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>2014-05-~~18 14~~:02:23 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-24 17:07:50 UTC</tt>.<br>

: The original revision id was <tt>~~509655860~~</tt>.<br>

: The original revision id was <tt>511016430</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>

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

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

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.

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<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><br />

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 />

<img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" align="left" /><br />

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>2014-05-18 14:02:23 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-24 17:07:50 UTC</tt>.<br>
-: The original revision id was <tt>509655860</tt>.<br>
+: The original revision id was <tt>511016430</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">
-[[image:mathhazard.jpg align="center"]]
+[[image:mathhazard.jpg align="left"]]
 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.
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 <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;&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;!-- ws:start:WikiTextLocalImageRule:6:&amp;lt;img src=&amp;quot;/file/view/mathhazard.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; align=&amp;quot;left&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" align="left" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:6 --&gt;&lt;br /&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;