Music comprises two core structural components, melody and rhythm, that vary widely across cultures. Whether these components coevolve in a coupled way or follow independent trajectories remains unclear. We introduce a novel computational pipeline to extract vocal melodic pitch-interval and percussive inter-onset timing distributions from 27,628 popular songs across 59 countries, enabling large-scale cross-cultural comparison that bypasses traditional music annotations. Musical similarities between countries aligned with geographic and linguistic relationships, validating our approach. Substantial variation emerged in both melodic and rhythmic structures across countries, yet the diversity of the two components was not significantly correlated, challenging assumptions of coupled evolution. Only rhythmic diversity was significantly associated with ethnic and linguistic heterogeneity, while melodic diversity showed no such association. These findings suggest that melody and rhythm constitute partially independent systems shaped by distinct cultural and evolutionary pressures, rather than components of a single monolithic musical style.