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Mathematics and Computation in Music - First International Conference, MCM 2007, Berlin, Germany, May 18-20, 2007. Revised Selected Papers
Preface
5
Table of Contents
Table of Contents
Rhythm and Transforms, Perception and Mathematics
Rhythm and Transforms, Perception and Mathematics
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
Visible Humour – Seeing P.D.Q. Bach's Musical Humour Devices in The Short-Tempered Clavier on the Spiral Array Space
1 Introduction
21
2 MuSA.RT and Visualization
2 MuSA.RT and Visualization
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
Category-Theoretic Consequences of Denotators as a Universal Data Format
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
Normal Form, Successive Interval Arrays, Transformations and Set Classes: A Re-evaluation and Reintegration
References
59
Appendix Rahn/Morris/Scotto Normal Form Algorithm
Appendix Rahn/Morris/Scotto Normal Form Algorithm
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
2 Topological Model of Motivic Structure
2.1 Melodic Clustering within Motivic Spaces
71
3 Model Implementation and Visualization in OpenMusic
71
4 Application to Schumann’s Traumerei
4 Application to Schumann’s Traumerei
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
Comparing Computational Approaches to Rhythmic and Melodic Similarity in Folksong Research
1 Introduction
88
2 Two Computational Approaches to Rhythmic Similarity
2 Two Computational Approaches to Rhythmic Similarity
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
3 Evaluation of the Rhythmic Similarity Approaches
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
Automatic Modulation Finding Using Convex Sets of Notes
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
On Pitch and Chord Stability in Folk Song Variation Retrieval
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
Bayesian Model Selection for Harmonic Labelling
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
3 The Composer
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
Calculating Tonal Fusion by the Generalized Coincidence Function
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
3.1.3 Autocorrelation Function of an Interval Represented by Rectangular Sequences
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
Predicting Music Therapy Clients’ Type of Mental Disorder Using Computational Feature Extraction and Statistical Modelling Techniques
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
Nonlinear Dynamics, the Missing Fundamental, and Harmony
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
8 Playing and Transposing with Golden Scales in Equal Temperament
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
Dynamic Excitation Impulse Modification as a Foundation of a Synthesis and Analysis System for Wind Instrument Sounds
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
Non-linear Circles and the Triple Harp: Creating a Microtonal Harp
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
Applying Inner Metric Analysis to 20th Century Compositions
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
Tracking Features with Comparison Sets in Scriabin’s Study op. 65/3
1 Comparison Set Analysis
221
2 About the Tail Segmentation and Similarity Measures Used in the Analyses
2 About the Tail Segmentation and Similarity Measures Used in the Analyses
3 The Occurrences of the ’Mystic Chord’ among Scriabin’s Piano Pieces
3 The Occurrences of the ’Mystic Chord’ among Scriabin’s Piano Pieces
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
Automated Extraction of Motivic Patterns and Application to the Analysis of Debussy’s Syrinx
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
Pitch Symmetry and Invariants in Webern's Sehr Schnell from Variations Op.27
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
Computational AnalysisWorkshop: Comparing Four Approaches to Melodic Analysis
1 Comparing Four Approaches to Melodic Analysis
257
References
259
Computer-Aided Investigation of Chord Vocabularies: Statistical Fingerprints of Mozart and Schubert
Computer-Aided Investigation of Chord Vocabularies: Statistical Fingerprints of Mozart and Schubert
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*
Mathematics and the Twelve-Tone System: Past, Present, and Future*
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*
A Transformational Space for Elliott Carter's Recent Complement-Union Music*
References
320
Networks
321
From Mathematica to Live Performance: Mapping Simple Programs to Music
From Mathematica to Live Performance: Mapping Simple Programs to Music
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
Nonlinear Dynamics of Networks: Applications to Mathematical Music Theory
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
Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3
References
355
A Local Maximum Phrase Detection Method for Analyzing Phrasing Strategies in Expressive Performances
A Local Maximum Phrase Detection Method for Analyzing Phrasing Strategies in Expressive Performances
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
Subgroup Relations among Pitch-Class Sets within Tetrachordal K-Families
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
Webern’s Twelve-Tone Rows through the Medium of Klumpenhouwer Networks
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)*
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)*
Bibliography
411
Combinatorial and Transformational Aspects of Euler's Speculum Musicum
Combinatorial and Transformational Aspects of Euler's Speculum Musicum
References
420
Structures Ia Pour Deux Pianos by Boulez: Towards Creative Analysis Using Open Musicand Rubato
Structures Ia Pour Deux Pianos by Boulez: Towards Creative Analysis Using Open Musicand Rubato
1 Introduction
422
2 Compositional Process in Structures Ia
423
2.1 Analysis of Constructional and Serial Principles: Decision and Automatism
2.1 Analysis of Constructional and Serial Principles: Decision and Automatism
2.2 How to Create from an Analysis
423
3 An Implementation in OpenMusic: A Visual and Functional Environment
3 An Implementation in OpenMusic: A Visual and Functional Environment
3.1 Patches and Circularity
424
3.2 Composing Following the Model with the Benefit of a Graphical Composition Environment
3.2 Composing Following the Model with the Benefit of a Graphical Composition Environment
4 Rubato: A Higher Level of Abstraction with a Categorical View
4 Rubato: A Higher Level of Abstraction with a Categorical View
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
2 Applying Pitch-Class Set Theory on Sets with Cardinality (Pitch-Classes) Other Than 12
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
Using Mathematica to Compose Music and Analyze Music with Information Theory
1 Composition of Music Using Mathematica
451
2 Nonlinear Time Series Analysis of Musical Compositions
2 Nonlinear Time Series Analysis of Musical Compositions
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
A Diatonic Chord with Unusual Voice-Leading Capabilities
References
469
Mathematical and Musical Properties of Pairwise Well-Formed Scales
Mathematical and Musical Properties of Pairwise Well-Formed Scales
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
1 DFT of a pc Set
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
4 Christoffel Words, Maximally Even Sets and Musical Modes
5 Christoffel Tree and the Monoid SL(2, N)
494
6 Final Remarks
6 Final Remarks
References
497
Interval Preservation in Group- and Graph-Theoretical Music Theories: A Comparative Study
Interval Preservation in Group- and Graph-Theoretical Music Theories: A Comparative Study
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
A Linear Algebraic Approach to Pitch-Class Set Genera
1 ‘Corner-Stone Set-Classes’
531
2 Applying Cosine Distance and the Determinant of a Matrix with Musical Set Classes
2 Applying Cosine Distance and the Determinant of a Matrix with Musical Set Classes
3 Volume Tests with Interval-Class Vectors
533
4 ‘Strangest’ Hexachords
535
5 Principal Component Analysis: A Flexible Approach for Mapping ICV-Space
5 Principal Component Analysis: A Flexible Approach for Mapping ICV-Space
6 Using Corner-Stone Vectors for Producing a System of Genera
6 Using Corner-Stone Vectors for Producing a System of Genera
7 Harmonic Space in Composition
538
8 Conclusions
539
References
539
Author Index
541
Index
542
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