Tenney–Euclidean temperament measures: Difference between revisions
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<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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | <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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | ||
=Introduction= | =Introduction= | ||
Given a [[Wedgies and Multivals|multival]] or multimonzo which is a [[http://en.wikipedia.org/wiki/Exterior_algebra|wedge product]] of weighted vals or monzos, we may define a norm by means of the usual Euclidean norm. We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ([[http://en.wikipedia.org/wiki/Root_mean_square|root mean square]]) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||. | Given a [[Wedgies and Multivals|multival]] or multimonzo which is a [[http://en.wikipedia.org/wiki/Exterior_algebra|wedge product]] of weighted vals or monzos, we may define a norm by means of the usual Euclidean norm. We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ([[http://en.wikipedia.org/wiki/Root_mean_square|root mean square]]) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||. | ||
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<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>Tenney-Euclidean temperament measures</title></head><body><!-- ws:start:WikiTextTocRule:13:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#TE Complexity">TE Complexity</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#TE simple badness">TE simple badness</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#TE error">TE error</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <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>Tenney-Euclidean temperament measures</title></head><body><!-- ws:start:WikiTextTocRule:13:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#TE Complexity">TE Complexity</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#TE simple badness">TE simple badness</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#TE error">TE error</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | ||
<!-- ws:end:WikiTextTocRule:18 --><br /> | <!-- ws:end:WikiTextTocRule:18 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:5 -->Introduction</h1> | <!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:5 -->Introduction</h1> | ||
Given a <a class="wiki_link" href="/Wedgies%20and%20Multivals">multival</a> or multimonzo which is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">wedge product</a> of weighted vals or monzos, we may define a norm by means of the usual Euclidean norm. We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Root_mean_square" rel="nofollow">root mean square</a>) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||.<br /> | Given a <a class="wiki_link" href="/Wedgies%20and%20Multivals">multival</a> or multimonzo which is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">wedge product</a> of weighted vals or monzos, we may define a norm by means of the usual Euclidean norm. We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Root_mean_square" rel="nofollow">root mean square</a>) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||.<br /> | ||