Tenney–Euclidean tuning: Difference between revisions

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<h4>Original Wikitext content:</h4>

If we have k linearly independent [[Vals and Tuning Space|vals]]  of dimension n, they will span a subspace of [[Vals and Tuning Space|tuning space]]. This subspace defines a regular temperament of rank k in the prime limit p, where p is the nth prime. Similarly, starting from n-k independent commas for the same regular temperament, the corresponding monzos span an n-k dimensional subspace of [[Monzos and Interval Space|interval space]]. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely.

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.

If we put the (weighted) Euclidean metric on tuning space, leading to TE tuning space in weighted coordinates, it is easy to find the nearest point in the subspace to the JI point &lt;1 1 ... 1|, and this closest point will define a tuning map which is called TE tuning (or TOP-RMS), a tuning which has been extensively studied by [[Graham Breed]]. We may also keep unweighted coordinates and use the TE norm on tuning space; in these coordinates the JI point is &lt;1 log2(3) ... log2(p)|. The two approaches are equivalent.

One way to find this tuning 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]].  

If A is an m×n matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the n×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.

* 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.

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 the vals are free of [Saturation|contorsion]. If J is the JI point, &lt;1 1 ... 1|, then JV` gives the TE 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 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.

If T = JP is the TE tuning map, then a corresponding pure-octaves map can be found by [[http://en.wikipedia.org/wiki/Scalar_multiplication|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. We may call pure-octaves Tenney-Euclidean tuning the [[POTE tuning]].

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 than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning 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 Frobenius 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 does not depend on whether the monzos or vals from which it is computed are [[Saturation|saturated]]. It 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.

If the vals defining V are linearly independent, then P = V*(VV*)^(-1)V. I the rows of M are independent, then we likewise have P = I - M*(MM*)^(-1)M.

The val for 5-limit 12et is &lt;12 19 28|. In weighted coordinates, that becomes v12 = &lt;12 19/log2(3) 28/log2(5)| ~ &lt;12.0 11.988 12.059|. If we take this to be a 1x3 matrix and take the pseudoinverse, we get the 3x1 matrix v12` ~ [|0.027706 0.027677 0.027842|]. Then P = v12`v12 is a projection map onto the one-dimensional subspace whose single basis vector is v12. We find that v12P equals v12; on the other hand, if we take the monzo for 81/80, which is |-4 4 -1&gt; and monzo-weight it to &lt;-4 4log2(3) - log2(5)| and multiply (either side, the matrix is symmetric) by P, we get the zero vector, corresponding to the unison.

This sends monzos for 50/49, 64/63 etc to the unison monzo, and vals for 10et, 12et and 22et to themselves.

</pre></div>

<h4>Original HTML content:</h4>

&lt;br /&gt;

If we have k linearly independent &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;vals&lt;/a&gt;  of dimension n, they will span a subspace of &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;tuning space&lt;/a&gt;. This subspace defines a regular temperament of rank k in the prime limit p, where p is the nth prime. Similarly, starting from n-k independent commas for the same regular temperament, the corresponding monzos span an n-k dimensional subspace of &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely.&lt;br /&gt;

&lt;br /&gt;

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.&lt;br /&gt;

&lt;br /&gt;

If we put the (weighted) Euclidean metric on tuning space, leading to TE tuning space in weighted coordinates, it is easy to find the nearest point in the subspace to the JI point &amp;lt;1 1 ... 1|, and this closest point will define a tuning map which is called TE tuning (or TOP-RMS), a tuning which has been extensively studied by &lt;a class="wiki_link" href="/Graham%20Breed"&gt;Graham Breed&lt;/a&gt;. We may also keep unweighted coordinates and use the TE norm on tuning space; in these coordinates the JI point is &amp;lt;1 log2(3) ... log2(p)|. The two approaches are equivalent.&lt;br /&gt;

&lt;br /&gt;

One way to find this tuning 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse" rel="nofollow"&gt;Moore-Penrose pseudoinverse&lt;/a&gt;. &lt;br /&gt;

&lt;br /&gt;

If A is an m×n matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the n×m matrix such that&lt;br /&gt;

&lt;ol&gt;&lt;li&gt;AA`A = A. Hence, AA` maps the rows of A to itself and A`A the columns of A to itself.&lt;/li&gt;&lt;li&gt;A`AA` = A&lt;/li&gt;&lt;li&gt;A`A and AA` are symmetric matrices&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;

From these properties it can be deduced that&lt;br /&gt;

&lt;ul&gt;&lt;li&gt;If A is invertible, its inverse is A`&lt;/li&gt;&lt;li&gt;If A has rational entries, so does A`&lt;/li&gt;&lt;li&gt;A`` = A&lt;/li&gt;&lt;li&gt;The pseudoinverse of the transpose is the transpose of the pseudoinverse&lt;/li&gt;&lt;li&gt;AA` is the orthogonal projection map onto the space spanned by the columns of A&lt;/li&gt;&lt;li&gt;A`A is the orthogonal projection map onto the space spanned by the rows of A&lt;/li&gt;&lt;li&gt;I - A`A, where I is the identity matrix, is the orthogonal projection map onto the kernel, or null space, of A&lt;/li&gt;&lt;li&gt;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.&lt;/li&gt;&lt;li&gt;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.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

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 the vals are free of [Saturation|contorsion]. If J is the JI point, &amp;lt;1 1 ... 1|, then JV` gives the TE 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.&lt;br /&gt;

&lt;br /&gt;

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.&lt;br /&gt;

&lt;br /&gt;

If T = JP is the TE tuning map, then a corresponding pure-octaves map can be found by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Scalar_multiplication" rel="nofollow"&gt;scalar multiplication&lt;/a&gt;, 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. We may call pure-octaves Tenney-Euclidean tuning the &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE tuning&lt;/a&gt;.&lt;br /&gt;

&lt;br /&gt;

We may also do the same things starting from unweighted vals. This leads to a different tuning, the &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;Frobenius tuning&lt;/a&gt;, which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning 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 &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;fractional monzos&lt;/a&gt;. The Frobenius projection map therefore, like the &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;, 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 does not depend on whether the monzos or vals from which it is computed are &lt;a class="wiki_link" href="/Saturation"&gt;saturated&lt;/a&gt;. It 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.&lt;br /&gt;

&lt;br /&gt;

If the vals defining V are linearly independent, then P = V*(VV*)^(-1)V. I the rows of M are independent, then we likewise have P = I - M*(MM*)^(-1)M.&lt;br /&gt;

&lt;br /&gt;

The val for 5-limit 12et is &amp;lt;12 19 28|. In weighted coordinates, that becomes v12 = &amp;lt;12 19/log2(3) 28/log2(5)| ~ &amp;lt;12.0 11.988 12.059|. If we take this to be a 1x3 matrix and take the pseudoinverse, we get the 3x1 matrix v12` ~ [|0.027706 0.027677 0.027842|]. Then P = v12`v12 is a projection map onto the one-dimensional subspace whose single basis vector is v12. We find that v12P equals v12; on the other hand, if we take the monzo for 81/80, which is |-4 4 -1&amp;gt; and monzo-weight it to &amp;lt;-4 4log2(3) - log2(5)| and multiply (either side, the matrix is symmetric) by P, we get the zero vector, corresponding to the unison.&lt;br /&gt;

&lt;br /&gt;