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