Computational Music Science (recopilación)

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A First Introduction to Mathematics for Music Theorists (2016) Preface Contents Part I Introduction and Short History 1 The ‘Counterpoint’ of Mathematics and Music 1.1 The Idea of a Contrapuntal Interaction 1.2 Formulas and Gestures 1.3 Mathematics and Technology for Music 1.4 Musical Creativity with Mathematics 2 Short History of the Relation Between Mathematics and Music 2.1 Pythagoras 2.2 Artes Liberales 2.3 Zarlino 2.4 Zaiyu Zhu 2.5 Mathematics in Counterpoint 2.5.1 An Example for Music Theorists 2.6 Athanasius Kircher 2.7 Leonhard Euler 2.8 Joseph Fourier 2.9 Hermann von Helmholtz 2.10 Wolfgang Graeser 2.11 Iannis Xenakis 2.12 Pierre Boulez and the IRCAM 2.13 American Set Theory 2.13.1 Genealogy 2.13.2 Comments 2.14 David Lewin 2.15 Guerino Mazzola and the IFM 2.15.1 Preparatory Work: First Steps in Darmstadt and Zürich (1985-1992) 2.15.2 The IFM Association: The Period Preceding the General Proliferation of the Internet (1992-1999) 2.15.3 The Virtual Institute: Pure Virtuality (1999-2003) 2.15.4 Dissolution of the IFM Association (2004) 2.16 The Society for Mathematics and Computation in Music Part II Sets and Functions 3 The Architecture of Sets 3.1 Some Preliminaries in Logic 3.2 Pure Sets 3.2.1 Boolean Algebra 3.2.2 Xenakis’ Herma 4 Functions and Relations 4.1 Ordered Pairs and Graphs 4.2 Functions 4.2.1 Equipollence 4.3 Relations 5 Universal Properties 5.1 Final and Initial Sets 5.2 The Cartesian Product 5.3 The Coproduct 5.4 Exponentials 5.5 Subobject Classifier 5.6 Cartesian Product of a Family of Sets Part III Numbers 6 Natural Numbers 6.1 Ordinal Numbers 6.2 Natural Numbers 6.3 Finite Sets 7 Recursion 8 Natural Arithmetic 9 Euclid and Normal Forms 9.1 The Infinity of Prime Numbers 10 Integers 10.1 Arithmetic of Integers 11 Rationals 11.1 Arithmetic of Rationals 12 Real Numbers 13 Roots, Logarithms, and Normal Forms 13.1 Roots, and Logarithms 13.2 Adic Representations 14 Complex Numbers Part IV Graphs and Nerves 15 Directed and Undirected Graphs 15.1 Directed Graphs 15.2 Undirected Graphs 15.3 Cycles 16 Nerves 16.1 A Nervous Sonata Construction 16.1.1 Infinity of Nervous Interpretations 16.1.2 Nerves and Musical Complexity Part V Monoids and Groups 17 Monoids 18 Groups 19 Group Actions, Subgroups, Quotients, and Products 19.1 Actions 19.2 Subgroups and Quotients 19.2.1 Classification of Chords of Pitch Classes 19.3 Products 20 Permutation Groups 20.1 Two Composition Methods Using Permutations 20.1.1 Mozart’s Musical Dice Game 20.1.2 Mannone’s Cubharmonic 21 The Third Torus and Counterpoint 21.1 The Third Torus 21.1.1 Geometry on T3×4 21.2 Music Theory 21.2.1 Chord Classification 21.2.2 Key Signatures 21.2.3 Counterpoint 22 Coltrane’s Giant Steps 22.1 The Analysis 22.2 The Composition 23 Modulation Theory 23.1 The Concept of a Tonal Modulation 23.2 The Modulation Theorem 23.3 Nerves for Modulation 23.4 Modulations in Beethoven’s op. 106 23.5 Quanta and and Fundamental Degrees for the Modulations Between Diatonic Major Scales (Dia(3)) Part VI Rings and Modules 24 Rings and Fields 24.1 Monoid Algebras and Polynomials 24.2 Fields 25 Primes 26 Matrices 26.1 Generalities on Matrices 26.2 Determinants 26.3 Linear Equations 27 Modules 27.1 Affine Homomorphisms 27.2 Free Modules and Vector Spaces 27.3 Sonification and Visualization in Modules 27.3.1 Creative Ideas from Math: A Mapping Between Images and Sounds 28 Just Tuning 28.1 Major and Minor Scales: Zarlino’s Versus Hindemith’s Explanation 28.2 Comparisons between Pythagorean, Just, and 12-tempered Tuning 28.3 Chinese Tuning Theory 28.3.1 The Original System 28.3.2 A System that Is Completely Based on Fifths 29 Categories 29.1 The Yoneda Philosophy Part VII Continuity and Calculus 30 Continuity 30.1 Generators for Topologies 30.2 Euler’s Substitution Theory 31 Differentiability 32 Performance 32.1 Mathematical and Musical Precision 32.2 Musical Notation for Performance 32.3 Structure Theory of Performance 32.4 Expressive Performance 33 Gestures 33.1 Western Notation and Gestures 33.2 Chinese Gestural Music Notation 33.3 Some Remarks on Gestural Performance 33.4 Philosophy of Gestures 33.5 Mathematical Theory of Gestures in Music 33.6 Hypergestures 33.7 Hypergestures in Complex Time Part VIII Solutions, References, Index 34 Solutions of Exercises 34.1 Solutions of Mathematical Exercises 34.2 Solutions of Musical Exercises References Index All About Music The Complete Ontology Realities, Semiotics, Communication, and Embodiment (2016) Preface Contents Part I Introduction 1 General Introduction by Guerino Mazzola 2 Ontology and Oniontology 2.1 Ontology: Where, Why, and How 2.2 Oniontology: Facts, Processes, and Gestures Part II Realities 3 Physical Reality 3.1 Physical Sound Anatomy 3.1.1 Acoustics 3.1.1.1 Standard Sound Representation 3.1.1.2 Pitch vs Frequency, Loudness vs Pressure 3.1.2 Fourier 3.1.3 Frequency Modulation 3.1.4 Wavelets 3.1.5 Physical Modeling 3.2 Hearing with Ear and Brain 3.2.1 Hearing with the Ear 3.2.2 Hearing with the Brain 3.2.3 Neuroplasticity 3.2.4 Music and the Brain Lobes 4 Psychological Reality 4.1 Emotions and Music 4.1.1 Defining Emotions 4.1.1.1 Lazarus and His School [35]: Categories of Emotion 4.1.1.2 Russell and Barrett: Core Affect 4.1.1.3 Mazzola: Neurotransmitter Model 4.1.2 Langer’s and Gabrielsson’s Thesis 4.2 Measuring Electrical Responses to Music 4.3 Some Physiological Evidences 4.4 Psychopathology and Music: van Gogh, Wölfli, Harrell, Tchaikovsky 4.5 Renate Wieland on Gestures and Emotions 5 Mental Reality Summary. 5.1 The Role of the Mental Reality 5.1.1 The Musical Score 5.1.2 The Pitch Space 5.1.3 Euler Space 5.1.4 Zarlino’s Symmetry 5.1.5 The Hidden Symmetry of Counterpoint 5.1.6 Euler’s Gradus Suavitatis Function Part III Semiotics 6 Generalities about Signs, Neumes, Periods and Development Sentences 6.1 Definition of Signs 6.2 Neumes 6.3 Musical Signs as Language 7 De Saussure and Peirce: the Semiotic Architecture 7.1 Pierce 7.2 de Saussure 7.3 Hjelmslev 7.4 Barthes 8 Riemannian Harmony and The HarmoRubette Software 8.1 The Semiotic Structure of Music 9 De Saussure’s Six Dichotomies 9.1 Defining the Dichotomies 9.1.1 Signifier/Signified 9.1.2 Arbitrary/Motivated 9.1.3 The Digital Approach, Sampling 9.1.4 Syntagm/Paradigm 9.1.5 Speech/Language 9.1.6 Synchrony/Diachrony 9.1.7 Lexem/Shifter 9.2 Speech and Language Examples: Bach and Schönberg 9.3 Semiotics in Music Performance: the Example of Celibidache’s Ideas 10 The Babushka Principle in Semiotics: Connotation, Motivation, and Metatheory Part IV Communication 11 What Is Art? 11.1 John Cage 11.2 Stockhausen’s 11.3 Molino’s Tripartition and Realities. 11.4 The Babushka Principle in Communication 11.5 Examples of the Poetic Ego in Artistic Communication 11.5.1 Art Ensemble of Chicago 11.5.2 Alanis Morissette 11.5.3 Angel Haze 11.5.4 Michael Jackson 11.5.5 Jackson Pollock 11.5.6 François Villon 11.5.7 Stolberg-Schubert 11.5.8 Raffaello’s School of Athens 11.6 The Opera and Music for Movies 11.6.1 Opera 11.6.2 Garden State 11.6.3 Satyricon 11.6.4 Fellini’s 11.6.5 Onibaba 11.7 The Infinite Production 11.7.1 Miles Davis’ 11.7.2 Mazzola’s Tetrade Group Recording 12 The MIDI Code 12.1 A Short History of MIDI 12.2 MIDI Networks: MIDI Devices, Ports, and Cables 12.3 Acoustics, Instruments, Music Software, and Creativity 12.4 Time in MIDI, Standard MIDI Files 13 Global Music 13.1 The Synthesis Project on the Presto Software 13.2 Time Hierarchies for Chopin’s Impromptu op. 29 13.3 Mystery Child 13.4 The Global Architecture of the Rubato Software 13.5 Braxton’s Cosmic Compositions 13.6 Machover’s Brain Opera 13.7 The iPod and Tanaka’s Malleable Mobile Music 13.8 Wolfram’s Cellular Music Automata 13.9 Mazzola’s and Armangil’s Transcultural Morphing Software Part V Embodiment 14 Recapitulation of the First Three Dimensions: Realities, Semiotics, Communication 14.1 Realities 14.2 Semiotics 14.3 Communication 15 The Need for a Gesture Theory in Music 15.1 Neumes and Musical Notation 15.2 Lewin, Adorno, and Hatten 15.2.1 David Lewin 15.2.2 Theodor W. Adorno 15.2.3 Robert S. Hatten 15.3 Lang Lang and Marquese ‘Nonstop’ ScottPerforming Chopin’s Etude no. 12 15.4 Mazzola’s Contributions 15.4.1 Teak Leaves at the Temples Movie 15.4.2 Books About Gesture Theory in Music 16 Frege’s Prison of Functions Summary. 16.1 Matrix Encapsulation of Geometric Rotation 16.2 Dracula and the Imaginary Numbers: How to Solve the Singularity of Real Number Negation 16.3 Imaginary Time 16.4 Other Hidden Concepts in Math 16.4.1 The Density of Real Numbers 16.4.2 Zeno’s Paradox 17 Music Without Scores Summary. 17.1 Cecil Taylor: Burning Poles 17.2 Artificial Embodied Intelligence: Cheap Design 17.3 The Hand Computer Graphics as a Gestural Challenge 17.4 Robotics for (Musical) Gestures: Asimo & Co. 18 Neuroscience and Gestures Summary. 18.1 Vilayanur S. Ramachandran and Merlin Donald 18.2 Mirror Neurons, Speech, and Hand Gestures 18.3 The Neuroscience of Imagination 18.4 Mazzola’s Gestural Dancing Project: “Dancing the Violent Body of Sound” 19 Mathematical Gesture Theory Summary. 19.1 Historical Roots Summary. 19.1.1 Tommaso Campanella 19.1.2 Hugues de Saint Victor 19.1.3 Paul Valéry 19.1.4 Jean Cavaillès 19.1.5 Maurice Merleau-Ponty 19.2 Definition of a Gesture 19.3 Hypergestures 19.4 A Gesture Suite for Piano 20 Creativity Theory Summary. 20.1 Defining Creativity 20.1.1 Example of Creative Processes in Beethoven’s op. 109, Third Movement 20.2 The Creative Strategy in “¡Ornette!” from the Movie Part VI References, Index References Index Basic Music Technology An Introduction (2018) Preface Contents Part I Introduction 1 General Introduction 2 Ontology and Oniontology 2.1 Ontology: Where, Why, and How 2.2 Oniontology: Facts, Processes, and Gestures Part II Acoustic Reality 3 Sound 3.1 Acoustic Reality 3.2 Sound Anatomy 3.3 The Communicative Dimension of Sound 3.3.1 Poiesis, Neutral Level, Esthesis 3.4 Hearing with Ear and Brain 4 Standard Sound Synthesis 4.1 Fourier Theory 4.1.1 Fourier’s Theorem 4.2 Simple Waves, Spectra, Noise, and Envelopes 4.3 Frequency Modulation 4.4 Wavelets 4.5 Physical Modeling 5 Musical Instruments 5.1 Classification of Instruments 5.2 Flutes 5.3 Reed Instruments 5.4 Brass 5.5 Strings 5.6 Percussion 5.7 Piano 5.8 Voice 5.9 Electronic Instruments 5.9.1 Theremin 5.9.2 Trautonium 5.9.3 U.P.I.C. 5.9.4 Telharmonium or Dynamophone 5.9.5 MUTABOR 6 The Euler Space 6.1 Tuning 6.1.1 An Introduction to Euler Space and Tuning 6.1.2 Euler’s Theory of Tuning 6.1.2.1 Equal Temperament 6.1.2.2 Pythagorean Tuning 6.1.2.3 Just Tuning 6.2 Contrapuntal Symmetries 6.2.1 The Third Torus 6.2.2 Counterpoint Part III Electromagnetic Encoding of Music: Hard- and Software 7 Analog and Digital Sound Encoding 7.1 General Picture of Analog/Digital Sound Encoding 7.2 LP and Tape Technologies 7.3 The Digital Approach and Sampling 8 Finite Fourier 8.1 Finite Fourier Analysis 8.2 Fast Fourier Transform (FFT) 8.2.1 Fourier via Complex Numbers 8.2.2 The FFT Algorithm 8.3 Compression 8.4 MP3, MP4, AIFF 9 Audio Effects 9.1 Filters 9.2 Equalizers 9.3 Reverberation 9.4 Time and Pitch Stretching Part IV Musical Instrument Digital Interface (MIDI) 10 Western Notation and Performance 10.1 Abstraction and Neumes 10.2 Western Notation and Ambiguity 11 A Short History of MIDI 12 MIDI Networks 12.1 Devices 12.2 Ports and Connectors 13 Messages 13.1 Anatomy 13.2 Hierarchy 14 Standard MIDI Files 14.1 Time 14.2 Standard MIDI Files Part V Software Environments 15 Denotators 16 Rubato 16.1 Introduction 16.2 Rubettes 16.3 The Software Architecture 17 The BigBang Rubette 18 Max 18.1 Introduction 18.2 Short History Environments 18.4 Some Technical Details 18.5 Max Part VI Global Music 19 Manifolds in Time and Space 19.1 Time Hierarchies in Chopin’s Op. 29 19.2 Braxton’s Cosmic Compositions 20 Music Transportation 20.1 Peer-to-Peer Networking 20.2 Downloads for Purchase 20.2.1 A Simple Example of Encryption 20.2.2 FairPlay: Fair or Unfair? 20.3 The Streaming Model 20.3.1 Effects on Consumers and Industry 21 Cultural Music Translation 21.1 Mystery Child 21.2 Mazzola’s and Armangil’s Transcultural Morphing Software 22 New Means of Creation 22.1 The Synthesis Project on the Presto Software 22.2 Wolfram’s Cellular Automata Music 22.3 Machover’s Brain Opera 22.4 The VOCALOID™ Software 22.4.1 Introducing VOCALOID™: History 22.4.2 VOCALOID™ Technologies BRE (Breathiness): BRE (Breathiness): BRI (Brightness): BRI (Brightness): CLE (Clearness): CLE (Clearness): OPE (Opening): OPE (Opening): GEN (Gender Factor): GEN (Gender Factor): POR (Portamento Timing): POR (Portamento Timing): PIT (Pitch Bend) PIT (Pitch Bend) PBS (Pitch Bend Sensitivity): PBS (Pitch Bend Sensitivity): 22.5 The iPod and Tanaka’s Malleable Mobile Music References Index Computational Counterpoint Worlds - Mathematical Theory, Software, and Experiments (2015) Preface Contents Chapter 1 Prolegomena on Counterpoint 1.1 Counterpoint’s Many Voices 1.2 Consonances, Dissonances, and the Fourth 1.3 Point Against Point 1.4 First-Species Counterpoint 1.5 Three Creativity Walls Chapter 2 First-Species Model 2.1 Dichotomies 2.2 Counterpoint Dichotomies 2.2.1 Musical Meaning of the Operations with Counterpoint Intervals 2.3 Counterpoint Symmetries 2.4 The Counterpoint Theorem 2.4.1 Some Preliminary Calculations 2.4.2 Hichert’s Algorithm Chapter 3 The Case of the Twelve-Tone Scale 3.1 Neuronal Evidence for the Polarity Function 3.1.1 The EEG Test 3.1.2 Analysis by Spectral Participation Vectors 3.1.3 Isolated Successive Intervals 3.1.4 Polarity 3.1.5 Music and the Hippocampal Gate Function 3.2 The Counterpoint Theorem Revisited 3.3 The Perspective from the Reduced Strict Style 3.4 The Antipodality of Fuxian and Ionian Dichotomies 3.5 The Fuxian and Riemann Dichotomies Chapter 4 Graphs 4.1 Counterpoint Worlds 4.2 Strict Digraphs 4.3 Quotient Digraphs 4.3.1 Vertex Partitions 4.3.2 Component Connections 4.3.2.1 Full Partitions 4.3.2.2 Weak Partitions Definition 4.9. 4.3.2.3 Strong Partitions 4.3.2.4 Homogeneous Partitions 4.3.2.5 Null Partitions 4.3.2.6 Inclusion Hierarchy 4.3.3 Homogeneous Digraphs 4.3.3.1 Homogeneous Components Form Maximal Preimages 4.3.3.2 Homogeneous Morphisms Are Strict 4.3.3.3 Homogeneous Morphisms Determine the Existence of Strict Morphisms 4.3.3.4 Homogeneous Components Do Not Split 4.3.3.5 Homogeneous Morphisms Do Not Merge Components 4.3.3.6 Gallery of Homogeneous Digraphs Chapter 5 Morphism Enumeration 5.1 Backtracking 5.2 Reducing the Problem Size 5.2.1 Avoiding Redundancy 5.2.2 Limiting Combinations 5.2.3 Pruning the Search Tree 5.2.3.1 Homogeneous Embedding Condition 5.2.3.2 Homogeneous Component Embedding Condition 5.2.3.3 Homogeneous Complement Embedding Condition 5.2.3.4 No Weak MappingsWithout Children 5.2.3.5 No Weak MappingsWithout Local Combinations 5.3 Procedure 5.3.1 Constructing the Strict Digraphs 5.3.2 Constructing the Quotient Digraphs 5.3.3 Constructing the Component Trees 5.3.4 Populating the Mapping Tree 5.3.5 Populating the Combinations Map 5.3.5.1 Collecting Parent Combinations 5.3.5.2 Computing Child Combinations 5.3.5.3 FilteringWeak Combinations 5.3.5.4 Filtering Homogeneous Combinations 5.3.6 Filtering the Quotient Structures 5.3.6.1 Iteration Verifications 5.3.6.2 Detecting Invalid Mappings 5.3.6.3 Removing Invalid Mappings and Combinations 5.3.7 Listing Valid Mappings 5.3.8 Generating Strict Mappings 5.4 Discussion 5.4.1 Complexity 5.4.2 Global Morphisms Chapter 6 Experimentation 6.1 Rubato 6.1.1 Installation 6.1.1.1 Java Virtual Machine 6.1.1.2 RUBATO Platform 6.1.1.3 BollyFux Plug-ins 6.1.2 Quick Start 6.1.2.1 How to Choose a Rubette 6.1.2.2 How to Connect Rubettes 6.1.2.3 How to Specify Parameters in a Rubette 6.1.2.4 How to Run a Network 6.1.2.5 Saving and Retrieving Rubato Machines 6.2 Recipes 6.2.1 Random Generation 6.2.1.1 Input score 6.2.1.2 Consonances 6.2.1.3 Trial and Error 6.2.1.4 Pitches 6.2.1.5 Random Counterpoint 6.2.2 Composition 6.2.2.1 Consonances 6.2.2.2 Scale 6.2.2.3 Intervals 6.2.2.4 Notes 6.2.2.5 Score 6.2.3 Transformation 6.2.3.1 Source World 6.2.3.2 Target World 6.2.3.3 Source Counterpoint 6.2.3.4 Interval Mapping 6.2.3.5 Target Pitches 6.2.4 Extensions 6.2.4.1 Complexity 6.2.4.2 Microtonality 6.2.4.3 Timbre 6.3 Rubettes 6.3.1 Counterpointiser 6.3.2 DeCounterpointiser 6.3.3 BollyWorld 6.3.4 BollyCarlo 6.3.5 BollyComposer 6.3.6 BollyMorpher 6.3.7 AnaBollyser 6.3.8 Midi File In 6.3.9 Midi File Out 6.3.10 Score Play 6.3.11 Voice Splitter 6.3.12 Voice Merger Chapter 7 Quasipolarities and Interval Dichotomies 7.1 Introductory Remarks 7.2 Characterization of Quasipolarities 7.3 Calculation of Strong Dichotomies Chapter 8 Towers of Counterpoint 8.1 The Category of Strong Dichotomies 8.2 Towers of Counterpoint 8.3 Dense Consonances and Dissonances Chapter 9 A Categorical Look at Gesture Theory 9.1 Gestures over Topological Categories 9.1.1 Digraphs Associated with Topological Categories 9.1.2 Toward Hypergestures: The Topological Category of Gestures with Body in a Topological Category 9.1.3 Functoriality with Respect to the Underlying Topological Category 9.2 Constructing Gestures from Morphisms 9.2.1 Interpreting Diagrams as Gestures 9.2.2 Gestures with Bodies in Factorization Categories 9.2.3 Homological Extensions Are Gestures Chapter 10 Hypergesture Homology for Counterpoint 10.1 Singular Homology for Hypergestures 10.1.1 Chain Modules for Singular Hypergesture Homology 10.1.2 Boundary Homomorphisms 10.2 Homological Interpretation of the Counterpoint Model 10.2.1 Hypergestural Singular Homology 10.2.2 A Classical Example of a Topological Category from Counterpoint 10.2.2.1 Generators of H1(GX) for the Groupoid GX Defined by a Group Action 10.2.3 The Meaning of H1 for Counterpoint 10.2.4 Concluding Comments Appendix A Mathematical Basics A.1 Sets and Relations A.2 Graph Theory A.3 Groups and Rings A.4 Modules A.5 Topology A.6 Categories A.6.1 Basic Definitions A.6.2 Subfunctors and Sieves A.6.3 Subobjects and Object Classifiers A.6.4 Adjoint Functors A.6.5 Topoi Appendix B A Guide to Counterpoint Worlds B.1 Discrete Digraphs B.2 Forests B.3 Stars B.4 Grids B.5 Unknown Digraphs Appendix C Strict Digraphs References Index Computational Musicology in Hindustani Music (2014) Preface References Contents Chapter 1: An Introduction to Indian Classical Music 1.1 A Critical Comparison Between Indian and Western Music 1.2 Terminologies Used in Hindustani Classical Music 1.2.1 Raga 1.2.2 Notation Used in Describing Ragas 1.3 Systematic Presentation of Ragas 1.4 Remarks References Chapter 2: The Role of Statistics in Computational Musicology 2.1 Modeling 2.2 Similarity Analysis 2.3 Rhythm Analysis 2.4 Entropy Analysis 2.5 Multivariate Statistical Analysis 2.6 Study of Varnalankars Through Graphical Features of Musical Data 2.7 The Link Between Raga and Probability 2.8 Statistical Pitch Stability Versus Psychological Pitch Stability 2.9 Statistical Analysis of Percussion Instruments References Chapter 3: Introduction to RUBATO: The Music Software for Statistical Analysis 3.1 Architecture 3.1.1 The Overall Modularity 3.1.2 Frame and Modules 3.2 The RUBETTE Family 3.2.1 MetroRUBETTE 3.2.2 MeloRUBETTE 3.2.3 HarmoRUBETTE 3.2.4 PerformanceRUBETTE 3.2.5 PrimavistaRUBETTE References Chapter 4: Modeling the Structure of Raga Bhimpalashree: A Statistical Approach 4.1 Introduction 4.2 Methodology 4.2.1 Getting the Musical Data for Structure Analysis 4.3 Statistical Analysis 4.4 Discussion Conclusion Appendix References Chapter 5: Analysis of Lengths and Similarity of Melodies in Raga Bhimpalashree 5.1 Introduction 5.2 Statistical Analysis of Melody Groups Conclusion References Chapter 6: Raga Analysis Using Entropy 6.1 Introduction 6.1.1 Mean entropy of raga Bhimpalashree 6.2 Discussion: Information on a Possible Event E with P(E)=0 Conclusion References Chapter 7: Modeling Musical Performance Data with Statistics 7.1 Statistics and Music 7.2 Time Series Analysis and Music 7.2.1 Time Series Data 7.2.2 Goal of Time Series Analysis 7.3 Autoregressive Integrated Moving Average 7.3.1 Autoregressive Process 7.3.2 Moving Average Process 7.3.3 ARIMA Methodology 7.3.4 Modeling Musical Data 7.3.5 Analysis of Bihag Musical Features (Dutta 2006) Abbreviations 7.4 Identification of ARIMA (p, d, q) Models 7.4.1 Autoregressive Components 7.4.2 Moving Average Components 7.4.3 Mixed Models 7.5 ACFs and PACFs 7.6 Estimating Model Parameters 7.7 Model Diagnostics 7.7.1 Ljung-Box (Q) Statistic for Diagnostic Checking 7.8 Modeling: Finding Fitted Model 7.9 Results and Discussions 7.9.1 Results for Night Raga Bihag Conclusion References Chapter 8: A Statistical Comparison of Bhairav (a Morning Raga) and Bihag (a Night Raga) 8.1 IOI Graph 8.2 Duration Graphs 8.3 RUBATO Analysis 8.4 RUBATO Analysis of Bhairav 8.5 RUBATO Analysis for Raga Bihag Conclusion References Chapter 9: Seminatural Composition 9.1 Introduction 9.2 Experimental Results Conclusion Appendix References Chapter 10: Concluding Remarks References Flow, Gesture, and Spaces in Free Jazz - Towards a Theory of Collaboration (2009) Part I Getting off Ground What Is Free Jazz? The Social, and Political, and Musical Origins of the Movement A Provisional Positive Characterization Jazz in Transition Archie Shepp in Donaueschingen John Coltrane's A Love Supreme Cecil Taylor and Buell Neidlinger Bill Evans: Gestural Dialogs Part II The Landscape of Free Jazz Out of this World Sun Ra: An Extraterrestrial Romantic Coltrane's Om Mythologies of The Art Ensemble of Chicago The Art of Collaboration A Short Overview of the Classical Ontological Landscape of Music The Oniontological Extension to The Fourth Dimension What Is The Art of Collaboration? Part III Collaborative Spaces in Free Jazz Which Collaboratories? Ornette Coleman's Melodic Spaces in Free Jazz John Coltrane's Harmonic Spaces in Ascension The Innards of Time Cecil Taylor: Unit Structures Trance Spaces: Archie Shepp's The Magic of Ju-Ju Dervish Dances: Albert Ayler's Love Cry Part IV Gestural Creativity From Philosophy to Thought Experiments Philosophy, Performance, Music Theory The French Approach to Gestures Châtelet's Gestural Thought Experiments Geometry of Gestures Gestures Are Diagrams of Curves Definition of Gestures and Hypergestures Hypergestures, Cognitive Science, and Cavaillès The Escher Theorem and Gestural Creativity The Escher Theorem Group Creativity and Categories of Hypergestures Rebecca Lazier's Vanish: Lawvere, Escher, Schoenberg Musical Poetology Part V What Group Flow Generates What Is Flow? Mihaly Csikszentmihalyi's Flow Concept Keith Sawyer's Group Flow Miles Davis' Bitches Brew Gestures in Geisser's and Mazzola's Chronotomy What Does Group Flow Produce? The Symbolic Axis of Distributed Identity Groups from Gestures The Fourier Ballet Passion Archie Shepp's Coral Rock Part VI Epilogue From Pre- to Postproduction: The Infinite Listening Global Strategies for Free Jazz The Future of Free Jazz References Making Musical Time (2021) Preface Contents Part I Ontological Orientation 1 Ontology, Oniontology, and the Artistic Presence 1.1 Ontology and Oniontology 1.2 Ontology: Where, Why, and How 1.3 Oniontology: Facts, Processes, and Gestures 1.4 A Short Characterization 1.5 The Artistic Presence: A Processual Unfolding of Oniontology Part II General Time Concepts 2 Time in Philosophy 2.1 Plato 2.2 Aristoteles 2.3 Kant 2.4 Valéry 2.5 Chinese Philosophical Time Concepts 2.6 John M. E. McTaggart: The Unreality of Time 2.7 Edmund Husserl: Vorlesungen zur Phänomenologie des inneren Zeitbewusstseins 2.8 Henri Bergson: Mathematical vs. Pure Time 2.9 Jean Wahl: Quality of Events 2.10 Maurice Merleau-Ponty: La Phénoménologie de la Perception 3 Time in Physics 3.1 Newton’s Divine Time 3.2 Relativity Theory 3.3 Quantum Mechanics (QM) 3.4 Complex Time 3.5 A Physical Interpretation 3.6 Leo Smolin’s “Time Reborn” 4 Time in Cultures, History and Present 4.1 Time in Languages 4.1.1 Time in Literature 4.2 Time in Upbringing and Social Grouping 4.3 The Musicality of Everyday Life 4.4 Geographical Location 4.4.1 Europe 4.4.1.1 Time Was Introduced by Clocks in the Church 4.4.1.2 Mensural Notation and Clocks 4.4.2 Asia 4.4.2.1 Understanding the Chinese Concept of Time From a Cultural Perspective 4.4.2.2 The Art of Ma Time in Noh Theatre 4.4.2.3 Time Concept in Japanese Composer Joe Hisaishi’s Music 4.4.3 Importance of Indian Classical Music 4.4.3.1 Chronology 4.4.3.2 Essence of Spirituality 4.4.4 Africa 4.4.4.1 The Sankofa Allegory 4.4.5 Americas 4.4.5.1 North America 4.4.5.2 South America 5 Genealogy and Ontology of Human Time Perception 5.1 Time Perception in Children 5.2 Neurological Localization of Time Processing 5.2.1 Pathological Temporal Aspects of Embodiment 5.2.1.1 Time Conception in Music and Dance Therapy for Parkinson’s Disease 5.2.1.2 Alzheimer: Two Times 5.2.2 Psychology of Metrical Perception 5.3 Handedness 5.3.1 Handedness in Culture and Science 5.3.2 Handedness in Music 5.3.3 Left-Handedness in Creative Versus Rote Performance 5.3.4 Left-Handers Who Play Right-Handed 5.3.5 Conclusions Part III Musical Time Concepts 6 Meters and Rhythm 6.1 The Genealogy of Meter and Rhythm 6.1.1 Biological Genealogy 6.1.2 Sociological Genealogy 6.2 Local Meters 6.2.1 Riemann’s Weights 6.2.2 Jackendoff-Lerdahl: Intrinsic Versus Extrinsic Time Structures 6.2.3 The Formal Theory of Local Meters 6.3 Global Meters 6.4 Metrical Topologies 6.5 Rhythms 7 Structures of Organized Time 7.1 Harald Krebs 7.2 Guerino Mazzola 7.3 Gérard Assayag 7.4 Jason Yust 8 Musical Gestures 8.1 Gestures and Hypergestures 8.2 The Escher Theorem 8.3 Natural Gestures 8.4 Gestures in Performance 8.4.1 Roger Sessions: Questions about Musical Performance 8.4.2 Theodor Wiesengrund Adorno’s Performance Theory 8.4.3 Renate Wieland’s and Jürgen Uhde’s Theory of Pianist’s Practicing 8.4.4 Gestures in Performance of Written Music vs. Gestures in Improvisation 8.4.4.1 Definition of Composition 8.4.4.2 Definition of Improvisation 8.4.4.3 Threefold Time in Composition 8.4.4.4 Characterization of creativity in composition vs. creativity in improvisation 8.4.4.5 Summary of the differences, especially the gestural aspect thereof 8.4.4.6 Performance in the European Tradition of Composition 8.4.4.7 Uniformization by Recordings 8.4.5 Cecil Taylor’s Burning Ploles 8.4.5.1 Time as a Central Existential Category 8.4.5.2 The Lonely Time 8.4.5.3 Body Interaction 8.4.5.4 Silence for Cecil Taylor 8.4.5.5 Performance of Cecil Taylor, Black Body Revolution 8.4.5.6 Dystopia 9 Kramer’s Time Concepts 9.1 Kramer’s Time Variety 9.2 Gestural Time 9.3 Moment Time 9.4 Linear/Absolute Time 9.5 Multiply-directed (Linear) Time 9.6 Vertical Time 9.7 The Twentieth Century Technological Time Revolution 10 Distributed Identity in Music 10.1 Collaboration in Free Jazz 10.2 When is Free Jazz Successful? 10.3 Distributed Identity of Passion 10.4 Distributed Identity outside of Free Jazz Part IV New Developments on Musical Time Concepts 11 Limits of Gestural Diagrams 11.1 The Role of Limits 11.1.1 Representable Functors and Limits 11.1.2 Existence of Limits 11.2 Temporal Interpretation of Limits 12 Imaginary Time 12.1 Complex Time in Music 12.2 Imaginary Time 13 3.3. Modeling Kramer’s Time Concepts 13.1 Interpreting Gestural Time 13.2 Interpreting Vertical Time 14 Semiotics of Time 14.1 Summary of Functorial Semiotics 14.2 Gestures in Semiotics 14.3 Time Gestures: From Gestures to Signs 14.4 Time in Creative Semiotics 15 Goebel, Pang, and Rochester: Applications 15.1 Jordon Goebel: Exploring the Construction of Time through Music Composition and Compositional Techniques 15.1.1 Introduction 15.1.2 Explanation of the Piece 15.1.3 Twelve Moments 15.1.4 Case Study 15.1.5 Layout of the experiment and emphasize on the importance of various performance notes and the significance 15.1.6 Score Reference and Performance Notes 15.2 Yan Pang’s Application/Composition–Dancing Upstream 15.2.1 Left-Centric Interpretation 15.2.1.1 Introduction 15.2.1.2 Interpretation as Creative Performance 15.2.1.3 The Mountain Dulcimer 15.2.1.4 Left-Centric Play and the Mountain Dulcimer 15.2.1.5 East-West Aesthetics of Musical Time: Exchanges and Collaborations 15.2.1.6 Teachers, Students, Collaborators 15.2.1.7 Standing Alone 15.2.2 Performer Time/Composer Time 15.2.2.1 Coda: Culture and/is Time 15.3 Christopher Rochester: Manipulating the Subconscious Gravity in Musical Time 15.3.1 Gravity in all Music Theories 15.3.2 The Black American 8th Note 15.3.3 Conclusion 16 Experiments with Local and Global Rhythms Part V Conclusions 17 Musical Time Constructs, the Art of Time and Human Creativity 17.1 Time Constructs as a Deeply Musical Endeavor 17.2 Art of Time 17.3 Humans Create their Own Time References Index Music Through Fourier Space Discrete Fourier Transform in Music Theory (2016) Introduction Historical Survey and Contents A Couple of Examples Public Acknowledgements Notations Contents 1 Discrete Fourier Transform of Distributions 1.1 Mathematical definitions and preliminary results 1.1.1 From pc-sets to an algebra of distributions 1.1.2 Introducing the Fourier transform 1.1.3 Basic notions 1.2 DFT of subsets 1.2.1 What stems from the general definition 1.2.2 Application to intervallic structure 1.2.3 Circulant matrixes 1.2.4 Polynomials Exercises 2 Homometry and the Phase Retrieval Problem 2.1 Spectral units 2.1.1 Moving between two homometric distributions 2.1.2 Chosen spectral units 2.1.3 Rational spectral units with finite order 2.1.4 Orbits for homometric sets 2.2 Extensions and generalisations 2.2.1 Hexachordal theorems 2.2.2 Phase retrieval even for some singular cases 2.2.3 Higher order homometry Exercises 3 Nil Fourier Coefficients and Tilings Cyclotomic polynomials 3.1 The Fourier nil set of a subset of 3.1.1 The original caveat 3.1.2 Singular circulating matrixes 3.1.3 Structure of the zero set of the DFT of a pc-set 3.2 Tilings of Zn by translation 3.2.1 Rhythmic canons in general 3.2.2 Characterisation of tiling sets 3.2.3 The Coven-Meyerowitz conditions 3.2.4 Inner periodicities 3.2.5 Transformations 3.2.6 Some conjectures and routes to solve them 3.3 Algorithms 3.3.1 Computing a DFT 3.3.2 Phase retrieval 3.3.3 Linear programming 3.3.4 Searching for Vuza canons Exercises 4 Saliency 4.1 Generated scales 4.1.1 Saturation in one interval 4.1.2 DFT of a generated scale 4.1.3 Alternative generators 4.2 Maximal evenness 4.2.1 Some regularity features 4.2.2 Three types of ME sets 4.2.3 DFT definition of ME sets 4.3 Pc-sets with large Fourier coefficients 4.3.1 Maximal values 4.3.2 Musical meaning 4.3.3 Flat distributions Exercises 5 Continuous Spaces, Continuous FT 5.1 Getting continuous 5.2 A DFT for ordered collections of pcs on the continuous circle 5.3 ‘Diatonicity’ of temperaments in archeo-musicology 5.4 Fourier vs. voice leading distances 5.5 Playing in Fourier space 5.5.1 Fourier scratching 5.5.2 Creation in Fourier space 5.5.3 Psycho-acoustic experimentation Exercises 6 Phases of Fourier Coefficients 6.1 Moving one Fourier coefficient 6.2 Focusing on phases 6.2.1 Defining the torus of phases 6.2.2 Phases between tonal or atonal music 6.3 Central symmetry in the torus of phases 6.3.1 Linear embedding of the T/I group 6.3.2 Topological implications 6.3.3 Explanation of the quasi-alignment of major and minor triads Exercises 7 Conclusion 8 Annexes and Tables 8.1 Solutions to some exercises 8.2 Lewin’s ‘special cases’ 8.3 Some pc-sets profiles 8.4 Phases of major/minor triads 8.5 Very symmetrically decomposable hexachords 8.6 Major Scales Similarity References Index Musical Creativity - Strategies and Tools in Composition and Improvisation (2011) Cover Computational Music Science Musical Creativity ISBN 9783642245169 Preface Contents Part I: Introduction 1 What the Book Is About 2 Oniontology: Realities, Communication, Semiotics, and Embodiment of Music 2.1 Realities 2.2 Communication 2.3 Semiotics 2.4 Embodiment 2.5 The Baboushka Principle Part II: Practice 3 The Tutorial 4 The General Method of Creativity 5 Getting Off the Ground 6 Motivational Aspects 6.1 What Is Your Open Question? 6.2 Let Us Describe the Context! 6.3 Find the Critical Concept! 6.4 We Inspect the Concept's Walls! 6.5 Try to Soften and Open the Walls! 6.6 How Can We Extend Opened Walls? 6.7 Final Step: Testing Our Extension 7 Rhythmical Aspects 7.1 What Is Your Open Question? 7.2 Let Us Describe the Context! 7.3 Find the Critical Concept! 7.4 We Inspect the Concept's Walls! 7.5 Try to Soften and Open the Walls! 7.6 How Can We Extend Opened Walls? 7.7 Final Step: Testing Our Extension 8 The Pitch Aspect 8.1 What Is Your Open Question? 8.2 Let Us Describe the Context! 8.3 Find the Critical Concept! 8.4 We Inspect the Concept's Walls! 8.5 Try to Soften and Open the Walls! 8.6 How Can We Extend Opened Walls? 8.7 Final Step: Testing Our Extension 9 The Harmonic Aspect 9.1 What Is Your Open Question? 9.2 Let Us Describe the Context! 9.3 Find the Critical Concept! 9.4 We Inspect the Concept's Walls! 9.5 Try to Soften and Open the Walls! 9.6 How Can We Extend Opened Walls? 9.7 Final Step: Testing Our Extension 10 Melodic Aspects 10.1 What Is Your Open Question? 10.2 Let Us Describe the Context! 10.3 Find the Critical Concept! 10.4 We Inspect the Concept's Walls! 10.5 Try to Soften and Open the Walls! 10.6 How Can We Extend Opened Walls? 10.7 Final Step: Testing Our Extension 11 The Contrapuntal Aspect 11.1 What Is Your Open Question? 11.2 Let Us Describe the Context! 11.3 Find the Critical Concept! 11.4 We Inspect the Concept's Walls! 11.5 Try to Soften and Open the Walls! 11.6 How Can We Extend Opened Walls? 11.7 Final Step: Testing Our Extension 12 Instrumental Aspects 12.1 What Is Your Open Question? 12.2 Let Us Describe the Context! 12.3 Find the Critical Concept! 12.4 We Inspect the Concept's Walls! 12.5 Try to Soften and Open the Walls! 12.6 How Can We Extend Opened Walls? 12.7 Final Step: Testing Our Extension 13 Creative Aspects of Musical Systems: The Case of Serialism 13.1 What Is Your Open Question? 13.2 Let Us Describe the Context! 13.3 Find the Critical Concept! 13.4 We Inspect the Concept's Walls! 13.5 Try to Soften and Open the Walls! 13.6 How Can We Extend Opened Walls? 13.6.1 Another Extension 13.7 Final Step: Testing Our Extension 14 Large Form Aspects 14.1 What Is Your Open Question? 14.2 Let Us Describe the Context! 14.3 Find the Critical Concept! 14.4 We Inspect the Concept's Walls! 14.5 Try to Soften and Open the Walls! 14.6 How Can We Extend Opened Walls? 14.7 Final Step: Testing Our Extension 15 Community Aspects 15.1 What is Your Open Question? 15.2 Let Us Describe the Context! 15.3 Find the Critical Concept! 15.4 We Inspect the Concept's Walls! 15.5 Try to Soften and Open the Walls! 15.6 How Can We Extend Opened Walls? 15.7 Final Step: Testing Our Extension 16 Commercial Aspects 16.1 What Is Your Open Question? 16.2 Let Us Describe the Context! 16.3 Find the Critical Concept! 16.4 We Inspect the Concept's Walls! 16.5 Try to Soften and Open the Walls! 16.6 How Can We Extend Opened Walls? 16.7 Final Step: Testing Our Extension Part III: Theory 17 Historical Approaches 17.1 The Concept of Creativity through (Western) History 17.2 Creativity in Early Psychology 17.3 Creativity Research in Recent Years 18 Present Approaches 18.1 The Creative Process Today 18.1.1 The Four P's of Creativity 18.1.2 The Creative Process 18.2 Musical Creativity 19 Our Approach 19.1 Approach to Creativity: A Semiotic Presentation 19.1.1 The Open Question's Context in Creativity 19.1.2 Motivation for a Semiotic Extension 19.1.3 The Critical Sign 19.1.4 Identifying a Concept's Walls 19.1.5 Opening a Wall and Displaying Its New Perspectives 19.1.6 Visual Representation of the Wall Paradigm 19.1.7 Evaluating the Extended Walls 19.2 Approach to Creativity: A Mathematical Model 19.3 The List of the Creativity Process 20 Principles of Creative Pedagogy 20.1 Origins of Creative Pedagogy 20.2 Applying Our Concept of Creativity to Creative Pedagogy 20.3 Creative Pedagogy for Musical Creativity 20.3.1 Conceiving Our Tutorial in Creative Pedagogy for Musical Creativity 21 Acoustics, Instruments, Music Software, and Creativity 21.1 Acoustic Reality 21.1.1 First Sound Anatomy 21.1.2 Making Sound 21.1.3 Fourier 21.1.4 FM, Wavelets, Physical Modeling 21.2 Electromagnetic Encoding of Music: Audio HW and SW 21.2.1 General Picture of Analog/Digital Sound Encoding 21.2.2 LP and Tape Technologies, Some History 21.2.3 The Digital Approach, Sampling 21.2.4 Finite Fourier Analysis 21.2.5 Fast Fourier Analysis (FFT) 21.2.6 Compression 21.2.7 MP3, MP4, AIFF 21.2.8 Filters and EQ 21.3 Symbolic Formats: Notes, MIDI, Denotators 21.3.1 Western Notation and Performance 21.3.2 MIDI: What It Is About, Short History 21.3.3 MIDI Networks: MIDI Devices, Ports, and Cables 21.3.4 MIDI Messages: Hierarchy and Anatomy 21.3.5 Time in MIDI, Standard MIDI Files 21.3.6 Short Introduction to Denotators 21.4 Creativity in Electronic Music: Languages and Theories 22 Creativity in Composition and Improvisation 22.1 Defining Composition and Improvisation 22.2 Creativity in Composition 22.2.1 Composition by Objectivation 22.2.2 Creativity in Composition with Symbolic Objects 22.3 Creativity in Improvisation 22.3.1 Improvisational Creativity in the Imaginary Time-Space 22.3.2 Improvisational Creativity with Gestural Embodiment 22.4 Instant Composition and Slow-Motion Improvisation Part IV: Case Studies 23 The CD Passionate Message 23.1 The General Background of This Production 23.1.1 The Overall Strategy 23.1.2 Joomi's Compositional Approach 23.1.3 Guerino's Improvisational Approach 23.2 Softening One's Boundaries in Creativity 23.2.1 Embodied Creation and the Crisis of Contemporary Composition 23.3 The Problem of Creativity in a Dense Cultural Heritage of Compositions 23.3.1 First Wall: Composition, an Object? 23.3.2 Second Wall: Originality 24 The Escher Theorem 24.1 A Short Review of the Escher Theorem 24.1.1 Gestures and Hypergestures 24.1.2 The Escher Theorem 24.2 The Escher Theorem and Creativity in Free Jazz 24.3 Applying the Escher Theorem to Open Walls of Critical Concepts 25 Boulez: Structures Recomposed 25.1 Boulez's Idea of a Creative Analysis 25.2 Ligeti's Analysis 25.3 A First Creative Analysis of Structure Ia from Ligeti's Perspective 25.3.1 Address Change Instead of Parameter Transformations 25.3.2 The System of Address Changes for the Primary Parameters 25.3.3 The System of Address Changes for the Secondary Parameters 25.3.4 The First Creative Analysis 25.4 Implementing Creative Analysis on RUBATO 25.4.1 The System of Boulettes 25.5 A Second More Creative Analysis and Reconstruction 25.5.1 The Conceptual Extensions 25.5.2 The BigBang Rubette for Computational Composition 25.5.3 A Composition Using the BigBang Rubette and the Boulettes 25.5.4 Was This ``Creative Analysis'' a Creative Success? 26 Ludwig van Beethoven's Sonata opus 109: Six Variations 26.1 Uhde's Perspective Metaphor 26.2 Why a Sixth Variation? Part V References, Index References Index The languages of western tonality (2013) Preface Theory; History; Cognition Reference Contents List of Figures List of Tables List of Definitions List of Notation Chapter 1: Proto-tonal Theory: Tapping into Ninth-Century Insights References Part I: Proto-tonality Chapter 2: Preliminaries 2.1 Descriptive and Explanatory Proto-tonal Adequacy: A Lesson from Linguistics 2.2 The Communication Principle 2.3 Three Additional Guiding Ideas 2.3.1 The Economical Principle 2.3.2 The Categorical Principle 2.3.3 The Maximalist Principle 2.4 Event Sequences References Chapter 3: Communicating Pitches and Transmitting Notes 3.1 Octave-Endowed Note Systems 3.2 Bases of the Interval Space 3.3 Pitch-Communication Systems 3.4 Absolute, Relative, and Reflexive Pitch Communication 3.4.1 Two Postscripts 3.4.1.1 A Possible Biology for Harmonic Templates 3.4.1.2 Pitch as a High-Level Mental Construct 3.5 Composite Tone Systems References Chapter 4: The Conventional Nomenclatures for Notes and Intervals 4.1 The Conventional Nomenclatures for Notes and Intervals 4.2 Staff Notation and Its Idiosyncrasies Chapter 5: Communicating the Primary Intervals 5.1 Efficient Tone Systems 5.2 Coherent Tone Systems 5.3 Categorical Equal Temperament References Chapter 6: Receiving Notes 6.1 Note Reception: A Lesson by Bartók 6.2 Note-Reception Systems 6.3 Proto-diatonic Systems 6.4 Diatonic Systems 6.5 Properties of Diatonic Systems References Chapter 7: Harmonic Systems 7.1 The Grammatical Basis of Harmonic Communication 7.2 Generic Klang Systems 7.3 Functional Klangs and Klang Classes 7.4 Harmonic Systems, Voice-Leading Enabled 7.5 Efficient Harmonic Systems References Chapter 8: Proto-tonality 8.1 Proto-tonal Systems 8.2 Categorical ET: Theory Lagging (Far) Behind Practice? 8.3 A Possible Alternative to the Theory of Proto-tonal Systems References Part II: The Languages of Western Tonality Chapter 9: Tonal Preliminaries 9.1 Dyadic and Triadic Consonance and Stability 9.2 The Chromatic Content of the Cluster References Chapter 10: Modal Communication 10.1 Modes, Semi-keys, and Keys 10.2 Modal Communication Systems 10.3 Scale Degrees 10.4 Robust and Semi-robust Communication Systems 10.5 Congruent and Standard Modes References Chapter 11: Topics in Dyadic and Triadic Theory 11.1 Glarean, Lippius, and Modal Theory 11.2 Aspects of Triadic Consonance and Stability 11.3 Relative and Parallel Triadic Keys 11.4 Robust Triadic Keys and Schenker´s ``Mixture´´ References Chapter 12: Modes, Semi-keys, and Keys: A Reality Check 12.1 The Octenary Doctrine and the ``Reality of Mode´´ Claim 1: ``Mode´´ Is a Theoretical Fabrication Claim 2: Tonal Types Are Real Claim 3: TTs in Renaissance Culture ``Represent´´ Modes Claim 4: The Relation from TTs to Modes Is Not a Function 12.2 The Seventeenth-Century ``Church Keys´´ as Triadic Semi-keys 12.3 On the Reality of Triadic Keys References Chapter 13: A Neo-Riepelian Key-Distance Theory 13.1 Key-Distance Theories of the Eighteenth and Nineteenth Centuries 13.2 The Krumhansl/Kessler Torus and Its Relation to Weber´s 13.3 A Neo-Riepelian Key-Distance Theory References Chapter 14: Tonal Communication 14.1 Dyadic and Triadic Heptads 14.2 Scales and Tonalities References Chapter 15: The Tonal Game 15.1 The Tonal Game 15.2 Chopin´s Mazurka, Op. 24, No. 2, and Fétis´s ``Tonal Perfection´´ References Appendix A: Mathematical Preliminaries Appendix B: Z Modules and Their Homomorphisms Index The Musical-Mathematical Mind - Patterns and Transformations (2017) Foreword Preface Acknowledgements Contents Contributors Acronyms Introduction References Extended Counterpoint Symmetries and Continuous Counterpoint 1 Introduction 2 Some Definitions and Notations 3 Extending Counterpoint Symmetries 4 A More Detailed Example 5 A Possible Continuous Counterpoint 6 Some Final Remarks References Gödel-Vector and Gödel-Address as Tools for Genealogical Determination of Genetically-Produced Musical Variants 1 Theoretical Foundations of the Research 2 The Gr-System and the GeneMus Complex 3 The Gödel-Vector and the Gödel-Address 4 Conclusions References A Survey of Applications of the Discrete Fourier Transform in Music Theory 1 Introduction 2 Basics 2.1 What is DFT? 2.2 Convolution and Lewin's Problem 2.3 Circulating Matrices 3 Homometry and Spectral Units 4 Tilings 5 Saliency 5.1 Measuring ``fifthishness'' 5.2 A Better Approximation of Peaks 6 A Torus of Phases References Gestures on Locales and Localic Topoi 1 Introduction 2 Gestures on Topological Spaces 2.1 Sober Spaces 3 Gestures on Locales 3.1 Locales and Frames 3.2 Motivation 3.3 Construction 3.4 Points and Gestures 4 Gestures on Localic Topoi 5 Comments About Gestures on Sites and Topoi, and Conclusions References On the Structural and the Abstract in My Compositional Work 1 Cheltrovype (1968--71) for Cello, Trombone, Vibraphone and Percussion 2 Sinophony II (1969--72) for Eight-Channel Electronics 3 Stochroma (1972) for Solo Piano 4 Bachanal for Jim Tenney and Tom Johnson (1990) for Solo Piano 5 Piano Concerto #2 (1961--1998) for Piano and Orchestra 6 Les Ciseaux de Tom Johnson (1998) for Chamber Ensemble 7 ``...or a Cherish'd Bard...'' (1999) for Solo Piano 8 Approximating Pi (2007) for up to 16 Channels of Electronics References A Proposal for a Music Writing for the Visually Impaired 1 Introduction 2 Braille Code 2.1 Literary Braille 2.2 Numbers in Braille 2.3 Music Braille 2.4 Braille Alternatives 3 Music Braille Problems 4 The Need of a New Musicography for the Blind 5 Proposed Methodology 6 Conclusions References Group Theory for Pitch Sequence Representation: From the Obvious to the Emergent Complexity 1 Introduction 2 (L(Sn),°) 3 Piph Music for Algorithmic Composition 4 Translating a Piece of Music into a Single Number References Mazzola's Escher Theorem 1 Basic Concepts 2 The Category of Gestures 3 Hypergestures with an Approach to Escher's Theorem 4 Topological Categories and Mazzola's Escher Theorem References The Mechanics of Tipping Points: A Case of Extreme Elasticity in Expressive Timing 1 Introduction 2 Tipping Points: A Definition 3 Three Case Studies 3.1 Case Study I: Puccini's O Mio Babbino Caro 3.2 Case Study II: Strauss' Burleske 3.3 Case Study III: Kreisler's Schon Rosmarin 4 Discussion and Conclusions References Lexicographic Orderings of Modes and Morphisms 1 Scale Theory Concepts and Notations 2 From Scales to Modes: Word Theory 3 Lexicographic Orderings References Music of Quantum Circles 1 Introduction 2 Circles, Classical and Quantum 3 Universal Harmony Partiture 4 Concluding Remarks 5 Further Reading References Partitiogram, Mnet, Vnet and Tnet: Embedded Abstractions Inside Compositional Games 1 Partitional Analysis 2 Textural Nets 3 Homology Between Textural Fields 4 Conclusions References Algebraic Combinatorics on Modes 1 Introduction 2 Progressive Transposition Scales 3 Deep Scales 4 Microtonal Diatonic Scales 5 Microtonal Modes of Limited Transposition 6 Plactic Modes Classification 7 Conclusion References Proportion, Perception, Speculation: Relationship Between Numbers and Music in the Construction of a Contemporary Pythagoreanism 1 Qualitative Numbers 2 Harmonic Duality 3 Time Scales 4 Levels of Perception 5 Objective Phenomenology 6 Analysis and Synthesis References Topos Echóchromas Hórou (The Place of the Tone of Space). On the Relationship Between Geometry, Sound and Auditory Cognition 1 Introduction 2 Spatial Composition Method 2.1 Binaurality 2.2 Echolocation 2.3 Evanescent Perception 3 Materia Oscura (Dark Matter) 4 Conclusion References Models and Algorithms for Music Generated by Physiological Processes 1 Introduction 2 The Model 3 Numerical Implementation 4 Generation of Musical Structures 5 Compositional Application 6 Conclusions References Music, Expectation, and Information Theory 1 When is Music Successful? 2 Information Theory References Gestural Dynamics in Modulation: (Towards) a Musical String Theory 1 Introduction 2 Hypergestures Between Triadic Degrees that are Parallel to Vector Fields 3 Lie Brackets Generate Vector Fields that Connect Symmetry-Related Degrees 4 Selecting Parallel Hypergestures that are Admissible for Modulation 5 The Other Direct Modulations 6 Stokes' Theorem for Hypergestures 7 Almost Regular Manifolds, Differential Forms, and Integration for Hypergestures 7.1 Locally Almost Regular Manifolds 7.2 Differential Forms 7.3 Integration 8 Stokes' Theorem References Manuel M. Ponce's Piano Sonata No. 2 (1916): An Analysis Using Signature Transformations and Spelled Heptachords 1 Introduction 2 Signature Transformations 3 Proper Spelled Heptachords 4 Ponce's Sonata No. 2 5 One Approach to these Transitions References Textural Contour: A Proposal for Textural Hierarchy Through the Ranking of Partitions lexset 1 Introduction 2 Musical Contour Theory 3 Partitional Analysis 4 Ranking Partitions for the Textural Contour 5 Conclusions References The Sense of Subdominant: A Fregean Perspective on Music-Theoretical Conceptualization 1 Competing Motivations for the Term `Subdominant' 2 Agmon's Diatonic Property: Gradus Ad Parnassum 3 Regener Transformations and Rameau's Equation References How Learned Patterns Allow Artist-Level Improvisers to Focus on Planning and Interaction During Improvisation 1 Patterns in Music Improvisation 2 An Algorithmic Research on Improvisation References Tuning Systems Nested Within the Arnold Tongues: Musicological and Structural Interpretations 1 Theoretical-Philosophical Framework 1.1 Why Carbon? 1.2 Cardiorespiratory Performance and Its Inheritance in Music 2 Arnold Tongues: Self-similarity in Music and Physiology 3 Self-contained Histories of Harmony and the Ear-Brain-Mind Complexity 4 Conclusions References Wooden Idiophones: Classification Through Phase Synchronization Analysis 1 Introduction 1.1 A Model of Wooden Idiophone Instrument 1.2 Cultural Implications on the Harmonic Model 1.3 Idiophone Timbral Continuum Study and Classification 1.4 Synchronous Motion of a Continuous Oscillatory Medium 2 Experimental Development 2.1 Experimental Set up 2.2 Results 2.3 Discussion 3 Conclusions References A Fuzzy Rule Model for High Level Musical Features on Automated Composition Systems 1 Exposition 2 Development 2.1 Implementation 3 Recapitulation References The Musical Experience Between Measurement and Computation: From Symbolic Description to Morphodynamical Unfolding 1 Introduction 2 Experience and Computation: Internal and External Semantics 3 Measurement and Symbolic Play 4 Music as an Algebraic Structure: The Concept of Musical Space 5 Musical Space as Topological Space 6 From Static Description to Morphodynamical Unfolding References Generic Additive Synthesis. Hints from the Early Foundational Crisis in Mathematics for Experiments in Sound Ontology 1 Spectres of Accumulation 2 The ``Birth Place of Set Theory'' and Its Potential Relevance to the Ontology of Sound 3 The Epistemic Value of Base Functions 4 Generic Additive Synthesis 5 Musique Axiomatique 6 Experiments in Partial Understanding 6.1 A Comparison of Two Examples 6.2 Comments 7 One More Step: Two Meanings of `Concatenating Combinators' 8 A Final Note on the Ontology of Sound References Dynamical Virtual Sounding Networks 1 Introduction 2 Basic Definitions 2.1 Algebraic Rhythmic Structures 3 Automataplex 3.1 Mapping Data to Sound Realm 4 Conclusions References Melodic Pattern Segmentation of Polyphonic Music as a Set Partitioning Problem 1 Motif Division 2 Transformation Group and Equivalence Classes of Motif 3 Formulation as a Set Partitioning Problem 3.1 Condition of Motif Division 3.2 Objective Function 3.3 Controlling the Number of Equivalence Classes 4 Result 5 Conclusion References Diagrams, Games and Time (Towards the Analysis of Open Form Scores) 1 On the Concept of Action GrammarŽ 2 On the Concept of Situation FieldŽ 3 Conclusions References On Minimal Change Musical Morphologies 1 Introduction and Preliminaries 1.1 Scope and Complexity of Morphological Constraints 1.2 Precedence of Musical Thinking with Respect to Morphological Constraints 2 Minimal Change Musical Morphologies: Applications and Resulting Mathematical Problems 2.1 Gray Codes 2.2 De Bruijn Sequences 2.3 Aperiodic Necklaces 2.4 Disjoint Subset Pairs 3 Conclusion References Restoring the Structural Status of Keys Through DFT Phase Space 1 Long-Range Voice-Leading Structure Without Reduction 1.1 Schenker's Implicit Premise 1.2 Triadic Orbits 2 Beethoven's Heiliger Dankgesang 2.1 Tonal Contexts and Triadic Orbits 2.2 Strength and Weakness References Mazzola, Galois, Peirce, Riemann, and Merleau-Ponty: A Triadic, Spatial Framework for Gesture Theory 1 Introduction 2 Mazzola: The Problem and the Triangular Set-Up (Sounds, Scores, Gestures) 3 Galois: How the (Bilateral) Dialectic Pairs Become Natural Adjunctions for Horotics 4 Peirce: How the (Degenerated) Triangle Becomes a True Triad 5 Riemann: How the (Triadic) Horos Becomes Ramified in a Multilayered Surface 6 Merleau-Ponty: How the (Visual) Entrelacs Becomes a Chiasmatic Musical Experience References The Rubato Composer Music Software_ Component-Based Implementation of a Functorial Concept Architecture (2009) Cover.jpg front-matter.pdf fulltext.pdf fulltext_2.pdf fulltext_3.pdf fulltext_4.pdf fulltext_5.pdf fulltext_6.pdf fulltext_7.pdf fulltext_8.pdf fulltext_9.pdf fulltext_10.pdf fulltext_11.pdf fulltext_12.pdf fulltext_13.pdf fulltext_14.pdf fulltext_15.pdf fulltext_16.pdf fulltext_17.pdf fulltext_18.pdf fulltext_19.pdf fulltext_20.pdf back-matter.pdf The Topos of Music Geometric Logic of Concepts, Theory, and Performance (2002) I Introduction and Orientation What is Music About? Fundamental Activities Fundamental Scientific Domains Topography Layers of Reality Physical Reality Mental Reality Psychological Reality Molino's Communication Stream Creator and Poietic Level Work and Neutral Level Listener and Esthesic Level Semiosis Expressions Content The Process of Signification A Short Overview of Music Semiotics The Cube of Local Topography Topographical Navigation Musical Ontology Where is Music? Depth and Complexity Models and Experiments in Musicology Interior and Exterior Nature What Is a Musicological Experiment? Questions---Experiments of the Mind New Scientific Paradigms and Collaboratories II Navigation on Concept Spaces Navigation Music in the EncycloSpace Receptive Navigation Productive Navigation Denotators Universal Concept Formats First Naive Approach To Denotators Interpretations and Comments Ordering Denotators and `Concept Leafing' Forms Variable Addresses Formal Definition Discussion of the Form Typology Denotators Formal Definition of a Denotator Anchoring Forms in Modules First Examples and Comments on Modules in Music Regular and Circular Forms Regular Denotators Circular Denotators Ordering on Forms and Denotators Concretizations and Applications Concept Surgery and Denotator Semantics III Local Theory Local Compositions The Objects of Local Theory First Local Music Objects Chords and Scales Local Meters and Local Rhythms Motives Functorial Local Compositions First Elements of Local Theory Alterations Are Tangents The Theorem of Mason--Mazzola Symmetries and Morphisms Symmetries in Music Elementary Examples Morphisms of Local Compositions Categories of Local Compositions Commenting the Concatenation Principle Embedding and Addressed Adjointness Universal Constructions on Local Compositions The Address Question Categories of Commutative Local Compositions Yoneda Perspectives Morphisms Are Points Yoneda's Fundamental Lemma The Yoneda Philosophy Understanding Fine and Other Arts Painting and Music The Art of Object-Oriented Programming Paradigmatic Classification Paradigmata in Musicology, Linguistics, and Mathematics Transformation Similarity Fuzzy Concepts in the Humanities Orbits Gestalt and Symmetry Groups The Framework for Local Classification Orbits of Elementary Structures Classification Techniques The Local Classification Theorem The Finite Case Dimension Chords Empirical Harmonic Vocabularies Self-addressed Chords Motives Enumeration Theory Pólya and de Bruijn Theory Big Science for Big Numbers Group-theoretical Methods in Composition and Theory Aspects of Serialism The American Tradition Esthetic Implications of Classification Jakobson's Poetic Function Motivic Analysis: Schubert/Stolberg ``Lied auf dem Wasser zu singen...'' Composition: Mazzola/Baudelaire ``La mort des artistes'' Mathematical Reflections on Historicity in Music Jean-Jacques Nattiez' Paradigmatic Theme Groups as a Parameter of Historicity Topological Specialization What Ehrenfels Neglected Topology Metrical Comparison Specialization Morphisms of Local Compositions The Problem of Sound Classification Topographic Determinants of Sound Descriptions Varieties of Sounds Semiotics of Sound Classification Making the Vague Precise IV Global Theory Global Compositions The Local-Global Dichotomy in Music Musical and Mathematical Manifolds What Are Global Compositions? The Nerve of an Objective Global Composition Functorial Global Compositions Interpretations and the Vocabulary of Global Concepts Iterated Interpretations The Pitch Domain: Chains of Thirds, Ecclesiastical Modes, Triadic and Quaternary Degrees Interpreting Time: Global Meters and Rhythms Motivic Interpretations: Melodies and Themes Global Perspectives Musical Motivation Global Morphisms Local Domains Nerves Simplicial Weights Categories of Commutative Global Compositions Global Classification Module Complexes Global Affine Functions Bilinear and Exterior Forms Deviation: Compositions vs. ``Molecules'' The Resolution of a Global Composition Global Standard Compositions Compositions from Module Complexes Orbits of Module Complexes Are Classifying Combinatorial Group Actions Classifying Spaces Classifying Interpretations Characterization of Interpretable Compositions Automorphism Groups of Interpretable Compositions A Cohomological Criterion Global Enumeration Theory Tesselation Mosaics Classifying Rational Rhythms and Canons Global American Set Theory Interpretable ``Molecules'' Esthetics and Classification Understanding by Resolution: An Illustrative Example Varèse's Program and Yoneda's Lemma Predicates What Is the Case: The Existence Problem Merging Systematic and Historical Musicology Textual and Paratextual Semiosis Textual and Paratextual Signification Textuality The Category of Denotators Textual Semiosis Atomic Predicates Logical and Geometric Motivation Paratextuality Topoi of Music The Grothendieck Topology Cohomology Marginalia on Presheaves The Topos of Music: An Overview Visualization Principles Problems Folding Dimensions R2 R Rn R An Explicit Construction of with Special Values. Folding Denotators Folding Limits Folding Colimits Folding Powersets Folding Circular Denotators Compound Parametrized Objects Examples V Topologies for Rhythm and Motives Metrics and Rhythmics Review of Riemann and Jackendoff--Lerdahl Theories Riemann's Weights Jackendoff--Lerdahl: Intrinsic Versus Extrinsic Time Structures Topologies of Global Meters and Associated Weights Macro-Events in the Time Domain Motif Gestalts Motivic Interpretation Shape Types Examples of Shape Types Metrical Similarity Examples of Distance Functions Paradigmatic Groups Examples of Paradigmatic Groups Pseudo-metrics on Orbits Topologies on Gestalts The Inheritance Property Cognitive Aspects of Inheritance Epsilon Topologies First Properties of the Epsilon Topologies Toroidal Topologies Rudolph Reti's Motivic Analysis Revisited Review of Concepts Reconstruction Motivic Weights VI Harmony Critical Preliminaries Hugo Riemann Paul Hindemith Heinrich Schenker and Friedrich Salzer Harmonic Topology Chord Perspectives Euler Perspectives 12-tempered Perspectives Enharmonic Projection Chord Topologies Extension and Intension Extension and Intension Topologies Faithful Addresses The Saturation Sheaf Harmonic Semantics Harmonic Signs---Overview Degree Theory Chains of Thirds American Jazz Theory Hans Straub: General Degrees in General Scales Function Theory Canonical Morphemes for European Harmony Riemann Matrices Chains of Thirds Tonal Functions from Absorbing Addresses Cadence Making the Concept Precise Classical Cadences Relating to 12-tempered Intonation Cadences in Triadic Interpretations of Diatonic Scales Cadences in More General Interpretations Cadences in Self-addressed Tonalities of Morphology Self-addressed Cadences by Symmetries and Morphisms Cadences for Just Intonation Tonalities in Third-Fifth Intonation Tonalities in Pythagorean Intonation Modulation Modeling Modulation by Particle Interaction Models and the Anthropic Principle Classical Motivation and Heuristics The General Background The Well-Tempered Case Reconstructing the Diatonic Scale from Modulation The Case of Just Tuning Quantized Modulations and Modulation Domains for Selected Scales Harmonic Tension The Riemann Algebra Weights on the Riemann Algebra Harmonic Tensions from Classical Harmony? Optimizing Harmonic Paths Applications First Examples Johann Sebastian Bach: Choral from ``Himmelfahrtsoratorium'' Wolfgang Amadeus Mozart: ``Zauberflöte'', Choir of Priests Claude Debussy: ``Préludes'', Livre 1, No.4 Modulation in Beethoven's Sonata op.106, 1st Movement Introduction The Fundamental Theses of Erwin Ratz and J Overview of the Modulation Structure Modulation B G via e-3 in W Modulation G E via Ug in W Modulation E D/b from W to W* Modulation D/b B via Ud/d=Ug/a within W* Modulation B B from W* to W Modulation B G via Ub within W Modulation G G via Ua/a within W Modulation G B via e3 within W Rhythmical Modulation in ``Synthesis'' Rhythmic Modes Composition for Percussion Ensemble VII Counterpoint Melodic Variation by Arrows Arrows and Alterations The Contrapuntal Interval Concept The Algebra of Intervals The Third Torus Musical Interpretation of the Interval Ring Self-addressed Arrows Change of Orientation Interval Dichotomies as a Contrast Dichotomies and Polarity The Consonance and Dissonance Dichotomy Fux and Riemann Consonances Are Isomorphic Induced Polarities Empirical Evidence for the Polarity Function Music and the Hippocampal Gate Function Modeling Counterpoint by Local Symmetries Deformations of the Strong Dichotomies Contrapuntal Symmetries Are Local The Counterpoint Theorem Some Preliminary Calculations Two Lemmata on Cardinalities of Intersections An Algorithm for Exhibiting the Contrapuntal Symmetries Transfer of the Counterpoint Rules to General Representatives of Strong Dichotomies The Classical Case: Consonances and Dissonances Discussion of the Counterpoint Theorem in the Light of Reduced Strict Style The Major Dichotomy---A Cultural Antipode? VIII Structure Theory of Performance Local and Global Performance Transformations Performance as a Reality Switch Why Do We Need Infinite Performance of the Same Piece? Local Structure The Coherence of Local Performance Transformations Differential Morphisms of Local Compositions Global Structure Modeling Performance Syntax The Formal Setup Performance qua Interpretation of Interpretation Performance Fields Classics: Tempo, Intonation, and Dynamics Tempo Intonation Dynamics Genesis of the General Formalism The Question of Articulation The Formalism of Performance Fields What Performance Fields Signify Th.W. Adorno, W. Benjamin, and D. Raffman Towards Composition of Performance Initial Sets and Initial Performances Taking off with a Shifter Anchoring Onset The Concert Pitch Dynamical Anchors Initializing Articulation Hit Point Theory Distances Flow Interpolation Hierarchies and Performance Scores Performance Cells The Category of Performance Cells Hierarchies Operations on Hierarchies Classification Issues Example: The Piano and Violin Hierarchies Local Performance Scores Global Performance Scores Instrumental Fibers IX Expressive Semantics Taxonomy of Expressive Performance Feelings: Emotional Semantics Motion: Gestural Semantics Understanding: Rational Semantics Cross-semantical Relations Performance Grammars Rule-based Grammars The KTH School Neil P. McAgnus Todd The Zurich School Remarks on Learning Grammars Stemma Theory Motivation from Practising and Rehearsing Does Reproducibility of Performances Help Understanding? Tempo Curves Are Inadequate The Stemma Concept The General Setup of Matrilineal Sexual Propagation The Primary Mother---Taking Off Mono- and Polygamy---Local and Global Actions Family Life---Cross-Correlations Operator Theory Why Weights? Discrete and Continuous Weights Weight Recombination Primavista Weights Dynamics Agogics Tuning and Intonation Articulation Ornaments Analytical Weights Taxonomy of Operators Splitting Operators Symbolic Operators Physical Operators Field Operators Tempo Operator Scalar Operator The Theory of Basis-Pianola Operators Basis Specialization Pianola Specialization Locally Linear Grammars X RUBATO"472 Architecture The Overall Modularity Frame and Modules The RUBETTE"472 Family MetroRUBETTE"472 MeloRUBETTE"472 HarmoRUBETTE"472 PerformanceRUBETTE"472 PrimavistaRUBETTE"472 Performance Experiments A Preliminary Experiment: Robert Schumann's ``Kuriose Geschichte'' Full Experiment: J.S. Bach's ``Kunst der Fuge'' Analysis Metric Analysis Motif Analysis Omission of Harmonic Analysis Stemma Constructions Performance Setup Instrumental Setup Global Discussion XI Statistics of Analysis and Performance Analysis of Analysis Hierarchical Decomposition General Motivation Hierarchical Smoothing Hierarchical Decomposition Comparing Analyses of Bach, Schumann, and Webern Differential Operators and Regression Analytical Data The Beran Operator The Concept The Formalism The Method of Regression Analysis The Full Model Step Forward Selection The Results of Regression Analysis Relations between Tempo and Analysis Complex Relationships Commonalities and Diversities Overview of Statistical Results XII Inverse Performance Theory Principles of Music Critique Boiling down Infinity---Is Feuilletonism Inevitable? ``Political Correctness'' in Performance---Reviewing Gould Transversal Ethnomusicology Critical Fibers The Stemma Model of Critique Fibers for Locally Linear Grammars Algorithmic Extraction of Performance Fields The Infinitesimal View on Expression Real-time Processing of Expressive Performance Score--Performance Matching Performance Field Calculation Visualization The EspressoRUBETTE"472: An Interactive Tool for Expression Extraction Local Sections Comparing Argerich and Horowitz XIII Operationalization of Poiesis Unfolding Geometry and Logic in Time Performance of Logic and Geometry Constructing Time from Geometry Discourse and Insight Local and Global Strategies in Composition Local Paradigmatic Instances Transformations Variations Global Poetical Syntax Roman Jakobson's Horizontal Function Roland Posner's Vertical Function Structure and Process The Paradigmatic Discourse on presto"472 The presto"472 Functional Scheme Modular Affine Transformations Ornaments and Variations Problems of Abstraction Case Study I:``Synthesis'' by Guerino Mazzola The Overall Organization The Material: 26 Classes of Three-Element Motives Principles of the Four Movements and Instrumentation 1st Movement: Sonata Form 2nd Movement: Variations 3rd Movement: Scherzo 4th Movement: Fractal Syntax Object-Oriented Programming in OpenMusic Object-Oriented Language Patches Objects Classes Methods Generic Functions Message Passing Inheritance Boxes and Evaluation Instantiation Musical Object Framework Internal Representation Interface Maquettes: Objects in Time Meta-object Protocol Reification of Temporal Boxes A Musical Example XIV String Quartet Theory Historical and Theoretical Prerequisites History Theory of the String Quartet Following Ludwig Finscher Four Part Texture The Topos of Conversation Among Four Humanists The Family of Violins Estimation of Resolution Parameters Parameter Spaces for Violins Estimation The Case of Counterpoint and Harmony Counterpoint Harmony Effective Selection XV Appendix: Sound Common Parameter Spaces Physical Spaces Neutral Data Sound Analysis and Synthesis Mathematical and Symbolic Spaces Onset and Duration Amplitude and Crescendo Frequency and Glissando Auditory Physiology and Psychology Physiology: From the Auricle to Heschl's Gyri Outer Ear Middle Ear Inner Ear (Cochlea) Cochlear Hydrodynamics: The Travelling Wave Active Amplification of the Traveling Wave Motion Neural Processing Discriminating Tones: Werner Meyer-Eppler's Valence Theory Aspects of Consonance and Dissonance Euler's Gradus Function von Helmholtz' Beat Model Psychometric Investigations by Plomp and Levelt Counterpoint Consonance and Dissonance: A Conceptual Field XVI Appendix: Mathematical Basics Sets, Relations, Monoids, Groups Sets Examples of Sets Relations Universal Constructions Graphs and Quivers Monoids Groups Homomorphisms of Groups Direct, Semi-direct, and Wreath Products Sylow Theorems on p-groups Classification of Groups General Affine Groups Permutation Groups Rings and Algebras Basic Definitions and Constructions Universal Constructions Prime Factorization Euclidean Algorithm Approximation of Real Numbers by Fractions Some Special Issues Integers, Rationals, and Real Numbers Modules, Linear, and Affine Transformations Modules and Linear Transformations Examples Module Classification Dimension Endomorphisms on Dual Numbers Semi-Simple Modules Jacobson Radical and Socle Theorem of Krull--Remak--Schmidt Categories of Modules and Affine Transformations Direct Sums Affine Forms and Tensors Biaffine Maps Symmetries of the Affine Plane Symmetries on Z2 Symmetries on Zn Complements on the Module of a Local Composition Fiber Products and Fiber Sums in Mod Complements of Commutative Algebra Localization Projective Modules Injective Modules Lie Algebras Algebraic Geometry Locally Ringed Spaces Spectra of Commutative Rings Sober Spaces Schemes and Functors Algebraic and Geometric Structures on Schemes The Zariski Tangent Space Grassmannians Quotients Categories, Topoi, and Logic Categories Instead of Sets Examples Functors Natural Transformations The Yoneda Lemma Universal Constructions: Adjoints, Limits, and Colimits Limit and Colimit Characterizations Topoi Subobject Classifiers Exponentiation Definition of Topoi Grothendieck Topologies Sheaves Formal Logic Propositional Calculus Predicate Logic A Formal Setup for Consistent Domains of Forms Complements on General and Algebraic Topology Topology General The Category of Topological Spaces Uniform Spaces Special Issues Algebraic Topology Simplicial Complexes Geometric Realization of a Simplicial Complex Contiguity Simplicial Coefficient Systems Cohomology Complements on Calculus Abstract on Calculus Norms and Metrics Completeness Differentiation Ordinary Differential Equations (ODEs) The Fundamental Theorem: Local Case The Fundamental Theorem: Global Case Flows and Differential Equations Vector Fields and Derivations Partial Differential Equations XVII Appendix: Tables Euler's Gradus Function Just and Well-Tempered Tuning Chord and Third Chain Classes Chord Classes Third Chain Classes Two, Three, and Four Tone Motif Classes Two Tone Motifs in OnPiMod12,12 Two Tone Motifs in OnPiMod5,12 Three Tone Motifs in OnPiMod12,12 Four Tone Motifs in OnPiMod12,12 Three Tone Motifs in OnPiMod5,12 Well-Tempered and Just Modulation Steps 12-Tempered Modulation Steps Scale Orbits and Number of Quantized Modulations Quanta and Pivots for the Modulations Between Diatonic Major Scales (No.38.1) Quanta and Pivots for the Modulations Between Melodic Minor Scales (No.47.1) Quanta and Pivots for the Modulations Between Harmonic Minor Scales (No.54.1) Examples of 12-Tempered Modulations for all Fourth Relations 2-3-5-Just Modulation Steps Modulation Steps between Just Major Scales Modulation Steps between Natural Minor Scales Modulation Steps From Natural Minor to Major Scales Modulation Steps From Major to Natural Minor Scales Modulation Steps Between Harmonic Minor Scales Modulation Steps Between Melodic Minor Scales General Modulation Behaviour for 32 Alterated Scales Counterpoint Steps Contrapuntal Symmetries Class Nr. 64 Class Nr. 68 Class Nr. 71 Class Nr. 75 Class Nr. 78 Class Nr. 82 Permitted Successors for the Major Scale XVIII References Bibliography Index The Topos of Music I Theory (2002, 2017) The Topos of Music I Theory (Guerino Mazzola) Preface to the Second Edition Preface Volume I Contents Book Set Contents Leitfaden Leitfaden I & II Leitfaden III Tom_CD Part I Introduction and Orientation 1 What Is Music About? 1.1 Fundamental Activities 1.2 Fundamental Scientific Domains 2 Topography 2.1 Layers of Reality 2.1.1 Physical Reality 2.1.2 Mental Reality 2.1.3 Psychological Reality 2.2 Molino's Communication Stream 2.2.1 Creator and Poietic Level 2.2.2 Work and Neutral Level 2.2.3 Listener and Esthesic Level 2.3 Semiosis 2.3.1 Expressions 2.2.3.1 The Problem of Identity 2.3.2 Content 2.3.3 The Process of Signification 2.3.4 A Short Overview of Music Semiotics 2.4 The Cube of Local Topography 2.5 Topographical Navigation 3 Musical Ontology 3.1 Where Is Music? 3.2 Depth and Complexity 4 Models and Experiments in Musicology 4.1 Interior and Exterior Nature 4.2 What Is a Musicological Experiment? 4.3 Questions-Experiments of the Mind 4.4 New Scientific Paradigms and Collaboratories Part II Navigation on Concept Spaces 5 Navigation 5.1 Music in the EncycloSpace 5.2 Receptive Navigation 5.3 Productive Navigation 6 Denotators 6.1 Universal Concept Formats 6.1.1 First Naive Approach to Denotators 6.1.2 Interpretations and Comments 6.1.3 Ordering Denotators and `Concept Leang' 6.2 Forms 6.2.1 Variable Addresses 6.2.2 Formal Definition 6.2.3 Discussion of the Form Typology 6.3 Denotators 6.3.1 Formal Definition of a Denotator 6.4 Anchoring Forms in Modules 6.4.1 First Examples and Comments on Modules in Music 6.5 Regular and Circular Forms 6.6 Regular Denotators 6.7 Circular Denotators 6.8 Ordering on Forms and Denotators 6.8.1 Concretizations and Applications 6.9 Concept Surgery and Denotator Semantics Part III Local Theory 7 Local Compositions 7.1 The Objects of Local Theory 7.2 First Local Music Objects 7.2.1 Chords and Scales 7.2.1.1 Chords 7.2.1.2 Scales 7.2.1.3 w-Tempered Scales 7.2.1.4 Just Scales 7.2.2 Local Meters and Local Rhythms 7.2.3 Motives 7.3 Functorial Local Compositions 7.4 First Elements of Local Theory 7.5 Alterations Are Tangents 7.5.1 The Theorem of Mason-Mazzola 8 Symmetries and Morphisms 8.1 Symmetries in Music 8.1.1 Elementary Examples 8.2 Morphisms of Local Compositions 8.3 Categories of Local Compositions 8.3.1 Commenting on the Concatenation Principle 8.3.2 Embedding and Addressed Adjointness 8.3.3 Universal Constructions on Local Compositions 8.3.4 The Address Question 8.3.5 Categories of Commutative Local Compositions 9 Yoneda Perspectives 9.1 Morphisms Are Points 9.2 Yoneda's Fundamental Lemma 9.3 The Yoneda Philosophy 9.4 Understanding Fine and Other Arts 9.4.1 Painting and Music 9.4.2 The Art of Object-Oriented Programming 10 Paradigmatic Classification 10.1 Paradigmata in Musicology, Linguistics, and Mathematics 10.2 Transformation 10.3 Similarity 10.4 Fuzzy Concepts in the Humanities 11 Orbits 11.1 Gestalt and Symmetry Groups 11.2 The Framework for Local Classication 11.3 Orbits of Elementary Structures 11.3.1 Classification Techniques 11.3.2 The Local Classification Theorem 11.3.3 The Finite Case 11.3.4 Dimension 11.3.5 Chords 11.3.6 Empirical Harmonic Vocabularies 11.3.7 Self-addressed Chords 11.3.8 Motives 11.4 Enumeration Theory 11.4.1 Polya and de Bruijn Theory 11.4.1.2 Enumeration of Series 11.4.1.3 Enumeration of Motives 11.4.2 Big Science for Big Numbers 11.5 Group-Theoretical Methods in Composition and Theory 11.5.1 Aspects of Serialism 11.5.2 The American Tradition 11.5.2.1 Genealogy 11.5.2.2 Concepts and Theory—A Vocabulary Switch 11.5.2.3 Software for Musical Set Theory 11.5.2.4 Comments 11.6 Esthetic Implications of Classification 11.6.1 Jakobson's Poetic Function 11.6.2 Motivic Analysis: Schubert/Stolberg "Lied auf dem Wasser zu singen..." 11.7 Mathematical Reections on Historicity in Music 11.7.1 Jean-Jacques Nattiez' Paradigmatic Theme 11.7.2 Groups as a Parameter of Historicity 12 Topological Specialization 12.1 What Ehrenfels Neglected 12.2 Topology 12.2.1 Metrical Comparison 12.2.2 Specialization Morphisms of Local Compositions 12.3 The Problem of Sound Classification 12.3.1 Topographic Determinants of Sound Descriptions 12.3.1.1 Communication 12.3.1.2 Reality 12.3.2 Varieties of Sounds 12.3.3 Semiotics of Sound Classification 12.4 Making the Vague Precise Part IV Global Theory 13 Global Compositions 13.1 The Local-Global Dichotomy in Music 13.1.1 Musical and Mathematical Manifolds 13.2 What Are Global Compositions? 13.2.1 The Nerve of an Objective Global Composition 13.3 Functorial Global Compositions 13.4 Interpretations and the Vocabulary of Global Concepts 13.4.1 Iterated Interpretations 13.4.2 The Pitch Domain: Chains of Thirds, Ecclesiastical Modes, Triadic and QuaternaryDegrees 13.4.2.1 Orientation in Riemann Function Theory 13.4.2.2 Just Triadic Degree Interpretations 13.4.3 Interpreting Time: Global Meters and Rhythms 13.4.4 Motivic Interpretations: Melodies and Themes 14 Global Perspectives 14.1 Musical Motivation 14.2 Global Morphisms 14.3 Local Domains 14.4 Nerves 14.5 Simplicial Weights 14.6 Categories of Commutative Global Compositions 15 Global Classification 15.1 Module Complexes 15.1.1 Global Affine Functions 15.1.2 Bilinear and Exterior Forms 15.1.3 Deviation: Compositions vs. \Molecules" 15.2 The Resolution of a Global Composition 15.2.1 Global Standard Compositions 15.2.2 Compositions from Module Complexes 15.3 Orbits of Module Complexes Are Classifying 15.3.1 Combinatorial Group Actions 15.3.2 Classifying Spaces 16 Classifying Interpretations 16.1 Characterization of Interpretable Compositions 16.1.2 A Cohomological Criterion 16.2 Global Enumeration Theory 16.2.1 Tesselation 16.2.2 Mosaics 16.2.3 Classifying Rational Rhythms and Canons 16.3 Global American Set Theory 16.4 Interpretable "Molecules" 17 Esthetics and Classification 17.1 Understanding by Resolution: An Illustrative Example 17.2 Varese's Program and Yoneda's Lemma 18 Predicates 18.1 What Is the Case: The Existence Problem 18.1.1 Merging Systematic and Historical Musicology 18.2 Textual and Paratextual Semiosis 18.2.1 Textual and Paratextual Signification 18.3 Textuality 18.3.1 The Category of Denotators 18.3.1.1 Morphisms as Denotators 18.3.2 Textual Semiosis 18.3.2.1 Predicates as Denotators 18.3.3 Atomic Predicates 18.3.3.1 Mathematical Predicates 18.3.3.2 Primavista Predicates 18.3.3.3 Shifter Predicates 18.3.4 Logical and Geometric Motivation 18.4 Paratextuality 19 Topoi of Music 19.1 The Grothendieck Topology 19.1.1 Cohomology 19.1.2 Marginalia on Presheaves 19.1.2.1 Function Presheaves 19.1.2.2 The Subobject Classifier 19.2 The Topos of Music: An Overview 20 Visualization Principles 20.1 Problems 20.2 Folding Dimensions 20.2.1 R2 Ñ R 20.2.2 Rn Ñ R 20.2.3 An Explicit Construction of u with Special Values. 20.3 Folding Denotators 20.3.1 Folding Limits 20.3.2 Folding Colimits 20.3.3 Folding Powersets 20.3.4 Folding Circular Denotators 20.4 Compound Parametrized Objects 20.5 Examples Part V Topologies for Rhythm and Motives 21 Metrics and Rhythmics 21.1 Review of Riemann and Jackendo-Lerdahl Theories 21.1.1 Riemann's Weights 21.1.2 Jackendo-Lerdahl: Intrinsic Versus Extrinsic Time Structures 21.2 Topologies of Global Meters and Associated Weights 21.3 Macro-events in the Time Domain 22 Motif Gestalts 22.1 Motivic Interpretation 22.2 Shape Types 22.2.1 Examples of Shape Types 22.2.1.1 Rigid Types 22.2.1.2 Diastematic Types 22.2.1.3 Elastic Type 22.2.1.4 Toroidal Type 22.3 Metrical Similarity 22.3.1 Examples of Distance Functions 22.3.1.1 Distances for Rigid Types 22.3.1.2 Distances for Diastematic Types 22.3.1.3 Distances for Elastic Type 22.3.1.4 Distances for Toroidal Types 22.4 Paradigmatic Groups 22.4.1 Examples of Paradigmatic Groups 22.4.1.1 Paradigmatic Groups for Rigid Types 22.4.1.2 Paradigmatic Groups for Diastematic Types 22.4.1.3 Paradigmatic Groups for Elastic Type 22.4.1.4 Paradigmatic Groups for Toroidal Types 22.5 Pseudo-metrics on Orbits 22.6 Topologies on Gestalts 22.6.1 The Inheritance Property 22.6.2 Cognitive Aspects of Inheritance 22.6.3 Epsilon Topologies 22.7 First Properties of the Epsilon Topologies 22.7.1 Toroidal Topologies 22.7.0.1 Relative Topologies 22.7.1.1 Dominance Topology 22.7.1.2 Specialization Inheritance and Specialization Topology 22.8 Rudolph Reti's Motivic Analysis Revisited 22.8.1 Review of Concepts 22.8.2 Reconstruction 22.8.2.1 Choice of Parameters 22.8.2.2 Shapes, Imitations and Transformations 22.8.2.3 Reti's Identity Relation Revisited 22.9 Motivic Weights Part VI Harmony 23 Critical Preliminaries 23.1 Hugo Riemann 23.2 Paul Hindemith 23.3 Heinrich Schenker and Friedrich Salzer 24 Harmonic Topology 24.1 Chord Perspectives 24.1.1 Euler Perspectives 24.1.1.1 Just Mutation 24.1.2 12-Tempered Perspectives 24.1.3 Enharmonic Projection 24.2 Chord Topologies 24.2.1 Extension and Intension 24.2.2 Extension and Intension Topologies 24.2.3 Faithful Addresses 25 Harmonic Semantics 25.1 Harmonic Signs|Overview 25.2 Degree Theory 25.2.1 Chains of Thirds 25.2.2 American Jazz Theory 25.2.3 Hans Straub: General Degrees in General Scales 25.3 Function Theory 25.3.3 Chains of Thirds 25.3.4 Tonal Functions from Absorbing Addresses 26 Cadence 26.1 Making the Concept Precise 26.2 Classical Cadences Relating to 12-Tempered Intonation 26.2.1 Cadences in Triadic Interpretations of Diatonic Scales 26.2.2 Cadences in More General Interpretations 26.3 Cadences in Self-addressed Tonalities of Morphology 26.4 Self-addressed Cadences by Symmetries and Morphisms 26.5 Cadences for Just Intonation 26.5.1 Tonalities in Third-Fifth Intonation 26.5.2 Tonalities in Pythagorean Intonation 27 Modulation 27.1 Modeling Modulation by Particle Interaction 27.1.1 Models and the Anthropic Principle 27.1.2 Classical Motivation and Heuristics 27.1.3 The General Background 27.1.4 The Well-Tempered Case 27.1.5 Reconstructing the Diatonic Scale from Modulation 27.1.6 The Case of Just Tuning 27.1.6.1 Just Scales and their Triadic Interpretations 27.1.6.2 Modulations and Quanta 27.1.6.3 Automorphisms of Triadic Interpretations of Seven-Element Scales 27.1.6.4 Finiteness of Modulation Domains 27.1.7 Quantized Modulations and Modulation Domains for Selected Scales 27.1.7.1 Modulation Between Major Tonalities 27.1.7.2 Modulation Between Natural Minor Tonalities 27.1.7.3 Modulation From Natural Minor to Major Tonalities 27.1.7.4 Modulation Steps From Major to Natural Minor Scales 27.1.7.5 Modulation Steps Between Harmonic Minor Scales 27.1.7.6 Modulation Steps Between Melodic Minor Scales 27.1.7.7 General Modulation Behavior for 32 Altered Scales 27.2 Harmonic Tension 27.2.1 The Riemann Algebra 27.2.2 Weights on the Riemann Algebra 27.2.3 Harmonic Tensions from Classical Harmony? 27.2.4 Optimizing Harmonic Paths 28 Applications 28.1 First Examples 28.1.1 Johann Sebastian Bach: Choral from "Himmelfahrtsoratorium" 28.1.2 Wolfgang Amadeus Mozart: "Zauberflote", Choir of Priests 28.1.3 Claude Debussy: "Preludes", Livre 1, No.4 28.2 Modulation in Beethoven's Sonata op.106, 1st Movement 28.2.1 Introduction 28.2.2 The Fundamental Theses of Erwin Ratz and Jurgen Uhde 28.2.3 Overview of the Modulation Structure 28.2.4 Modulation B5ùG via e-3 in W 28.2.5 Modulation GùE5 via Ug in W 28.2.6 Modulation E5ùD{b from W to W* 28.2.7 Modulation D{bùB via Ud{d7 Ug7{a within W* 28.2.8 Modulation BùB5 from W* to W 28.2.9 Modulation B5ùG5 via Ub5within W 28.2.10 Modulation G5ùG via Ua5{a within W 28.2.10.1 Modulation GùB5 via e3 within W 28.3 Rhythmical Modulation in "Synthesis" 28.3.1 Rhythmic Modes 28.3.2 Composition for Percussion Ensemble Part VII Counterpoint 29 Melodic Variation by Arrows 29.1 Arrows and Alterations 29.2 The Contrapuntal Interval Concept 29.3 The Algebra of Intervals 29.3.1 The Third Torus 29.4 Musical Interpretation of the Interval Ring 29.5 Self-addressed Arrows 29.6 Change of Orientation 30 Interval Dichotomies as an Expression of Contrast 30.1 Dichotomies and Polarity 30.2 The Consonance and Dissonance Dichotomy 30.2.1 Fux and Riemann Consonances Are Isomorphic 30.2.2 Induced Polarities 30.2.3 Empirical Evidence for the Polarity Function 30.2.3.1 The EEG Test 30.2.3.2 Analysis by Spectral Participation Vectors 30.2.3.3 Isolated Successive Intervals 30.2.3.4 Polarity 30.2.4 Music and the Hippocampal Gate Function 31 Modeling Counterpoint by Local Symmetries 31.1 Deformations of the Strong Dichotomies by Contrapuntal Symmetries on IntMod12;qr"s 31.2 Contrapuntal Symmetries Are Local 31.3 The Counterpoint Theorem 31.3.1 Some Preliminary Calculations 31.3.2 Two Lemmata on Cardinalities of Intersections 31.3.3 An Algorithm for Exhibiting the Contrapuntal Symmetries 31.3.4 Transfer of the Counterpoint Rules to General Representatives of Strong Dichotomies 31.4 The Classical Case: Consonances and Dissonances 31.4.1 Discussion of the Counterpoint Theorem in the Light of Reduced Strict Style 31.4.2 The Major Dichotomy-A Cultural Antipode? 31.4.3 Software for Counterpoint and Theoretical Extentions Part XXIV References and Index References Index The Topos of Music II Performance (Guerino Mazzola) Preface to the Second Edition Preface Volume II Contents Book Set Contents Leitfaden Leitfaden I & II Leitfaden III Tom_CD Part VIII Structure Theory of Performance Chapter 32 Local and Global Performance Transformations 32.1 Performance as a Reality Switch 32.2 Why Do We Need Infinite Performance of the Same Piece? 32.3 Local Structure 32.3.1 The Coherence of Local Performance Transformations 32.3.2 Differential Morphisms of Local Compositions 32.3.2.1 A Recursive Interpolation Algorithm 32.4 Global Structure 32.4.1 Modeling Performance Syntax 32.4.2 The Formal Setup 32.4.3 Performance qua Interpretation of Interpretation Chapter 33 Performance Fields 33.1 Classics: Tempo, Intonation, and Dynamics 33.1.1 Tempo 33.1.2 Intonation 33.1.3 Dynamics 33.2 Genesis of the General Formalism 33.2.1 The Question of Articulation 33.2.2 The Formalism of Performance Fields 33.3 What Performance Fields Signify 33.3.1 Th.W. Adorno, W. Benjamin, and D. Raffman 33.3.2 Towards Composition of Performance Chapter 34 Initial Sets and Initial Performances 34.1 Taking Off with a Shifter 34.2 Anchoring Onset 34.3 The Concert Pitch 34.4 Dynamical Anchors 34.5 Initializing Articulation 34.6 Hit Point Theory 34.6.1 Distances 34.6.2 Flow Interpolation Chapter 35 Hierarchies and Performance Scores 35.1 Performance Cells 35.2 The Category of Performance Cells 35.3 Hierarchies 35.3.1 Operations on Hierarchies 35.3.2 Classification Issues 35.3.3 Example: The Piano and Violin Hierarchies 35.4 Local Performance Scores 35.5 Global Performance Scores 35.5.1 Instrumental Fibers Part IX Expressive Semantics Chapter 36 Taxonomy of Expressive PerformanceThis 36.1 Feelings: Emotional Semantics 36.2 Motion: Gestural Semantics 36.3 Understanding: Rational Semantics 36.4 Cross-semantical Relations Chapter 37 Performance Grammars 37.1 Rule-Based Grammars 37.1.1 The KTH School 37.1.2 Neil P. McAngus Todd 37.1.3 The Zurich School 37.2 Remarks on Learning Grammars Chapter 38 Stemma Theory 38.1 Motivation from Practising and Rehearsing 38.1.1 Does Reproducibility of Performances Help Understanding? 38.2 Tempo Curves Are Inadequate 38.3 The Stemma Concept 38.3.1 The General Setup of Matrilineal Sexual Propagation 38.3.2 The Primary Mother—Taking Off 38.3.3 Mono- and Polygamy—Local and Global Actions 38.3.4 Family Life—Cross-correlations Chapter 39 Operator Theory 39.1 Why Weights? 39.1.1 Discrete and Continuous Weights 39.1.2 Weight Recombination 39.2 Primavista Weights 39.2.1 Dynamics 39.2.2 Agogics 39.2.3 Tuning and Intonation 39.2.4 Articulation 39.2.5 Ornaments 39.3 Analytical Weights 39.4 Taxonomy of Operators 39.4.1 Splitting Operators 39.4.2 Symbolic Operators 39.4.3 Physical Operators 39.4.4 Field Operators 39.5 Tempo Operator 39.6 Scalar Operator 39.7 The Theory of Basis—Pianola Operators 39.7.1 Basis Specialization 39.7.1.1 Deforming Hierarchies 39.7.1.2 Lie Derivatives 39.7.2 Pianola Specialization 39.8 Locally Linear Grammars Part X RUBATO Chapter 40 Architecture 40.1 The Overall Modularity 40.2 Frame and Modules 40.3 Postscriptum: The Rubato Composer Environment Chapter 41 The RUBETTE Family 41.1 MetroRUBETTE 41.2 MeloRUBETTE 41.3 HarmoRUBETTE 41.3.1 A Set of New Harmonic Analysis Rubettes on RUBATO Composer 41.4 PerformanceRUBETTE 41.5 PrimavistaRUBETTE Chapter 42 Performance Experiments 42.1 A Preliminary Experiment: Robert Schumann's "Kuriose Geschichte" 42.2 Full Experiment: J.S. Bach's "Kunst der Fuge" 42.3 Analysis 42.3.1 Metric Analysis 42.3.1.1 Single Voices 42.3.1.2 Weight Sums of All Voices 42.3.1.3 Union of All Voices 42.3.2 Motif Analysis 42.3.3 Omission of Harmonic Analysis 42.4 Stemma Constructions 42.4.1 Performance Setup 42.4.1.1 Results From First Performance Parcours 42.4.1.2 Construction of Second Performance Parcours 42.4.1.3 Construction of Third Performance Parcours 42.4.1.4 Local Discussion 42.4.2 Instrumental Setup 42.4.3 Global Discussion Part XI Statistics of Analysis and Performance Chapter 43 Analysis of Analysis 43.1 Hierarchical Decomposition 43.1.1 General Motivation 43.1.2 Hierarchical Smoothing 43.1.3 Hierarchical Decomposition 43.2 Comparing Analyses of Bach, Schumann, and Webern Chapter 44 Differential Operators and Regression 44.0.1 Analytical Data 44.1 The Beran Operator 44.1.1 The Concept 44.1.2 The Formalism 44.1.2.1 Tempo Information 44.1.2.2 The Explanatory Variables 44.2 The Method of Regression Analysis 44.2.1 The Full Model 44.2.2 Step Forward Selection 44.3 The Results of Regression Analysis 44.3.1 Relations Between Tempo and Analysis 44.3.2 Complex Relationships 44.3.3 Commonalities and Diversities 44.3.3.1 Signs of Coefficients 44.3.3.2 Frequency of Variable Inclusion 44.3.3.3 Largest Coefficients 44.3.3.4 Argerich "Versus" Horowitz 44.3.4 Overview of Statistical Results 45 Relating Tempo to Metric, Melodic and Harmonic Analyses in Chopin's Prélude op. 28, No. 4 45.1 Introduction 45.2 Data 45.2.1 Analytical Data 45.2.2 Tempo Data 45.3 Short Summary of the Results 45.4 Some Philosophical Comments Part XII Inverse Performance Theory Chapter 46 Principles of Music Critique 46.1 Boiling Down Infinity—Is Feuilletonism Inevitable? 46.2 "Political Correctness" in Performance—Reviewing Gould 46.3 Transversal Ethnomusicology Chapter 47 Critical Fibers 47.1 The Stemma Model of Critique 47.2 Fibers for Locally Linear Grammars 47.3 Algorithmic Extraction of Performance Fields 47.3.1 The Infinitesimal View on Expression 47.3.2 Real-Time Processing of Expressive Performance 47.3.3 Score-Performance Matching 47.3.4 Performance Field Calculation 47.3.4.1 Obtaining the Bases 47.3.5 Visualization 47.3.5.1 Field Interpolation 47.3.6 The EspressoRUBETTE: An Interactive Tool for Expression Extraction 47.3.6.1 Example 1: Tempo Field of a Chromatic Scale 47.3.6.2 Example 2: Excerpt from Czerny's Piano School 47.4 Local Sections 47.4.1 Comparing Argerich and Horowitz Part XIII Operationalization of Poiesis Chapter 48 Unfolding Geometry and Logic in Time 48.1 Performance of Logic and Geometry 48.2 Constructing Time from Geometry 48.3 Discourse and Insight Chapter 49 Local and Global Strategies in Composition 49.1 Local Paradigmatic Instances 49.1.1 Transformations 49.1.2 Variations 49.2 Global Poetical Syntax 49.2.1 Roman Jakobson's Horizontal Function 49.2.2 Roland Posner's Vertical Function 49.3 Structure and Process Chapter 50 The Paradigmatic Discourse on presto 50.1 The prestor Functional Scheme 50.2 Modular Ane Transformations 50.3 Ornaments and Variations 50.4 Problems of Abstraction Chapter 51 Case Study I: "Synthesis" by Guerino Mazzola 51.1 The Overall Organization 51.1.1 The Material: 26 Classes of Three-Element Motives 51.1.2 Principles of the Four Movements and Instrumentation 51.2 1st Movement: Sonata Form 51.3 2nd Movement: Variations 51.4 3rd Movement: Scherzo 51.5 4th Movement: Fractal Syntax Chapter 52 Object-Oriented Programming in OpenMusic 52.1 Object-Oriented Language 52.1.1 Patches 52.1.2 Objects 52.1.3 Classes 52.1.4 Methods 52.1.5 Generic Functions 52.1.6 Message Passing 52.1.7 Inheritance 52.1.8 Boxes and Evaluation 52.1.9 Instantiation 52.2 Musical Object Framework 52.2.1 Internal Representation 52.2.2 Interface 52.2.2.1 Rhythmic Trees 52.3 Maquettes: Objects in Time 52.4 Meta-object Protocol 52.4.1 Reification of Temporal Boxes 52.5 A Musical Example Part XIV String Quartet Theory Chapter 53 Historical and Theoretical Prerequisites 53.1 History 53.2 Theory of the String Quartet Following Ludwig Finscher 53.2.1 Four Part Texture 53.2.2 The Topos of Conversation Among Four Humanists 53.2.3 The Family of Violins Chapter 54 Estimation of Resolution Parameters 54.1 Parameter Spaces for Violins 54.2 Estimation Chapter 55 The Case of Counterpoint and Harmony 55.1 Counterpoint 55.2 Harmony 55.3 Effective Selection Part XXIV References and Index References Index The Topos of Music III Gestures (Guerino Mazzola, René Guitart, Jocelyn Ho, Alex Lubet, Maria Mannone, Matt Rahaim, Florian Thalmann) Preface to the Second Edition Preface Volume III Contents Book Set Contents Leitfaden Leitfalden I & II Leitfaden III Tom_CD Part XV Gesture Philosophy for Music 56 The Topos of Gestures 57 Gesture Philosophy: Phenomenology, Ontology, and Semiotics 57.1 A Short Recapitulation of Musical Ontology 57.1.1 Ontology: Where, Why, and How 57.1.2 Oniontology: Facts, Processes, and Gestures 57.2 Jean-Claude Schmitt’s Historiographic and Philosophical Treatise “Laraison des gestes dans l’Occident m´edi´eval” 57.2.1 Comments 57.3 Vil´em Flusser’s Gesten: Versuch einer Ph¨anomenologie 57.3.1 A Short Introduction to Flusser’s Essay 57.3.2 The Semiotic Neurosis 57.4 Michel Guérin’s philosophie des gestes 57.4.1 The Essay’s Structure 57.4.2 Gestural Ontology and Four Elementary Gestures 57.5 Flusser and Gu´erin: Some Consequences 57.6 A Program 57.6.1 Circularity 57.7 The Semiotic Gesture Concept of Adam Kendon and David McNeill 57.7.1 Comments 57.8 Juhani Pallasmaa and Andr´e Chastel: The Thinking Hand in Architectureand the Arts 57.9 ´Emile Benveniste and Marie-Dominique Popelard/Anthony Wall:Gestures as a Dialogical Category 58 The French Presemiotic Approach 58.1 Maurice Merleau-Ponty 58.2 Francis Bacon and Gilles Deleuze 58.3 Jean Cavaill`es and Charles Alunni 58.4 Gilles Chˆatelet 59 Cognitive Science 59.1 Embodiment 59.1.1 Embodiment Science 59.1.1.1 The Cognitive Layer 59.2 Neuroscience 59.2.1 Embodied AI 59.3 Anthropology 59.4 Dance 59.5 Disabled Gestures Versus Gestures Disabled: Parlan’s Versus Peterson’sPianism 59.5.1 Performative Gestures: Disabled Jazz Pianists 59.5.2 Horace Parlan: Disabled Gestures 59.5.3 Parlan with Bass (and Drums) 59.5.4 Parlan with Rhythm Section 59.5.5 Parlan as Soloist 59.5.6 Parlan’s Duets with Archie Shepp 59.5.7 Disabled Gestures 59.5.8 Gestures Disabled: Oscar Peterson 59.5.9 Conclusion 59.6 Aristotle, Blind Lemon Jefferson, and Vilayanur S. Ramachandran Walkinto a Bar: Blues, Blindness, Politics, and Mirror Neurons 59.6.1 Introduction 59.6.2 Division by (Almost) Zero: Many Blind Bluesmen but Few Blind Blues 59.6.3 Seeing Blind Blues: Gesture, Flow, Circuitry, and Amplification 59.6.4 Epilogue: Puns as Gestures 60 Models from Music 60.1 Wolfgang Graeser 60.2 Adorno, Wieland, Sessions, Clynes 60.2.1 Theodor Wiesengrund Adorno 60.2.2 Renate Wieland 60.2.3 Roger Sessions 60.2.4 Manfred Clynes 60.3 Johan Sundberg and Neil P. McAngus Todd 60.4 David Lewin and Robert S. Hatten 60.5 Marcelo Wanderley and Claude Cadoz, Rolf Inge Godøy and Marc Leman Part XVI Mathematics of Gestures 61 Fundamental Concepts and Associated Categories 61.1 Introduction 61.2 Towards a Musical String Theory 61.3 Initial Investigations: Diagrams of Curves 61.4 Modeling a Pianist’s Hand 61.4.1 The Hand’s Model 61.4.2 Transforming Abstract Note Symbols into Symbolic Gestures 61.4.3 From Symbolic Hand Gestures to Physical Gestures 61.5 The Mathematical Definition of Gestures 61.6 Hypergestures 61.6.1 Spatial Hypergestures 61.7 Categorically Natural Gestures 61.8 Connecting to Algebraic Topology: Hypergestures Generalize Homotopy 61.9 Gestoids 61.9.1 The Fundamental Group, Klumpenhouwer Networks, and Fourier Representation 61.10 Gabriel’s Spectroids and Natural Formulas 61.10.1 Solutions of Representations of Natural Formulas by Local Networks 61.11 The Tangent Category 61.12 The Diamond Conjecture 61.13 Topos Logic for Gestures 61.14 The Escher Theorem for Hypergestures 61.14.1 The Hypergestures and the Escher Theorem for Fux Counterpoint 61.14.2 Rebecca Lazier’s Vanish: Lawvere, Escher, Schoenberg 62 Categories of Gestures over Topological Categories 62.1 Gestures over Topological Categories 62.1.1 The Categorical Digraph of a Topological Category 62.1.2 Gestures with Body in a Topological Category 62.1.3 Varying the Underlying Topological Category 62.2 From Morphisms to Gestures 62.2.1 Diagrams as Gestures 62.2.2 Gestures in Factorization Categories 62.2.3 Extensions from Homological Algebra Are Gestures 62.2.4 The Bicategory of Gestures 62.2.5 Entering the Diamond Space 62.3 Diagrams in Topological Groups for Gestures 62.4 Modulations in Beethoven’s “Hammerklavier” Sonata op.106/Allegro: AGestural Interpretation 62.4.1 Recapitulation of the Results from Section 28.2 62.4.2 The Modulation B5-majorùG-major Between Measure 31 and Measure 44 62.4.3 Lewin’s Characteristic Gestures Identified? 62.4.4 Modulation E5-majorùD-major{B-minor from W to W˚ 62.4.5 The Fanfare 62.5 Conclusion for the Categorial Gesture Approach 62.6 Functorial Gestures: General Addresses 62.7 Yoneda’s Lemma for Gestures 62.8 Examples from Music 62.8.1 Collections of Acoustical Waves 62.8.2 Collections of Spectral Music Data 62.8.3 MIDI-Type ON-OFF Transformations 63 Singular Homology of Hypergestures 63.1 An Introductory Example 63.2 Chain Modules for Singular Hypergestural Homology 63.3 The Boundary Homomorphism 64 Stokes’ Theorem for Hypergestures 64.1 The Need for Stokes’ Theorem for Hypergestures 64.2 Almost Regular Manifolds, Differential Forms, and Integration forHypergestures 64.2.1 Locally Almost Regular Manifolds 64.2.2 Differential Forms 64.2.3 Integration 64.3 Stokes’ Theorem 65 Local Facts, Processes, and Gestures 65.1 Categories of Local Compositions 65.2 Categories of Local Networks 65.3 Categories of Local Gestures 65.3.1 Local Gestures on Topological Categories of Points 65.4 Connecting Functors 65.5 Hypernetworks and Hypergestures 65.5.1 Escher Theorems 65.6 Singular Homology of Hypernetworks and Hypergestures 66 Global Categories 66.1 Categories of Global Compositions 66.1.1 Simplicial Methods 66.2 Classification of Global Compositions 66.3 Non-interpretable Global Compositions 66.4 Categories of Global Networks 66.4.1 Non-interpretable Global Networks 66.5 Categories of Global Gestures 66.6 Globalizing Topological Categories: Categorical Manifolds 66.7 Globalizing Skeleta 66.8 Functorial Global Gestures 67 Mathematical Models of Creativity 67.1 Forewarning: Invention of Gestures in Mathematics 67.1.1 Thinking Exactness, Like a Rolling Mind 67.1.2 Thought as an Algebra of Gestures 67.2 Method and Objects, Summarily Explained: I—Preamble 67.2.1 Prelude to a Discourse of a Method: “Caminos”, “Aletheia”, Irreverence 67.2.1.1 Categorical Modeling, Method, Estrangement, Intellectuality 67.2.1.2 With Ren´e Descartes 67.2.1.3 In the School of the Mathematicians, According to John Locke 67.2.1.4 Methods and Creativity, with Giambattista Vico 67.2.2 Our Posture 67.2.2.1 Towards the True and the Being, Mathematically: On the Road Again 67.2.2.2 Calculo, Ergo Sum: Mathˆema and Doubt 67.3 Method and Objects, Summarily Explained: II—Data 67.3.1 Simple Objects, Structures and Invariants in Mathematics 67.3.1.1 Multiplicity, Ambiguity, Alterity of Objects, Varying Elements of Objects 67.3.1.2 The Hexagram of Pascal 67.3.1.3 A Formula of Frye 67.3.1.4 Finite Configurations: Example of Latin Squares of Euler 67.3.1.5 Structures or Recreational Mathematics: Same Recourses for Solving 67.3.1.6 Undirectness, Synthetic Thinking and Intuitions 67.3.1.7 Categories, Sets, Groups, Lattices, Structures, out of Logical Concern 67.3.2 Complete Frameworks, Computations and Representations 67.3.2.1 Do We Need Universes as Complete Global Foundations, or Completions as LocallyAchieved Frameworks? 67.3.2.2 Calculations and Sketches of Gestures 67.3.2.3 What About Applications, Implements, and Representations? 67.4 Creativity in Mathematics: Gestures in Historical Contexts 67.4.1 Creativity: Phenomenology, Psychology and Skills, and Life 67.4.1.1 At the Beginning of Our Creations Are Our Imaginary Gestures 67.4.1.2 Gestures, Diagrams, Computations, Detours, Pulsations 67.4.1.3 Three Pulsations Which Are Internal to Any Mathematical Commitment 67.4.1.4 Creative Mathematics into a Peculiar Notional Living Scenery 67.4.1.5 Style and Notional Sceneries in Mathematics as a Natural Language 67.4.1.6 Creativity with Mathematics, in Mathematics: To Prove, to Understand 67.4.1.7 Creativity from the Double-Sided Point of View of Categories 67.4.2 Determination of Mathematics as a History of Its Gestures 67.4.2.1 Gestures as Transits, Pulsation Among Diagrams, and Machines 67.4.2.2 To Do and to Apply Mathematics: Mathematical Gestures 67.4.2.3 History as Series of Analytico-Synthetical Gestures: Doubt, Obviousness 67.4.2.4 Rigor and Subjectivity, High Level Gestures 67.4.2.5 Problems and Mathematical Pulsation in the Production of Forms 67.4.2.6 History as Imaginary Resource of Necessities for Mathematicians 67.4.2.7 Fashion, Successes and Errors, Scruples 67.4.2.8 Toward Categorical Modeling 67.4.3. Invention in the Art of Mathematics 67.4.3.1 The Truly Creative Mathematician Lives in the Real No-Reality World 67.4.3.2 Method of Invention Towarde an Art of Functional Modeling 67.5 On the Mathematical Invention of Coordinations 67.5.1 Emergence of Coordinations 67.5.1.1 Sympton, Characteristic Equations, Linear coordinates 67.5.1.2 Curvilinear Coordinates as Families of Surfaces or Curves 67.5.1.3. Tripolar Coordinates, from a Symptom of the plane 67.5.2 Arrows 67.5.2.1 Semiotics and Hermeneutics 67.5.2.2. The Case of a Mathematical Discourse 67.5.2.3. Coordinations, Diagrams, Abbreviations 67.5.2.4 The Concrete Map as an Abstract Arrow Abridging a System of Arrows 67.5.2.5 Functional Spaces, Algebras of Functions, Duality 67.5.3. Bodies, Implicit Surfaces, Abstract Relations 67.5.3.1. Relational Coordinations 67.5.3.2. Implicit Surfaces and Spaces 67.5.4 Sketches 67.5.4.1 Coordinations as Categorical Diagrams 67.5.4.2 Projective and Mixed Sketches 67.6 Pulsation in the Living Process of Invention Among Shapes 67.6.1 Production: Objects and Relations, Problems, Pulsation 67.6.1.1 Historical Transfers of Meanings in the course of Research 67.6.1.2. The Fundamental Gesture of Pulsation 67.6.1.3 Mathematics Invent Effective Transitions Between Possible-Objects 67.6.1.4 Diagrams: Sketches and Sites, Topoi and Algebraic Universe 67.6.1.5 The Dialectic Resides in Mathematical Acts 67.6.2 Creativity in the Mathematical world seen as a Living System of Shapes, in a Categorical Framework 67..2.1 Living System 67.6.2.2 Axiomatic Modeling of Mathematical Creativity? 67.6.2.3 Shape Theory and Models, Cohomology, Differentials 67.7 Conclusion: Categorial Presentation of Pulsations 67.8 The Hegel Group Action on a Critical concept's Walls 67.9 Introduction 67.10 The Hegel Concept Group G 67.10.1 Hegel’s Initial Thought Movement in Wissenschaft der Logik 67.10.2 The Implicit Group Structure 67.10.3 The conceptual Box Structure 67.11 The G Action on the Yoneda Model of Creativity 67.12 The Hegel Body B in the Concept Architecture of Forms and Denotators 67.13 The Usage of G fir the Dynamics of Creativity 67.13.1 Two Preliminary Examples 67.13.2 The Challenge: Creating a Spectrum of Conceptual Extensions 67.13.3 Escher"s Theoren for Beethoven's Fanfare in the "Hammerklavier" Sonata op. 67.13.4 The Rotation S@N as a Driving Creative Force in the Incipit of Liszt's Mephisto Walzer No.1 67.14 An Experimental Composition 67.15 Still more Symmetries? Future Developments Part XVII Concept Architecture and Software for Gesture Theory 68 Forms and Denotators over topological Categories 68.1 The General Topos—Theoritical Framework 68.1.1 The category Topcat of small Topological Categories 68.2 Forms and Denotators 68.3 Mathematics of Objects, Structures, and Concepts 68.4 Galois Theory of Concepts 68.4.1 Introduction 68.4.2 Form Semiotics 68.4.3 The Category of Form Semiotics 68.4.4 Galois Correspondence of form Semiotics 69 The Rubato Composer Architecture 69.1 The Software Architecture 69.2 The Rubette World 69.2.1 Rubette for Counterpoint 69.2.2 Rubettes for Harmony 69.2.3 MetroRubettes 70 The BigBang Rubette and the Ontological Dimension of Embodiment 71 Facts: Denotators and Their Visualization and Sonification 71.1 Some Earlier Visualization of Denotators 71.1.1 Göller's Priman Vista Browser 71.1.2 Milmeister's ScorePlay and Select2D Rubettes 71.2 An Early Score-Based Version of BigBang 71.2.1 The Early BigBang Rubette's View Configurations 71.2.2 BigBangObjects And Visualization of Arbitrary Mod@ Denotators 71.2.3 Sonifying Score-Based Denotators 71.3 BigBangObjects and VIsualization of M od@ 71.3.1 A Look at Potential Visual Characteristics of Form Types 71.3.1.1 Simple Denotators 71.3.1.2 Limit Denotators 71.3.1.3 Colimit Denotators 71.3.1.4 Power and List Denotators 71.3.2 From a General View Concept to BigBang Objects 71.3.2.1 Implication for Satellites 71.3.3 New Visual Dimensions 71.4 The Sonification of BigBangObjects 71.5 Examples of Forms and the Visualiazation of Their Denotators 71.5.1 Some Set-Theoectical Structures 71.5.2 Tonal and Transformation Theory 71.5.3 Synthesizers and Sound Design 72 Processes: BigBang's Operation Graph 72.1 Temporal BigBangObjects, Object Selection, and Layers 72.1.1 Selecting None and Lewin's Transformation Graphs 72.1.2 The Temporal Existence of BigBang Objects 72.1.3 BigBang Layers 72.2 Operations and Transformation in BIgBang 72.2.1 Non-transformational Operations 72.2.1.1 AddObjects and Delete Objects 72.2.1.2 InputComposition 72.2.1.3 BuidSatellite and Flatten 72.2.1.4 Shaping 72.2.1.5 Wallpaper Operations 72.2.1.6 Alteration 72.2.2 Transformations 72.2.2.1 Transformation in Arbitrary Spaces 72.3 BigBang's Process View 72.3.1 Visualization of Processess 72.3.2 Selecting States and Modifying Operations 72.3.3 Alternative and Parallel Processess 72.3.4 Structurally Modifying the Graph 72.3.4.1 Removing Operations 72.3.4.2. Inserting Operations 72.3.4.3. Splitting Operations 72.3.5 Undo/Redo 73 Gestures: Interaction and Gesturalization 73.1 Formalizing: From Gestures to Operation 73.1.1 Modes, Gestural Operations, and the Mouse 73.1.1.1 Gestural Transformations 73.1.1.2 Other Gestural Operations 73.1.1.3 Non-gesturalo Operations 73.1.2 Affine Transformations and Multi-touch 73.1.3 Dynamic Motives, Sound Synthesis, and Leap Motion 73.1.4 Recording, Modifying Operations and MIDI Controllers 73.2 Gesturalizing and the Real BigBang:Animated Composition History 73.2.1 Gesturalizing Transformations 73.2.1.1 Translation 73.2.1.2 Rotation 73.2.1.3 Scaling 73.2.1.4 Shearing 73.2.1.5 Reflection 73.2.1.6 Affine Transformations 73.2.1.7 Gesturalizing Beyond the Transformation 73.2.2 Gesturalizing other operations 73.2.3 Using Gesturalization as a Compositonal Tool 74 Musical Examples 74.1 Some Example Compositions 74.1.1 Transforming an Existing Composition 74.1.2 Gesturalizating and Looping with a Simple Graph 74.1.3 Drawing UPIC-like Motives and Transforming 74.1.4 Drawing Time-Slices 74.1.5 Converting Forms, Tricks for Gesturalizing 74.1.6 Gesturalizing A Spectrum 74.1.7 Using Wallpapers to-create Rhythmical Structures 74.2 Improvisation and Performance with BIgBang 74.2.1 Improvising by Selecting States and Modifying Transformations 74.2.2 Playing Sounds with a MIDI Keyboard and Modifying Them 74.2.3 Playing A MIDI Grand Piano with Leap Motion 74.2.4 Playing a MIDI Grand Piano with the Ableton Push 74.2.5 Improvising with 12-Tone Rows Part XVIII The Multiverse Perspective 75 Gesture Theoryand String Theory 76 Physical and Musical Multiverse 77 Hesse's Melting Beads: A Multiverse Game with Strings and Gestures 77.1 Review of Hesse's Glass Bead Game 77.2 Frozen Glass Beads of Facticity 77.3 The Revolution of Functors 77.4 Gestures in Philosophy and Science 77.5 Gesture Theory in Music 77.6 A Remark on Gestural Creativity 77.7 Gestures and Strings 77.8 Playing the Multiversed Game in a Pre-semiotic Ontology 78 Euler-Lagrange Equations for Hypergestures 78.1 The Problem in Performance Theory with the Physical Nambu-Goto Lagrangian 78.1.1 Complex Time and Descartes's Dualistic Ontology 78.2 Lagrangian Density for Complex time 78.2.1 The Lagrangian Action for Performance 78.2.2 the World-Sheet of complex Time 78.2.3 The Space for a Hand's Gestures 78.2.4 the World-sheet for a Simple Case 78.2.5 The Elementry Gesture of A Pianist 78.2.6 The Overarching Framework Between Note Performance and Gesture Performance 78.2.7 Examples of Functional Relations Between Potential and Physical Gesture 78.2.7 Examples of Funcrional Relations Between Potential and Physical Gesture 78.2.7.1 Solving the Poisson Equation for Rectangular Boundary conditions 78.2.7.2 Three Examples of Potentials 78.2.7.3 Examples of Lagrangian Action 78.2.8 Calculus of Variations for the Physical Gesture 78.2.9 A First Solution, World-Sheet Potentials Determine A Pianist's Gesture: Calculus of Variations and Fourier Analysis 78.2.10 The Calculus with Vanishing Potential 78.2.10.1 The Variation Calculus on s1 78.2.10.2 the Fourier Calculus 78.2.10.3 The Non-singular Matrix 78.2.10.4 A Second Fourier Calculus 78.2.11 The Calculus with General Potential 78.2.12 solution of the Differential Equations Using 2D Fourier Series 78.2.12.1 Funtional Dependence of the Physical Gesture 78.2.13 Parallels Between Performance Operators for scores and for Gestures 78.2.13.1 Some Detailed Calculation Regarding the Variations Calculous of s1 with Potential 78.2.14 Complex Time and the Artistic Effort 78.2.15 Opening the Aesthetic Question that Is Quantified in Lagrange Potentials 78.2.16 A Musical Composition by Maria Mannome Realized Using These Ideas 78.2.16.1 First Movement 78.2.16.2 Second Movement 78.2.11.1 The Variational Calculus of s1 with Potential 78.3 Global Performance Hypergestures 78.3.1 The Musical Situation: An Intuitive Introduction 78.4 Categorical Gestures and Global Performance Hypergestures 78.4.1 Categorical Gestures: The Case of Potentials 78.4.2 The Mathematics of Global Performance Hypergestures 78.5 World-Sheet Hypergestures for General Skeleta 78.6 A Global Variational Principle for the Lagrange formalism Part XIX Gestures in Music and Performance Theory, and in Ethnomusicology 79 Gesture Homology for counterpoint 79.1 Summary of Mathematical Theory of counterpoint: What It Is About and What Is Missing 79.2 Hypergestural Singular Homology 79.3 A Classical Example of a Topological Category from Counterpoint 79.3.1 Generators of H1pGXq for a Groupoid GX Defined by a Group Action 79.4 The Meaning of H1 for Counterpoint 79.5 Concluding Comments 80 Modulation Theory and Lie Brackets of Vector Fields 80.1 Introduction 80.1.1 Short Recapitulation of the Classical Model's Structure 80.2 Hypergestures Between Triadic Degree That Arc Parallel to Vector Fields 80.3 Lie Brackets Generate Vector Fields That Connect Symmetry-Related Degrees 80.4 Selecting Parallel Hypergestures That Arc Admissible for Modulation 80.5 The Other Direct Modulations 81 Hypergestures for Performance Stemmata 81.1 Motivation, Terminology, and Previous Results 81.1.1 Performance Stemmata and Performance Gestures of Locally Compact Points 81.2 Gestures with Lie Operators in Stemma Theory 81.3 Connecting Stemmatic Gestures for Weights and Performance Fields 81.4 Hamology of Weights Parameter Stemmata 81.5 A Concrete Example 81.6 A Final Comment 82. Composing and Analyzing with the Performing Body 82.1 Gesture: A Sign or a Totality? 82.2 A Gesture-Based Structural Reading in Rain Tree Sketch II by Torn 82.2.1 Process I: Synergy of Mirroring and Parallel Gestures 82.2.2 Process II: Towards Relaxation, Balance, and Weightfulness 82.3 The Last Leg of a Bodily Journey 82.3.1 Sheng for Piano, Smartphones, and Fixed Playback 82.3.2 Cross-modality of Gestures 82.3.3 Learning the Smartphone Instrument 82.3.4 Kinesthetic Awareness and Modes of Listening 82.4 Conclusion: Foregrounding the Performer’s Body 83 Gestural Analysis and Classification of a Conductor’s Movements 83.1 Gestures and Communication in Orchestral Conducting: A Case Study 83.1.1 Problematics and Solving Methods 83.1.2 Results, Consequences, Applications 83.1.3 Some Remarks 83.2 Hints for a Mathematical Description 83.3 Data Analysis 83.4 Conclusion 83.5 Addendum 84 Reviewing Flow, Gesture, and Spaces in Free Jazz 84.1 Improvisation: Defining Time 84.2 Flow, Gestures, Imaginary Time and Spaces in the Music Movie 84.2.1 The Compositional Character of the Pieces 84.2.2 Large Forms 84.2.3 Precision of Attacks 84.2.4 Co-presence of Different Time Layers 84.2.5 The Reality of Imaginary Time 84.2.6 Measuring Flow 84.2.7 Explicit Perception of Gestures 85 Gesture and Vocalization 85.1 Vocal Gesture 85.2 Vocal and Manual Motion 85.3 Gait 85.4 Hindustani Vocal Music 85.5 Notic Models and Kinetic Models 85.6 The Realist Pitfall 85.7 The Subjectivist Pitfall 85.8 Speech Gesture 86 Elements of a Future Vocal Gesture Theory 86.1 Why a Theory of Vocal Gestures? 86.1.1 Studying the Voice Without the Singer? 86.1.2 Parts of the Phonatory System and Their Functions 86.1.3 Imaginary Gestures in Real Time? 86.1.4 Space of Voice Parameters Gestures 86.1.5 About the Importance of Breathing and of Laryngeal Movements 86.1.6 Mathematical Description of Vocal Gestures 86.1.6.1 Why Such a Formalism? 86.1.6.2 Other Comments on Vocal Hypergestures 86.1.6.3 Branching 86.1.7 Gestures Thought by Singers 86.1.7.1 Cultures of the Voice: An Example from Ethnomusicology 86.1.7.2 Gregorian Chant and Gauls 86.2 A Powerful Tool from the Past for the Mathematical/Physical Theory of the Future: The Neumes of Gregorian Chant 86.2.1 Gestures in Gregorian Chant Didactics 86.2.2 Concept of Rhythm and Time 86.2.2.1 The Chironomic Game 86.2.2.2 Voice in Imaginary Time, Silence in Physical Time? 86.2.3 The Neumes 86.3 Connecting Physiology, Gestures and Notation. Toward New Neumes? 86.3.0.1 A New Score 86.3.1 New Neumes Part XXIV References and Index References Index The Topos of Music IV Roots (Guerino Mazzola) 3573596) Preface to the Second Edition Preface Volume IV Contents Book Set Volume Leitfaden Leitfaden I & II Leitfaden III Tom_CD Part XX Appendix: Sound A Common Parameter Spaces A.1 Physical Spaces A.1.1 Neutral Data A.1.1.1 Room Acoustics A.1.2 Sound Analysis and Synthesis A.1.2.1 Fourier A.1.2.2 Frequency Modulation A.1.2.3 Wavelets A.1.2.4 Some Remarks on Physical Modeling A.2 Mathematical and Symbolic Spaces A.2.1 Onset and Duration A.2.2 Amplitude and Crescendo A.2.3 Frequency and Glissando B Auditory Physiology and Psychology B.1 Physiology: From the Auricle to Heschl’s Gyri B.1.1 Outer Ear B.1.2 Middle Ear B.1.3 Inner Ear (Cochlea) B.1.4 Cochlear Hydrodynamics: The Travelling Wave B.1.5 Active Amplification of the Traveling Wave Motion B.1.6 Neural Processing B.2 Discriminating Tones: Werner Meyer-Eppler’s Valence Theory B.3 Symbolic, Physiological, and Psychological Aspects of Consonance and Dissonance B.3.1 Euler’s Gradus Function B.3.2 von Helmholtz’ Beat Model B.3.3 Psychometric Investigations by Plomp and Levelt B.3.4 Counterpoint B.3.5 Consonance and Dissonance: A Conceptual Field Part XXI Appendix: Mathematical Basics C Sets, Relations, Monoids, Groups C.1 Sets C.1.1 Examples of Sets C.2 Relations C.2.1 Universal Constructions C.2.2 Graphs and Quivers C.2.3 Monoids C.3 Groups C.3.1 Homomorphisms of Groups C.3.2 Direct, Semi-direct, and Wreath Products C.3.3 Sylow Theorems on C.3.4 Classification of Groups C3.4.1 Classification of Cyclic Groups C.3.4.2 Classification of Finitely Generated Abelian Groups C.3.5 General Affine Groups C.3.6 Permutation Groups D Rings and Algebras D.1 Basic Definitions and Constructions D.1.1 Universal Constructions D.1.1.1 Quiver Algebras D.2 Prime Factorization D.3 Euclidean Algorithm D.4 Approximation of Real Numbers by Fractions D.5 Some Special Issues D.5.1 Integers, Rationals, and Real Numbers E Modules, Linear, and Affine Transformations E.1 Modules and Linear Transformations E.1.1 Examples E.1.1 Examples E.2 Module Classification E.2.1 Dimension E.2.2 Endomorphisms on Dual Numbers E.2.3 Semi-simple Modules E.2.4 Jacobson Radical and Socle E.2.5 Theorem of Krull-Remak-Schmidt E.3 Categories of Modules and Affine Transformations E.3.1 Direct Sums E.3.2 Affine Forms and Tensors E.3.3 Biaffine Maps E.3.4 Symmetries of the Affine Plane E.3.5 Symmetries on Z2 E.3.6 Symmetries on Zn E.3.7 Complements on the Module of a Local Composition E.3.8 Fiber Products and Fiber Sums in Mod E.4 Complements of Commutative Algebra E.4.1 Localization E.4.2 Projective Modules E.4.3 Injective Modules E.4.4 Lie Algebras F Algebraic Geometry F.1 Locally Ringed Spaces F.2 Spectra of Commutative Rings F.2.1 Sober Spaces F.3 Schemes and Functors F.4 Algebraic and Geometric Structures on Schemes F.4.1 The Zariski Tangent Space F.5 Grassmannians F.6 Quotients G Categories, Topoi, and Logic G.1 Categories Instead of Sets G.1.1 Examples G.1.2 Functors G.1.3 Natural Transformations G.2 The Yoneda Lemma G.2.1 Universal Constructions: Adjoints, Limits, and Colimits G.2.2 Limit and Colimit Characterizations G.2.2.1 Special Results for Mod G.3 Topoi G.3.1 Subobject Classifiers G.3.2 Exponentiation G.3.3 Definition of Topoi G.4 Grothendieck Topologies G.4.1 Sheaves G.5 Formal Logic G.5.1 Propositional Calculus G.5.2 Predicate Logic G.5.3 A Formal Setup for Consistent Domains of Forms G.5.3.1 Morphisms Between Semiotics of Forms G.5.3.2 Local and Global Form Semiotics G.5.3.3 Connotator Form Semiotics H Complements on General and Algebraic Topology H.1 Topology H.1.1 General H.1.2 The Category of Topological Spaces H.1.3 Uniform Spaces H.1.4 Special Issues H.2 Algebraic Topology H.2.1 Simplicial Complexes H.2.2 Geometric Realization of a Simplicial Complex H.2.3 Contiguity H.3 Simplicial Coefficient Systems H.3.1 Cohomology I Complements on Calculus I.1 Abstract on Calculus I.1.1 Norms and Metrics I.1.2 Completeness I.1.3 Differentiation I.2 Ordinary Differential Equations (ODEs) I.2.1 The Fundamental Theorem: Local Case I.2.2 The Fundamental Theorem: Global Case I.2.3 Flows and Differential Equations I.2.4 Vector Fields and Derivations I.3 Partial Differential Equations J More Complements on Mathematics J.1 Directed Graphs J.1.1 The Category of Directed Graphs (Digraphs) J.1.1.1 Unordered Graphs J.1.2 Two Standard Constructions in Graph Theory J.1.3 The Topos of Digraphs J.2 Galois Theory J.3 Splines J.3.1 Some Simplex Constructions for Splines J.3.2 Definition of General Splines J.4 Topology and Topological Categories J.4.1 Topology J.4.1.1 Generators for Topologies J.4.1.2 Compact-Open Topology J.4.2 Topological Categories J.5 Complex Analysis J.6 Differentiable Manifolds J.6.1 Manifolds with Boundary J.6.2 The Tangent Manifold J.7 Tensor Fields J.7.1 Alternating Tensors J.7.2 Tangent Tensors J.8 Stokes’ Theorem J.9 Calculus of Variations J.10 Partial Differential Equations J.10.1 Explicit Calculation J.11 Algebraic Topology J.11.1 Homotopy Theory J.11.2 The Fundamental Group(oid) J.12 Homology J.12.1 Singular Homology J.13 Cohomology Part XXII Appendix: Complements in Physics K Complements on Physics K.1 Hamilton’s Variational Principle K.1.1 Euler-Lagrange Equations for a Non-relativistic Particle K.2 String Theory K.3 Duality and Supersymmetry K.4 Quantum Mechanics K.4.1 Banach and Hilbert Spaces K.4.1.1 Bounded Operators K.4.1.2 Lebesque Integration K.4.1.3 Lebesgue K.4.2 Geometry on Hilbert Spaces K.4.2.1 The K.4.3 Axioms for Quantum Mechanics K.4.3.1 Resolvents and Spectra K.4.4 The Spectral Theorem K.4.4.1 Projection-valued Measures Part XXIII Appendix: Tables L Euler’s Gradus Function M Just and Well-Tempered Tuning N Chord and Third Chain Classes N.1 Chord Classes N.2 Third Chain Classes O Two, Three, and Four Tone Motif Classes O.1 Two Tone Motifs in O.2 Two Tone Motifs in O.3 Three Tone Motifs in O.4 Four Tone Motifs in O.5 Three Tone Motifs in P Well-Tempered and Just Modulation Steps P.1 12-Tempered Modulation Steps P.1.1 Scale Orbits and Number of Quantized Modulations P.1.2 Quanta and Pivots for the Modulations Between Diatonic Major Scales (No.38.1) P.1.3 Quanta and Pivots for the Modulations Between Melodic Minor Scales (No.47.1) P.1.4 Quanta and Pivots for the Modulations Between Harmonic Minor Scales (No.54.1) P.1.5 Examples of 12-Tempered Modulations for All Fourth Relations P.2 2-3-5-Just Modulation Steps P.2.1 Modulation Steps Between Just Major Scales P.2.2 Modulation Steps Between Natural Minor Scales P.2.3 Modulation Steps from Natural Minor to Major Scales P.2.4 Modulation Steps from Major to Natural Minor Scales P.2.5 Modulation Steps Between Harmonic Minor Scales P.2.6 Modulation Steps Between Melodic Minor Scales P.2.7 General Modulation Behaviour for 32 Alterated Scales Q Counterpoint Steps Q.1 Contrapuntal Symmetries Q.1.1 Class No. 64 Q.1.2 Class No. 68 Q.1.3 Class No. 71 Q.1.4 Class No. 75 Q.1.5 Class No. 78 Q.1.6 Class No. 82 Q.2 Permitted Successors for the Major Scale Part XXIV References and Index References Index The Topos of Music II Performance Preface to the Second Edition Preface Volume II Contents Book Set Contents Leitfaden Leitfaden I & II Leitfaden III Tom_CD Part VIII Structure Theory of Performance Chapter 32 Local and Global Performance Transformations 32.1 Performance as a Reality Switch 32.2 Why Do We Need Infinite Performance of the Same Piece? 32.3 Local Structure 32.3.1 The Coherence of Local Performance Transformations 32.3.2 Differential Morphisms of Local Compositions 32.3.2.1 A Recursive Interpolation Algorithm 32.4 Global Structure 32.4.1 Modeling Performance Syntax 32.4.2 The Formal Setup 32.4.3 Performance qua Interpretation of Interpretation Chapter 33 Performance Fields 33.1 Classics: Tempo, Intonation, and Dynamics 33.1.1 Tempo 33.1.2 Intonation 33.1.3 Dynamics 33.2 Genesis of the General Formalism 33.2.1 The Question of Articulation 33.2.2 The Formalism of Performance Fields 33.3 What Performance Fields Signify 33.3.1 Th.W. Adorno, W. Benjamin, and D. Raffman 33.3.2 Towards Composition of Performance Chapter 34 Initial Sets and Initial Performances 34.1 Taking Off with a Shifter 34.2 Anchoring Onset 34.3 The Concert Pitch 34.4 Dynamical Anchors 34.5 Initializing Articulation 34.6 Hit Point Theory 34.6.1 Distances 34.6.2 Flow Interpolation Chapter 35 Hierarchies and Performance Scores 35.1 Performance Cells 35.2 The Category of Performance Cells 35.3 Hierarchies 35.3.1 Operations on Hierarchies 35.3.2 Classification Issues 35.3.3 Example: The Piano and Violin Hierarchies 35.4 Local Performance Scores 35.5 Global Performance Scores 35.5.1 Instrumental Fibers Part IX Expressive Semantics Chapter 36 Taxonomy of Expressive PerformanceThis 36.1 Feelings: Emotional Semantics 36.2 Motion: Gestural Semantics 36.3 Understanding: Rational Semantics 36.4 Cross-semantical Relations Chapter 37 Performance Grammars 37.1 Rule-Based Grammars 37.1.1 The KTH School 37.1.2 Neil P. McAngus Todd 37.1.3 The Zurich School 37.2 Remarks on Learning Grammars Chapter 38 Stemma Theory 38.1 Motivation from Practising and Rehearsing 38.1.1 Does Reproducibility of Performances Help Understanding? 38.2 Tempo Curves Are Inadequate 38.3 The Stemma Concept 38.3.1 The General Setup of Matrilineal Sexual Propagation 38.3.2 The Primary Mother—Taking Off 38.3.3 Mono- and Polygamy—Local and Global Actions 38.3.4 Family Life—Cross-correlations Chapter 39 Operator Theory 39.1 Why Weights? 39.1.1 Discrete and Continuous Weights 39.1.2 Weight Recombination 39.2 Primavista Weights 39.2.1 Dynamics 39.2.2 Agogics 39.2.3 Tuning and Intonation 39.2.4 Articulation 39.2.5 Ornaments 39.3 Analytical Weights 39.4 Taxonomy of Operators 39.4.1 Splitting Operators 39.4.2 Symbolic Operators 39.4.3 Physical Operators 39.4.4 Field Operators 39.5 Tempo Operator 39.6 Scalar Operator 39.7 The Theory of Basis—Pianola Operators 39.7.1 Basis Specialization 39.7.1.1 Deforming Hierarchies 39.7.1.2 Lie Derivatives 39.7.2 Pianola Specialization 39.8 Locally Linear Grammars Part X RUBATO Chapter 40 Architecture 40.1 The Overall Modularity 40.2 Frame and Modules 40.3 Postscriptum: The Rubato Composer Environment Chapter 41 The RUBETTE Family 41.1 MetroRUBETTE 41.2 MeloRUBETTE 41.3 HarmoRUBETTE 41.3.1 A Set of New Harmonic Analysis Rubettes on RUBATO Composer 41.4 PerformanceRUBETTE 41.5 PrimavistaRUBETTE Chapter 42 Performance Experiments 42.1 A Preliminary Experiment: Robert Schumann's "Kuriose Geschichte" 42.2 Full Experiment: J.S. Bach's "Kunst der Fuge" 42.3 Analysis 42.3.1 Metric Analysis 42.3.1.1 Single Voices 42.3.1.2 Weight Sums of All Voices 42.3.1.3 Union of All Voices 42.3.2 Motif Analysis 42.3.3 Omission of Harmonic Analysis 42.4 Stemma Constructions 42.4.1 Performance Setup 42.4.1.1 Results From First Performance Parcours 42.4.1.2 Construction of Second Performance Parcours 42.4.1.3 Construction of Third Performance Parcours 42.4.1.4 Local Discussion 42.4.2 Instrumental Setup 42.4.3 Global Discussion Part XI Statistics of Analysis and Performance Chapter 43 Analysis of Analysis 43.1 Hierarchical Decomposition 43.1.1 General Motivation 43.1.2 Hierarchical Smoothing 43.1.3 Hierarchical Decomposition 43.2 Comparing Analyses of Bach, Schumann, and Webern Chapter 44 Differential Operators and Regression 44.0.1 Analytical Data 44.1 The Beran Operator 44.1.1 The Concept 44.1.2 The Formalism 44.1.2.1 Tempo Information 44.1.2.2 The Explanatory Variables 44.2 The Method of Regression Analysis 44.2.1 The Full Model 44.2.2 Step Forward Selection 44.3 The Results of Regression Analysis 44.3.1 Relations Between Tempo and Analysis 44.3.2 Complex Relationships 44.3.3 Commonalities and Diversities 44.3.3.1 Signs of Coefficients 44.3.3.2 Frequency of Variable Inclusion 44.3.3.3 Largest Coefficients 44.3.3.4 Argerich "Versus" Horowitz 44.3.4 Overview of Statistical Results 45 Relating Tempo to Metric, Melodic and Harmonic Analyses in Chopin's Prélude op. 28, No. 4 45.1 Introduction 45.2 Data 45.2.1 Analytical Data 45.2.2 Tempo Data 45.3 Short Summary of the Results 45.4 Some Philosophical Comments Part XII Inverse Performance Theory Chapter 46 Principles of Music Critique 46.1 Boiling Down Infinity—Is Feuilletonism Inevitable? 46.2 "Political Correctness" in Performance—Reviewing Gould 46.3 Transversal Ethnomusicology Chapter 47 Critical Fibers 47.1 The Stemma Model of Critique 47.2 Fibers for Locally Linear Grammars 47.3 Algorithmic Extraction of Performance Fields 47.3.1 The Infinitesimal View on Expression 47.3.2 Real-Time Processing of Expressive Performance 47.3.3 Score-Performance Matching 47.3.4 Performance Field Calculation 47.3.4.1 Obtaining the Bases 47.3.5 Visualization 47.3.5.1 Field Interpolation 47.3.6 The EspressoRUBETTE: An Interactive Tool for Expression Extraction 47.3.6.1 Example 1: Tempo Field of a Chromatic Scale 47.3.6.2 Example 2: Excerpt from Czerny's Piano School 47.4 Local Sections 47.4.1 Comparing Argerich and Horowitz Part XIII Operationalization of Poiesis Chapter 48 Unfolding Geometry and Logic in Time 48.1 Performance of Logic and Geometry 48.2 Constructing Time from Geometry 48.3 Discourse and Insight Chapter 49 Local and Global Strategies in Composition 49.1 Local Paradigmatic Instances 49.1.1 Transformations 49.1.2 Variations 49.2 Global Poetical Syntax 49.2.1 Roman Jakobson's Horizontal Function 49.2.2 Roland Posner's Vertical Function 49.3 Structure and Process Chapter 50 The Paradigmatic Discourse on presto 50.1 The prestor Functional Scheme 50.2 Modular Ane Transformations 50.3 Ornaments and Variations 50.4 Problems of Abstraction Chapter 51 Case Study I: "Synthesis" by Guerino Mazzola 51.1 The Overall Organization 51.1.1 The Material: 26 Classes of Three-Element Motives 51.1.2 Principles of the Four Movements and Instrumentation 51.2 1st Movement: Sonata Form 51.3 2nd Movement: Variations 51.4 3rd Movement: Scherzo 51.5 4th Movement: Fractal Syntax Chapter 52 Object-Oriented Programming in OpenMusic 52.1 Object-Oriented Language 52.1.1 Patches 52.1.2 Objects 52.1.3 Classes 52.1.4 Methods 52.1.5 Generic Functions 52.1.6 Message Passing 52.1.7 Inheritance 52.1.8 Boxes and Evaluation 52.1.9 Instantiation 52.2 Musical Object Framework 52.2.1 Internal Representation 52.2.2 Interface 52.2.2.1 Rhythmic Trees 52.3 Maquettes: Objects in Time 52.4 Meta-object Protocol 52.4.1 Reification of Temporal Boxes 52.5 A Musical Example Part XIV String Quartet Theory Chapter 53 Historical and Theoretical Prerequisites 53.1 History 53.2 Theory of the String Quartet Following Ludwig Finscher 53.2.1 Four Part Texture 53.2.2 The Topos of Conversation Among Four Humanists 53.2.3 The Family of Violins Chapter 54 Estimation of Resolution Parameters 54.1 Parameter Spaces for Violins 54.2 Estimation Chapter 55 The Case of Counterpoint and Harmony 55.1 Counterpoint 55.2 Harmony 55.3 Effective Selection Part XXIV References and Index References Index The Topos of Music III Gestures Preface to the Second Edition Preface Volume III Contents Book Set Contents Leitfaden Leitfalden I & II Leitfaden III Tom_CD Part XV Gesture Philosophy for Music 56 The Topos of Gestures 57 Gesture Philosophy: Phenomenology, Ontology, and Semiotics 57.1 A Short Recapitulation of Musical Ontology 57.1.1 Ontology: Where, Why, and How 57.1.2 Oniontology: Facts, Processes, and Gestures 57.2 Jean-Claude Schmitt’s Historiographic and Philosophical Treatise “Laraison des gestes dans l’Occident m´edi´eval” 57.2.1 Comments 57.3 Vil´em Flusser’s Gesten: Versuch einer Ph¨anomenologie 57.3.1 A Short Introduction to Flusser’s Essay 57.3.2 The Semiotic Neurosis 57.4 Michel Guérin’s philosophie des gestes 57.4.1 The Essay’s Structure 57.4.2 Gestural Ontology and Four Elementary Gestures 57.5 Flusser and Gu´erin: Some Consequences 57.6 A Program 57.6.1 Circularity 57.7 The Semiotic Gesture Concept of Adam Kendon and David McNeill 57.7.1 Comments 57.8 Juhani Pallasmaa and Andr´e Chastel: The Thinking Hand in Architectureand the Arts 57.9 ´Emile Benveniste and Marie-Dominique Popelard/Anthony Wall:Gestures as a Dialogical Category 58 The French Presemiotic Approach 58.1 Maurice Merleau-Ponty 58.2 Francis Bacon and Gilles Deleuze 58.3 Jean Cavaill`es and Charles Alunni 58.4 Gilles Chˆatelet 59 Cognitive Science 59.1 Embodiment 59.1.1 Embodiment Science 59.1.1.1 The Cognitive Layer 59.2 Neuroscience 59.2.1 Embodied AI 59.3 Anthropology 59.4 Dance 59.5 Disabled Gestures Versus Gestures Disabled: Parlan’s Versus Peterson’sPianism 59.5.1 Performative Gestures: Disabled Jazz Pianists 59.5.2 Horace Parlan: Disabled Gestures 59.5.3 Parlan with Bass (and Drums) 59.5.4 Parlan with Rhythm Section 59.5.5 Parlan as Soloist 59.5.6 Parlan’s Duets with Archie Shepp 59.5.7 Disabled Gestures 59.5.8 Gestures Disabled: Oscar Peterson 59.5.9 Conclusion 59.6 Aristotle, Blind Lemon Jefferson, and Vilayanur S. Ramachandran Walkinto a Bar: Blues, Blindness, Politics, and Mirror Neurons 59.6.1 Introduction 59.6.2 Division by (Almost) Zero: Many Blind Bluesmen but Few Blind Blues 59.6.3 Seeing Blind Blues: Gesture, Flow, Circuitry, and Amplification 59.6.4 Epilogue: Puns as Gestures 60 Models from Music 60.1 Wolfgang Graeser 60.2 Adorno, Wieland, Sessions, Clynes 60.2.1 Theodor Wiesengrund Adorno 60.2.2 Renate Wieland 60.2.3 Roger Sessions 60.2.4 Manfred Clynes 60.3 Johan Sundberg and Neil P. McAngus Todd 60.4 David Lewin and Robert S. Hatten 60.5 Marcelo Wanderley and Claude Cadoz, Rolf Inge Godøy and Marc Leman Part XVI Mathematics of Gestures 61 Fundamental Concepts and Associated Categories 61.1 Introduction 61.2 Towards a Musical String Theory 61.3 Initial Investigations: Diagrams of Curves 61.4 Modeling a Pianist’s Hand 61.4.1 The Hand’s Model 61.4.2 Transforming Abstract Note Symbols into Symbolic Gestures 61.4.3 From Symbolic Hand Gestures to Physical Gestures 61.5 The Mathematical Definition of Gestures 61.6 Hypergestures 61.6.1 Spatial Hypergestures 61.7 Categorically Natural Gestures 61.8 Connecting to Algebraic Topology: Hypergestures Generalize Homotopy 61.9 Gestoids 61.9.1 The Fundamental Group, Klumpenhouwer Networks, and Fourier Representation 61.10 Gabriel’s Spectroids and Natural Formulas 61.10.1 Solutions of Representations of Natural Formulas by Local Networks 61.11 The Tangent Category 61.12 The Diamond Conjecture 61.13 Topos Logic for Gestures 61.14 The Escher Theorem for Hypergestures 61.14.1 The Hypergestures and the Escher Theorem for Fux Counterpoint 61.14.2 Rebecca Lazier’s Vanish: Lawvere, Escher, Schoenberg 62 Categories of Gestures over Topological Categories 62.1 Gestures over Topological Categories 62.1.1 The Categorical Digraph of a Topological Category 62.1.2 Gestures with Body in a Topological Category 62.1.3 Varying the Underlying Topological Category 62.2 From Morphisms to Gestures 62.2.1 Diagrams as Gestures 62.2.2 Gestures in Factorization Categories 62.2.3 Extensions from Homological Algebra Are Gestures 62.2.4 The Bicategory of Gestures 62.2.5 Entering the Diamond Space 62.3 Diagrams in Topological Groups for Gestures 62.4 Modulations in Beethoven’s “Hammerklavier” Sonata op.106/Allegro: AGestural Interpretation 62.4.1 Recapitulation of the Results from Section 28.2 62.4.2 The Modulation B5-majorùG-major Between Measure 31 and Measure 44 62.4.3 Lewin’s Characteristic Gestures Identified? 62.4.4 Modulation E5-majorùD-major{B-minor from W to W˚ 62.4.5 The Fanfare 62.5 Conclusion for the Categorial Gesture Approach 62.6 Functorial Gestures: General Addresses 62.7 Yoneda’s Lemma for Gestures 62.8 Examples from Music 62.8.1 Collections of Acoustical Waves 62.8.2 Collections of Spectral Music Data 62.8.3 MIDI-Type ON-OFF Transformations 63 Singular Homology of Hypergestures 63.1 An Introductory Example 63.2 Chain Modules for Singular Hypergestural Homology 63.3 The Boundary Homomorphism 64 Stokes’ Theorem for Hypergestures 64.1 The Need for Stokes’ Theorem for Hypergestures 64.2 Almost Regular Manifolds, Differential Forms, and Integration forHypergestures 64.2.1 Locally Almost Regular Manifolds 64.2.2 Differential Forms 64.2.3 Integration 64.3 Stokes’ Theorem 65 Local Facts, Processes, and Gestures 65.1 Categories of Local Compositions 65.2 Categories of Local Networks 65.3 Categories of Local Gestures 65.3.1 Local Gestures on Topological Categories of Points 65.4 Connecting Functors 65.5 Hypernetworks and Hypergestures 65.5.1 Escher Theorems 65.6 Singular Homology of Hypernetworks and Hypergestures 66 Global Categories 66.1 Categories of Global Compositions 66.1.1 Simplicial Methods 66.2 Classification of Global Compositions 66.3 Non-interpretable Global Compositions 66.4 Categories of Global Networks 66.4.1 Non-interpretable Global Networks 66.5 Categories of Global Gestures 66.6 Globalizing Topological Categories: Categorical Manifolds 66.7 Globalizing Skeleta 66.8 Functorial Global Gestures 67 Mathematical Models of Creativity 67.1 Forewarning: Invention of Gestures in Mathematics 67.1.1 Thinking Exactness, Like a Rolling Mind 67.1.2 Thought as an Algebra of Gestures 67.2 Method and Objects, Summarily Explained: I—Preamble 67.2.1 Prelude to a Discourse of a Method: “Caminos”, “Aletheia”, Irreverence 67.2.1.1 Categorical Modeling, Method, Estrangement, Intellectuality 67.2.1.2 With Ren´e Descartes 67.2.1.3 In the School of the Mathematicians, According to John Locke 67.2.1.4 Methods and Creativity, with Giambattista Vico 67.2.2 Our Posture 67.2.2.1 Towards the True and the Being, Mathematically: On the Road Again 67.2.2.2 Calculo, Ergo Sum: Mathˆema and Doubt 67.3 Method and Objects, Summarily Explained: II—Data 67.3.1 Simple Objects, Structures and Invariants in Mathematics 67.3.1.1 Multiplicity, Ambiguity, Alterity of Objects, Varying Elements of Objects 67.3.1.2 The Hexagram of Pascal 67.3.1.3 A Formula of Frye 67.3.1.4 Finite Configurations: Example of Latin Squares of Euler 67.3.1.5 Structures or Recreational Mathematics: Same Recourses for Solving 67.3.1.6 Undirectness, Synthetic Thinking and Intuitions 67.3.1.7 Categories, Sets, Groups, Lattices, Structures, out of Logical Concern 67.3.2 Complete Frameworks, Computations and Representations 67.3.2.1 Do We Need Universes as Complete Global Foundations, or Completions as LocallyAchieved Frameworks? 67.3.2.2 Calculations and Sketches of Gestures 67.3.2.3 What About Applications, Implements, and Representations? 67.4 Creativity in Mathematics: Gestures in Historical Contexts 67.4.1 Creativity: Phenomenology, Psychology and Skills, and Life 67.4.1.1 At the Beginning of Our Creations Are Our Imaginary Gestures 67.4.1.2 Gestures, Diagrams, Computations, Detours, Pulsations 67.4.1.3 Three Pulsations Which Are Internal to Any Mathematical Commitment 67.4.1.4 Creative Mathematics into a Peculiar Notional Living Scenery 67.4.1.5 Style and Notional Sceneries in Mathematics as a Natural Language 67.4.1.6 Creativity with Mathematics, in Mathematics: To Prove, to Understand 67.4.1.7 Creativity from the Double-Sided Point of View of Categories 67.4.2 Determination of Mathematics as a History of Its Gestures 67.4.2.1 Gestures as Transits, Pulsation Among Diagrams, and Machines 67.4.2.2 To Do and to Apply Mathematics: Mathematical Gestures 67.4.2.3 History as Series of Analytico-Synthetical Gestures: Doubt, Obviousness 67.4.2.4 Rigor and Subjectivity, High Level Gestures 67.4.2.5 Problems and Mathematical Pulsation in the Production of Forms 67.4.2.6 History as Imaginary Resource of Necessities for Mathematicians 67.4.2.7 Fashion, Successes and Errors, Scruples 67.4.2.8 Toward Categorical Modeling 67.4.3. Invention in the Art of Mathematics 67.4.3.1 The Truly Creative Mathematician Lives in the Real No-Reality World 67.4.3.2 Method of Invention Towarde an Art of Functional Modeling 67.5 On the Mathematical Invention of Coordinations 67.5.1 Emergence of Coordinations 67.5.1.1 Sympton, Characteristic Equations, Linear coordinates 67.5.1.2 Curvilinear Coordinates as Families of Surfaces or Curves 67.5.1.3. Tripolar Coordinates, from a Symptom of the plane 67.5.2 Arrows 67.5.2.1 Semiotics and Hermeneutics 67.5.2.2. The Case of a Mathematical Discourse 67.5.2.3. Coordinations, Diagrams, Abbreviations 67.5.2.4 The Concrete Map as an Abstract Arrow Abridging a System of Arrows 67.5.2.5 Functional Spaces, Algebras of Functions, Duality 67.5.3. Bodies, Implicit Surfaces, Abstract Relations 67.5.3.1. Relational Coordinations 67.5.3.2. Implicit Surfaces and Spaces 67.5.4 Sketches 67.5.4.1 Coordinations as Categorical Diagrams 67.5.4.2 Projective and Mixed Sketches 67.6 Pulsation in the Living Process of Invention Among Shapes 67.6.1 Production: Objects and Relations, Problems, Pulsation 67.6.1.1 Historical Transfers of Meanings in the course of Research 67.6.1.2. The Fundamental Gesture of Pulsation 67.6.1.3 Mathematics Invent Effective Transitions Between Possible-Objects 67.6.1.4 Diagrams: Sketches and Sites, Topoi and Algebraic Universe 67.6.1.5 The Dialectic Resides in Mathematical Acts 67.6.2 Creativity in the Mathematical world seen as a Living System of Shapes, in a Categorical Framework 67..2.1 Living System 67.6.2.2 Axiomatic Modeling of Mathematical Creativity? 67.6.2.3 Shape Theory and Models, Cohomology, Differentials 67.7 Conclusion: Categorial Presentation of Pulsations 67.8 The Hegel Group Action on a Critical concept's Walls 67.9 Introduction 67.10 The Hegel Concept Group G 67.10.1 Hegel’s Initial Thought Movement in Wissenschaft der Logik 67.10.2 The Implicit Group Structure 67.10.3 The conceptual Box Structure 67.11 The G Action on the Yoneda Model of Creativity 67.12 The Hegel Body B in the Concept Architecture of Forms and Denotators 67.13 The Usage of G fir the Dynamics of Creativity 67.13.1 Two Preliminary Examples 67.13.2 The Challenge: Creating a Spectrum of Conceptual Extensions 67.13.3 Escher"s Theoren for Beethoven's Fanfare in the "Hammerklavier" Sonata op. 67.13.4 The Rotation S@N as a Driving Creative Force in the Incipit of Liszt's Mephisto Walzer No.1 67.14 An Experimental Composition 67.15 Still more Symmetries? Future Developments Part XVII Concept Architecture and Software for Gesture Theory 68 Forms and Denotators over topological Categories 68.1 The General Topos—Theoritical Framework 68.1.1 The category Topcat of small Topological Categories 68.2 Forms and Denotators 68.3 Mathematics of Objects, Structures, and Concepts 68.4 Galois Theory of Concepts 68.4.1 Introduction 68.4.2 Form Semiotics 68.4.3 The Category of Form Semiotics 68.4.4 Galois Correspondence of form Semiotics 69 The Rubato Composer Architecture 69.1 The Software Architecture 69.2 The Rubette World 69.2.1 Rubette for Counterpoint 69.2.2 Rubettes for Harmony 69.2.3 MetroRubettes 70 The BigBang Rubette and the Ontological Dimension of Embodiment 71 Facts: Denotators and Their Visualization and Sonification 71.1 Some Earlier Visualization of Denotators 71.1.1 Göller's Priman Vista Browser 71.1.2 Milmeister's ScorePlay and Select2D Rubettes 71.2 An Early Score-Based Version of BigBang 71.2.1 The Early BigBang Rubette's View Configurations 71.2.2 BigBangObjects And Visualization of Arbitrary Mod@ Denotators 71.2.3 Sonifying Score-Based Denotators 71.3 BigBangObjects and VIsualization of M od@ 71.3.1 A Look at Potential Visual Characteristics of Form Types 71.3.1.1 Simple Denotators 71.3.1.2 Limit Denotators 71.3.1.3 Colimit Denotators 71.3.1.4 Power and List Denotators 71.3.2 From a General View Concept to BigBang Objects 71.3.2.1 Implication for Satellites 71.3.3 New Visual Dimensions 71.4 The Sonification of BigBangObjects 71.5 Examples of Forms and the Visualiazation of Their Denotators 71.5.1 Some Set-Theoectical Structures 71.5.2 Tonal and Transformation Theory 71.5.3 Synthesizers and Sound Design 72 Processes: BigBang's Operation Graph 72.1 Temporal BigBangObjects, Object Selection, and Layers 72.1.1 Selecting None and Lewin's Transformation Graphs 72.1.2 The Temporal Existence of BigBang Objects 72.1.3 BigBang Layers 72.2 Operations and Transformation in BIgBang 72.2.1 Non-transformational Operations 72.2.1.1 AddObjects and Delete Objects 72.2.1.2 InputComposition 72.2.1.3 BuidSatellite and Flatten 72.2.1.4 Shaping 72.2.1.5 Wallpaper Operations 72.2.1.6 Alteration 72.2.2 Transformations 72.2.2.1 Transformation in Arbitrary Spaces 72.3 BigBang's Process View 72.3.1 Visualization of Processess 72.3.2 Selecting States and Modifying Operations 72.3.3 Alternative and Parallel Processess 72.3.4 Structurally Modifying the Graph 72.3.4.1 Removing Operations 72.3.4.2. Inserting Operations 72.3.4.3. Splitting Operations 72.3.5 Undo/Redo 73 Gestures: Interaction and Gesturalization 73.1 Formalizing: From Gestures to Operation 73.1.1 Modes, Gestural Operations, and the Mouse 73.1.1.1 Gestural Transformations 73.1.1.2 Other Gestural Operations 73.1.1.3 Non-gesturalo Operations 73.1.2 Affine Transformations and Multi-touch 73.1.3 Dynamic Motives, Sound Synthesis, and Leap Motion 73.1.4 Recording, Modifying Operations and MIDI Controllers 73.2 Gesturalizing and the Real BigBang:Animated Composition History 73.2.1 Gesturalizing Transformations 73.2.1.1 Translation 73.2.1.2 Rotation 73.2.1.3 Scaling 73.2.1.4 Shearing 73.2.1.5 Reflection 73.2.1.6 Affine Transformations 73.2.1.7 Gesturalizing Beyond the Transformation 73.2.2 Gesturalizing other operations 73.2.3 Using Gesturalization as a Compositonal Tool 74 Musical Examples 74.1 Some Example Compositions 74.1.1 Transforming an Existing Composition 74.1.2 Gesturalizating and Looping with a Simple Graph 74.1.3 Drawing UPIC-like Motives and Transforming 74.1.4 Drawing Time-Slices 74.1.5 Converting Forms, Tricks for Gesturalizing 74.1.6 Gesturalizing A Spectrum 74.1.7 Using Wallpapers to-create Rhythmical Structures 74.2 Improvisation and Performance with BIgBang 74.2.1 Improvising by Selecting States and Modifying Transformations 74.2.2 Playing Sounds with a MIDI Keyboard and Modifying Them 74.2.3 Playing A MIDI Grand Piano with Leap Motion 74.2.4 Playing a MIDI Grand Piano with the Ableton Push 74.2.5 Improvising with 12-Tone Rows Part XVIII The Multiverse Perspective 75 Gesture Theoryand String Theory 76 Physical and Musical Multiverse 77 Hesse's Melting Beads: A Multiverse Game with Strings and Gestures 77.1 Review of Hesse's Glass Bead Game 77.2 Frozen Glass Beads of Facticity 77.3 The Revolution of Functors 77.4 Gestures in Philosophy and Science 77.5 Gesture Theory in Music 77.6 A Remark on Gestural Creativity 77.7 Gestures and Strings 77.8 Playing the Multiversed Game in a Pre-semiotic Ontology 78 Euler-Lagrange Equations for Hypergestures 78.1 The Problem in Performance Theory with the Physical Nambu-Goto Lagrangian 78.1.1 Complex Time and Descartes's Dualistic Ontology 78.2 Lagrangian Density for Complex time 78.2.1 The Lagrangian Action for Performance 78.2.2 the World-Sheet of complex Time 78.2.3 The Space for a Hand's Gestures 78.2.4 the World-sheet for a Simple Case 78.2.5 The Elementry Gesture of A Pianist 78.2.6 The Overarching Framework Between Note Performance and Gesture Performance 78.2.7 Examples of Functional Relations Between Potential and Physical Gesture 78.2.7 Examples of Funcrional Relations Between Potential and Physical Gesture 78.2.7.1 Solving the Poisson Equation for Rectangular Boundary conditions 78.2.7.2 Three Examples of Potentials 78.2.7.3 Examples of Lagrangian Action 78.2.8 Calculus of Variations for the Physical Gesture 78.2.9 A First Solution, World-Sheet Potentials Determine A Pianist's Gesture: Calculus of Variations and Fourier Analysis 78.2.10 The Calculus with Vanishing Potential 78.2.10.1 The Variation Calculus on s1 78.2.10.2 the Fourier Calculus 78.2.10.3 The Non-singular Matrix 78.2.10.4 A Second Fourier Calculus 78.2.11 The Calculus with General Potential 78.2.12 solution of the Differential Equations Using 2D Fourier Series 78.2.12.1 Funtional Dependence of the Physical Gesture 78.2.13 Parallels Between Performance Operators for scores and for Gestures 78.2.13.1 Some Detailed Calculation Regarding the Variations Calculous of s1 with Potential 78.2.14 Complex Time and the Artistic Effort 78.2.15 Opening the Aesthetic Question that Is Quantified in Lagrange Potentials 78.2.16 A Musical Composition by Maria Mannome Realized Using These Ideas 78.2.16.1 First Movement 78.2.16.2 Second Movement 78.2.11.1 The Variational Calculus of s1 with Potential 78.3 Global Performance Hypergestures 78.3.1 The Musical Situation: An Intuitive Introduction 78.4 Categorical Gestures and Global Performance Hypergestures 78.4.1 Categorical Gestures: The Case of Potentials 78.4.2 The Mathematics of Global Performance Hypergestures 78.5 World-Sheet Hypergestures for General Skeleta 78.6 A Global Variational Principle for the Lagrange formalism Part XIX Gestures in Music and Performance Theory, and in Ethnomusicology 79 Gesture Homology for counterpoint 79.1 Summary of Mathematical Theory of counterpoint: What It Is About and What Is Missing 79.2 Hypergestural Singular Homology 79.3 A Classical Example of a Topological Category from Counterpoint 79.3.1 Generators of H1pGXq for a Groupoid GX Defined by a Group Action 79.4 The Meaning of H1 for Counterpoint 79.5 Concluding Comments 80 Modulation Theory and Lie Brackets of Vector Fields 80.1 Introduction 80.1.1 Short Recapitulation of the Classical Model's Structure 80.2 Hypergestures Between Triadic Degree That Arc Parallel to Vector Fields 80.3 Lie Brackets Generate Vector Fields That Connect Symmetry-Related Degrees 80.4 Selecting Parallel Hypergestures That Arc Admissible for Modulation 80.5 The Other Direct Modulations 81 Hypergestures for Performance Stemmata 81.1 Motivation, Terminology, and Previous Results 81.1.1 Performance Stemmata and Performance Gestures of Locally Compact Points 81.2 Gestures with Lie Operators in Stemma Theory 81.3 Connecting Stemmatic Gestures for Weights and Performance Fields 81.4 Hamology of Weights Parameter Stemmata 81.5 A Concrete Example 81.6 A Final Comment 82. Composing and Analyzing with the Performing Body 82.1 Gesture: A Sign or a Totality? 82.2 A Gesture-Based Structural Reading in Rain Tree Sketch II by Torn 82.2.1 Process I: Synergy of Mirroring and Parallel Gestures 82.2.2 Process II: Towards Relaxation, Balance, and Weightfulness 82.3 The Last Leg of a Bodily Journey 82.3.1 Sheng for Piano, Smartphones, and Fixed Playback 82.3.2 Cross-modality of Gestures 82.3.3 Learning the Smartphone Instrument 82.3.4 Kinesthetic Awareness and Modes of Listening 82.4 Conclusion: Foregrounding the Performer’s Body 83 Gestural Analysis and Classification of a Conductor’s Movements 83.1 Gestures and Communication in Orchestral Conducting: A Case Study 83.1.1 Problematics and Solving Methods 83.1.2 Results, Consequences, Applications 83.1.3 Some Remarks 83.2 Hints for a Mathematical Description 83.3 Data Analysis 83.4 Conclusion 83.5 Addendum 84 Reviewing Flow, Gesture, and Spaces in Free Jazz 84.1 Improvisation: Defining Time 84.2 Flow, Gestures, Imaginary Time and Spaces in the Music Movie 84.2.1 The Compositional Character of the Pieces 84.2.2 Large Forms 84.2.3 Precision of Attacks 84.2.4 Co-presence of Different Time Layers 84.2.5 The Reality of Imaginary Time 84.2.6 Measuring Flow 84.2.7 Explicit Perception of Gestures 85 Gesture and Vocalization 85.1 Vocal Gesture 85.2 Vocal and Manual Motion 85.3 Gait 85.4 Hindustani Vocal Music 85.5 Notic Models and Kinetic Models 85.6 The Realist Pitfall 85.7 The Subjectivist Pitfall 85.8 Speech Gesture 86 Elements of a Future Vocal Gesture Theory 86.1 Why a Theory of Vocal Gestures? 86.1.1 Studying the Voice Without the Singer? 86.1.2 Parts of the Phonatory System and Their Functions 86.1.3 Imaginary Gestures in Real Time? 86.1.4 Space of Voice Parameters Gestures 86.1.5 About the Importance of Breathing and of Laryngeal Movements 86.1.6 Mathematical Description of Vocal Gestures 86.1.6.1 Why Such a Formalism? 86.1.6.2 Other Comments on Vocal Hypergestures 86.1.6.3 Branching 86.1.7 Gestures Thought by Singers 86.1.7.1 Cultures of the Voice: An Example from Ethnomusicology 86.1.7.2 Gregorian Chant and Gauls 86.2 A Powerful Tool from the Past for the Mathematical/Physical Theory of the Future: The Neumes of Gregorian Chant 86.2.1 Gestures in Gregorian Chant Didactics 86.2.2 Concept of Rhythm and Time 86.2.2.1 The Chironomic Game 86.2.2.2 Voice in Imaginary Time, Silence in Physical Time? 86.2.3 The Neumes 86.3 Connecting Physiology, Gestures and Notation. Toward New Neumes? 86.3.0.1 A New Score 86.3.1 New Neumes Part XXIV References and Index References Index The Topos of Music IV Roots Preface to the Second Edition Preface Volume IV Contents Book Set Volume Leitfaden Leitfaden I & II Leitfaden III Tom_CD Part XX Appendix: Sound A Common Parameter Spaces A.1 Physical Spaces A.1.1 Neutral Data A.1.1.1 Room Acoustics A.1.2 Sound Analysis and Synthesis A.1.2.1 Fourier A.1.2.2 Frequency Modulation A.1.2.3 Wavelets A.1.2.4 Some Remarks on Physical Modeling A.2 Mathematical and Symbolic Spaces A.2.1 Onset and Duration A.2.2 Amplitude and Crescendo A.2.3 Frequency and Glissando B Auditory Physiology and Psychology B.1 Physiology: From the Auricle to Heschl’s Gyri B.1.1 Outer Ear B.1.2 Middle Ear B.1.3 Inner Ear (Cochlea) B.1.4 Cochlear Hydrodynamics: The Travelling Wave B.1.5 Active Amplification of the Traveling Wave Motion B.1.6 Neural Processing B.2 Discriminating Tones: Werner Meyer-Eppler’s Valence Theory B.3 Symbolic, Physiological, and Psychological Aspects of Consonance and Dissonance B.3.1 Euler’s Gradus Function B.3.2 von Helmholtz’ Beat Model B.3.3 Psychometric Investigations by Plomp and Levelt B.3.4 Counterpoint B.3.5 Consonance and Dissonance: A Conceptual Field Part XXI Appendix: Mathematical Basics C Sets, Relations, Monoids, Groups C.1 Sets C.1.1 Examples of Sets C.2 Relations C.2.1 Universal Constructions C.2.2 Graphs and Quivers C.2.3 Monoids C.3 Groups C.3.1 Homomorphisms of Groups C.3.2 Direct, Semi-direct, and Wreath Products C.3.3 Sylow Theorems on C.3.4 Classification of Groups C3.4.1 Classification of Cyclic Groups C.3.4.2 Classification of Finitely Generated Abelian Groups C.3.5 General Affine Groups C.3.6 Permutation Groups D Rings and Algebras D.1 Basic Definitions and Constructions D.1.1 Universal Constructions D.1.1.1 Quiver Algebras D.2 Prime Factorization D.3 Euclidean Algorithm D.4 Approximation of Real Numbers by Fractions D.5 Some Special Issues D.5.1 Integers, Rationals, and Real Numbers E Modules, Linear, and Affine Transformations E.1 Modules and Linear Transformations E.1.1 Examples E.1.1 Examples E.2 Module Classification E.2.1 Dimension E.2.2 Endomorphisms on Dual Numbers E.2.3 Semi-simple Modules E.2.4 Jacobson Radical and Socle E.2.5 Theorem of Krull-Remak-Schmidt E.3 Categories of Modules and Affine Transformations E.3.1 Direct Sums E.3.2 Affine Forms and Tensors E.3.3 Biaffine Maps E.3.4 Symmetries of the Affine Plane E.3.5 Symmetries on Z2 E.3.6 Symmetries on Zn E.3.7 Complements on the Module of a Local Composition E.3.8 Fiber Products and Fiber Sums in Mod E.4 Complements of Commutative Algebra E.4.1 Localization E.4.2 Projective Modules E.4.3 Injective Modules E.4.4 Lie Algebras F Algebraic Geometry F.1 Locally Ringed Spaces F.2 Spectra of Commutative Rings F.2.1 Sober Spaces F.3 Schemes and Functors F.4 Algebraic and Geometric Structures on Schemes F.4.1 The Zariski Tangent Space F.5 Grassmannians F.6 Quotients G Categories, Topoi, and Logic G.1 Categories Instead of Sets G.1.1 Examples G.1.2 Functors G.1.3 Natural Transformations G.2 The Yoneda Lemma G.2.1 Universal Constructions: Adjoints, Limits, and Colimits G.2.2 Limit and Colimit Characterizations G.2.2.1 Special Results for Mod G.3 Topoi G.3.1 Subobject Classifiers G.3.2 Exponentiation G.3.3 Definition of Topoi G.4 Grothendieck Topologies G.4.1 Sheaves G.5 Formal Logic G.5.1 Propositional Calculus G.5.2 Predicate Logic G.5.3 A Formal Setup for Consistent Domains of Forms G.5.3.1 Morphisms Between Semiotics of Forms G.5.3.2 Local and Global Form Semiotics G.5.3.3 Connotator Form Semiotics H Complements on General and Algebraic Topology H.1 Topology H.1.1 General H.1.2 The Category of Topological Spaces H.1.3 Uniform Spaces H.1.4 Special Issues H.2 Algebraic Topology H.2.1 Simplicial Complexes H.2.2 Geometric Realization of a Simplicial Complex H.2.3 Contiguity H.3 Simplicial Coefficient Systems H.3.1 Cohomology I Complements on Calculus I.1 Abstract on Calculus I.1.1 Norms and Metrics I.1.2 Completeness I.1.3 Differentiation I.2 Ordinary Differential Equations (ODEs) I.2.1 The Fundamental Theorem: Local Case I.2.2 The Fundamental Theorem: Global Case I.2.3 Flows and Differential Equations I.2.4 Vector Fields and Derivations I.3 Partial Differential Equations J More Complements on Mathematics J.1 Directed Graphs J.1.1 The Category of Directed Graphs (Digraphs) J.1.1.1 Unordered Graphs J.1.2 Two Standard Constructions in Graph Theory J.1.3 The Topos of Digraphs J.2 Galois Theory J.3 Splines J.3.1 Some Simplex Constructions for Splines J.3.2 Definition of General Splines J.4 Topology and Topological Categories J.4.1 Topology J.4.1.1 Generators for Topologies J.4.1.2 Compact-Open Topology J.4.2 Topological Categories J.5 Complex Analysis J.6 Differentiable Manifolds J.6.1 Manifolds with Boundary J.6.2 The Tangent Manifold J.7 Tensor Fields J.7.1 Alternating Tensors J.7.2 Tangent Tensors J.8 Stokes’ Theorem J.9 Calculus of Variations J.10 Partial Differential Equations J.10.1 Explicit Calculation J.11 Algebraic Topology J.11.1 Homotopy Theory J.11.2 The Fundamental Group(oid) J.12 Homology J.12.1 Singular Homology J.13 Cohomology Part XXII Appendix: Complements in Physics K Complements on Physics K.1 Hamilton’s Variational Principle K.1.1 Euler-Lagrange Equations for a Non-relativistic Particle K.2 String Theory K.3 Duality and Supersymmetry K.4 Quantum Mechanics K.4.1 Banach and Hilbert Spaces K.4.1.1 Bounded Operators K.4.1.2 Lebesque Integration K.4.1.3 Lebesgue K.4.2 Geometry on Hilbert Spaces K.4.2.1 The K.4.3 Axioms for Quantum Mechanics K.4.3.1 Resolvents and Spectra K.4.4 The Spectral Theorem K.4.4.1 Projection-valued Measures Part XXIII Appendix: Tables L Euler’s Gradus Function M Just and Well-Tempered Tuning N Chord and Third Chain Classes N.1 Chord Classes N.2 Third Chain Classes O Two, Three, and Four Tone Motif Classes O.1 Two Tone Motifs in O.2 Two Tone Motifs in O.3 Three Tone Motifs in O.4 Four Tone Motifs in O.5 Three Tone Motifs in P Well-Tempered and Just Modulation Steps P.1 12-Tempered Modulation Steps P.1.1 Scale Orbits and Number of Quantized Modulations P.1.2 Quanta and Pivots for the Modulations Between Diatonic Major Scales (No.38.1) P.1.3 Quanta and Pivots for the Modulations Between Melodic Minor Scales (No.47.1) P.1.4 Quanta and Pivots for the Modulations Between Harmonic Minor Scales (No.54.1) P.1.5 Examples of 12-Tempered Modulations for All Fourth Relations P.2 2-3-5-Just Modulation Steps P.2.1 Modulation Steps Between Just Major Scales P.2.2 Modulation Steps Between Natural Minor Scales P.2.3 Modulation Steps from Natural Minor to Major Scales P.2.4 Modulation Steps from Major to Natural Minor Scales P.2.5 Modulation Steps Between Harmonic Minor Scales P.2.6 Modulation Steps Between Melodic Minor Scales P.2.7 General Modulation Behaviour for 32 Alterated Scales Q Counterpoint Steps Q.1 Contrapuntal Symmetries Q.1.1 Class No. 64 Q.1.2 Class No. 68 Q.1.3 Class No. 71 Q.1.4 Class No. 75 Q.1.5 Class No. 78 Q.1.6 Class No. 82 Q.2 Permitted Successors for the Major Scale Part XXIV References and Index References Index Ubiquitous Music (2014) Preface Acknowledgements Prologue—Ubiquitous Music: A Manifesto 1 Introduction 1.1 Computer Music and Ubiquitous Computing 1.2 Contributions 2 Concepts of Ubiquitous Music 3 Metaphors and Patterns for Ubiquitous Music 3.1 Design Patterns 3.2 Interaction and Mobile Devices Conclusions References Contents Contributors Part I Theory 1 Ubimus Through the Lens of Creativity Theories 1.1 Introduction 1.2 General Creativity Frameworks 1.3 Domain-Specific Creativity Models 1.3.1 Summary and Implications of Domain-Specific Models for Ubimus Research 1.4 Ecologically Grounded Creative Practice 1.4.1 Ecologically Grounded Practice and Ubimus Experimental Work 1.5 Targeting the Needs of Little-c Musical Creativity Conclusions References 2 Methods in Creativity-Centred Design for Ubiquitous Musical Activities 2.1 Introduction 2.2 Ubiquitous Music and Everyday Creativity 2.3 Mobility and Connectivity in Ubiquitous Music 2.4 Creativity-Centred Design 2.5 Phase 1: Defining Strategies for Ubimus Design 2.5.1 Avoid Early Domain Restriction 2.5.2 Support Rapid Prototyping 2.5.2.1 Prototyping Creative Interaction 2.5.2.2 Prototyping Signal Processing 2.5.3 Foster Social Interaction 2.6 Phase 2: Planning 2.6.1 First Workshop: Procedures 2.6.2 First Workshop: Results 2.6.3 Second Workshop: Procedures 2.6.4 Second Workshop: Results 2.7 Phase 3: Prototyping 2.7.1 Definition of Mixing 2.7.2 Definition of Time Tagging 2.7.3 MixDroid Prototype 2.7.4 Preliminary Experimental Results 2.8 Phase 4: Creativity Assessment of Time Tagging 2.8.1 Procedures 2.8.2 Creativity Assessment Results 2.8.3 Discussion of the Time-Tagging Study Conclusions Perspectives for Future Work References Part II Applications 3 Repertoire Remix in the Context of Festival City 3.1 Repertoire Remix in the Context of Festival City 3.2 Audience Integration in Network Music Environments 3.3 Design and Implementation 3.3.1 Shared Music Style Arranging Environment 3.3.2 Slider and Button Interface for Director 3.3.3 Web Streaming 3.3.4 Comment/Feedback System 3.4 Interpretation of Shared Visual Score by Performers 3.5 Pilot Run 3.6 Activity Assessment of the Pilot Run 3.7 Comments from Participants During the Pilot Run Conclusion References 4 Making Meaningful Musical Experiences Accessible Using the iPad 4.1 Introduction 4.1.1 Brief Description of the Project 4.2 Accessibility via Mobile Technologies 4.3 Meaningful Engagement 4.4 Resilience 4.5 Collaboration and Sustainability 4.6 Case Study: iPads and Music at the Murri School 4.6.1 Designing Music-Based Activities 4.6.2 Measuring Resilience and Engagement 4.7 Survey Results Summary 4.8 Qualitative Results Summary 4.8.1 Pre-intervention Results 4.9 Post-intervention Results 4.9.1 Classroom Management 4.9.2 Student Engagement 4.9.3 Teacher Engagement 4.9.4 The ``Signature'' Event 4.10 Findings 4.11 Lessons Learnt Conclusion References 5 Analogue Audio Recording Using Remote Servers 5.1 Introduction 5.1.1 Proposed Architecture and Access Model 5.2 A System for Remote Recording of Analogue Synthesisers 5.2.1 Motivation 5.2.2 Working Prototype 5.2.3 Sound Selection and Preview 5.3 Usage and Operation Opportunities 5.3.1 Typical Uses 5.3.2 Aggregate Value and Market Span 5.3.3 Relationship with the Traditional Options 5.3.4 Simple Cost Estimation 5.3.5 Ubiquitous Access with Uncompromised Quality 5.3.6 Other Business Models 5.4 Audio Processes Suitable for Access as Remote Servers 5.4.1 What Cannot Be Done 5.4.2 More About Synthesisers 5.4.3 Acoustic Instruments 5.4.4 Analogue Mixing 5.4.5 Effect Processors Conclusions References Part III Technology 6 Development Tools for Ubiquitous Music on the World Wide Web 6.1 Introduction 6.2 Audio Technologies for the Web 6.3 Csound-Based Web Application Design 6.4 Emscripten 6.4.1 CsoundEmscripten 6.4.1.1 Wrapping the Csound C API for Use with JavaScript 6.4.1.2 The CsoundEmscripten JavaScript Interface 6.4.1.3 A Simple Example 6.4.1.4 Limitations 6.5 Beyond Web Audio: Creating Audio Applications with PNaCl 6.5.1 The Pepper Plugin API 6.5.2 PNaCl 6.5.3 Csound for PNaCl 6.5.3.1 The JavaScript Interface 6.5.3.2 An Introductory Example 6.5.3.3 Limitations Conclusions References 7 Ubiquitous Music Ecosystems: Faust Programs in Csound 7.1 Introduction 7.2 Csound 7.2.1 Csound Programming 7.2.2 The Csound API 7.3 Faust 7.3.1 Faust Programming 7.3.2 Architectures 7.3.3 The Faust Library 7.4 The Faust Csound Unit Generators 7.4.1 Faustcompile 7.4.1.1 Syntax 7.4.1.2 Example 7.4.2 Faustaudio 7.4.2.1 Syntax 7.4.2.2 Example 7.4.3 Faustctl 7.4.3.1 Syntax 7.4.3.2 Example 7.4.4 Faustgen 7.4.4.1 Syntax 7.4.4.2 Example 7.5 Examples 7.5.1 A Sine Wave Generator 7.5.2 Karplus–Strong Synthesizer 7.5.3 Effects 7.6 Music Programming in a Multi-Language Environment 7.6.1 Separation of Concerns in Ubiquitous Music 7.6.2 Ubiquitous Music Ecosystems Conclusions References Index The Topos of Music-Errata (2008)