Homometry in the Dihedral Groups: Lifting Sets from $\mathbb {Z}_{n}$ to $D_{n}$

Abstract

The paper deals with the question of homometry in the dihedral groups $D_{n}$ of order 2n. These groups are non-commutative, leading to new and challenging definitions of homometry, as compared to the well-known case of homometry in the commutative group $\mathbb {Z}_{n}$. We give here a musical interpretation of homometry in $D_{12}$ using the well-known neo-Riemannian groups, some results on a complete enumeration of homometric sets for small values of n, and some properties disclosing the deep links between homometry in $\mathbb {Z}_{n}$ and homometry in $D_{n}$.