Mathematics and Computation in Music (eBook)

Preface 5
Organization 7
Foreword 10
Table of Contents 13
Hamiltonian Cycles in the Topological Dual of the Tonnetz 15
Introduction 15
Tone-Networks, the Tonnetz and Its Topological Dual 16
Definition 16
Definition 16
Definition 16
Theorem 16
Definition 17
Definition 17
Definition 17
Lemma 18
Theorem 19
Hamiltonian Cycles in $D(Ton)$ 19
The Hamiltonian Cycle $#41$ and Beethoven's Ninth Symphony 21
Definition 21
Example 21
Hamiltonian Cycles as a Compositional Tool 23
References 24
The Continuous Hexachordal Theorem 25
Introduction 25
Basic Definitions 25
History 27
Outline 28
The Continuous Hexachordal Theorem 28
Weighted Rhythms 28
The Continuous Generalizations 29
Continuous Hexachordal Theorem and Proof 30
Discrete Theorem as Corollary 31
Double-Counting Diameter Intervals 32
Patterson’s First Theorem 33
OpenProblems 34
References 34
Speech Rhythms and Metric Frames 36
Introduction 36
Assessing Local Meter 37
Glued at the Subdivision 39
Beyond the Eighth-Note 41
The Tuplets 41
Conclusion 45
References 45
Temporal Patterns in Polyphony 46
Motivation 46
Relational Patterns 49
Humdrum 49
Structured Polyphonic Patterns 51
{/mathcal H} and {/mathcal SPP} are Distinct 52
The Common Denominator {/mathcal SPP}$_{seq}$ 53
Discussion 55
References 55
Maximally Smooth Diatonic Trichord Cycles 57
Introduction 57
Maximally Smooth Cycles and Parsimonious Triads 57
Useful Scales 58
Trichord Species and Their Multiplicities 60
Trichord Cycles 62
Conclusions 69
References 70
Towards a Symbolic Approach to Sound Analysis 71
The Levels of Representation 71
Sound Types 73
Simple Type Theory 73
Models for Simple Type Theory 74
Low-Level Features and Audio-Indexing 75
The $Typed$ Model 76
Properties of Sound Types 78
Conclusions and Perspectives 78
References 78
Plain and Twisted Adjoints of Well-Formed Words 79
Geometrical Motivations 79
Well-Formed Words 81
Plain and Twisted Adjoints 84
Divider Incidence 88
Final Remarks 93
References 93
Regions and Standard Modes 95
Regions 95
$Ut-Re-Mi-Fa-Sol-La$ 96
Central Words 98
Duality for Central Words 99
Standard Modes 101
$Do-Re-Mi-Fa-Sol-La-Ti-(Do')$ 101
Standard Pairs and Their Duality 103
References 105
Compatibility of the Different Tuning Systems in an Orchestra 107
Introduction 107
Some Concepts and Notation 108
Introducing Fuzzy Logic 109
Fuzzy Musical Notes 111
Measuring Compatibility 112
Computational Results 114
Conclusions 116
References 116
Formal Diatonic Intervallic Notation 118
Introduction 118
Quality Modifiers 119
Group Structures 121
Group Actions 123
Generalized Interval Systems 125
Coda 126
References 127
Determining Feature Relevance in Subject Responses to Musical Stimuli 129
Introduction 129
Prior Work 130
Feature Relevance Measured by Polynomial Least-Mean Square Estimation 131
Extension to General Nonlinear Estimators and Probabilistic Models 134
Kullback-Leibler Distance 136
Experimental Results 138
Data Set 138
Results 138
References 142
Sequential Association Rules in Atonal Music 144
Introduction 144
Atonal Music and Pitch Class Set Theory 145
Pitch Class Set Categories in Atonal Music 146
Sequential Association Rules 148
The Method 148
Results 149
Concluding Remarks 151
References 152
Badness of Serial Fit Revisited 153
Introduction 153
Badness of Serial Fit and Partial Orders 154
Logarithmic BSF and the Metric 155
Transformational Similarity and Presortedness of Permutations 157
Conclusions 159
References 159
Estimating the Tonalness of Transpositional Type Pitch-Class Sets Using Learned Tonal Key Spaces 160
Introduction 160
Low Dimensional Tonal Key Space 162
Mapping Tn-Type Sets to the Tonal Key Space 163
Evaluation 164
Conclusions 166
References 166
Musical Experiences with Block Designs 168
t-Designs: A Brief Survey 168
Drawingt-Designs 170
Cyclic Representations 172
Pcsets and Designs 175
A Compositional Application 176
References 178
A Generalisation of Diatonicism and the Discrete Fourier Transform as a Mean for Classifying and Characterising Musical Scales 180
Introduction 180
The Diatonic Bell 181
Input Parameters 181
Find All Scales 182
Find All Centred Scales 182
Find the Reference Scale 182
Find the Reference Mode 183
Find All Centred Modes 183
Construct All Representations 184
Order All Scales 185
Modal Transposition 186
The DFT Analysis of Scales 187
Phases 189
Modules 190
Periodicity 190
Chord Quality 191
Conclusion 192
Symmetry 192
Measuring the Diatonic Character of a Scale 192
References 193
The Geometry of Melodic, Harmonic, and Metrical Hierarchy 194
General Characteristics of Musical Hierarchy 194
Harmonic, Melodic, and Metrical Forms of Hierarchy 196
Harmonic Hierarchy 196
Melodic Hierarchy 197
Metric Hierarchy 198
Musical Realizations of the Stasheff Polytope 198
Relating Hierarchies on Different Musical Parameters 201
Conflict between Melodic and Metric Structures 201
Melodic Structures in Counterpoint 203
Relationships between Melodic and Harmonic Structure 203
Conclusion 205
References 205
A Multi-tiered Approach for Analyzing Expressive Timing in Music Performance 207
Introduction 207
Related Previous Research 208
Tempo Curve Calculation Using a Non-parametric Regression Model 209
The Hierarchy of Metric Deviations 212
Conclusions and Future Directions 212
References 218
HMM Analysis of Musical Structure: Identification of Latent Variables Through Topology-Sensitive Model Selection 219
Introduction 219
HMM Training and Topology Identification 221
Case Study I: Statistical Segmentation of Symbolic Sequences 222
Case Study II: Meter Induction from Rhythmic Patterns 224
Conclusions 227
References 231
A Declarative Language for Dynamic Multimedia Interaction Systems 232
Introduction 232
Preliminaries 234
A Model for Dynamic Interactive Scores 235
A Model for Music Improvisation 238
Concluding Remarks 240
References 241
Generalized Voice Exchange 242
Introduction 242
Connection to Contextual Inversion 243
Generalized Voice Exchange 243
Generalized Chromatic Voice Exchange: The Variable $i /epsilon {/mathbb Z}_{12}$ 244
Permutations of the Orbits: The Variable $p /epsilon {/mathbb Z}_{5}$ 246
Initial Harmonic Intervals: The Variable $q /epsilon {/mathbb Z}_{12}$ 247
Conclusions: The Group $R$ and Transformational Networks 248
References 249
Representing and Estimating Musical Expression in Melody 250
Introduction 250
The Theremin 252
Representing Musical Interpretation 252
From Labeling to Audio 253
Does the Labeling Capture Musicality? 254
Estimating the Interpretation 255
Results 256
References 258
Evaluating Tonal Distances between Pitch-Class Sets and Predicting Their Tonal Centres by Computational Models 259
Introduction 259
Algorithmic Models 261
Training the Weights of a Linear Polynomial with Tonal Constraints 261
Circle-of-Fifths-Based Algorithm 262
Training a Neural Network with Empirical Results 263
The Tonal Profile of a PCS as a Weighted Mean of KK-Profiles 265
Comparing and Combining Predictions 266
Is Alban Berg’s $Invention on a Key$ in D Minor? 267
Conclusions 269
References 269
APPENDIX A 270
APPENDIX B 270
APPENDIX C 271
Three Conceptions of Musical Distance 272
Introduction 272
Voice-Leading Lattices and Acoustic Affinity 275
Tuning Lattices as Approximate Models of Voice Leading 278
Voice Leading, “Quality Space,” and the Fourier Transform 281
Conclusion 285
References 286
Pairwise Well-Formed Scales and a Bestiary of Animals on the Hexagonal Lattice 287
Well-Formed Scales 287
Pairwise Well-Formed Scales 287
Animals on the Hexagonal Lattice—The Heptatonic Case 289
Three-Stepped Animals on the Tonnetz of Fifths and Thirds 290
Pwwf Animals on the $Tonnetz$ of Fifths and Thirds 290
Pwwf Animals on Other Lattices 293
Arbitrary Heptatonic Pwwf Scales 296
Pwwf Scales of Other Cardinalities 298
References 299
Generalized $Tonnetz$ and Well-Formed GTS: A Scale Theory Inspired by the Neo-Riemannians 300
Inspirations 300
Generalized $Tonnetz$ 301
Unpitched Generated Tone System 303
Generated Tone System 305
The Main Theorem 309
A Case Study: The System of $/'{S}rutis$ 309
References 312
Author Index 313