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WHAT IS MUSIC? Solving a…
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WHAT IS MUSIC?
Solving a Scientific Mystery
The science of music started more than 2000 years ago, when
Pythagoras made his observations about consonant intervals
and ratios of string lengths.
But despite all the advances made in acoustics, psychology,
neuroscience and evolutionary biology, scientists still have no
idea what music is.
The theory in this book is the result of more than 20 years
of research by the author. It explains in detail many of the
familiar features of music: notes, scales, melody, harmony,
chords, home chords, bass, rhythm and repetition.
It also explains the symmetries of music. These symmetries in-
clude invariances under pitch translation, octave translation,
time translation, time scaling, amplitude scaling and pitch
reflection.
Most importantly, the theory explains the emotional effects of
music, and this explanation sits firmly within the framework of
modern evolutionary theory. For the benefit of those not fully
familiar with the concepts of theoretical biology, what this
means is that the theory explains how our ability to respond
to music helps us have more grandchildren.
Copyright c 2004, 2005 Philip Dorrell
Published by Philip Dorrell, 2005.
All rights reserved. This online copy of the book "What is Music" may be
downloaded and printed for personal use only.
While every precaution has been taken in the preparation of this book, the
publisher assumes no responsibilities for errors or omissions, or for
damages resulting from the use of information contained herein.
Philip Dorrell asserts his moral right to be identified as the author of this
book.
Revision Date: 22 March 2005
ISBN 1-4116-2117-4
The official website for this book is http://whatismusic.info/.
The author's personal website is http://www.1729.com/, and current
contact details may be found at http://www.1729.com/email.html.
WHAT IS MUSIC?
Solving a Scientific Mystery
by Philip Dorrell
Dedicated to
Amanda and Natalie.
Contents
Acknowledgements 8
1 Introduction 9
1.1 An Autobiographical History . . . . . . . . . . . . . . . . . 9
1.1.1 The Facts of Life . . . . . . . . . . . . . . . . . . . 9
1.1.2 The Mathematics of the Universe . . . . . . . . . . 10
1.2 The Science and Mathematics of Music . . . . . . . . . . . 11
1.3 A First Breakthrough: 2D/3D . . . . . . . . . . . . . . . . 12
1.4 A Second Breakthrough: Super-Stimulus . . . . . . . . . . 13
1.5 The Rest of This Book . . . . . . . . . . . . . . . . . . . . 14
1.5.1 Background Concepts . . . . . . . . . . . . . . . . . 14
1.5.2 The Super-Stimulus Theory . . . . . . . . . . . . . 14
1.5.3 Questions, Review and the Future . . . . . . . . . . 16
2 What is Music? 18
2.1 Music is Something We Like . . . . . . . . . . . . . . . . . 18
2.2 The Biology of Feeling Good . . . . . . . . . . . . . . . . . 19
2.2.1 Having More Grandchildren . . . . . . . . . . . . . 19
2.2.2 Charles Darwin and His Theory . . . . . . . . . . . 20
2.3 Explaining Purposeful Behaviour . . . . . . . . . . . . . . . 23
2.3.1 Incorrect or Apparently Incorrect Sub-Goals . . . . 25
2.4 Proof of our Ignorance About Music . . . . . . . . . . . . . 27
2.4.1 Subjective and Objective . . . . . . . . . . . . . . . 28
2.4.2 The Martian Scientist . . . . . . . . . . . . . . . . . 29
2.4.3 The Incompleteness of Music Theory . . . . . . . . 30
2.4.4 Musical Formulae . . . . . . . . . . . . . . . . . . . 32
2.4.5 The Economics of Musical Composition . . . . . . . 33
2.5 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.1 Author's Declaration . . . . . . . . . . . . . . . . . 38
2.6 Scientific Theories . . . . . . . . . . . . . . . . . . . . . . . 38
2.6.1 Testability and Falsifiability . . . . . . . . . . . . . 38
2.6.2 Simplicity and Complexity . . . . . . . . . . . . . . 41
3 Existing Music Science 44
3.1 Existing Literature . . . . . . . . . . . . . . . . . . . . . . 44
3.2 The Origins of Music . . . . . . . . . . . . . . . . . . . . . 45
1
CONTENTS
3.3 The Archaeology of Music . . . . . . . . . . . . . . . . . . 46
3.4 Common Assumptions . . . . . . . . . . . . . . . . . . . . 48
3.4.1 The Evolutionary Assumption . . . . . . . . . . . . 48
3.4.2 The Music Assumption . . . . . . . . . . . . . . . . 49
3.4.3 The Communication Hypothesis . . . . . . . . . . . 50
3.4.4 The Social Assumption . . . . . . . . . . . . . . . . 51
3.4.5 The "In the Past" Assumption . . . . . . . . . . . . 52
3.4.6 The Music-Language Assumption . . . . . . . . . . 53
3.4.7 The Cultural Assumption . . . . . . . . . . . . . . 53
3.4.8 The Cortical Plasticity Assumption . . . . . . . . . 54
3.4.9 The Simultaneous Pitch Assumption . . . . . . . . 55
3.4.10 Other Musical Aspect Assumptions . . . . . . . . . 57
3.5 Questions That Have to be Answered . . . . . . . . . . . . 58
3.6 Approaches to Studying Music . . . . . . . . . . . . . . . . 61
4 Sound and Music 63
4.1 Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 Vibrations Travelling Through a Medium . . . . . . 63
4.1.2 Linearity, Frequency and Fourier Analysis . . . . . 64
4.2 Music: Pitch and Frequency . . . . . . . . . . . . . . . . . 71
4.2.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.2 Intervals . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.3 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.4 Consonant Intervals . . . . . . . . . . . . . . . . . . 75
4.2.5 Harmony and Chords . . . . . . . . . . . . . . . . . 76
4.2.6 Home Chords and Dominant Sevenths . . . . . . . 77
4.3 Musical Time . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.1 Tempo . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Melody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 Accompaniments . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5.1 Harmonic Accompaniment . . . . . . . . . . . . . . 83
4.5.2 Rhythmic Accompaniment . . . . . . . . . . . . . . 84
4.5.3 Bass . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 Other Aspects of Music . . . . . . . . . . . . . . . . . . . . 84
4.6.1 Repetition . . . . . . . . . . . . . . . . . . . . . . . 84
4.6.2 Songs, Lyrics and Poetry . . . . . . . . . . . . . . . 85
4.6.3 Dance . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Vector Analysis of Musical Intervals 87
5.1 Three Different Vector Representations . . . . . . . . . . . 87
5.1.1 What is a Vector Space? . . . . . . . . . . . . . . . 88
5.1.2 1D Semitones Representation . . . . . . . . . . . . 91
5.1.3 2D Tones/Semitones Representation . . . . . . . . . 92
5.1.4 3D Consonant Interval Representation . . . . . . . 92
5.2 Bases and Linear Mappings . . . . . . . . . . . . . . . . . . 94
5.2.1 2D to 1D Natural Mapping . . . . . . . . . . . . . . 95
2
5.2.2 3D to 1D Natural Mapping . . . . . . . . . . . . . . 99
5.2.3 3D to 2D Natural Mapping . . . . . . . . . . . . . . 99
5.2.4 Images and Kernels . . . . . . . . . . . . . . . . . . 100
5.2.5 Visualising the Syntonic Comma . . . . . . . . . . . 103
5.3 The Harmonic Heptagon . . . . . . . . . . . . . . . . . . . 105
6 The Brain 107
6.1 An Information Processing System . . . . . . . . . . . . . . 107
6.1.1 Analogy with Computers . . . . . . . . . . . . . . . 108
6.2 The Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2.1 Comparison to Computer Components . . . . . . . 113
6.2.2 How Many Connections? . . . . . . . . . . . . . . . 114
6.3 Modularity in the Brain . . . . . . . . . . . . . . . . . . . . 115
6.3.1 The Representation of Meaning . . . . . . . . . . . 118
6.3.2 Temporal Coding . . . . . . . . . . . . . . . . . . . 120
6.3.3 Localisation and Functional Maps . . . . . . . . . . 122
6.4 Separation and Binding . . . . . . . . . . . . . . . . . . . . 123
6.4.1 Colour Perception . . . . . . . . . . . . . . . . . . . 124
6.4.2 The Binding Problem . . . . . . . . . . . . . . . . . 125
6.5 Population Encoding . . . . . . . . . . . . . . . . . . . . . 127
7 2D/3D Theory of Music 131
7.1 More Vector Space Mappings . . . . . . . . . . . . . . . . . 131
7.1.1 Another Mapping from 2D to 1D . . . . . . . . . . 131
7.1.2 Another Perceptual 3D to 2D Mapping . . . . . . . 132
7.2 The Looping Theory . . . . . . . . . . . . . . . . . . . . . . 134
7.3 Outlook for the 2D/3D Theory . . . . . . . . . . . . . . . . 135
8 The Perception of Musicality 137
8.1 Where is the Purpose? . . . . . . . . . . . . . . . . . . . . 137
8.2 That Which is Like Music . . . . . . . . . . . . . . . . . . 138
8.3 Corollaries to the Hypothesis . . . . . . . . . . . . . . . . . 142
8.3.1 What is Musicality? . . . . . . . . . . . . . . . . . . 143
8.3.2 The Dimensionality of Musicality . . . . . . . . . . 144
8.3.3 Subjective Awareness of Musicality . . . . . . . . . 144
8.3.4 Double Dissociation . . . . . . . . . . . . . . . . . . 145
8.3.5 Differences in Melody and Rhythm . . . . . . . . . 146
8.3.6 Attributes Apparently Absent in Speech . . . . . . 147
8.3.7 Implications for Cortical Maps . . . . . . . . . . . . 148
8.4 Explaining Musical Behaviours . . . . . . . . . . . . . . . . 148
8.4.1 Dance . . . . . . . . . . . . . . . . . . . . . . . . . 150
9 Symmetries 151
9.1 Definition of Symmetry . . . . . . . . . . . . . . . . . . . . 151
9.1.1 Symmetries of Physics . . . . . . . . . . . . . . . . 153
9.2 A Little More Mathematics . . . . . . . . . . . . . . . . . . 155
9.2.1 Discrete and Continuous . . . . . . . . . . . . . . . 155
3
CONTENTS
9.2.2 Generators . . . . . . . . . . . . . . . . . . . . . . . 156
9.2.3 Stronger and Weaker Symmetries . . . . . . . . . . 156
9.3 Musical Symmetries . . . . . . . . . . . . . . . . . . . . . . 157
9.3.1 Pitch Translation Invariance . . . . . . . . . . . . . 158
9.3.2 Octave Translation Invariance . . . . . . . . . . . . 160
9.3.3 Octave Translation and Pitch Translation . . . . . 161
9.3.4 Time Scaling Invariance . . . . . . . . . . . . . . . 162
9.3.5 Time Translation Invariance . . . . . . . . . . . . . 162
9.3.6 Amplitude Scaling Invariance . . . . . . . . . . . . 163
9.3.7 Pitch Reflection Invariance . . . . . . . . . . . . . . 164
9.4 Invariant Characterisations . . . . . . . . . . . . . . . . . . 165
9.4.1 Application to Biology . . . . . . . . . . . . . . . . 167
9.4.2 Frames of Reference . . . . . . . . . . . . . . . . . . 169
9.4.3 Complete and Incomplete Representations . . . . . 169
10 Musical Cortical Maps 172
10.1 Cortical Plasticity . . . . . . . . . . . . . . . . . . . . . . . 172
10.1.1 Plasticity and Theories of Music . . . . . . . . . . . 176
10.2 Musicality in Cortical Maps . . . . . . . . . . . . . . . . . 177
10.3 The Regular Beat Cortical Map . . . . . . . . . . . . . . . 178
10.3.1 Symmetries of Regular Beat Perception . . . . . . . 182
10.3.2 Unification . . . . . . . . . . . . . . . . . . . . . . . 183
10.4 The Harmonic Cortical Map . . . . . . . . . . . . . . . . . 183
10.4.1 Active Zones . . . . . . . . . . . . . . . . . . . . . . 187
10.4.2 Octave Translation Invariant Representations . . . 187
10.4.3 Intensity Invariance . . . . . . . . . . . . . . . . . . 187
10.5 The Bass Cortical Map . . . . . . . . . . . . . . . . . . . . 188
10.6 The Scale Cortical Map . . . . . . . . . . . . . . . . . . . . 189
10.7 The Home Chord Cortical Map . . . . . . . . . . . . . . . 193
10.7.1 Why Reflective Symmetry? . . . . . . . . . . . . . . 196
10.7.2 Alternative Theory: The Dominant 7th . . . . . . . 196
10.7.3 The Evolution of Cortical Maps . . . . . . . . . . . 197
10.8 The Note Duration Cortical Map . . . . . . . . . . . . . . 198
10.9 The Melodic Contour Cortical Map . . . . . . . . . . . . . 199
11 Octave Translation Invariance 200
11.1 Octave Translation Invariant Aspects of Music . . . . . . . 200
11.2 Separation of Concerns . . . . . . . . . . . . . . . . . . . . 201
11.3 Digital versus Analogue . . . . . . . . . . . . . . . . . . . . 201
11.4 Digital Representations in the Brain . . . . . . . . . . . . . 203
11.5 Split Representation of Pitch . . . . . . . . . . . . . . . . . 205
11.6 Octaves and Consonant Intervals . . . . . . . . . . . . . . . 209
12 Calibration 210
12.1 A Four-Way Relationship . . . . . . . . . . . . . . . . . . . 210
12.2 Making Measurement Accurate . . . . . . . . . . . . . . . . 211
4
12.2.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . 213
12.2.2 Complex Fractions . . . . . . . . . . . . . . . . . . 214
12.2.3 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . 214
12.2.4 Not Measuring Non-Harmonic Intervals . . . . . . . 215
12.3 Calibration Experiments . . . . . . . . . . . . . . . . . . . 217
12.4 Temporal Coding . . . . . . . . . . . . . . . . . . . . . . . 218
12.5 Other Calibrations . . . . . . . . . . . . . . . . . . . . . . . 219
12.5.1 Calibration of Octave Perception . . . . . . . . . . 219
12.5.2 Calibrating Ratios of Durations . . . . . . . . . . . 219
12.5.3 Calibrating Against Regular Beats . . . . . . . . . 220
13 Repetition 222
13.1 Repetition as a Super-Stimulus . . . . . . . . . . . . . . . . 222
13.2 Reasons for Perception of Repetition . . . . . . . . . . . . 224
13.3 Perceptual State Machines . . . . . . . . . . . . . . . . . . 225
13.3.1 A Neuronal State Machine . . . . . . . . . . . . . . 226
13.4 The Flow Model . . . . . . . . . . . . . . . . . . . . . . . . 226
13.4.1 Breaking Out of the Loop . . . . . . . . . . . . . . 228
13.4.2 Almost Exact Repetitions . . . . . . . . . . . . . . 228
13.4.3 Faking n Dimensions in 2-Dimensional Maps . . . . 229
13.5 Non-Free Repetition: Summary . . . . . . . . . . . . . . . 231
13.6 Free Repetition and Home Chords . . . . . . . . . . . . . . 232
13.7 Reduplication . . . . . . . . . . . . . . . . . . . . . . . . . 234
14 Final Theory 235
14.1 The Story So Far . . . . . . . . . . . . . . . . . . . . . . . 235
14.2 So What is Musicality? . . . . . . . . . . . . . . . . . . . . 236
14.2.1 A List of Clues . . . . . . . . . . . . . . . . . . . . 237
14.2.2 Musicality is an Attribute of Speech . . . . . . . . . 237
14.2.3 The Emotional Effect of Music . . . . . . . . . . . . 238
14.2.4 Different Aspects and Genres . . . . . . . . . . . . 239
14.2.5 Constant Activity Patterns . . . . . . . . . . . . . . 240
14.3 The Musicality Neuron . . . . . . . . . . . . . . . . . . . . 242
14.4 Discount Factors . . . . . . . . . . . . . . . . . . . . . . . . 246
14.5 The Meaning of Musicality . . . . . . . . . . . . . . . . . . 248
14.5.1 The Conscious Arousal Hypothesis . . . . . . . . . 249
14.5.2 Arousal, Emotion and Emphasis . . . . . . . . . . . 252
14.6 Other Cortical Maps . . . . . . . . . . . . . . . . . . . . . . 253
14.7 Implication of Identified CAP . . . . . . . . . . . . . . . . 254
14.8 Can CAP be Consciously Controlled? . . . . . . . . . . . . 255
14.9 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
14.9.1 The Implications of Constraint . . . . . . . . . . . . 258
14.10 Compromises and Rule-Breaking . . . . . . . . . . . . . . . 260
14.11 Aspectual Cross-Talk . . . . . . . . . . . . . . . . . . . . . 262
14.12 Music/Speech Specialisation . . . . . . . . . . . . . . . . . 263
14.12.1 Double Dissociation Revisited . . . . . . . . . . . . 265
5
CONTENTS
14.12.2 The Implied Importance of Musicality . . . . . . . 265
15 Questions and Further Research 267
15.1 Questions Answered by the Theory . . . . . . . . . . . . . 267
15.2 Outstanding Questions . . . . . . . . . . . . . . . . . . . . 269
15.2.1 The Effect of Loudness . . . . . . . . . . . . . . . . 269
15.2.2 Stereo versus Mono . . . . . . . . . . . . . . . . . . 270
15.2.3 Rhyme . . . . . . . . . . . . . . . . . . . . . . . . . 270
15.2.4 Timbre . . . . . . . . . . . . . . . . . . . . . . . . . 270
15.2.5 Home Chords . . . . . . . . . . . . . . . . . . . . . 274
15.3 Further Research . . . . . . . . . . . . . . . . . . . . . . . . 274
15.3.1 Brain Studies . . . . . . . . . . . . . . . . . . . . . 274
15.3.2 Musical Brain Studies . . . . . . . . . . . . . . . . . 275
15.3.3 Constant Activity Patterns . . . . . . . . . . . . . . 275
15.3.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . 276
15.3.5 Symmetries . . . . . . . . . . . . . . . . . . . . . . 276
15.3.6 Repetition: Free and Non-Free . . . . . . . . . . . . 277
15.3.7 Cortical Maps . . . . . . . . . . . . . . . . . . . . . 277
15.3.8 Musicality . . . . . . . . . . . . . . . . . . . . . . . 277
15.3.9 Non-Typical Musical Aspects . . . . . . . . . . . . 278
15.3.10 Mathematical Models . . . . . . . . . . . . . . . . . 279
15.4 Musical Taste . . . . . . . . . . . . . . . . . . . . . . . . . 280
15.4.1 Why Does Musical Taste Vary? . . . . . . . . . . . 280
15.4.2 Variation in Super-Stimuli . . . . . . . . . . . . . . 280
15.4.3 Variation in Musicality Perception . . . . . . . . . . 280
15.4.4 Dependence on Exposure to Language . . . . . . . 282
15.4.5 Dependence on Exposure to Music . . . . . . . . . 282
15.4.6 Adaptation and CAP-Detectors . . . . . . . . . . . 284
15.4.7 Why Language Makes Little Difference . . . . . . . 284
15.5 Intensity/Position Conversion . . . . . . . . . . . . . . . . 285
15.6 Choruses and Verses . . . . . . . . . . . . . . . . . . . . . . 286
15.7 The Pleasure of Music . . . . . . . . . . . . . . . . . . . . . 288
16 Review of Assumptions 289
16.1 General Assumptions . . . . . . . . . . . . . . . . . . . . . 289
16.1.1 Information Processing . . . . . . . . . . . . . . . . 289
16.1.2 The Importance of Musicality . . . . . . . . . . . . 290
16.1.3 We Need to Explain Perception of Musicality . . . 291
16.1.4 Musicality of Speech . . . . . . . . . . . . . . . . . 291
16.1.5 Music is a Super-Stimulus . . . . . . . . . . . . . . 292
16.1.6 Emotions . . . . . . . . . . . . . . . . . . . . . . . . 293
16.1.7 Our Emotions, Not the Speaker's . . . . . . . . . . 293
16.1.8 Musicality is Not Emotion-Specific . . . . . . . . . 293
16.1.9 Musical Cortical Maps . . . . . . . . . . . . . . . . 294
16.1.10 Symmetries . . . . . . . . . . . . . . . . . . . . . . 295
16.2 Individual Cortical Maps . . . . . . . . . . . . . . . . . . . 298
6
CONTENTS
16.2.1 Scale Map . . . . . . . . . . . . . . . . . . . . . . . 298
16.2.2 Harmonic Map . . . . . . . . . . . . . . . . . . . . . 298
16.2.3 Home Chord Map . . . . . . . . . . . . . . . . . . . 299
16.2.4 Regular Beat Map . . . . . . . . . . . . . . . . . . . 300
16.2.5 Note Duration Map . . . . . . . . . . . . . . . . . . 300
16.2.6 Melodic Contour Map . . . . . . . . . . . . . . . . . 300
16.3 Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
16.4 Assumptions of the Final Theory . . . . . . . . . . . . . . . 300
16.4.1 General Principle of Music . . . . . . . . . . . . . . 300
16.4.2 Echoing . . . . . . . . . . . . . . . . . . . . . . . . 301
16.4.3 General Principle and Conscious Arousal . . . . . . 301
16.4.4 Constant Activity Patterns . . . . . . . . . . . . . . 301
17 The Future of Music 303
17.1 Music as a Commercial Enterprise . . . . . . . . . . . . . . 303
17.1.1 Composition Technology . . . . . . . . . . . . . . . 305
17.1.2 Profiting from a Complete Theory . . . . . . . . . . 306
17.2 A Post-Music-Theory World . . . . . . . . . . . . . . . . . 307
17.2.1 Music Junkies? . . . . . . . . . . . . . . . . . . . . 310
17.2.2 The Future . . . . . . . . . . . . . . . . . . . . . . . 311
Bibliography 312
Index 314
7
Acknowledgements
I would like to thank my wife Marcelina and my children Amanda and Natalie,
for putting up with my efforts to write this book.
Thanks to my sister Jean who edited the book, and then, after I had done
a substantial rewrite, edited the book a second time.
Also thanks to my Mum who read the book and made some useful sug-
gestions, to Sean Broadley who made a remark about the musical quality
of purely rhythmical music, and to Vasil Dimitrievski who told me about
Macedonian dance music.
Any errors of style, grammar or content remain my own responsibility.
8 Copyright c 2004, 2005 Philip Dorrell