Tenney–Euclidean tuning
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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 m**x**n matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the n**x**m 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 symmetric 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` = A*(AA*)^(-1), where A* is the transpose of A. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.
* 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 TOP-RMS 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 TOP-RMS 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.
We may find the same projection map starting from a list of weighted monzos rather than vals. If M is a rank n matrix whose rows are weighted monzos, and I is the nxn identity matrix, then P = I - M`M is the same projection map as V`V so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.
===Pure octaves RMS tuning===
If T = JP is the TOP-RMS tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.
==The rational projection map==
We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification. However, if greater weight needs to be attached to the larger primes than RMS-TOP attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[Fractional monzos|fractional monzos]]. The projection map therefore, like the [[Wedgies and Multivals|wedgie]], defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which automatically takes care of "torsion problems". It also may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I-Q is the same projection map as would be found if starting from a set of vals defining the same temperament.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 m<strong>x</strong>n matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the n<strong>x</strong>m 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 symmetric 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` = A*(AA*)^(-1), where A* is the transpose of A. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.</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 TOP-RMS 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 TOP-RMS 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.<br /> <br /> We may find the same projection map starting from a list of weighted monzos rather than vals. If M is a rank n matrix whose rows are weighted monzos, and I is the nxn identity matrix, then P = I - M`M is the same projection map as V`V so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x-RMS tuning-Pure octaves RMS tuning"></a><!-- ws:end:WikiTextHeadingRule:6 -->Pure octaves RMS tuning</h3> If T = JP is the TOP-RMS tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x-The rational projection map"></a><!-- ws:end:WikiTextHeadingRule:8 -->The rational projection map</h2> We may also do the same things starting from unweighted vals. This leads to a different tuning, the <a class="wiki_link" href="/Fractional%20monzos">Frobenius tuning</a>, which is perfectly functional but has less theoretical justification. However, if greater weight needs to be attached to the larger primes than RMS-TOP attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are <a class="wiki_link" href="/Fractional%20monzos">fractional monzos</a>. The projection map therefore, like the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which automatically takes care of "torsion problems". It also may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I-Q is the same projection map as would be found if starting from a set of vals defining the same temperament.</body></html>