Convex scale: Difference between revisions
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Wikispaces>keenanpepper **Imported revision 266010180 - Original comment: ** |
Wikispaces>clumma **Imported revision 266099472 - Original comment: Couldn't fix this paragraph, and doesn't seem to belong here anyway.** |
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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: | : This revision was by author [[User:clumma|clumma]] and made on <tt>2011-10-18 15:30:22 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>266099472</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>Couldn't fix this paragraph, and doesn't seem to belong here anyway.</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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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. | 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== | ==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]]. | 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=== | ===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 | 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 | ||
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[[math]] | [[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.) | (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. | A convex set is a set that includes all convex combinations of its elements. | ||
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<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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:1:&lt;h2&gt; --><h2 id="toc0"><a name="x-Formal definition"></a><!-- ws:end:WikiTextHeadingRule:1 -->Formal definition</h2> | <!-- ws:start:WikiTextHeadingRule:1:&lt;h2&gt; --><h2 id="toc0"><a name="x-Formal definition"></a><!-- ws:end:WikiTextHeadingRule:1 -->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 /> | 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:3:&lt;h3&gt; --><h3 id="toc1"><a name="x-Formal definition-Convex combination"></a><!-- ws:end:WikiTextHeadingRule:3 -->Convex combination</h3> | <!-- ws:start:WikiTextHeadingRule:3:&lt;h3&gt; --><h3 id="toc1"><a name="x-Formal definition-Convex combination"></a><!-- ws:end:WikiTextHeadingRule:3 -->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 /> | 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 /> | ||
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--><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 /> | --><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 /> | ||
(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 /> | ||
<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.<br /> | A convex set is a set that includes all convex combinations of its elements.<br /> | ||
Revision as of 15:30, 18 October 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author clumma and made on 2011-10-18 15:30:22 UTC.
- The original revision id was 266099472.
- The revision comment was: Couldn't fix this paragraph, and doesn't seem to belong here anyway.
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
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.) ===Convex set=== 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]]
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:1:<h2> --><h2 id="toc0"><a name="x-Formal definition"></a><!-- ws:end:WikiTextHeadingRule:1 -->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:3:<h3> --><h3 id="toc1"><a name="x-Formal definition-Convex combination"></a><!-- ws:end:WikiTextHeadingRule:3 -->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 /> (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 /> <br /> <!-- ws:start:WikiTextHeadingRule:5:<h3> --><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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:7:<h2> --><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>