- Music and Probability
In 2003, I reviewedDavid Temperley's previous book, The Cognition of Basic Music Structures (The MIT Press, 2001) for this journal (CMJ 27:2, pp. 113–115). Primary concerns expressed in that review were over limitations in the computational approach used to represent music, the experimental assumptions made about music cognition based on the author's intuitions about how music is received, and the way these two elements were related to illustrate the core of the argument. Mr. Temperley's current book takes substantial steps toward addressing some of these issues by linking cognitive experience represented in notation to directly test models of how the perceptual process of music might work.
Although the book has the title "Music and Probability," a more accurate one, given the substantial focus of the book, might be "Probability in Western Tonal MusicMeter and Key." Intended for a general audience, the writing is clear, concise, and accessible, although it does require readers to have a basic knowledge of Western music notation and mathematics. Given the density and focus of the text, itwill also engage awide range of specialists in the fields that it covers.
The introductory chapter starts with a quote by Leonard B. Meyer who indicated "the fundamental link between musical style, perception and probability" (p. 2) over 50 years ago, reflecting Mr. Temperley's sense of the essence of musical communication. The goal of the book, then, is to work toward an understanding of "how probabilities shape music perception, and indeed music itself" (p. 2). The promise is one of combining the fields of music cognition/perception and musicology/music theory.
Anyone who has had to learn a second language will sense familiarity with the initial assumptions set out in the opening pages about perception being "an inferential, multileveled, uncertain process," knowledge of probabilities coming largely from regularities in the environment, and "producers of communication [that] are sensitive to, and affected by, its probabilistic nature" (p. 3). In his new book, Mr. Temperley takes a step forward in extending his previous work on preference rule models by drawing on recent work in cognitive science, toward exploring the logical inferences that can be made, given certain assumptions, about the relationship between musical surfaces and musical structures.
Concepts of probability that are needed to understand the author's arguments are introduced in the second chapter, which assumes no prior knowledge, starting from Bayes's Rule, a theory that relates the probability of specific events taking place to the probability that events that are conditional upon them have occurred. Applied to music perception, when one hears a note pattern and wants to know the background/structure it relates to, Bayes's Rule can be used to decipher this, given prior knowledge of structures. The second main concept used comes from informational theory, that of cross-entropy, which illustrates quantitatively the degree to which a model can predict a body of data. The chapter is well set out and examples lead the non-specialist reader through the topic comfortably.
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Chapter 3, "Melody 1: The Rhythm Model," covers rhythmic understanding of melody from a listener's perspective, the notion being that people understand rhythm by imposing a complex hierarchical structure on it. The task is then to put forward a computational model to simulate the process. The argument is that one can discern the metrical grid or metrical structure from the surface (melody) based on the likelihood of it being one thing above another based on previous experience, which Mr. Temperley's model attempts to simulate. The discussion begins from a survey of prior literature, then develops a new probabilistic model following a Bayesian approach, which leads to a generative model of rhythm that creates a three-leveled metrical [End Page 102] grid to look at how listeners might guess at meter.
Multiple probabilistic parameters are...