Tenney–Euclidean tuning: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-~~16 19~~:09:31 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-04 04:25:49 UTC</tt>.<br>

: The original revision id was <tt>~~163292193~~</tt>.<br>

: The original revision id was <tt>167505323</tt>.<br>

: The revision comment was: <tt></tt><br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

Line 12:

===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

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

Line 25:

* 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. 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.

* 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.

Line 47:

<br />

<h3 id="toc1"><a name="x-RMS tuning-The pseudoinverse"></a>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 />

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` = At(~~AAt~~)^(-1), where At 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 />

<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 />

<h3 id="toc2"><a name="x-RMS tuning-Computing the TOP-RMS tuning"></a>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 (&quot;torsion problems&quot;) cannot be found in some combination of the unweighted vals. If J is the JI point, &lt;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 />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-16 19:09:31 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-04 04:25:49 UTC</tt>.<br>
-: The original revision id was <tt>163292193</tt>.<br>
+: The original revision id was <tt>167505323</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 12: / Line 12: @@
 ===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
+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
@@ Line 25: / Line 25: @@
 * 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. 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.
+* 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.
@@ Line 47: / Line 47: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-RMS tuning-The pseudoinverse"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;The pseudoinverse&lt;/h3&gt;
-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&lt;br /&gt;
+If A is an m&lt;strong&gt;x&lt;/strong&gt;n matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the n&lt;strong&gt;x&lt;/strong&gt;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 matricies&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` = At(AAt)^(-1), where At 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;
+&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;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-RMS tuning-Computing the TOP-RMS tuning"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Computing the TOP-RMS tuning&lt;/h3&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 common factors (&amp;quot;torsion problems&amp;quot;) cannot be found in some combination of the unweighted vals. If J is the JI point, &amp;lt;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.&lt;br /&gt;