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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

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

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

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

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.