Hahn distance: Difference between revisions
Wikispaces>guest **Imported revision 356818512 - Original comment: again switched to "math" syntax** |
Wikispaces>genewardsmith **Imported revision 357266516 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-10 22:07:31 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>357266516</tt>.<br> | ||
: The revision comment was: <tt> | : 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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)} | ||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)} | ||
[[math]] | [[math]] | ||
and discussed [[The Seven Limit Symmetrical Lattices|here]]. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.</pre></div> | and discussed [[The Seven Limit Symmetrical Lattices|here]]. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice. | ||
In the 13-limit the formula for Hahn distance can be given as | |||
[[math]] | |||
|| |x1 x2 x3 x4 x5 x6> ||_{hahn} = (|y|+}x3|+|x4|+|x5|+|x6|+|y+x3+x4+x5+x6|)/2 | |||
[[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 5, 7, 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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||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}&lt;br/&gt;[[math]] | ||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}</script><!-- ws:end:WikiTextMathRule:2 --><br /> | --><script type="math/tex">||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
and discussed <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">here</a>. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.</body></html></pre></div> | and discussed <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">here</a>. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.<br /> | ||
<br /> | |||
In the 13-limit the formula for Hahn distance can be given as<br /> | |||
<!-- ws:start:WikiTextMathRule:3: | |||
[[math]]&lt;br/&gt; | |||
|| |x1 x2 x3 x4 x5 x6&gt; ||_{hahn} = (|y|+}x3|+|x4|+|x5|+|x6|+|y+x3+x4+x5+x6|)/2&lt;br/&gt;[[math]] | |||
--><script type="math/tex">|| |x1 x2 x3 x4 x5 x6> ||_{hahn} = (|y|+}x3|+|x4|+|x5|+|x6|+|y+x3+x4+x5+x6|)/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 5, 7, 9 or 11 limit can all be found from this formula also.</body></html></pre></div> | |||