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~~-11-~~22 20~~:42:19 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-18 14:02:23 UTC</tt>.<br>

: The original revision id was <tt>~~278339880~~</tt>.<br>

: The original revision id was <tt>509655860</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>

<h4>Original Wikitext content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">By a //target tuning// is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.

[[image:mathhazard.jpg align="center"]]

By a //target tuning// is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.

==Least squares tunings==

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Starting from the same six intervals of the 5-limit diamond and adding 2, we find after computing the normal interval lists that the three subgroups are [2, 3], [2, 5], and [2, 5/3]. Computing the projection matrix and from thence the tuning in each case, we find that the minimax tuning is [|1 0 0>, |1 0 1/4>, |0 0 1>], the projection matrix for 1/4-comma meantone.

</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Target tunings</title></head><body>By a <em>target tuning</em> is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.<br />

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Target tunings</title></head><body><br />

<div style="text-align: center"><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /></div>By a <em>target tuning</em> is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.<br />

<br />

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

@@ 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>2011-11-22 20:42:19 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-18 14:02:23 UTC</tt>.<br>
-: The original revision id was <tt>278339880</tt>.<br>
+: The original revision id was <tt>509655860</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>
 <h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">By a //target tuning// is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">
+[[image:mathhazard.jpg align="center"]]
+By a //target tuning// is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.
 ==Least squares tunings==
@@ Line 30: / Line 32: @@
 Starting from the same six intervals of the 5-limit diamond and adding 2, we find after computing the normal interval lists that the three subgroups are [2, 3], [2, 5], and [2, 5/3]. Computing the projection matrix and from thence the tuning in each case, we find that the minimax tuning is [|1 0 0&gt;, |1 0 1/4&gt;, |0 0 1&gt;], the projection matrix for 1/4-comma meantone.
 </pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Target tunings&lt;/title&gt;&lt;/head&gt;&lt;body&gt;By a &lt;em&gt;target tuning&lt;/em&gt; is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.&lt;br /&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Target tunings&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:6:&amp;lt;div style=&amp;quot;text-align: center&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/mathhazard.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/div&amp;gt; --&gt;&lt;div style="text-align: center"&gt;&lt;img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextLocalImageRule:6 --&gt;By a &lt;em&gt;target tuning&lt;/em&gt; is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.&lt;br /&gt;
 &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;