Hahn distance

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]\displaystyle \begin{align} & \lVert 3^a \cdot 5^b \cdot 7^c \rVert_\text {hahn} \\ =& (\lvert a \rvert + \lvert b \rvert + \lvert c \rvert + \lvert a + b + c \rvert)/2 \\ =& \max(\lvert a \rvert, \lvert b \rvert, \lvert c \rvert, \lvert a + b \rvert, \lvert b + c \rvert, \lvert c + a \rvert, \lvert a + b + c \rvert) \end{align} [/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]\displaystyle \left\lVert (a, b, c) \right\rVert_\text {sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}[/math]

and discussed in The Seven Limit Symmetrical Lattices. 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]\displaystyle \begin{align} & \left\lVert \lvert x_1\ x_2\ x_3\ x_4\ x_5\ x_6 \rangle \right\rVert_\text{hahn} \\ =& (\lvert y \rvert + \lvert x_3 \rvert + \lvert x_4 \rvert + \lvert x_5 \rvert + \lvert x_6 \rvert + \lvert y + x_3 + x_4 + x_5 + x_6 \rvert)/2 \end{align} [/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]\displaystyle \begin{align} & \left\lVert \lvert x_1\ x_2\ x_3\ x_4\ x_5\ x_6 \rangle \right\rVert \\ =& \lvert x_2/2 \rvert + \lvert x_3 \rvert + \lvert x_4 \rvert + \lvert x_5 \rvert + \lvert x_6 \rvert + \lvert x_2/2 + x_3 + x_4 + x_5 + x_6 \rvert \end{align} [/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.