# Non-Commutative Homometry in the Dihedral Groups

## TL;DR

This paper explores homometry within non-commutative dihedral groups, offering new theoretical insights, computational enumeration results, and musical applications, contrasting with traditional commutative group approaches.

## Contribution

It introduces a novel non-commutative framework for homometry in dihedral groups, linking it to musical theory and extending previous commutative methods.

## Key findings

- Computational enumeration of homometric sets for small n
- Identification of links between homometry in Z_n and D_n
- Musical interpretation of homometry in D_{12}

## Abstract

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.