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 and apply it to any triple of real numbers ||(a, b, c)||_hahn = (|a|+|b|+|c|+|a+b+c|)/2.
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; it also defines a seminorm on 7-limit [[Monzos and Interval Space|interval space]]. 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} =
[[math]]
[[math]]
(|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 also be found from this formula.
It should be noted that this formula does not define a norm and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to
[[math]]
|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> || =
[[math]]
[[math]]
|x_2/2|+|x_3|+|x_4|+|x_5|+|x_6|+|x_2/2+x_3+x_4+x_5+x_6|
[[math]]
This makes the 9.5.7.11.13 sublattice symmetrical, corresponded to even distance values from the origin, with the full lattice corresponding to all positive integer distances.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 and apply it to any triple of real numbers ||(a, b, c)||_hahn = (|a|+|b|+|c|+|a+b+c|)/2.<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; it also defines a seminorm on 7-limit <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>. 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} = <br/>[[math]]
--><script type="math/tex">|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> ||_{hahn} = </script><!-- ws:end:WikiTextMathRule:3 --><br />
<!-- ws:start:WikiTextMathRule:4:
[[math]]<br/>
(|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">(|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2</script><!-- ws:end:WikiTextMathRule:4 --><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 also be found from this formula. <br />
<br />
It should be noted that this formula does not define a norm and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to<br />
<!-- ws:start:WikiTextMathRule:5:
[[math]]<br/>
|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> || = <br/>[[math]]
--><script type="math/tex">|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> || = </script><!-- ws:end:WikiTextMathRule:5 --><br />
<!-- ws:start:WikiTextMathRule:6:
[[math]]<br/>
|x_2/2|+|x_3|+|x_4|+|x_5|+|x_6|+|x_2/2+x_3+x_4+x_5+x_6|<br/>[[math]]
--><script type="math/tex">|x_2/2|+|x_3|+|x_4|+|x_5|+|x_6|+|x_2/2+x_3+x_4+x_5+x_6|</script><!-- ws:end:WikiTextMathRule:6 --><br />
This makes the 9.5.7.11.13 sublattice symmetrical, corresponded to even distance values from the origin, with the full lattice corresponding to all positive integer distances.</body></html>