Can music be represented as a meaningful geometric and topological object? In this paper, we propose a strategy to describe some music features as a polyhedral surface obtained by a simplicial interpretation of the \textit{Tonnetz}. The \textit{Tonnetz} is a graph largely used in computational musicology to describe the harmonic relationships of notes in equal tuning. In particular, we use persistent homology in order to describe the \textit{persistent} properties of music encoded in the aforementioned model. Both the relevance and the characteristics of this approach are discussed by analyzing some paradigmatic compositional styles. Eventually, the task of automatic music style classification is addressed by computing the hierarchical clustering of the topological fingerprints associated with some collections of compositions.