Target tuning: Difference between revisions

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<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>2011-05-~~19 16~~:21:32 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 20:42:19 UTC</tt>.<br>

: The original revision id was <tt>~~230119444~~</tt>.<br>

: The original revision id was <tt>278339880</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>

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==Least squares tunings==

By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0<r<1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ~~sum_i~~ (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.

By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0<r<1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑_i (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.

We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are vals spanning the temperament. From U we form the matrix V by taking the [[normal lists|normal val list]] for U and removing the first ("period") row. A list of [[fractional monzos|eigenmonzos]] E = VR*R, where the asterisk denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The [[fractional monzos|projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E, including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.

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<br />

<h2 id="toc0"><a name="x-Least squares tunings"></a>Least squares tunings</h2>

By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0&lt;r&lt;1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ~~sum_i~~ (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.<br />

By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0&lt;r&lt;1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑_i (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.<br />

<br />

We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are vals spanning the temperament. From U we form the matrix V by taking the <a class="wiki_link" href="/normal%20lists">normal val list</a> for U and removing the first (&quot;period&quot;) row. A list of <a class="wiki_link" href="/fractional%20monzos">eigenmonzos</a> E = VR*R, where the asterisk denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The <a class="wiki_link" href="/fractional%20monzos">projection matrix</a> for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E, including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero. <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>2011-05-19 16:21:32 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 20:42:19 UTC</tt>.<br>
-: The original revision id was <tt>230119444</tt>.<br>
+: The original revision id was <tt>278339880</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 9: / Line 9: @@
 ==Least squares tunings==
-By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0&lt;r&lt;1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, sum_i (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.
+By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0&lt;r&lt;1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑_i (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.
 We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are vals spanning the temperament. From U we form the matrix V by taking the [[normal lists|normal val list]] for U and removing the first ("period") row. A list of [[fractional monzos|eigenmonzos]] E = VR*R, where the asterisk denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The [[fractional monzos|projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E, including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Least squares tunings"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Least squares tunings&lt;/h2&gt;
-By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0&amp;lt;r&amp;lt;1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, sum_i (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.&lt;br /&gt;
+By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0&amp;lt;r&amp;lt;1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑_i (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.&lt;br /&gt;
 &lt;br /&gt;
 We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are vals spanning the temperament. From U we form the matrix V by taking the &lt;a class="wiki_link" href="/normal%20lists"&gt;normal val list&lt;/a&gt; for U and removing the first (&amp;quot;period&amp;quot;) row. A list of &lt;a class="wiki_link" href="/fractional%20monzos"&gt;eigenmonzos&lt;/a&gt; E = VR*R, where the asterisk denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The &lt;a class="wiki_link" href="/fractional%20monzos"&gt;projection matrix&lt;/a&gt; for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E, including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero. &lt;br /&gt;