Non-Commutative Homometry in the Dihedral Groups

The paper deals with the question of homometry in the dihedral groups $D_{n}$ of order $2n$. These groups have the specificity to be non-commutative. It leads to a new approach as compared as the one used in the traditional framework of 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 computational results concerning enumeration of homometric sets for small values of $n$, and some properties disclosing important links between homometry in $\mathbb{Z}_{n}$ and homometry in $D_{n}$. Finally we propose an extension of musical applications for this non-commutative homometry.