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As an undeniably cultural artifact, music has been subject to humanistic inquiry for centuries. How does this square with the equally ancient, yet conflicting, fascination with music as a scientific object—a fascination that has yielded important insights into the physics of musical sound and, more recently, the biology and evolution of musical behavior? This dissertation develops a cognitive, theoretical answer to this question by considering similar issues in language research, specifically ideas from the Minimalist Program in generative linguistics. In particular, it explores the unique, innate ability of the human mind to compute grammar in music in a manner likewise to that proposed for language by Minimalism. It proceeds from there to argue that the grammatical, musical mind is optimally suited to its various aesthetic functions, such as its ability to create meaning in both language and music. The dissertation makes this argument in two ways; first, by examining a deep, historical and philosophical link between the music theory of Heinrich Schenker and the generative linguistics tradition, and secondly, by using ideas and methods from Minimalism to explore various facets of musical grammar, including its computational structure, its cross-cultural invariance across Western tonal and North Indian Classical music and its ability to govern musical phenomena often considered extra-grammatical, such as musical meaning and rhythm. The dissertation explores these issues with many analytical examples, primarily from the works of Beethoven and Chopin, and from North Indian Classical instrumental performance.