Connectionist Representations of Tonal Music CONNECTIONIST REPRESENTATIONS OF TONAL MUSIC Discovering Musical Patterns by Interpreting Artificial Neural Networks Michael R. W. Dawson Copyright © 2018 Michael Dawson Published by AU Press, Athabasca University 1200, 10011 – 109 Street, Edmonton, AB T5J 3S8 ISBN 978-1-77199-220-6 (pbk.) 978-1-77199-221-3 (PDF) 978-1-77199-222-0 (epub) doi: 10.15215/aupress/9781771992206.01 Cover and interior design by Sergiy Kozakov Printed and bound in Canada by Friesens Library and Archives Canada Cataloguing in Publication Dawson, Michael Robert William, 1959-, author Connectionist representations of tonal music : discovering musical patterns by interpreting artificial neural networks / Michael R.W. Dawson. Includes bibliographical references and index. Issued in print and electronic formats. 1. Music—Psychological aspects—Case studies. 2. Music—Physiological aspects—Case studies. 3. Music—Philosophy and aesthetics—Case studies. 4. Neural networks (Computer science)—Case studies. I. Title. ML3830.D39 2018 781.1’1 C2017-907867-4 C2017-907868-2 We acknowledge the financial support of the Government of Canada through the Canada Book Fund (CBF) for our publishing activities and the assistance provided by the Government of Alberta through the Alberta Media Fund. This publication is licensed under a Creative Commons License, Attribution– Noncommercial–NoDerivative Works 4.0 International: see www.creativecommons.org. The text may be reproduced for non-commercial purposes, provided that credit is given to the original author. To obtain permission for uses beyond those outlined in the Creative Commons license, please contact AU Press, Athabasca University, at [email protected]. Contents List of Figures ix List of Tables xiii Acknowledgements xv Overture: Alien Music 3 Chapter 1: Science, Music, and Cognitivism 1.1 Mechanical Philosophy, Mathematics, and Music 9 1.2 Mechanical Philosophy and Tuning 10 1.3 Psychophysics of Music 13 1.4 From Rationalism to Classical Cognitive Science 15 1.5 Musical Cognitivism 17 1.6 Summary 26 Chapter 2: Artificial Neural Networks and Music 2.1 Some Connectionist Basics 29 2.2 Romanticism and Connectionism 36 2.3 Against Connectionist Romanticism 38 2.4 The Value Unit Architecture 42 2.5 Summary and Implications 45 Chapter 3: The Scale Tonic Perceptron 3.1 Pitch-Class Representations of Scales 49 3.2 Identifying the Tonics of Musical Scales 56 3.3 Interpreting the Scale Tonic Perceptron 58 3.4 Summary and Implications 66 Chapter 4: The Scale Mode Network 4.1 The Multilayer Perceptron 69 4.2 Identifying Scale Mode 72 4.3 Interpreting the Scale Mode Network 74 4.4 Tritone Imbalance and Key Mode 79 4.5 Further Network Analysis 80 4.6 Summary and Implications 89 Chapter 5: Networks for Key-Finding 5.1 Key-Finding 91 5.2 Key-Finding with Multilayered Perceptrons 93 5.3 Interpreting the Network 95 5.4 Coarse Codes for Key-Finding 98 5.5 Key-Finding with Perceptrons 104 5.6 Network Interpretation 112 5.7 Summary and Implications 115 Chapter 6: Classifying Chords with Strange Circles 6.1 Four Types of Triads 119 6.2 Triad Classification Networks 121 6.3 Interval Cycles and Strange Circles 128 6.4 Added Note Tetrachords 143 6.5 Classifying Tetrachords 146 6.6 Interpreting the Tetrachord Network 148 6.7 Summary and Implications 165 Chapter 7: Classifying Extended Tetrachords 7.1 Extended Tetrachords 169 7.2 Classifying Extended Tetrachords 173 7.3 Interpreting the Extended Tetrachord Network 175 7.4 Bands and Coarse Coding 200 7.5 Summary and Implications 206 Chapter 8: Jazz Progression Networks 8.1 The ii-V-I Progression 209 8.2 The Importance of Encodings 212 8.3 Four Encodings of the ii-V-I Problem 213 8.4 Complexity, Encoding, and Training Time 219 8.5 Interpreting a Pitch-class Perceptron 222 8.6 The Coltrane Changes 232 8.7 Learning the Coltrane Changes 237 8.8 Interpreting a Coltrane Perceptron 240 8.9 Strange Circles and Coltrane Changes 244 8.10 Summary and Implications 248 Chapter 9: Connectionist Reflections 9.1 A Less Romantic Connectionism 251 9.2 Synthetic Psychology 0f Music 254 9.3 Musical Implications 261 9.4 Implications for Musical Cognition 267 9.5 Future Directions 271 References 273 Index 293 Figures Figure 1-1 An illustration of “relatedness ratings” for each pitch-class in the context of the musical key A major. Figure 1-2 A three-dimensional spiral can simultaneously capture the linear arrangement of pitch and the circular arrangement of pitch-class on a piano keyboard. Figure 1-3 Using circles of minor seconds to explain the tritone paradox. Figure 2-1 An example artificial neural network that, when presented a stimulus chord, responds with another chord. Figure 2-2 The logistic activation function used by an integration device to convert net input into activity. Figure 2-3 The Gaussian activation function used by a value unit to convert net input into activity. Figure 3-1 An example major scale, and an example harmonic minor scale, represented using multiple staffs. Figure 3-2 The circle of minor seconds can be used to represent the pitch-classes found in the A major and the A harmonic minor scales. Figure 3-3 Architecture of a perceptron trained to identify the tonic notes of input patterns of pitch-classes. Figure 3-4 The connection weights between the 12 input units and any output unit in the scale tonic perceptron. Figure 3-5 The connection weights between the 12 input units and an output unit in the scale tonic perceptron, showing only those connections that have a signal sent through them when the output unit’s major scale pattern is presented to it. Figure 3-6 The connection weights between the 12 input units and an output unit, showing only those connections that have a signal sent through them when the output unit’s harmonic minor scale pattern is presented to it. Figure 3-7 The active connections to the output unit for pitch-class A when the network is presented the G major scale. Figure 4-1 A multilayer perceptron, with two hidden units, that detects whether a presented scale is major or minor. Figure 4-2 The hidden unit space for the scale mode network. Figure 4-3 The connection weights between the 12 input units and each hidden unit in the scale mode network. ix doi: 10.15215/aupress/9781771992206.01 Figure 4-4 Connection weights between input units and each hidden unit. Figure 4-5 Spokes in a circle of minor seconds used to represent the pitch-classes that define major and minor keys. Figure 4-6 A two-dimensional and a three-dimensional multidimensional scaling solution that arranges scales associated with different keys in a spatial map. Figure 4-7 The structure of Hook’s (2006) Tonnetz for triads. Figure 5-1 A multilayer perceptron that uses four hidden units to detect both the mode and the tonic of presented major or harmonic minor scales. Figure 5-2 A hypothetical two-dimensional hidden unit space for key-finding. Figure 5-3 A three-dimensional projection of the four-dimensional hidden unit space for the key-finding multilayer perceptron. Figure 5-4 A different three-dimensional projection of the four-dimensional hidden unit space for the key-finding multilayer perceptron. Figure 5-5 The results of wiretapping each hidden unit, using the 24 input patterns as stimuli. Figure 5-6 A perceptron that can be used to map key profiles onto musical key. Figure 6-1 The top staff provides examples of four different types of triads built on the root note of A: A major (A), A minor (Am), A diminished (Adim) and A augmented (Aaug). Figure 6-2 A multilayer perceptron with local pitch encoding that learns to identify four types of triads, ignoring a triad’s key and inversion. Figure 6-3 The connection weights from the 28 input units for pitch to Hidden Unit 1 in the network trained to classify triad types for different keys and inversions. Figure 6-4 The connection weights from the 28 input units for pitch to Hidden Unit 2 in the network trained to classify triad types for different keys and inversions. Figure 6-5 The connection weights from the 28 input units for pitch to Hidden Unit 3 in the network trained to classify triad types for different keys and inversions. Figure 6-6 The connection weights from the 28 input units for pitch to Hidden Unit 4 in the network trained to classify triad types for different keys and inversions. Figure 6-7 The geography of the piano. Figure 6-8 Using the number of piano keys as a measure of the distance between pitches. Figure 6-9 The circle of minor seconds. x doi: 10.15215/aupress/9781771992206.01 Figure 6-10 The circle of major sevenths. Figure 6-11 The circle of perfect fourths. Figure 6-12 The circle of perfect fifths. Figure 6-13 The two circles of major seconds. Figure 6-14 The two circles of minor sevenths. Figure 6-15 The three circles of minor thirds. Figure 6-16 The three circles of major sixths. Figure 6-17 The four circles of major thirds. Figure 6-18 The four circles of minor sixths. Figure 6-19 The six circles of tritones. Figure 6-20 The 12 circles of octaves, or circles of unison. Figure 6-21 Added note tetrachords in the key of C major. Figure 6-22 A multilayer perceptron that classifies tetrachords into four different types. Figure 6-23 The hidden unit space for a multilayer perceptron trained to identify the four types of tetrachords.