Exploring the rhythm of the Palestrina style: A case study in probabilistic grammar induction

This article presents a general-purpose formalism for modeling musical syntax as a probabilistic musical grammar. The formal probabilistic framework offers a precise yet flexible characterization of musical style as structure and process. Moreover, the grammar can be built algorithmically from a sample of musical examples, using a statistical grammar induction technique known as hidden Markov models (HMMs). The two fundamental assumptions of HMMs-termed finite memory and stationarity - are analyzed to show that the framework is expressive enough to capture a broad range of syntactic constraints in music. It is argued that the HMM technique draws its power from the ability to identify hidden structures that are important for shaping the musical surface. Thus, HMM-based grammar induction offers a practical, accurate, and methodologically sound tool for fine-grained modeling of musical style. As a concrete illustration, this article builds a formal grammar of rhythm for the Palestrina style. The grammar's structure and components are carefully explained, and the formalism is compared with existing approaches to style characterization. Many traditional counterpoint rules are shown to naturally correspond with the grammar's formal structure and are thus supported or refined. Other rules are disconfirmed or shown to lie outside the formalism's scope. The long-standing problem of Renaissance meter is discussed in light of these results. Thus, through the Palestrina case study, the problem of grammar induction is framed in terms of traditional concerns in music scholarship in order to motivate application of the technique, illustrate its usefulness, and place it in a historical and methodological context within music theory research.