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Eulerian scale

[Gene Ward Smith, Yahoo tuning-math message 11300, adapted by Joe Monzo]

Message 11300
From: "Gene Ward Smith"
Date: Tue Aug 10, 2004 2:59 pm
Subject: Eulerian and Hamiltonian scales

A traversable or Eulerian graph is a graph which has a circuit -- a path starting and ending at the same point -- which traverses each edge exactly once.

Given an octave-equivalent scale, we can form the q-odd-limit graph of the scale by making the degrees of the scale the nodes, and the concordances of the scale the edges. If the resulting graph is Eulerian, we can say the scale is Eulerian in the q-limit.

A graph is Eulerian if and only if each vertex is of even degree, which means in scale terms each degree connects with an even number of other degrees. An example of an Eulerian scale is the hexany, where each degree connects with four others.

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