Hahn distance: Difference between revisions
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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 /> | ||
| Line 56: | Line 56: | ||
|| |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> | ||
Revision as of 22:56, 10 August 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2012-08-10 22:56:03 UTC.
- The original revision id was 357270976.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
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.
Up to the 7-limit, Hahn distance has a very nice formula give by
[[math]]
||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2
[[math]]
[[math]]
= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)
[[math]]
We may take this formula (or the similar formulas we would obtain for higher odd limits) and apply it to any triple of real numbers||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)
If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by
[[math]]
||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}
[[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.
In the 13-limit the formula for Hahn distance can be given as
[[math]]
|| |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]]
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. Original HTML content:
<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 >= 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 />
<!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2<br/>[[math]]
--><script type="math/tex">||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2</script><!-- ws:end:WikiTextMathRule:0 --><br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]<br/>
= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)<br/>[[math]]
--><script type="math/tex">= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)</script><!-- ws:end:WikiTextMathRule:1 --><br />
We may take this formula (or the similar formulas we would obtain for higher odd limits) and apply it to any triple of real numbers||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)<br />
If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]<br/>
||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}<br/>[[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 />
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]]<br/>
|| |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<br/>[[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 />
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.</body></html>