Hahn distance
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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||(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.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||(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>