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<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.

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