Tenney–Euclidean temperament measures: Difference between revisions

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If J = &lt;1 1 ... 1| is the JI point, then the relative error of W is defined as ||J^W||. Relative error is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. Once again, if we have a list of vectors we may use a Gramian to compute relative error. First we note that ai = J.vi/n is the mean value of the entries of vi. Then note that J^(v1-a1*J)^(v2-a2*J)^...^(vr-ar*J) = J^v1^v2^...^vr, since wedge products with more than one term are zero. The Gram matrix of the vectors J and v1-J*ai will have n as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain

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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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tenney-Euclidean temperament measures&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Given a &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;multival&lt;/a&gt; or multimonzo which is a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow"&gt;wedge product&lt;/a&gt; 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 (&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Root_mean_square" rel="nofollow"&gt;root mean square&lt;/a&gt;) 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 W is a multivector, we denote the RMS norm as ||W||.&lt;br /&gt;

If J = &amp;lt;1 1 ... 1| is the JI point, then the relative error of W is defined as ||J^W||. Relative error is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. Once again, if we have a list of vectors we may use a Gramian to compute relative error. First we note that ai = J.vi/n is the mean value of the entries of vi. Then note that J^(v1-a1*J)^(v2-a2*J)^...^(vr-ar*J) = J^v1^v2^...^vr, since wedge products with more than one term are zero. The Gram matrix of the vectors J and v1-J*ai will have n as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain&lt;br /&gt;

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