Mathematics and Computation in Music - First International Conference, MCM 2007, Berlin, Germany, May 18-20, 2007. Revised Selected Papers

Preface

Table of Contents

Rhythm and Transforms, Perception and Mathematics

1 What Is Rhythm?

2 Auditory Perception

3 Transforms

4 Adaptive Oscillators

5 Statistical Models

6 Automated Rhythm Analysis

7 Beat-Based Signal Processing

8 Musical Composition and Recomposition

9 Musical Analysis via Feature Scores

10 Conclusions

References

Visible Humour – Seeing P.D.Q. Bach's Musical Humour Devices in The Short-Tempered Clavier on the Spiral Array Space

1 Introduction

2 MuSA.RT and Visualization

2.1 Seeing Style Differences

3 Expectations Violated

3.1 The Jazz Ending

3.2 Improbable Harmonies

3.3 Excessive Repetition

4 Conclusions

Acknowledgements

References

Category-Theoretic Consequences of Denotators as a Universal Data Format

1 Introduction

2 Diagrams in Category Theory

3 Limits

4 Colimits

5 Integration in RUBATO COMPOSER

References

Normal Form, Successive Interval Arrays, Transformations and Set Classes: A Re-evaluation and Reintegration

References

Appendix Rahn/Morris/Scotto Normal Form Algorithm

A Model of Musical Motifs

1 Introduction

2 The Formal Model

3 AnExample

4 Discussion

References

Melodic Clustering within Motivic Spaces: Visualization in OpenMusic and Application to Schumann’s Träumerei

1 Introduction

2 Topological Model of Motivic Structure

2.1 Melodic Clustering within Motivic Spaces

3 Model Implementation and Visualization in OpenMusic

4 Application to Schumann’s Traumerei

References

Topological Features of the Two-Voice Inventions

1 Introduction

2 The Similarity Neighbourhood Model

3 Inheritance Property

4 Redundant Melodies

5 Finding Subsequences

6 Melodic Topologies

6.1 Melodic Topologies on the Syntagms

6.2 Investigation of the Inventions

7 Conclusion

References

Comparing Computational Approaches to Rhythmic and Melodic Similarity in Folksong Research

1 Introduction

2 Two Computational Approaches to Rhythmic Similarity

2.1 Transportation Distances

2.2 Inner Metric Analysis

2.3 Defining Similarity Based on Inner Metric Analysis

3 Evaluation of the Rhythmic Similarity Approaches

3.1 A Detailed Comparison on the Melody Group Deze Morgen

3.2 Summary of Further Results

4 Conclusion

References

Automatic Modulation Finding Using Convex Sets of Notes

1 Introduction

2 Probability of Convex Sets in Music

2.1 Finding Modulations by Means of Convexity

3 Results

4 Conclusions

Acknowledgments

References

On Pitch and Chord Stability in Folk Song Variation Retrieval

1 Introduction

Overview

2 Modifications of the Retrieval System

3 Pitch Stability

3.1 Metrical Levels

3.2 Evaluation of Pitch Stability

3.3 Query Formulation

4 Implied Chord Stability

4.1 Harmonization

4.2 Evaluation of Implied Chord Stability

4.3 Contextualization

5 Excerpts from the Variation Group ‘Frankrijk B1’

6 Summary

Acknowledgements

References

Bayesian Model Selection for Harmonic Labelling

1 Introduction

2 Previous Work

3 Model

3.1 Dirichlet Distributions

3.2 The Chord Model

3.3 Bayesian Model Selection

4 Experiment

4.1 Parameter Estimation

4.2 Results

5 Conclusions

References

The Flow of Harmony as a Dynamical System

1 Dynamical Systems Applied to Harmony

2 Dynamical Systems Applied to Counterpoint

3 The Composer

4 Summary

References

Tonal Implications of Harmonic and Melodic Tn-Types

Tn-types of cardinality 3

The harmonic profile

The tonal profile

Perceptual profiles, consonance and prevalence

Conclusion

References

Calculating Tonal Fusion by the Generalized Coincidence Function

1 Background

1.1 Tonal Fusion and Roughness

1.2 Interspike Interval Distributions, Pitch Estimates and Harmony

1.2.1 Neuronal Code and Pitch

1.2.2 Interspike Intervals

1.2.3 Coinciding Periodicity Patterns for Intervals

1.2 Autocorrelation

1.2.1 Autocorrelation versus Fourier-Analysis

1.2.2 Hearing Theories and Autocorrelation

1.3 Langner’s Neuronal Correlator

2 Mathematical Model of Generalized Coincidence

2.1 Correlation Functions

2.2 Sequence Representation of a Tone

2.3 Sequence Representation of an Interval

2.4 Autocorrelation Function of an Interval

2.5 Definition of the Generalized Coincidence Function

3 Application of the Model to Rectangular Pulse Sequences

3.1 Correlation Functions of Rectangular Pulses

3.1.1 Autocorrelation Function of the Rectangular Pulse

3.1.2 Cross Correlation Function of the Rectangular Pulse

3.1.3 Autocorrelation Function of an Interval Represented by Rectangular Sequences

3.2 Calculation of the Generalized Coincidence Function

4 Conclusion

References

Predicting Music Therapy Clients’ Type of Mental Disorder Using Computational Feature Extraction and Statistical Modelling Techniques

1 Introduction

2 Previous Music Therapy Research

3 Computational Music Analysis

4 Method

5 Quantifying the Client-Therapist Interaction

6 Results

7 Discussion

References

Nonlinear Dynamics, the Missing Fundamental, and Harmony

1 Pitch Perception

2 Residue Behaviour

3 Nonlinear Dynamics of Forced Oscillators

3.1 n = 1

3.1.1 Synchronization

3.1.2 Quasiperiodicity

3.2 n = 2

3.2.1 Synchronization

3.2.2 Three-Frequency Resonances

4 A Nonlinear Theory for the Residue

5 The Golden Mean in Art and Science

6 The Need for Musical Scales

7 The Golden Scales

8 Playing and Transposing with Golden Scales in Equal Temperament

9 Can Our Senses Be Viewed as Generic Nonlinear Systems?

References

Dynamic Excitation Impulse Modification as a Foundation of a Synthesis and Analysis System for Wind Instrument Sounds

1 Introduction

2 Cyclical Spectra

3 Synthesis and Analysis Framework

3.1 The Digital Variophon

3.2 Formalisation

3.3 The Pulse Width Function

3.4 Application of the System

4 Discussion

Acknowledgement

References

Non-linear Circles and the Triple Harp: Creating a Microtonal Harp

1 Introduction

2 The Triple Harp

3 Non-linear Tuning Systems

4 Microtonal Triple Harp

5 Notation

6 Composing for Microtonal Triple Harp

7 Conclusion

References

Applying Inner Metric Analysis to 20th Century Compositions

1 Inner Metric Analysis

2 Analytic Results

2.1 Skrjabin’s op. 65 No. 3

2.2 Webern’s Op. 27, 2nd Movement

2.3 Xenakis’ Keren

2.4 Comparison of the Results

References

Tracking Features with Comparison Sets in Scriabin’s Study op. 65/3

1 Comparison Set Analysis

2 About the Tail Segmentation and Similarity Measures Used in the Analyses

3 The Occurrences of the ’Mystic Chord’ among Scriabin’s Piano Pieces

4 Detecting Op. 65/3 with Comparison Sets

5 Conclusions

References

Computer Aided Analysis of Xenakis-Keren

1 Introduction

2 Xenakis – Keren

References

Automated Extraction of Motivic Patterns and Application to the Analysis of Debussy’s Syrinx

1 General Framework

1.1 Motivic Pattern Extraction

1.2 Musical Dimensions

1.3 Matching Strategy

1.4 Analysis of Debussy’s Syrinx

2 Controlling the Combinatorial Redundancy

2.1 Maximal Patterns and Closed Patterns

2.2 Multidimensionality of Music

2.3 Formal Concept – Representation of Patterns

2.4 Specificity Relations

2.5 Cyclic Patterns

3 From Monody to Polyphony

References

Pitch Symmetry and Invariants in Webern's Sehr Schnell from Variations Op.27

1 Introduction

2 w = One Eighth Note

3 w = Two Eighth Notes

4 w = Three Eighth Notes

5 Center on A

Acknowledgements

References

Computational AnalysisWorkshop: Comparing Four Approaches to Melodic Analysis

1 Comparing Four Approaches to Melodic Analysis

References

Computer-Aided Investigation of Chord Vocabularies: Statistical Fingerprints of Mozart and Schubert

Presentation

References

The Irrelative System in Tonal Harmony

1 Introduction

2 Algorithm Enabling Classification of Chords

3 Chords

4 Metrical Units

5 Record Table

Acknowledgement

References

Mathematics and the Twelve-Tone System: Past, Present, and Future*

1 Introduction

2 The Introduction of Math into Twelve-Tone Music Research

3 Important Results and Trends

4 Present State of Research

5 Future

6 Conclusion

References

Approaching Musical Actions*

References

A Transformational Space for Elliott Carter's Recent Complement-Union Music*

References

Networks

From Mathematica to Live Performance: Mapping Simple Programs to Music

1 Background

2 Data Gathering

3 Large Scale Piece

3.1 Choice of a Rule

3.2 Partitioning

4 Initial Conditions

5 Choice of Musical Parameters

6 The Outcome

7 Generative Pitch Collections and Rhythmic Grouping

8 Mapping

8.1 Rule 90

8.2 Rule 30

8.3 Rule 110

9 New Ground

Acknowledgements

References

Nonlinear Dynamics of Networks: Applications to Mathematical Music Theory

1 Introduction and Musical Motivation

2 Nonlinear Dynamics of Networks

3 Discussion and Applications

3.1 Nonlinear Dynamics and Musical Ontology

3.2 Applications to Algorithmic Composition

References

Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3

References

A Local Maximum Phrase Detection Method for Analyzing Phrasing Strategies in Expressive Performances

1 Introduction

2 The Method

2.1 Data Extraction

2.2 The Case for Loudness

2.3 Local Maximum Phrase Detection

2.3.1 Phrase Strength and Volatility

2.3.2 Phrase Typicality

3 Conclusion and Discussion

Acknowledgements

References

Subgroup Relations among Pitch-Class Sets within Tetrachordal K-Families

References

K-Net Recursion in Perlean Hierarchical Structure

1 Introduction

2 K-Nets and Perle Cycles

3 K-Nets, Arrays, and Axis-Dyad Chords

4 K-Nets and Array Relationships

5 K-Nets, Interval Systems, Modes, and Keys

6 K-Nets and Synoptic Arrays

7 K-Nets and Tonality

8 Summary

References

Webern’s Twelve-Tone Rows through the Medium of Klumpenhouwer Networks

References

Isographies of Pitch-Class Sets and Set Classes

1 Introduction

2 Isography of Pitch-Class Sets and Set Classes

3 Tonality and Whole-Tone Scale Proportion

4 Relations of Set Classes

References

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

Combinatorial and Transformational Aspects of Euler's Speculum Musicum

References

Structures Ia Pour Deux Pianos by Boulez: Towards Creative Analysis Using Open Musicand Rubato

1 Introduction

2 Compositional Process in Structures Ia

2.1 Analysis of Constructional and Serial Principles: Decision and Automatism

2.2 How to Create from an Analysis

3 An Implementation in OpenMusic: A Visual and Functional Environment

3.1 Patches and Circularity

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

4.1 Different Perspectives Delivered by Rubato

4.2 Possibilities Brought by Rubato

4.3 Scheme of the Construction

5 Conclusion

References

The Sieves of Iannis Xenakis

1 Introduction

2 Types of Formulae

3 Symmetries/Periodicities

4 Inner-Periodic Formula

4.1 Inner Periodicities and Formulae Redundancy

4.2 Construction of the Inner-Periodic Simplified Formula

4.3 Analytical Algorithm: Early Stage

4.4 The Condition of Inner Periodicity

4.5 Analytical Algorithm: Final Stage

4.6 The Condition of Inner Symmetry

4.7 Inner-Symmetric Analysis

4.8 Modules and Degree of Symmetry

References

Tonal, Atonal and Microtonal Pitch-Class Categories

1 Introduction

2 Applying Pitch-Class Set Theory on Sets with Cardinality (Pitch-Classes) Other Than 12

3 Pitch-Class Set Theory within a Bit-Sequence

4 Pitch-Class Categories

5 Discussion and Future Work

6 Conclusion

References

Appendix

Using Mathematica to Compose Music and Analyze Music with Information Theory

1 Composition of Music Using Mathematica

2 Nonlinear Time Series Analysis of Musical Compositions

2.1 Creating Time Series from Sheet Music

2.2 Transfer Entropy and the Relationship between Physical Systems

2.3 The Application of the Transfer Entropy to a Symphony

3 Conclusions

References

A Diatonic Chord with Unusual Voice-Leading Capabilities

References

Mathematical and Musical Properties of Pairwise Well-Formed Scales

1 Pairwise Well-Formed and Well-Formed Scales

2 Some Properties of Pairwise Well-Formed Scales

3 Classification of Pairwise Well-Formed Scales

References

Eine Kleine Fourier Musik

Introduction

1 DFT of a pc Set

2 Maximal Values

2.1 Regular Polygons

2.2 The General Case

2.3 Other Maximal Values

3 Minimal Values

4 MeanValue(s)

5 Coda

References

WF Scales, ME Sets, and Christoffel Words

1 Well-Formed Scales

2 Christoffel Words

3 Well-Formed Classes and Christoffel Words, Duality

4 Christoffel Words, Maximally Even Sets and Musical Modes

5 Christoffel Tree and the Monoid SL(2, N)

6 Final Remarks

References

Interval Preservation in Group- and Graph-Theoretical Music Theories: A Comparative Study

References

Pseudo-diatonic Scales

1 Shuffled Stern-Brocot Tree

2 Construction of Pseudo-diatonic Scales

References

Affinity Spaces and Their Host Set Classes

1 Affinities in the Medieval Dasian Scale

2 The Dasian Space

3 Four Properties of the Dasian Space

4 Affinity Spaces

5 Three Properties of Host Set Classes

6 Generating Affinity Spaces

7 Conclusion

References

The Step-Class Automorphism Group in Tonal Analysis

Bibliography

A Linear Algebraic Approach to Pitch-Class Set Genera

1 ‘Corner-Stone Set-Classes’

2 Applying Cosine Distance and the Determinant of a Matrix with Musical Set Classes

3 Volume Tests with Interval-Class Vectors

4 ‘Strangest’ Hexachords

5 Principal Component Analysis: A Flexible Approach for Mapping ICV-Space

6 Using Corner-Stone Vectors for Producing a System of Genera

7 Harmonic Space in Composition

8 Conclusions

References

Author Index

Index