Hahn distance: Difference between revisions
Wikispaces>genewardsmith **Imported revision 357266856 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 357270976 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-10 22:56:03 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>357270976</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> ||_{hahn} = (|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2 | || |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> ||_{hahn} = (|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2 | ||
[[math]] | [[math]] | ||
where y = signum(x2)ceil(|x2/2|); here "signum" is +1 or -1 depending on the sign of x2 and "ceil" is the ceiling function. Hahn distance for the | where y = signum(x2)ceil(|x2/2|); here "signum" is +1 or -1 depending on the sign of x2 and "ceil" is the ceiling function. Hahn distance for the 9 or 11 limit can all be found from this formula also. </pre></div> | ||
<h4>Original HTML content:</h4> | <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>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 /> | <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 /> | ||
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|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6&gt; ||_{hahn} = (|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2&lt;br/&gt;[[math]] | || |x_1\ x_2\ x_3\ x_4\ x_5\ x_6&gt; ||_{hahn} = (|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2&lt;br/&gt;[[math]] | ||
--><script type="math/tex">|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> ||_{hahn} = (|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2</script><!-- ws:end:WikiTextMathRule:3 --><br /> | --><script type="math/tex">|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> ||_{hahn} = (|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2</script><!-- ws:end:WikiTextMathRule:3 --><br /> | ||
where y = signum(x2)ceil(|x2/2|); here &quot;signum&quot; is +1 or -1 depending on the sign of x2 and &quot;ceil&quot; is the ceiling function. Hahn distance for the | where y = signum(x2)ceil(|x2/2|); here &quot;signum&quot; is +1 or -1 depending on the sign of x2 and &quot;ceil&quot; is the ceiling function. Hahn distance for the 9 or 11 limit can all be found from this formula also.</body></html></pre></div> | ||