Mathematics and Computation in Music (eBook)

Preface 5
Table of Contents 
6 
Rhythm and Transforms, Perception and Mathematics 
11 
1 What Is Rhythm? 11
2 Auditory Perception 12
3 Transforms 13
4 Adaptive Oscillators 14
5 Statistical Models 14
6 Automated Rhythm Analysis 15
7 Beat-Based Signal Processing 16
8 Musical Composition and Recomposition 18
9 Musical Analysis via Feature Scores 19
10 Conclusions 19
References 20
Visible Humour – Seeing P.D.Q. Bach's Musical Humour Devices in The Short-Tempered Clavier on the Spiral Array Space 
21 
1 Introduction 21
2 MuSA.RT and Visualization 
22 
2.1 Seeing Style Differences 22
3 Expectations Violated 24
3.1 The Jazz Ending 24
3.2 Improbable Harmonies 25
3.3 Excessive Repetition 26
4 Conclusions 27
Acknowledgements 27
References 28
Category-Theoretic Consequences of Denotators as a Universal Data Format 
29 
1 Introduction 29
2 Diagrams in Category Theory 30
3 Limits 30
4 Colimits 32
5 Integration in RUBATO COMPOSER 33
References 34
Normal Form, Successive Interval Arrays, Transformations and Set Classes: A Re-evaluation and Reintegration 
35 
References 59
Appendix Rahn/Morris/Scotto Normal Form Algorithm 
59 
A Model of Musical Motifs 62
1 Introduction 62
2 The Formal Model 63
3 AnExample 65
4 Discussion 67
References 68
Melodic Clustering within Motivic Spaces: Visualization in OpenMusic and Application to Schumann’s Träumerei 69
1 Introduction 69
2 Topological Model of Motivic Structure 
70 
2.1 Melodic Clustering within Motivic Spaces 71
3 Model Implementation and Visualization in OpenMusic 71
4 Application to Schumann’s Traumerei 
74 
References 76
Topological Features of the Two-Voice Inventions 77
1 Introduction 77
2 The Similarity Neighbourhood Model 78
3 Inheritance Property 80
4 Redundant Melodies 81
5 Finding Subsequences 82
6 Melodic Topologies 84
6.1 Melodic Topologies on the Syntagms 85
6.2 Investigation of the Inventions 85
7 Conclusion 86
References 87
Comparing Computational Approaches to Rhythmic and Melodic Similarity in Folksong Research 
88 
1 Introduction 88
2 Two Computational Approaches to Rhythmic Similarity 
89 
2.1 Transportation Distances 89
2.2 Inner Metric Analysis 89
2.3 Defining Similarity Based on Inner Metric Analysis 91
3 Evaluation of the Rhythmic Similarity Approaches 
92 
3.1 A Detailed Comparison on the Melody Group Deze Morgen 93
3.2 Summary of Further Results 96
4 Conclusion 96
References 96
Automatic Modulation Finding Using Convex Sets of Notes 
98 
1 Introduction 98
2 Probability of Convex Sets in Music 98
2.1 Finding Modulations by Means of Convexity 102
3 Results 104
4 Conclusions 105
Acknowledgments 105
References 105
On Pitch and Chord Stability in Folk Song Variation Retrieval 
107 
1 Introduction 107
Overview 107
2 Modifications of the Retrieval System 108
3 Pitch Stability 109
3.1 Metrical Levels 109
3.2 Evaluation of Pitch Stability 110
3.3 Query Formulation 110
4 Implied Chord Stability 111
4.1 Harmonization 111
4.2 Evaluation of Implied Chord Stability 111
4.3 Contextualization 112
5 Excerpts from the Variation Group ‘Frankrijk B1’ 113
6 Summary 115
Acknowledgements 116
References 116
Bayesian Model Selection for Harmonic Labelling 
117 
1 Introduction 117
2 Previous Work 118
3 Model 120
3.1 Dirichlet Distributions 120
3.2 The Chord Model 120
3.3 Bayesian Model Selection 121
4 Experiment 122
4.1 Parameter Estimation 122
4.2 Results 123
5 Conclusions 125
References 125
The Flow of Harmony as a Dynamical System 127
1 Dynamical Systems Applied to Harmony 127
2 Dynamical Systems Applied to Counterpoint 129
3 The Composer 
130 
4 Summary 131
References 131
Tonal Implications of Harmonic and Melodic Tn-Types 134
Tn-types of cardinality 3 135
The harmonic profile 137
The tonal profile 142
Perceptual profiles, consonance and prevalence 144
Conclusion 145
References 146
Calculating Tonal Fusion by the Generalized Coincidence Function 
150 
1 Background 150
1.1 Tonal Fusion and Roughness 150
1.2 Interspike Interval Distributions, Pitch Estimates and Harmony 151
1.2.1 Neuronal Code and Pitch 151
1.2.2 Interspike Intervals 152
1.2.3 Coinciding Periodicity Patterns for Intervals 152
1.2 Autocorrelation 153
1.2.1 Autocorrelation versus Fourier-Analysis 153
1.2.2 Hearing Theories and Autocorrelation 153
1.3 Langner’s Neuronal Correlator 153
2 Mathematical Model of Generalized Coincidence 154
2.1 Correlation Functions 154
2.2 Sequence Representation of a Tone 155
2.3 Sequence Representation of an Interval 156
2.4 Autocorrelation Function of an Interval 158
2.5 Definition of the Generalized Coincidence Function 158
3 Application of the Model to Rectangular Pulse Sequences 158
3.1 Correlation Functions of Rectangular Pulses 158
3.1.1 Autocorrelation Function of the Rectangular Pulse 158
3.1.2 Cross Correlation Function of the Rectangular Pulse 159
3.1.3 Autocorrelation Function of an Interval Represented by Rectangular Sequences 
161 
3.2 Calculation of the Generalized Coincidence Function 162
4 Conclusion 163
References 163
Predicting Music Therapy Clients’ Type of Mental Disorder Using Computational Feature Extraction and Statistical Modelling Techniques 
166 
1 Introduction 166
2 Previous Music Therapy Research 167
3 Computational Music Analysis 168
4 Method 169
5 Quantifying the Client-Therapist Interaction 172
6 Results 174
7 Discussion 175
References 176
Nonlinear Dynamics, the Missing Fundamental, and Harmony 
178 
1 Pitch Perception 178
2 Residue Behaviour 179
3 Nonlinear Dynamics of Forced Oscillators 181
3.1 n = 1 181
3.1.1 Synchronization 181
3.1.2 Quasiperiodicity 182
3.2 n = 2 183
3.2.1 Synchronization 183
3.2.2 Three-Frequency Resonances 183
4 A Nonlinear Theory for the Residue 184
5 The Golden Mean in Art and Science 186
6 The Need for Musical Scales 188
7 The Golden Scales 189
8 Playing and Transposing with Golden Scales in Equal Temperament 
192 
9 Can Our Senses Be Viewed as Generic Nonlinear Systems? 194
References 196
Dynamic Excitation Impulse Modification as a Foundation of a Synthesis and Analysis System for Wind Instrument Sounds 
199 
1 Introduction 199
2 Cyclical Spectra 200
3 Synthesis and Analysis Framework 203
3.1 The Digital Variophon 203
3.2 Formalisation 204
3.3 The Pulse Width Function 205
3.4 Application of the System 206
4 Discussion 206
Acknowledgement 207
References 207
Non-linear Circles and the Triple Harp: Creating a Microtonal Harp 
208 
1 Introduction 208
2 The Triple Harp 209
3 Non-linear Tuning Systems 209
4 Microtonal Triple Harp 210
5 Notation 211
6 Composing for Microtonal Triple Harp 211
7 Conclusion 213
References 213
Applying Inner Metric Analysis to 20th Century Compositions 
214 
1 Inner Metric Analysis 214
2 Analytic Results 215
2.1 Skrjabin’s op. 65 No. 3 215
2.2 Webern’s Op. 27, 2nd Movement 216
2.3 Xenakis’ Keren 217
2.4 Comparison of the Results 220
References 220
Tracking Features with Comparison Sets in Scriabin’s Study op. 65/3 
221 
1 Comparison Set Analysis 221
2 About the Tail Segmentation and Similarity Measures Used in the Analyses 
223 
3 The Occurrences of the ’Mystic Chord’ among Scriabin’s Piano Pieces 
224 
4 Detecting Op. 65/3 with Comparison Sets 225
5 Conclusions 228
References 229
Computer Aided Analysis of Xenakis-Keren 230
1 Introduction 230
2 Xenakis – Keren 231
References 239
Automated Extraction of Motivic Patterns and Application to the Analysis of Debussy’s Syrinx 
240 
1 General Framework 240
1.1 Motivic Pattern Extraction 240
1.2 Musical Dimensions 241
1.3 Matching Strategy 241
1.4 Analysis of Debussy’s Syrinx 242
2 Controlling the Combinatorial Redundancy 242
2.1 Maximal Patterns and Closed Patterns 242
2.2 Multidimensionality of Music 244
2.3 Formal Concept – Representation of Patterns 245
2.4 Specificity Relations 246
2.5 Cyclic Patterns 247
3 From Monody to Polyphony 248
References 248
Pitch Symmetry and Invariants in Webern's Sehr Schnell from Variations Op.27 
250 
1 Introduction 250
2 w = One Eighth Note 251
3 w = Two Eighth Notes 253
4 w = Three Eighth Notes 254
5 Center on A 255
Acknowledgements 256
References 256
Computational AnalysisWorkshop: Comparing Four Approaches to Melodic Analysis 
257 
1 Comparing Four Approaches to Melodic Analysis 257
References 259
Computer-Aided Investigation of Chord Vocabularies: Statistical Fingerprints of Mozart and Schubert 
260 
Presentation 260
References 266
The Irrelative System in Tonal Harmony 267
1 Introduction 267
2 Algorithm Enabling Classification of Chords 267
3 Chords 270
4 Metrical Units 272
5 Record Table 273
Acknowledgement 275
References 275
Mathematics and the Twelve-Tone System: Past, Present, and Future* 
276 
1 Introduction 276
2 The Introduction of Math into Twelve-Tone Music Research 277
3 Important Results and Trends 283
4 Present State of Research 293
5 Future 294
6 Conclusion 295
References 295
Approaching Musical Actions* 299
References 311
A Transformational Space for Elliott Carter's Recent Complement-Union Music* 
313 
References 320
Networks 321
From Mathematica to Live Performance: Mapping Simple Programs to Music 
328 
1 Background 328
2 Data Gathering 330
3 Large Scale Piece 331
3.1 Choice of a Rule 331
3.2 Partitioning 331
4 Initial Conditions 332
5 Choice of Musical Parameters 332
6 The Outcome 332
7 Generative Pitch Collections and Rhythmic Grouping 333
8 Mapping 333
8.1 Rule 90 334
8.2 Rule 30 335
8.3 Rule 110 336
9 New Ground 338
Acknowledgements 338
References 338
Nonlinear Dynamics of Networks: Applications to Mathematical Music Theory 
340 
1 Introduction and Musical Motivation 340
2 Nonlinear Dynamics of Networks 341
3 Discussion and Applications 345
3.1 Nonlinear Dynamics and Musical Ontology 345
3.2 Applications to Algorithmic Composition 348
References 349
Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3 
350 
References 355
A Local Maximum Phrase Detection Method for Analyzing Phrasing Strategies in Expressive Performances 
357 
1 Introduction 357
2 The Method 358
2.1 Data Extraction 358
2.2 The Case for Loudness 358
2.3 Local Maximum Phrase Detection 360
2.3.1 Phrase Strength and Volatility 360
2.3.2 Phrase Typicality 362
3 Conclusion and Discussion 362
Acknowledgements 363
References 363
Subgroup Relations among Pitch-Class Sets within Tetrachordal K-Families 
364 
References 374
K-Net Recursion in Perlean Hierarchical Structure 375
1 Introduction 375
2 K-Nets and Perle Cycles 375
3 K-Nets, Arrays, and Axis-Dyad Chords 377
4 K-Nets and Array Relationships 378
5 K-Nets, Interval Systems, Modes, and Keys 379
6 K-Nets and Synoptic Arrays 380
7 K-Nets and Tonality 382
8 Summary 384
References 384
Webern’s Twelve-Tone Rows through the Medium of Klumpenhouwer Networks 
385 
References 395
Isographies of Pitch-Class Sets and Set Classes 396
1 Introduction 396
2 Isography of Pitch-Class Sets and Set Classes 397
3 Tonality and Whole-Tone Scale Proportion 398
4 Relations of Set Classes 399
References 401
The Transmission of Pythagorean Arithmetic in the Context of the Ancient Musical Tradition from the Greek to the Latin Orbits During the Renaissance: A Computational Approach of Identifying and Analyzing the Formation of Scales in the De Harmonia Musicorum Instrumentorum Opus (Milan, 1518) of Franchino Gaffurio (1451–1522)* 
402 
Bibliography 411
Combinatorial and Transformational Aspects of Euler's Speculum Musicum 
416 
References 420
Structures Ia Pour Deux Pianos by Boulez: Towards Creative Analysis Using Open Musicand Rubato 
422 
1 Introduction 422
2 Compositional Process in Structures Ia 423
2.1 Analysis of Constructional and Serial Principles: Decision and Automatism 
423 
2.2 How to Create from an Analysis 423
3 An Implementation in OpenMusic: A Visual and Functional Environment 
424 
3.1 Patches and Circularity 424
3.2 Composing Following the Model with the Benefit of a Graphical Composition Environment 
424 
4 Rubato: A Higher Level of Abstraction with a Categorical View 
425 
4.1 Different Perspectives Delivered by Rubato 425
4.2 Possibilities Brought by Rubato 426
4.3 Scheme of the Construction 426
5 Conclusion 427
References 427
The Sieves of Iannis Xenakis 429
1 Introduction 429
2 Types of Formulae 430
3 Symmetries/Periodicities 430
4 Inner-Periodic Formula 431
4.1 Inner Periodicities and Formulae Redundancy 431
4.2 Construction of the Inner-Periodic Simplified Formula 431
4.3 Analytical Algorithm: Early Stage 432
4.4 The Condition of Inner Periodicity 433
4.5 Analytical Algorithm: Final Stage 433
4.6 The Condition of Inner Symmetry 434
4.7 Inner-Symmetric Analysis 435
4.8 Modules and Degree of Symmetry 438
References 439
Tonal, Atonal and Microtonal Pitch-Class Categories 440
1 Introduction 440
2 Applying Pitch-Class Set Theory on Sets with Cardinality (Pitch-Classes) Other Than 12 
441 
3 Pitch-Class Set Theory within a Bit-Sequence 442
4 Pitch-Class Categories 444
5 Discussion and Future Work 446
6 Conclusion 447
References 447
Appendix 448
Using Mathematica to Compose Music and Analyze Music with Information Theory 
451 
1 Composition of Music Using Mathematica 451
2 Nonlinear Time Series Analysis of Musical Compositions 
453 
2.1 Creating Time Series from Sheet Music 453
2.2 Transfer Entropy and the Relationship between Physical Systems 455
2.3 The Application of the Transfer Entropy to a Symphony 455
3 Conclusions 458
References 458
A Diatonic Chord with Unusual Voice-Leading Capabilities 
459 
References 469
Mathematical and Musical Properties of Pairwise Well-Formed Scales 
474 
1 Pairwise Well-Formed and Well-Formed Scales 475
2 Some Properties of Pairwise Well-Formed Scales 476
3 Classification of Pairwise Well-Formed Scales 476
References 478
Eine Kleine Fourier Musik 479
Introduction 479
1 DFT of a pc Set 
479 
2 Maximal Values 480
2.1 Regular Polygons 481
2.2 The General Case 481
2.3 Other Maximal Values 482
3 Minimal Values 483
4 MeanValue(s) 484
5 Coda 485
References 486
WF Scales, ME Sets, and Christoffel Words 487
1 Well-Formed Scales 487
2 Christoffel Words 489
3 Well-Formed Classes and Christoffel Words, Duality 490
4 Christoffel Words, Maximally Even Sets and Musical Modes 
492 
5 Christoffel Tree and the Monoid SL(2, N) 494
6 Final Remarks 
496 
References 497
Interval Preservation in Group- and Graph-Theoretical Music Theories: A Comparative Study 
499 
References 502
Pseudo-diatonic Scales 503
1 Shuffled Stern-Brocot Tree 503
2 Construction of Pseudo-diatonic Scales 504
References 507
Affinity Spaces and Their Host Set Classes 509
1 Affinities in the Medieval Dasian Scale 509
2 The Dasian Space 511
3 Four Properties of the Dasian Space 513
4 Affinity Spaces 515
5 Three Properties of Host Set Classes 519
6 Generating Affinity Spaces 519
7 Conclusion 521
References 521
The Step-Class Automorphism Group in Tonal Analysis 522
Bibliography 530
A Linear Algebraic Approach to Pitch-Class Set Genera 
531 
1 ‘Corner-Stone Set-Classes’ 531
2 Applying Cosine Distance and the Determinant of a Matrix with Musical Set Classes 
532 
3 Volume Tests with Interval-Class Vectors 533
4 ‘Strangest’ Hexachords 535
5 Principal Component Analysis: A Flexible Approach for Mapping ICV-Space 
536 
6 Using Corner-Stone Vectors for Producing a System of Genera 
537 
7 Harmonic Space in Composition 538
8 Conclusions 539
References 539
Author Index 541
Index 542