Music is inherently complex, with structures and interactions that unfold across multiple layers. Complex networks have emerged as powerful structures for the quantitative analysis of Western classical music, revealing significant features of its harmonic and structural organization. Although notable works have used these approaches to study music, dyadic representations of interactions fall short in conveying the underlying complexity and depth. In recent years, the limitations of traditional graph representations have been questioned and challenged in the context of interactions that could be higher-dimensional. Effective musical analysis requires models that capture higher-order interactions and a framework that simultaneously captures transitions between them. Subsequently, in this paper, we present a topological framework for analyzing J. S. Bach's Solo Violin Sonatas and Partitas that uses higher-order networks where single notes are vertices, two-note chords are edges, three-notes are triangles, etc. We subsequently account for the flow of music, by modeling transitions between successive notes. We identify genre-specific patterns in the works' geometric and topological properties. In particular, we find signatures in the trends of the evolution of the Euler characteristic and curvature, as well as examining adherence to the Gauss-Bonnet theorem across different movement types. The distinctions are revealed between slow movements, Fugues, and Baroque dance movements through their simplicial complex representation.