Tenney–Euclidean tuning
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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]].
===The pseudoinverse===
If A is an mxn matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the nxm matrix such that
# AA`A = A. Hence, AA` maps the rows of A to itself and A`A the columns of A to itself.
# A`AA` = A
# A`A and AA` are symmetrical matricies
From these properties it can be deduced that
* If A is invertible, its inverse is A`
* If A has rational entries, so does A`
* A`` = A
* The pseudoinverse of the transpose is the transpose of the pseudoinverse
* AA` is the orthogonal projection map onto the space spanned by the columns of A
* A`A is the orthogonal projection map onto the space spanned by the rows of A
* I - A`A, where I is the identity matrix, is the orthogonal projection map onto the kernel, or null space, of A
* If the rows of A are linearly independent, then A` = At(AAt)^(-1), where At is the transpose of A. Since our primary interest is in the case where the rows of A are independent, this means the pseudoinverse can be found using a matrix inverse routine by people who don't have a pseudoinverse routine available.
* uA` is the nearest point to u in the subspace spanned by the rows of A; A`v is the nearest point to v in the space spanned by the columns of A.
===Computing the TOP-RMS tuning===
Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that common factors ("torsion problems") cannot be found in some combination of the unweighted vals. If J is the JI point, <1 1 ... 1|, then JV` gives the RMS-TOP tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number.
We may also obtain the RMS-TOP tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.
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>. <br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-RMS tuning-The pseudoinverse"></a><!-- ws:end:WikiTextHeadingRule:2 -->The pseudoinverse</h3> If A is an mxn matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the nxm matrix such that<br /> <ol><li>AA`A = A. Hence, AA` maps the rows of A to itself and A`A the columns of A to itself.</li><li>A`AA` = A</li><li>A`A and AA` are symmetrical matricies</li></ol><br /> From these properties it can be deduced that<br /> <ul><li>If A is invertible, its inverse is A`</li><li>If A has rational entries, so does A`</li><li>A`` = A</li><li>The pseudoinverse of the transpose is the transpose of the pseudoinverse</li><li>AA` is the orthogonal projection map onto the space spanned by the columns of A</li><li>A`A is the orthogonal projection map onto the space spanned by the rows of A</li><li>I - A`A, where I is the identity matrix, is the orthogonal projection map onto the kernel, or null space, of A</li><li>If the rows of A are linearly independent, then A` = At(AAt)^(-1), where At is the transpose of A. Since our primary interest is in the case where the rows of A are independent, this means the pseudoinverse can be found using a matrix inverse routine by people who don't have a pseudoinverse routine available.</li><li>uA` is the nearest point to u in the subspace spanned by the rows of A; A`v is the nearest point to v in the space spanned by the columns of A.</li></ul><br /> <!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x-RMS tuning-Computing the TOP-RMS tuning"></a><!-- ws:end:WikiTextHeadingRule:4 -->Computing the TOP-RMS tuning</h3> Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that common factors ("torsion problems") cannot be found in some combination of the unweighted vals. If J is the JI point, <1 1 ... 1|, then JV` gives the RMS-TOP tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number.<br /> <br /> We may also obtain the RMS-TOP tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.</body></html>