This study uses topological data analysis (TDA) to examine the harmonic and melodic patterns in Claude Debussy’s Clair de Lune. By applying tools such as persistent homology and Betti coefficients, we uncover how Debussy shapes his music through shifting tonal centers, harmonic ambiguity, and smooth, flowing progressions. These features, central to his Impressionist style, create the soundscapes which he is well known for. To place these findings in context, we compare Debussy’s work with that of Arnold Schoenberg, whose very different musical style produces contrasting topological results. The differences in the topological data generated from their works are then explained in musical terms, highlighting how Schoenberg’s use of atonality and structural discontinuity diverges from Debussy’s fluidity. By connecting mathematics with music, this study offers new ways to visualize and understand how composers build complex sound worlds. This analysis bridges the fields of mathematics and music, providing a novel perspective on Debussy’s music and offering new tools for understanding complex musical elements through mathematical lenses.
C. Wang (Tue,) studied this question.