As a universal expression of human creativity, music is capable of conveying great subtlety and complexity. Crucially, this complexity is not encoded in the score or in the sounds, but is rather construed in the mind of the listener in the form of nuanced perceptual experiences, commonly referred to as "structural hearing". While these experiences are to some extent accessible to introspection, which is made explicit in the music-theoretical discourse, the underlying cognitive mechanisms are elusive of empirical investigation. In this thesis, we conceptualise the experience of musical structure in the context of Bayesian cognition as a form of inference: namely, the inference of representations of structure as a way of making sense of music's sensory signals. Exploiting a computational analogy with linguistic processing, we model the emergence of structural interpretations in terms of grammar-based incremental parsing. In a series of behavioural experiments, we test some crucial implications of this modelling approach: (1) the existence of representations of structure abstracted from sensory information, which we test by adapting a structural-priming paradigm to the musical case, (2) the cognitive relevance of idiom-specific syntactic categories, exemplified by the notion of harmonic function in extended-tonal harmony, (3) the time-course of cognitive computations implementing incremental parsing in real time during listening, and (4) the existence of mechanisms of retrospective reanalysis by analogy with the linguistic garden-path effect. Overall, these results contribute proofs of existence for some cornerstones of a computational- and algorithmic-level theory of structural hearing. They are compatible with an inference process implemented through parsing computations including the integration of newly encountered events into a pre-existing representation, the projection of expected events in the future, and the retrospective revision of the interpretation of past events. Building on the proposed framework, future work may further test implications of different fine-grained algorithmic models of parsing, in order to distinguish between accounts of processing similarly to how models of sentence comprehension are disambiguated in psycholinguistics.