Convex scale: Difference between revisions
Wikispaces>clumma **Imported revision 266099472 - Original comment: Couldn't fix this paragraph, and doesn't seem to belong here anyway.** |
Wikispaces>genewardsmith **Imported revision 266449450 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-19 12:44:28 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>266449450</tt>.<br> | ||
: The revision comment was: <tt> | : 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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$ | $(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$ | ||
[[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. An equivalent defintion can be given in terms of the [[http://en.wikipedia.org/wiki/Injective_hull|injective hull]] of the Z-module, which extends monzos to fractional monzos 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]] | |||
[[math]] | |||
$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$ | |||
[[math]] | |||
where d_i = c_i/c. | |||
===Convex set=== | ===Convex set=== | ||
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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: | <!-- ws:start:WikiTextHeadingRule:3:&lt;h2&gt; --><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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:5:&lt;h3&gt; --><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 /> | 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: | <!-- ws:start:WikiTextMathRule:0: | ||
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$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$&lt;br/&gt;[[math]] | $(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$&lt;br/&gt;[[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 /> | --><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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | 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 defintion 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 monzos to fractional monzos and allows for the c_i to be rational numbers. By dividing through by <br /> | ||
<a class="wiki_link" href="/math">math</a> <br /> | |||
$c = c_1 + c_2 + \dots + c_k$<br /> | |||
<!-- ws:start:WikiTextMathRule:1: | |||
[[math]]&lt;br/&gt; | |||
we obtain&lt;br/&gt;[[math]] | |||
--><script type="math/tex">we obtain</script><!-- ws:end:WikiTextMathRule:1 --><br /> | |||
<!-- ws:start:WikiTextMathRule:2: | |||
[[math]]&lt;br/&gt; | |||
$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$&lt;br/&gt;[[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.<br /> | |||
<br /> | |||
<!-- 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 is a set that includes all convex combinations of its elements.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- 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> | ||