Computational Music Science (recopilación)

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)
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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 Lea ng'
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 Classi cation
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)