Convex scale: Difference between revisions
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.** |
||
| Line 1: | Line 1: | ||
<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> | ||
| Line 11: | Line 11: | ||
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 | ||
| Line 22: | Line 21: | ||
[[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. | ||
| Line 37: | Line 37: | ||
<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 /> | ||
| Line 49: | Line 48: | ||
--><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 /> | ||