Music captioning has gained significant attention in the wake of the rising prominence of streaming media platforms. Traditional approaches often prioritize either the audio or lyrics aspect of the music, inadvertently ignoring the intricate interplay between the two. However, a comprehensive understanding of music necessitates the integration of both these elements. In this study, we delve into this overlooked realm by introducing a method to systematically learn multimodal alignment between audio and lyrics through contrastive learning. This not only recognizes and emphasizes the synergy between audio and lyrics but also paves the way for models to achieve deeper cross-modal coherence, thereby producing high-quality captions. We provide both theoretical and empirical results demonstrating the advantage of the proposed method, which achieves new state-of-the-art on two music captioning datasets.