Graph-theoretic properties of scales: 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>2014-04-26 05:26:51 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-04-26 23:24:44 UTC</tt>.<br>

: The original revision id was <tt>~~504664768~~</tt>.<br>

: The original revision id was <tt>504758060</tt>.<br>

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

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=Edge-labeled tuning=

An //edge-labeled graph// is a graph with a function from its edges to a set of "labels". Ordinarily, this function is a bijection, but for our purposes we allow different edges to have the same label. Instead, we take the labels to be the intervals which are the intended targets for the scale tuning. The pitch classes denoted by a and b have representatives in the octave 1 < z ≤ 2, which by abuse of notation we will also call a and b, and assume 1 < a < b ≤ 2. The edge {a, b} then has a target interval 1 < z < 2, or 0 < z < 1200 in terms of cents, where b/a is approximately z (or b-a is approximately z if we are working additively in terms of cents.) In the simplest case, the target intervals are members of our consonance set. If we want to approximate more than one interval in our consonance set by a given edge, for instance both 10/9 and 9/8 in a temperament in the meantone family, we can replace the set of approximate consonances by a suitable average, such as the geometric mean.

An edge-labeled tuning is now definable as a tuning derived by optimizing the fit to the target interrvals of the edges. This could be, eg, a minimax optimization, but the simplest to compute and a good practical choice is to take the least squares edge-labeled tuning. In many cases this will result in a [[lesfip scales|lesfip]] tuning. This result is both a good practical tuning, and a reasonable choice for a canonical tuning if such a thing is needed.

=Seven note scales=

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<br />

<h1 id="toc6"><a name="Edge-labeled tuning"></a>Edge-labeled tuning</h1>

An <em>edge-labeled graph</em> is a graph with a function from its edges to a set of &quot;labels&quot;. Ordinarily, this function is a bijection, but for our purposes we allow different edges to have the same label. Instead, we take the labels to be the intervals which are the intended targets for the scale tuning. The pitch classes denoted by a and b have representatives in the octave 1 &lt; z ≤ 2, which by abuse of notation we will also call a and b, and assume 1 &lt; a &lt; b ≤ 2. The edge {a, b} then has a target interval 1 &lt; z &lt; 2, or 0 &lt; z &lt; 1200 in terms of cents, where b/a is approximately z (or b-a is approximately z if we are working additively in terms of cents.) In the simplest case, the target intervals are members of our consonance set. If we want to approximate more than one interval in our consonance set by a given edge, for instance both 10/9 and 9/8 in a temperament in the meantone family, we can replace the set of approximate consonances by a suitable average, such as the geometric mean.<br />

<br />

An edge-labeled tuning is now definable as a tuning derived by optimizing the fit to the target interrvals of the edges. This could be, eg, a minimax optimization, but the simplest to compute and a good practical choice is to take the least squares edge-labeled tuning. In many cases this will result in a <a class="wiki_link" href="/lesfip%20scales">lesfip</a> tuning. This result is both a good practical tuning, and a reasonable choice for a canonical tuning if such a thing is needed.<br />

<br />

<h1 id="toc7"><a name="Seven note scales"></a>Seven note scales</h1>

@@ 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>2014-04-26 05:26:51 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-04-26 23:24:44 UTC</tt>.<br>
-: The original revision id was <tt>504664768</tt>.<br>
+: The original revision id was <tt>504758060</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 56: / Line 56: @@
 =Edge-labeled tuning=
+An //edge-labeled graph// is a graph with a function from its edges to a set of "labels". Ordinarily, this function is a bijection, but for our purposes we allow different edges to have the same label. Instead, we take the labels to be the intervals which are the intended targets for the scale tuning. The pitch classes denoted by a and b have representatives in the octave 1 &lt; z ≤ 2, which by abuse of notation we will also call a and b, and assume 1 &lt; a &lt; b ≤ 2. The edge {a, b} then has a target interval 1 &lt; z &lt; 2, or 0 &lt; z &lt; 1200 in terms of cents, where b/a is approximately z (or b-a is approximately z if we are working additively in terms of cents.) In the simplest case, the target intervals are members of our consonance set. If we want to approximate more than one interval in our consonance set by a given edge, for instance both 10/9 and 9/8 in a temperament in the meantone family, we can replace the set of approximate consonances by a suitable average, such as the geometric mean.
+An edge-labeled tuning is now definable as a tuning derived by optimizing the fit to the target interrvals of the edges. This could be, eg, a minimax optimization, but the simplest to compute and a good practical choice is to take the least squares edge-labeled tuning. In many cases this will result in a [[lesfip scales|lesfip]] tuning. This result is both a good practical tuning, and a reasonable choice for a canonical tuning if such a thing is needed.
 =Seven note scales=
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Edge-labeled tuning"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Edge-labeled tuning&lt;/h1&gt;
+An &lt;em&gt;edge-labeled graph&lt;/em&gt; is a graph with a function from its edges to a set of &amp;quot;labels&amp;quot;. Ordinarily, this function is a bijection, but for our purposes we allow different edges to have the same label. Instead, we take the labels to be the intervals which are the intended targets for the scale tuning. The pitch classes denoted by a and b have representatives in the octave 1 &amp;lt; z ≤ 2, which by abuse of notation we will also call a and b, and assume 1 &amp;lt; a &amp;lt; b ≤ 2. The edge {a, b} then has a target interval 1 &amp;lt; z &amp;lt; 2, or 0 &amp;lt; z &amp;lt; 1200 in terms of cents, where b/a is approximately z (or b-a is approximately z if we are working additively in terms of cents.) In the simplest case, the target intervals are members of our consonance set. If we want to approximate more than one interval in our consonance set by a given edge, for instance both 10/9 and 9/8 in a temperament in the meantone family, we can replace the set of approximate consonances by a suitable average, such as the geometric mean.&lt;br /&gt;
+&lt;br /&gt;
+An edge-labeled tuning is now definable as a tuning derived by optimizing the fit to the target interrvals of the edges. This could be, eg, a minimax optimization, but the simplest to compute and a good practical choice is to take the least squares edge-labeled tuning. In many cases this will result in a &lt;a class="wiki_link" href="/lesfip%20scales"&gt;lesfip&lt;/a&gt; tuning. This result is both a good practical tuning, and a reasonable choice for a canonical tuning if such a thing is needed.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Seven note scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Seven note scales&lt;/h1&gt;