Hahn distance: 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>2012-08-~~10 22~~:56:03 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-11 02:25:04 UTC</tt>.<br>

: The original revision id was <tt>~~357270976~~</tt>.<br>

: The original revision id was <tt>357283838</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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<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">In graph theory, the distance between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path. Given a set of just intervals, or more usually, of classes of octave-equivalent intervals, we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a consonance. Normally the unison is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales.

If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, where q is an odd number q ~~>=~~ p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.

If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, where q is an odd number q ≥ p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.

Up to the 7-limit, Hahn distance has a very nice formula give by

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<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>Hahn distance</title></head><body>In graph theory, the distance between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path. Given a set of just intervals, or more usually, of classes of octave-equivalent intervals, we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a consonance. Normally the unison is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales.<br />

<br />

If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, where q is an odd number q ~~&gt;=~~ p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.<br />

If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, where q is an odd number q ≥ p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.<br />

<br />

Up to the 7-limit, Hahn distance has a very nice formula give by<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>2012-08-10 22:56:03 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-11 02:25:04 UTC</tt>.<br>
-: The original revision id was <tt>357270976</tt>.<br>
+: The original revision id was <tt>357283838</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 8: / Line 8: @@
 <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">In graph theory, the distance between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path. Given a set of just intervals, or more usually, of classes of octave-equivalent intervals, we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a consonance. Normally the unison is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales.
-If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, where q is an odd number q &gt;= p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.
+If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, where q is an odd number q ≥ p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.
 Up to the 7-limit, Hahn distance has a very nice formula give by
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 <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;Hahn distance&lt;/title&gt;&lt;/head&gt;&lt;body&gt;In graph theory, the distance between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path. Given a set of just intervals, or more usually, of classes of octave-equivalent intervals, we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a consonance. Normally the unison is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales.&lt;br /&gt;
 &lt;br /&gt;
-If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, where q is an odd number q &amp;gt;= p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.&lt;br /&gt;
+If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, where q is an odd number q ≥ p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.&lt;br /&gt;
 &lt;br /&gt;
 Up to the 7-limit, Hahn distance has a very nice formula give by&lt;br /&gt;