Hahn distance: Difference between revisions

Line 1:

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.

In [[wikipedia: Graph (mathematics)|graph theory]], the [[wikipedia: 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.

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.

Line 32:

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:Math]]

[[Category:measure]]

[[Category:Interval complexity measure]]

~~[[Category:todo:add_examples]]~~

@@ Line 1: / Line 1: @@
-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.
+In [[wikipedia: Graph (mathematics)|graph theory]], the [[wikipedia: 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.
 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.
@@ Line 32: / Line 32: @@
 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:Math]]
-[[Category:measure]]
+[[Category:Interval complexity measure]]
-[[Category:todo:add_examples]]
+{{Todo| add examples | cleanup }}