Unitary Group Representations and Applications to Musical Acoustics

unitary irreducible representations of finite dimension of the group $G$. Here the crucial point is highlighted by the fact that the unitary irreducible representations of an abelian group are of dimension 1 and therefore in the compact case are equivalent to homomorphisms from $G$ to $S^1$. In summary: the musical acoustic phenomenon described in the previous paragraph, the decomposition of an excitation into normal modes or harmonics, extends to all those phenomena that are mathematically modelable through smooth functions defined on a compact and abelian group and geometrically equivalent to a circumference. In the continuation of this article we will show a special case of application of this theorem of general character.