Tenney–Euclidean tuning
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- This revision was by author genewardsmith and made on 2010-07-27 06:24:43 UTC.
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Original Wikitext content:
If we have k linearly independent vals of dimension n, they will define a subspace of the n-dimensional [[http://en.wikipedia.org/wiki/Vector_space|vector space]] Q^n over the field of rational numbers, which extends to a subspace of [[Vals and Tuning Space|tuning space]] over the field of real numbers by "tensoring" or extending coefficients. Either subspace defines a regular temperament of rank k in the prime limit p, where p is the nth prime. A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is RMS tuning. ==RMS tuning== If we put the (weighted) Euclidean metric on tuning space, leading to Euclidean tuning space, it is easy to find the nearest point in the subspace to the JI point <1 1 ... 1|, and this closest point will define a tuning map which is called TOP-RMS tuning, a tuning which has been extensively studied by [[Graham Breed]]. One way to do this is to use k parameters times the vals, leading to a parametrization of the subspace, and then to find the nearest point by least squares, differentiating the square of the distance to the JI point and solving the resulting linear equations. Another is to use the [[http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse|Moore-Penrose pseudoinverse]].
Original HTML content:
<html><head><title>Tenney-Euclidean Tuning</title></head><body>If we have k linearly independent vals of dimension n, they will define a subspace of the n-dimensional <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">vector space</a> Q^n over the field of rational numbers, which extends to a subspace of <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">tuning space</a> over the field of real numbers by "tensoring" or extending coefficients. Either subspace defines a regular temperament of rank k in the prime limit p, where p is the nth prime. A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is RMS tuning.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-RMS tuning"></a><!-- ws:end:WikiTextHeadingRule:0 -->RMS tuning</h2> If we put the (weighted) Euclidean metric on tuning space, leading to Euclidean tuning space, it is easy to find the nearest point in the subspace to the JI point <1 1 ... 1|, and this closest point will define a tuning map which is called TOP-RMS tuning, a tuning which has been extensively studied by <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>. One way to do this is to use k parameters times the vals, leading to a parametrization of the subspace, and then to find the nearest point by least squares, differentiating the square of the distance to the JI point and solving the resulting linear equations. Another is to use the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse" rel="nofollow">Moore-Penrose pseudoinverse</a>.</body></html>