Hahn distance: Difference between revisions

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-~~07 22~~:55:29 UTC</tt>.<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2012-08-08 04:35:19 UTC</tt>.<br>

: The original revision id was <tt>~~356788584~~</tt>.<br>

: The original revision id was <tt>356818286</tt>.<br>

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

: The revision comment was: <tt>converted using "math" - the linebreak made problems</tt><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>

<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.

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

||3^a 5^b 7^c||~~_hahn~~ = (|a| + |b| + |c| + |a+b+c|)/2 = 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

||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2

||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)

[[math]]

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 = √(a^2 + b^2 + c^2 + ab + bc + ca)

= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)

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.

[[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|)

</pre></div>

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||(a, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca)

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>

<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>~~<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 />

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 />

<br />

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

||3^a 5^b 7^c||~~_hahn~~ = (|a| + |b| + |c| + |a+b+c|)/2 = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)<br />

<!-- ws:start:WikiTextMathRule:0:

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~~<br />~~

[[math]]&lt;br/&gt;

||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)<br />

||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2&lt;br/&gt;[[math]]

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 />~~

--><script type="math/tex">||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2</script><br />

||(a, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca)<br />

<!-- ws:start:WikiTextMathRule:1:

[[math]]&lt;br/&gt;

= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)&lt;br/&gt;[[math]]

--><script type="math/tex">= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)</script><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||(a, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca)<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>

@@ 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>2012-08-07 22:55:29 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2012-08-08 04:35:19 UTC</tt>.<br>
-: The original revision id was <tt>356788584</tt>.<br>
+: The original revision id was <tt>356818286</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>converted using "math" - the linebreak made problems</tt><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>
-<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">
+<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.
-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.
 Up to the 7-limit, Hahn distance has a very nice formula give by
-||3^a 5^b 7^c||_hahn = (|a| + |b| + |c| + |a+b+c|)/2 = 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
+||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2
-||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)
+[[math]]
-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 = √(a^2 + b^2 + c^2 + ab + bc + ca)
+= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)
-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.
+[[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|)
-</pre></div>
+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||(a, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca)
+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>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Hahn distance&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;br /&gt;
+<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;
-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;
 &lt;br /&gt;
 Up to the 7-limit, Hahn distance has a very nice formula give by&lt;br /&gt;
-||3^a 5^b 7^c||_hahn = (|a| + |b| + |c| + |a+b+c|)/2 = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:0:
-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&lt;br /&gt;
+[[math]]&amp;lt;br/&amp;gt;
-||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)&lt;br /&gt;
+||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2&amp;lt;br/&amp;gt;[[math]]
-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&lt;br /&gt;
+ --&gt;&lt;script type="math/tex"&gt;||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
-||(a, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca)&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:1:
+[[math]]&amp;lt;br/&amp;gt;
+= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:1 --&gt;&lt;br /&gt;
+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|)&lt;br /&gt;
+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||(a, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca)&lt;br /&gt;
 and discussed &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;here&lt;/a&gt;. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.&lt;/body&gt;&lt;/html&gt;</pre></div>