Hahn distance: Difference between revisions

In [http://en.wikipedia.org/wiki/Graph_(mathematics) graph theory], the [http://en.wikipedia.org/wiki/Distance_(graph_theory) 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.

 

<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>

 

 

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.

 

<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>

 

It should be noted that this formula defines a [http://en.wikipedia.org/wiki/Metric_space 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> || = </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.

[[Category:distance]]

[[Category:math]]

[[Category:measure]]

[[Category:todo:add_examples]]