Convex scale: Difference between revisions
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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>2011-10-19 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-19 15:23:03 UTC</tt>.<br> | ||
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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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===Convex set=== | ===Convex set=== | ||
A convex set is a set that includes all convex combinations of its elements. | A convex set (sometimes called a Z-polytope) is a set that includes all convex combinations of its elements. The convex closure of a set is the smallest convex set containing the set. | ||
==Examples== | ==Examples== | ||
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<!-- ws:start:WikiTextHeadingRule:7:&lt;h3&gt; --><h3 id="toc2"><a name="x-Formal definition-Convex set"></a><!-- ws:end:WikiTextHeadingRule:7 -->Convex set</h3> | <!-- ws:start:WikiTextHeadingRule:7:&lt;h3&gt; --><h3 id="toc2"><a name="x-Formal definition-Convex set"></a><!-- ws:end:WikiTextHeadingRule:7 -->Convex set</h3> | ||
A convex set is a set that includes all convex combinations of its elements.<br /> | A convex set (sometimes called a Z-polytope) is a set that includes all convex combinations of its elements. The convex closure of a set is the smallest convex set containing the set.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc3"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:9 -->Examples</h2> | <!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc3"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:9 -->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> | <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> | ||
Revision as of 15:23, 19 October 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
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Original Wikitext content:
In a [[Regular Temperaments|regular temperament]], a **convex scale** is a set of pitches that form a **convex set** in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament. A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [[http://en.wikipedia.org/wiki/Convex_set|convex region]] of continuous space. See below for a more formal definition. The **convex hull** or **convex closure** of a scale is the smallest convex scale that contains it. See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches. ==Formal definition== The following definitions make sense in the context of any Z-[[http://en.wikipedia.org/wiki/Module_%28mathematics%29|module]], which is the same concept as an [[http://en.wikipedia.org/wiki/Abelian_group|abelian group]]. ===Convex combination=== A **convex combination** of a set of vectors is, intuitively, a weighted average of the vectors where all the weights are non-negative. Formally, b is a convex combination of a1, a2... whenever there exist non-negative integers c1, c2... such that [[math]] $(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$ [[math]] 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. An equivalent definition can be given in terms of the [[http://en.wikipedia.org/wiki/Injective_hull|injective hull]] of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by [[math]] $c = c_1 + c_2 + \dots + c_k$ [[math]] we obtain [[math]] $b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$ [[math]] where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers. ===Convex set=== A convex set (sometimes called a Z-polytope) is a set that includes all convex combinations of its elements. The convex closure of a set is the smallest convex set containing the set. ==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]]
Original HTML content:
<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 "regular temperament" 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 /> <br /> A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_set" rel="nofollow">convex region</a> of continuous space. See below for a more formal definition.<br /> <br /> The <strong>convex hull</strong> or <strong>convex closure</strong> of a scale is the smallest convex scale that contains it. See <a class="wiki_link" href="/Gallery%20of%20Z-polygon%20transversals">Gallery of Z-polygon transversals</a> for many scales that are the convex closures of interesting sets of pitches.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:3:<h2> --><h2 id="toc0"><a name="x-Formal definition"></a><!-- ws:end:WikiTextHeadingRule:3 -->Formal definition</h2> The following definitions make sense in the context of any Z-<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Module_%28mathematics%29" rel="nofollow">module</a>, which is the same concept as an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian group</a>.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:5:<h3> --><h3 id="toc1"><a name="x-Formal definition-Convex combination"></a><!-- ws:end:WikiTextHeadingRule:5 -->Convex combination</h3> A <strong>convex combination</strong> of a set of vectors is, intuitively, a weighted average of the vectors where all the weights are non-negative. Formally, b is a convex combination of a1, a2... whenever there exist non-negative integers c1, c2... such that<br /> <!-- ws:start:WikiTextMathRule:0: [[math]]<br/> $(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$<br/>[[math]] --><script type="math/tex">$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$</script><!-- ws:end:WikiTextMathRule:0 --><br /> <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. An equivalent definition can be given in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow">injective hull</a> of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by <br /> <!-- ws:start:WikiTextMathRule:1: [[math]]<br/> $c = c_1 + c_2 + \dots + c_k$<br/>[[math]] --><script type="math/tex">$c = c_1 + c_2 + \dots + c_k$</script><!-- ws:end:WikiTextMathRule:1 --><br /> we obtain<br /> <!-- ws:start:WikiTextMathRule:2: [[math]]<br/> $b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$<br/>[[math]] --><script type="math/tex">$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$</script><!-- ws:end:WikiTextMathRule:2 --><br /> where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:7:<h3> --><h3 id="toc2"><a name="x-Formal definition-Convex set"></a><!-- ws:end:WikiTextHeadingRule:7 -->Convex set</h3> A convex set (sometimes called a Z-polytope) is a set that includes all convex combinations of its elements. The convex closure of a set is the smallest convex set containing the set.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:9:<h2> --><h2 id="toc3"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:9 -->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>