Tenney–Euclidean temperament measures: Difference between revisions
Wikispaces>guest **Imported revision 199013098 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 201605138 - Original comment: ** |
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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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-14 11:46:12 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>201605138</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : 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> | ||
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[[math]] | [[math]] | ||
\displaystyle ||J \wedge M|| = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j]) | \displaystyle ||J \wedge M|| = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j]) | ||
[[math]]</pre></div> | [[math]] | ||
===TE error=== | |||
TE simple badness can also be called and considered to be error relative to complexity. By dividing it by TE complexity, we get an error measurement, TE error. Multiplying this by 1200, we get a figure we can consider to be a weighted average with values in cents.</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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>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 /> | <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>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 /> | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\displaystyle ||J \wedge M|| = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])&lt;br/&gt;[[math]] | \displaystyle ||J \wedge M|| = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\displaystyle ||J \wedge M|| = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])</script><!-- ws:end:WikiTextMathRule:1 --></body></html></pre></div> | --><script type="math/tex">\displaystyle ||J \wedge M|| = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc2"><a name="x--TE error"></a><!-- ws:end:WikiTextHeadingRule:6 -->TE error</h3> | |||
TE simple badness can also be called and considered to be error relative to complexity. By dividing it by TE complexity, we get an error measurement, TE error. Multiplying this by 1200, we get a figure we can consider to be a weighted average with values in cents.</body></html></pre></div> | |||