Mathematical Foundations of Polyphonic Music Generation via Structural Inductive Bias

This monograph addresses the "Missing Middle" problem in AI music generation - the challenge of producing coherent, phrase-level musical structure. Using Beethoven's piano sonatas as a case study, I introduce the Smart Embedding architecture, a factorized representation grounded in the empirically verified independence of pitch and hand attributes (NMI=0.167). The architecture achieves a 48.3% reduction in embedding parameters while improving validation loss by 9.47%. Theoretically, I establish formal guarantees through information theory, Rademacher complexity analysis (yielding a 28.09% tighter generalization bound), and category-theoretic interpretation. These results are further supported by Singular Value Decomposition analysis and a blind expert listening study (N=53). Collectively, this work presents a dual contribution that combines architectural innovation with mathematical rigor, offering a principled framework for building more efficient, stable, and interpretable generative models for complex sequential data.