%20transversals">Gallery of Z-polygon transversals</a></li></ul></body></html></pre></div> | |||