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Music and Probability

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If music perception is largely probabilistic in nature this should not be surprising since probability pervades almost every aspect of mental life. Thus the author invokes a number of concepts from probability theory and probabilistic modeling, relying most heavily on Bayes Rule, an axiom of probability. In music perception, one is often confronted with a pattern of notes and wishes to know the underlying structure that gave rise to it. Bayes' Rule allows us to identify that underlying structure. The author also makes use of concepts from information theory such as the idea of cross-entropy. Cross-entropy shows in a quantitative way how well a model predicts a body of data. In chapter 2 the author surveys all the probability theory needed for the following chapters. He also shows a few simple examples, and discusses the applications of probability theory to other areas of study. In chapter three the author addresses a basic problem of music perception - the identification meter - and proposes a probabilistic model of this process.

Chapters 4 and 6 examine the problem of key perception from a probabilistic standpoint. The author first proposes a model of key perception in monophonic music (melodies). This model is then expanded to accommodate polyphonic music. With regard to both meter and key, the models proposed are not merely models of information retrieval, but also shed light on other aspects of perception. In particular they lead very naturally to ways of identifying the probability of actual note patterns. This in turn provides a way of modeling cognitive processes such as error detection, expectation, and pitch identification, as well as more subtle musical phenomena such as musical ambiguity, tension, and "tonalness". These issues are explored in chapter 5 with regard to monophonic music and chapter 7 with regard to polyphonic music. The final three chapters of the book explore a range of further issues in music and probability. Chapter eight surveys some recent work by other authors in which probabilistic methods are applied to a variety of problems in music perception and cognition: transcription, phrase perception, pattern perception, harmony, and improvisation. Chapter nine considers the idea of construing probabilistic models as descriptions of musical styles and thus as hypotheses about cognitive processes involved in composition.

In summary, this book is a good one in demonstrating that a probabilistic perspective opens the door to a new and powerful approach to the study of music creation. Highly recommended for all who have an interest in algorithmic composition. The following is the table of contents:

1. Introduction 1

2.Probabilistic Foundations and Background 7
2.1 Elementary Probability 7
2.2 Conditional Probability and Bayes' Rule 8
2.3 Other Probabilistic Concepts 14
2.4 Early Work on Music and Probability 19

3. Melody I: The Rhythm Model 23
3.1 Rhythm and Meter 23
3.2 Previous Models of Meter Perception 26
3.3 A Probabilistic Rhythm Model 30
3.4 The Generative Process 31
3.5 The Meter-Finding Process 36
3.6 Testing the Model on Meter-Finding 41
3.7 Problems and Possible Improvements 43

4. Melody II: The Pitch Model 49
4.1 Previous Models of Key-Finding 50
4.2 The Pitch Model 56
4.3 Testing the Model on Key-Finding 62

5. Melody III: Expectation and Error Detection 65
5.1 Calculating the Probability of a Melodic Surface 65
5.2 Pitch Expectation 66
5.3 Rhythmic Expectation 71
5.4 Error Detection 74
5.5 Further Issues 76

6. A Polyphonic Key-Finding Model 79
6.1 A Pitch-Class-Set Approach to Key-Finding 79
6.2 The Generative Process 83
6.3 The Key-Finding Process 85
6.4 Comparing Distributional Models of Key-Finding 89
6.5 Further Issues in Key-Finding 92

7. Applications of the Polyphonic Key-Finding Model 99
7.1 Key Relations 99
7.2 Tonalness 108
7.3 Tonal Ambiguity and Clarity 116
7.4 Another Look at Major and Minor 121
7.5 Ambiguous Pitch-Collections in Common-Practice Music 125
7.6 Explaining Common Strategies of Tonal Harmony 131

8. Bayesian Models of Other Aspects of Music 139
8.1 Probabilistic Transcription Models 139
8.2 Bod: The Perception of Phrase Structure 143
8.3 Raphael and Stoddard: Harmonic Analysis 147
8.4 Mavromatis: Modeling Greek Chant Improvisation 151
8.5 Saffran et al.: Statistical Learning of Melodic Patterns 156

9. Style and Composition 159
9.1 Some Simple Cross-Entropy Experiments 161
9.2 Modeling Stylistic Differences 166
9.3 Testing Schenkerian Theory 172

10. Communicative Pressure 181
10.1 Communicative Pressure in Rules of Voice-Leading 182
10.2 The Syncopation-Rubato Trade-Off 184
10.3 Other Examples of Communicative Pressure in Rhythm 191
10.4 "Trading Relationships" 197
10.5 Low-Probability Events in Constrained Contexts 202
10.6 Conclusions 205