Hahn distance

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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; if there is no path connection them, the distance is regarded as infinite. 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, excluding the unison and octaves, 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 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 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\gt ||_{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 defines a metric space distance function but not 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\gt || = [/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.