Mathematics and Computation in Music: First International Conference, MCM 2007, Berlin, Germany, May 18-20, 2007. Revised Selected Papers (Communications in Computer and Information Science)

Cover......Page 1
Mathematics
and Computation
in Music......Page 3
Communications
in Computer and Information Science 37......Page 2
ISBN-10 3642045782......Page 4
Preface......Page 5
Table of Contents
......Page 6
1 What Is
Rhythm?......Page 11
2 Auditory Perception......Page 12
3 Transforms......Page 13
5 Statistical Models......Page 14
6 Automated Rhythm Analysis......Page 15
7 Beat-Based Signal Processing......Page 16
8 Musical Composition and Recomposition......Page 18
10 Conclusions......Page 19
References......Page 20
1 Introduction......Page 21
2.1 Seeing Style Differences......Page 22
3.1 The Jazz Ending......Page 24
3.2 Improbable Harmonies......Page 25
3.3 Excessive Repetition......Page 26
Acknowledgements......Page 27
References......Page 28
1 Introduction......Page 29
3 Limits......Page 30
4 Colimits......Page 32
5 Integration in RUBATO COMPOSER......Page 33
References......Page 34
Normal Form, Successive Interval Arrays, Transformations and Set Classes: A Re-evaluation and Reintegration
......Page 35
Appendix Rahn/Morris/Scotto Normal Form Algorithm
......Page 59
1 Introduction......Page 62
2 The Formal Model......Page 63
3 AnExample......Page 65
4 Discussion......Page 67
References......Page 68
1 Introduction......Page 69
2 Topological Model of Motivic Structure
......Page 70
3 Model Implementation and Visualization in OpenMusic......Page 71
4 Application to Schumann’s Traumerei
......Page 74
References......Page 76
1 Introduction......Page 77
2 The Similarity Neighbourhood Model......Page 78
3 Inheritance Property......Page 80
4 Redundant Melodies......Page 81
5 Finding Subsequences......Page 82
6 Melodic Topologies......Page 84
6.2 Investigation of the Inventions......Page 85
7 Conclusion......Page 86
References......Page 87
1 Introduction......Page 88
2.2 Inner Metric Analysis......Page 89
2.3 Defining Similarity Based on Inner Metric Analysis......Page 91
3 Evaluation of the Rhythmic Similarity Approaches
......Page 92
3.1 A Detailed Comparison on the Melody Group Deze Morgen......Page 93
References......Page 96
2 Probability of Convex Sets in Music......Page 98
2.1 Finding Modulations by Means of Convexity......Page 102
3 Results......Page 104
References......Page 105
Overview......Page 107
2 Modifications of the Retrieval System......Page 108
3.1 Metrical Levels......Page 109
3.3 Query Formulation......Page 110
4.2 Evaluation of Implied Chord Stability......Page 111
4.3 Contextualization......Page 112
5 Excerpts from the Variation Group ‘Frankrijk B1’......Page 113
6 Summary......Page 115
References......Page 116
1 Introduction......Page 117
2 Previous Work......Page 118
3.2 The Chord Model......Page 120
3.3 Bayesian Model Selection......Page 121
4.1 Parameter Estimation......Page 122
4.2 Results......Page 123
References......Page 125
1 Dynamical Systems Applied to Harmony......Page 127
2 Dynamical Systems Applied to Counterpoint......Page 129
3 The Composer
......Page 130
References......Page 131
Tonal Implications of Harmonic and Melodic Tn-Types......Page 134
Tn-types of cardinality 3......Page 135
The harmonic profile......Page 137
The tonal profile......Page 142
Perceptual profiles, consonance and prevalence......Page 144
Conclusion......Page 145
References......Page 146
1.1 Tonal Fusion and Roughness......Page 150
1.2.1 Neuronal Code and Pitch......Page 151
1.2.3 Coinciding Periodicity Patterns for Intervals......Page 152
1.3 Langner’s Neuronal Correlator......Page 153
2.1 Correlation Functions......Page 154
2.2 Sequence Representation of a Tone......Page 155
2.3 Sequence Representation of an Interval......Page 156
3.1.1 Autocorrelation Function of the Rectangular Pulse......Page 158
3.1.2 Cross Correlation Function of the Rectangular Pulse......Page 159
3.1.3 Autocorrelation Function of an Interval Represented by Rectangular Sequences
......Page 161
3.2 Calculation of the Generalized Coincidence Function......Page 162
References......Page 163
1 Introduction......Page 166
2 Previous Music Therapy Research......Page 167
3 Computational Music Analysis......Page 168
4 Method......Page 169
5 Quantifying the Client-Therapist Interaction......Page 172
6 Results......Page 174
7 Discussion......Page 175
References......Page 176
1 Pitch Perception......Page 178
2 Residue Behaviour......Page 179
3.1.1 Synchronization......Page 181
3.1.2 Quasiperiodicity......Page 182
3.2.2 Three-Frequency Resonances......Page 183
4 A Nonlinear Theory for the Residue......Page 184
5 The Golden Mean in Art and Science......Page 186
6 The Need for Musical Scales......Page 188
7 The Golden Scales......Page 189
8 Playing and Transposing with Golden Scales in Equal Temperament
......Page 192
9 Can Our Senses Be Viewed as Generic Nonlinear Systems?......Page 194
References......Page 196
1 Introduction......Page 199
2 Cyclical Spectra......Page 200
3.1 The Digital Variophon......Page 203
3.2 Formalisation......Page 204
3.3 The Pulse Width Function......Page 205
4 Discussion......Page 206
References......Page 207
1 Introduction......Page 208
3 Non-linear Tuning Systems......Page 209
4 Microtonal Triple Harp......Page 210
6 Composing for Microtonal Triple Harp......Page 211
References......Page 213
1 Inner Metric Analysis......Page 214
2.1 Skrjabin’s op. 65 No. 3......Page 215
2.2 Webern’s Op. 27, 2nd Movement......Page 216
2.3 Xenakis’ Keren......Page 217
References......Page 220
1 Comparison Set Analysis......Page 221
2 About the Tail Segmentation and Similarity Measures Used in the Analyses
......Page 223
3 The Occurrences of the ’Mystic Chord’ among Scriabin’s Piano Pieces
......Page 224
4 Detecting Op. 65/3 with Comparison Sets......Page 225
5 Conclusions......Page 228
References......Page 229
1 Introduction......Page 230
2 Xenakis – Keren......Page 231
References......Page 239
1.1 Motivic Pattern Extraction......Page 240
1.3 Matching Strategy......Page 241
2.1 Maximal Patterns and Closed Patterns......Page 242
2.2 Multidimensionality of Music......Page 244
2.3 Formal Concept – Representation of Patterns......Page 245
2.4 Specificity Relations......Page 246
2.5 Cyclic Patterns......Page 247
References......Page 248
1 Introduction......Page 250
2 w = One Eighth Note......Page 251
3 w = Two Eighth Notes......Page 253
4 w = Three Eighth Notes......Page 254
5 Center on A......Page 255
References......Page 256
1 Comparing Four Approaches to Melodic Analysis......Page 257
References......Page 259
Presentation......Page 260
References......Page 266
2 Algorithm Enabling Classification of Chords......Page 267
3 Chords......Page 270
4 Metrical Units......Page 272
5 Record Table......Page 273
References......Page 275
1 Introduction......Page 276
2 The Introduction of Math into Twelve-Tone Music Research......Page 277
3 Important Results and Trends......Page 283
4 Present State of Research......Page 293
5 Future......Page 294
References......Page 295
Approaching Musical Actions*......Page 299
References......Page 311
A Transformational Space for Elliott Carter's Recent Complement-Union Music*
......Page 313
References......Page 320
Networks......Page 321
1 Background......Page 328
2 Data Gathering......Page 330
3.2 Partitioning......Page 331
6 The Outcome......Page 332
8 Mapping......Page 333
8.1 Rule 90......Page 334
8.2 Rule 30......Page 335
8.3 Rule 110......Page 336
References......Page 338
1 Introduction and Musical Motivation......Page 340
2 Nonlinear Dynamics of Networks......Page 341
3.1 Nonlinear Dynamics and Musical Ontology......Page 345
3.2 Applications to Algorithmic Composition......Page 348
References......Page 349
Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3
......Page 350
References......Page 355
1 Introduction......Page 357
2.2 The Case for Loudness......Page 358
2.3.1 Phrase Strength and Volatility......Page 360
3 Conclusion and Discussion......Page 362
References......Page 363
Subgroup Relations among Pitch-Class Sets within Tetrachordal K-Families
......Page 364
References......Page 374
2 K-Nets and Perle Cycles......Page 375
3 K-Nets, Arrays, and Axis-Dyad Chords......Page 377
4 K-Nets and Array Relationships......Page 378
5 K-Nets, Interval Systems, Modes, and Keys......Page 379
6 K-Nets and Synoptic Arrays......Page 380
7 K-Nets and Tonality......Page 382
References......Page 384
Webern’s Twelve-Tone Rows through the Medium of Klumpenhouwer Networks
......Page 385
References......Page 395
1 Introduction......Page 396
2 Isography of Pitch-Class Sets and Set Classes......Page 397
3 Tonality and Whole-Tone Scale Proportion......Page 398
4 Relations of Set Classes......Page 399
References......Page 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)*
......Page 402
Bibliography......Page 411
Combinatorial and Transformational Aspects of Euler's Speculum Musicum
......Page 416
References......Page 420
1 Introduction......Page 422
2.2 How to Create from an Analysis......Page 423
3.2 Composing Following the Model with the Benefit of a Graphical Composition Environment
......Page 424
4.1 Different Perspectives Delivered by Rubato......Page 425
4.3 Scheme of the Construction......Page 426
References......Page 427
1 Introduction......Page 429
3 Symmetries/Periodicities......Page 430
4.2 Construction of the Inner-Periodic Simplified Formula......Page 431
4.3 Analytical Algorithm: Early Stage......Page 432
4.5 Analytical Algorithm: Final Stage......Page 433
4.6 The Condition of Inner Symmetry......Page 434
4.7 Inner-Symmetric Analysis......Page 435
4.8 Modules and Degree of Symmetry......Page 438
References......Page 439
1 Introduction......Page 440
2 Applying Pitch-Class Set Theory on Sets with Cardinality (Pitch-Classes) Other Than 12
......Page 441
3 Pitch-Class Set Theory within a Bit-Sequence......Page 442
4 Pitch-Class Categories......Page 444
5 Discussion and Future Work......Page 446
References......Page 447
Appendix......Page 448
1 Composition of Music Using Mathematica......Page 451
2.1 Creating Time Series from Sheet Music......Page 453
2.3 The Application of the Transfer Entropy to a Symphony......Page 455
References......Page 458
A Diatonic Chord with Unusual Voice-Leading Capabilities
......Page 459
References......Page 469
Mathematical and Musical Properties of Pairwise Well-Formed Scales
......Page 474
1 Pairwise Well-Formed and Well-Formed Scales......Page 475
3 Classification of Pairwise Well-Formed Scales......Page 476
References......Page 478
1 DFT of a pc Set
......Page 479
2 Maximal Values......Page 480
2.2 The General Case......Page 481
2.3 Other Maximal Values......Page 482
3 Minimal Values......Page 483
4 MeanValue(s)......Page 484
5 Coda......Page 485
References......Page 486
1 Well-Formed Scales......Page 487
2 Christoffel Words......Page 489
3 Well-Formed Classes and Christoffel Words, Duality......Page 490
4 Christoffel Words, Maximally Even Sets and Musical Modes
......Page 492
5 Christoffel Tree and the Monoid SL(2, N)......Page 494
6 Final Remarks
......Page 496
References......Page 497
Interval Preservation in Group- and Graph-Theoretical Music Theories: A Comparative Study
......Page 499
References......Page 502
1 Shuffled Stern-Brocot Tree......Page 503
2 Construction of Pseudo-diatonic Scales......Page 504
References......Page 507
1 Affinities in the Medieval Dasian Scale......Page 509
2 The Dasian Space......Page 511
3 Four Properties of the Dasian Space......Page 513
4 Affinity Spaces......Page 515
6 Generating Affinity Spaces......Page 519
References......Page 521
The Step-Class Automorphism Group in Tonal Analysis......Page 522
Bibliography......Page 530
1 ‘Corner-Stone Set-Classes’......Page 531
2 Applying Cosine Distance and the Determinant of a Matrix with Musical Set Classes
......Page 532
3 Volume Tests with Interval-Class Vectors......Page 533
4 ‘Strangest’ Hexachords......Page 535
5 Principal Component Analysis: A Flexible Approach for Mapping ICV-Space
......Page 536
6 Using Corner-Stone Vectors for Producing a System of Genera
......Page 537
7 Harmonic Space in Composition......Page 538
References......Page 539
index.pdf......Page 541