The Topos of Music, w. CD-ROM
Geometric Logic of Concepts, Theory, and Performance
(Sprache: Englisch)
The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Töne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication...
Leider schon ausverkauft
versandkostenfrei
Buch
160.49 €
80 DeutschlandCard Punkte
sammeln
- Lastschrift, Kreditkarte, Paypal, Rechnung
- Kostenlose Rücksendung
- Ratenzahlung möglich
Produktdetails
Produktinformationen zu „The Topos of Music, w. CD-ROM “
The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Töne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of Geometrie der Töne in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a classification theory of musical objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies.
The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical music theory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances.
The book's formal language and models are currently being used by leading researchers in Europe and Northern America and have become a foundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples.
Klappentext zu „The Topos of Music, w. CD-ROM “
Man kann einen jeden BegrifJ, einen jeden Titel, darunter viele Erkenntnisse gehoren, einen logischen Ort nennen. Immanuel Kant [258, p. B 324] This book's title subject, The Topos of Music, has been chosen to communicate a double message: First, the Greek word "topos" (r01rex; = location, site) alludes to the logical and transcendental location of the concept of music in the sense of Aristotle's [20, 592] and Kant's [258, p. B 324] topic. This view deals with the question of where music is situated as a concept and hence with the underlying ontological problem: What is the type of being and existence of music? The second message is a more technical understanding insofar as the system of musical signs can be associated with the mathematical theory of topoi, which realizes a powerful synthesis of geometric and logical theories. It laid the foundation of a thorough geometrization of logic and has been successful in central issues of algebraic geometry (Grothendieck, Deligne), independence proofs and intuitionistic logic (Cohen, Lawvere, Kripke). But this second message is intimately entwined with the first since the present concept framework of the musical sign system is technically based on topos theory, so the topos of music receives its top os-theoretic foundation. In this perspective, the double message of the book's title in fact condenses to a unified intention: to unite philosophical insight with mathematical explicitness.
Inhaltsverzeichnis zu „The Topos of Music, w. CD-ROM “
I Introduction and Orientation1 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
2.3.2 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
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 Leafing'
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
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
7.2.2 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
... mehr
Local Compositions
8.3 Categories of Local Compositions
8.3.1 Commenting 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 Classification
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 Pólya and de Bruijn Theory
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
11.6 Esthetic Implications of Classification
11.6.1 Jakobson's Poetic Function
11.6.2 Motivic Analysis: Schubert/Stolberg "Lied auf dem Wasser zu singen..."
11.6.3 Composition: Mazzola/Baudelaire "La mort des artistes"
11.7 Mathematical Reflections 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
12.3.2 Varieties of Sounds
12.3.3 Semiotics of Sound Classification
12.4 Making the Vague Precise
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 Quaternary Degrees
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
16.1.1 Automorphism Groups 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
18.3.2 Textual Semiosis
18.3.3 Atomic 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
19.2 The Topos of Music: An Overview
20 Visualization Principles
20.1 Problems
20.2 Folding Dimensions
20.2.1 ?2 ? ?
20.2.1 ?n ? ?
20.2.3 An Explicit Construction of ? 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
V Topologies for Rhythm and Motives
21 Metrics and Rhythmics
21.1 Review of Riemann and Jackendoff-Lerdahl Theories
21.1.1 Riemann's Weights
21.1.2 Jackendoff-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
22.3 Metrical Similarity
22.3.1 Examples of Distance Functions
22.4 Paradigmatic Groups
22.4.1 Examples of Paradigmatic Groups
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
22.8 Rudolph Reti's Motivic Analysis Revisited
22.8.1 Review of Concepts
22.8.2 Reconstruction
22.9 Motivic Weights
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
24.1.2 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
24.2.4 The Saturation Sheaf
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
25.3.1 Canonical Morphemes for European Harmony
25.3.2 Riemann Matrices
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
27.1.7 Quantized Modulations and Modulation Domains for Selected 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: "Zauberflöte", Choir of Priests
28.1.3 Claude Debussy: "Préludes", Livre 1, No.4
28.2 Modulation in Beethoven's Sonata op.106, 1stMovement
28.2.1 Introduction
28.2.2 The Fundamental Theses of Erwin Ratz and Jrgen Uhde
28.2.3 Overview of the Modulation Structure
28.2.4 Modulation $${{B}_{\flat }} \rightsquigarrow G$$ via e?3 in W
28.2.5 Modulation $$G \rightsquigarrow {{E}_{\flat }}$$ via UginW
28.2.6 Modulation $${{E}_{\flat }} \rightsquigarrow D/b$$ from WtoW*
28.2.7 Modulation $$D/b \rightsquigarrow B via {{U}_{{d/{{d}_{\sharp }}}}} = {{U}_{{{{g}_{\sharp }}/a}}}$$ within W*
28.2.8 Modulation $$B \rightsquigarrow {{B}_{\flat }}$$ from W*toW
28.2.9 Modulation $${{B}_{\flat }} \rightsquigarrow {{G}_{\flat }} via {{U}_{{{{b}_{\flat }}}}}$$ within W
28.2.10 Modulation $${{G}_{\flat }} \rightsquigarrow G via {{U}_{{{{a}_{\flat }}/a}}}$$ within W
28.2.11 Modulation $$G \rightsquigarrow {{B}_{\flat }}$$ via e3withinW
28.3 Rhythmical Modulation in "Synthesis"
28.3.1 Rhythmic Modes
28.3.2 Composition for Percussion Ensemble
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 a 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
30.2.4 Music and the Hippocampal Gate Function
31 Modeling Counterpoint by Local Symmetries
31.1 Deformations of the Strong Dichotomies
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?
VIII Structure Theory of Performance
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
32.4 Global Structure
32.4.1 Modeling Performance Syntax
32.4.2 The Formal Setup
32.4.3 Performance qua Interpretation of Interpretation
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
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
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
IX Expressive Semantics
36 Taxonomy of Expressive Performance
36.1 Feelings: Emotional Semantics
36.2 Motion: Gestural Semantics
36.3 Understanding: Rational Semantics
36.4 Cross-semantical Relations
37 Performance Grammars
37.1 Rule-based Grammars
37.1.1 The KTH School
37.1.2 Neil P. McAgnus Todd
37.1.3 The Zurich School
37.2 Remarks on Learning Grammars
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
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
39.7.2 Pianola Specialization
39.8 Locally Linear Grammars
X RUBATO®
40 Architecture
40.1 The Overall Modularity
40.2 Frame and Modules
41 The RUBETTE®Family
41.1 MetroRUBETTE®
41.2 MeloRUBETTE®
41.3 HarmoRUBETTE®
41.4 PerformanceRUBETTE®
41.5 PrimavistaRUBETTE®
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
42.3.2 Motif Analysis
42.3.3 Omission of Harmonic Analysis
42.4 Stemma Constructions
42.4.1 Performance Setup
42.4.2 Instrumental Setup
42.4.3 Global Discussion
XI Statistics of Analysis and Performance
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
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
44.2 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
44.3.4 Overview of Statistical Results
XII Inverse Performance Theory
45 Principles of Music Critique
45.1 Boiling down Infinity-Is Feuilletonism Inevitable?
45.2 "Political Correctness" in Performance-Reviewing Gould
45.3 Transversal Ethnomusicology
46 Critical Fibers
46.1 The Stemma Model of Critique
46.2 Fibers for Locally Linear Grammars
46.3 Algorithmic Extraction of Performance Fields
46.3.1 The Infinitesimal View on Expression
46.3.2 Real-time Processing of Expressive Performance
46.3.3 Score-Performance Matching
46.3.4 Performance Field Calculation
46.3.5 Visualization
46.3.6 The EspressoRUBETTE®: An Interactive Tool for Expression Extraction
46.4 Local Sections
46.4.1 Comparing Argerich and Horowitz
XIII Operationalization of Poiesis
47 Unfolding Geometry and Logic in Time
47.1 Performance of Logic and Geometry
47.2 Constructing Time from Geometry
47.3 Discourse and Insight
48 Local and Global Strategies in Composition
48.1 Local Paradigmatic Instances
48.1.1 Transformations
48.1.2 Variations
48.2 Global Poetical Syntax
48.2.1 Roman Jakobson's Horizontal Function
48.2.2 Roland Posner's Vertical Function
48.3 Structure and Process
49 The Paradigmatic Discourse on presto®
49.1 The presto®Functional Scheme
49.2 Modular Affine Transformations
49.3 Ornaments and Variations
49.4 Problems of Abstraction
50 Case Study I:"Synthesis" by Guerino Mazzola
50.1 The Overall Organization
50.1.1 The Material: 26 Classes of Three-Element Motives
50.1.2 Principles of the Four Movements and Instrumentation
50.2 1st Movement: Sonata Form
50.3 2nd Movement: Variations
50.4 3rd Movement: Scherzo
50.5 4th Movement: Fractal Syntax
51 Object-Oriented Programming in OpenMusic
51.1 Object-Oriented Language
51.1.1 Patches
51.1.2 Objects
51.1.3 Classes
51.1.4 Methods
51.1.5 Generic Functions
51.1.6 Message Passing
51.1.7 Inheritance
51.1.8 Boxes and Evaluation
51.1.9 Instantiation
51.2 Musical Object Framework
51.2.1 Internal Representation
51.2.2 Interface
51.3 Maquettes: Objects in Time
51.4 Meta-object Protocol
51.4.1 Reification of Temporal Boxes
51.5 A Musical Example
XIV String Quartet Theory
52 Historical and Theoretical Prerequisites
52.1 History
52.2 Theory of the String Quartet Following Ludwig Finscher
52.2.1 Four Part Texture
52.2.2 The Topos of Conversation Among Four Humanists
52.2.3 The Family of Violins
53 Estimation of Resolution Parameters
53.1 Parameter Spaces for Violins
53.2 Estimation
54 The Case of Counterpoint and Harmony
54.1 Counterpoint
54.2 Harmony
54.3 Effective Selection
XV Appendix: Sound
- A Common Parameter Spaces
- A.1 Physical Spaces
- A.1.1 Neutral Data
- A.1.2 Sound Analysis and Synthesis
- 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 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
XVI 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 p-groups
- C.3.4 Classification of 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.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.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.3.7 Complements on the Module of a Local Composition
- 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.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
- 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
XVII Appendix: Tables
- J Euler's Gradus Function
- K Just and Well-Tempered Tuning
- L Chord and Third Chain Classes
- L.1 Chord Classes
- L.2 Third Chain Classes
- M Two, Three, and Four Tone Motif Classes
- M.1 Two Tone Motifs in OnPiMod12,12
- M.2 Two Tone Motifs in OnPiMod5,12
- M.3 Three Tone Motifs in OnPiMod12,12
- M.4 Four Tone Motifs in OnPiMod12,12
- M.5 Three Tone Motifs in OnPiMod5,12
- N Well-Tempered and Just Modulation Steps
- N.1 12-Tempered Modulation Steps
- N.1.1 Scale Orbits and Number of Quantized Modulations
- N.1.2 Quanta and Pivots for the Modulations Between Diatonic Major Scales (No.38.1)
- N.1.3 Quanta and Pivots for the Modulations Between Melodic Minor Scales (No.47.1)
- N.1.4 Quanta and Pivots for the Modulations Between Harmonic Minor Scales (No.54.1)
- N.1.5 Examples of 12-Tempered Modulations for all Fourth Relations
- N.2 2-3-5-Just Modulation Steps
- N.2.1 Modulation Steps between Just Major Scales
- N.2.2 Modulation Steps between Natural Minor Scales
- N.2.3 Modulation Steps From Natural Minor to Major Scales
- N.2.4 Modulation Steps From Major to Natural Minor Scales
- N.2.5 Modulation Steps Between Harmonic Minor Scales
- N.2.6 Modulation Steps Between Melodic Minor Scales
- N.2.7 General Modulation Behaviour for 32 Alterated Scales
- O Counterpoint Steps
- O.1 Contrapuntal Symmetries
- O.1.1 Class Nr. 64
- O.1.2 Class Nr. 68
- O.1.3 Class Nr. 71
- O.1.4 Class Nr. 75
- O.1.5 Class Nr. 78
- O.1.6 Class Nr. 82
- O.2 Permitted Successors for the Major Scale
XVIII References
8.3 Categories of Local Compositions
8.3.1 Commenting 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 Classification
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 Pólya and de Bruijn Theory
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
11.6 Esthetic Implications of Classification
11.6.1 Jakobson's Poetic Function
11.6.2 Motivic Analysis: Schubert/Stolberg "Lied auf dem Wasser zu singen..."
11.6.3 Composition: Mazzola/Baudelaire "La mort des artistes"
11.7 Mathematical Reflections 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
12.3.2 Varieties of Sounds
12.3.3 Semiotics of Sound Classification
12.4 Making the Vague Precise
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 Quaternary Degrees
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
16.1.1 Automorphism Groups 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
18.3.2 Textual Semiosis
18.3.3 Atomic 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
19.2 The Topos of Music: An Overview
20 Visualization Principles
20.1 Problems
20.2 Folding Dimensions
20.2.1 ?2 ? ?
20.2.1 ?n ? ?
20.2.3 An Explicit Construction of ? 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
V Topologies for Rhythm and Motives
21 Metrics and Rhythmics
21.1 Review of Riemann and Jackendoff-Lerdahl Theories
21.1.1 Riemann's Weights
21.1.2 Jackendoff-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
22.3 Metrical Similarity
22.3.1 Examples of Distance Functions
22.4 Paradigmatic Groups
22.4.1 Examples of Paradigmatic Groups
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
22.8 Rudolph Reti's Motivic Analysis Revisited
22.8.1 Review of Concepts
22.8.2 Reconstruction
22.9 Motivic Weights
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
24.1.2 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
24.2.4 The Saturation Sheaf
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
25.3.1 Canonical Morphemes for European Harmony
25.3.2 Riemann Matrices
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
27.1.7 Quantized Modulations and Modulation Domains for Selected 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: "Zauberflöte", Choir of Priests
28.1.3 Claude Debussy: "Préludes", Livre 1, No.4
28.2 Modulation in Beethoven's Sonata op.106, 1stMovement
28.2.1 Introduction
28.2.2 The Fundamental Theses of Erwin Ratz and Jrgen Uhde
28.2.3 Overview of the Modulation Structure
28.2.4 Modulation $${{B}_{\flat }} \rightsquigarrow G$$ via e?3 in W
28.2.5 Modulation $$G \rightsquigarrow {{E}_{\flat }}$$ via UginW
28.2.6 Modulation $${{E}_{\flat }} \rightsquigarrow D/b$$ from WtoW*
28.2.7 Modulation $$D/b \rightsquigarrow B via {{U}_{{d/{{d}_{\sharp }}}}} = {{U}_{{{{g}_{\sharp }}/a}}}$$ within W*
28.2.8 Modulation $$B \rightsquigarrow {{B}_{\flat }}$$ from W*toW
28.2.9 Modulation $${{B}_{\flat }} \rightsquigarrow {{G}_{\flat }} via {{U}_{{{{b}_{\flat }}}}}$$ within W
28.2.10 Modulation $${{G}_{\flat }} \rightsquigarrow G via {{U}_{{{{a}_{\flat }}/a}}}$$ within W
28.2.11 Modulation $$G \rightsquigarrow {{B}_{\flat }}$$ via e3withinW
28.3 Rhythmical Modulation in "Synthesis"
28.3.1 Rhythmic Modes
28.3.2 Composition for Percussion Ensemble
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 a 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
30.2.4 Music and the Hippocampal Gate Function
31 Modeling Counterpoint by Local Symmetries
31.1 Deformations of the Strong Dichotomies
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?
VIII Structure Theory of Performance
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
32.4 Global Structure
32.4.1 Modeling Performance Syntax
32.4.2 The Formal Setup
32.4.3 Performance qua Interpretation of Interpretation
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
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
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
IX Expressive Semantics
36 Taxonomy of Expressive Performance
36.1 Feelings: Emotional Semantics
36.2 Motion: Gestural Semantics
36.3 Understanding: Rational Semantics
36.4 Cross-semantical Relations
37 Performance Grammars
37.1 Rule-based Grammars
37.1.1 The KTH School
37.1.2 Neil P. McAgnus Todd
37.1.3 The Zurich School
37.2 Remarks on Learning Grammars
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
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
39.7.2 Pianola Specialization
39.8 Locally Linear Grammars
X RUBATO®
40 Architecture
40.1 The Overall Modularity
40.2 Frame and Modules
41 The RUBETTE®Family
41.1 MetroRUBETTE®
41.2 MeloRUBETTE®
41.3 HarmoRUBETTE®
41.4 PerformanceRUBETTE®
41.5 PrimavistaRUBETTE®
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
42.3.2 Motif Analysis
42.3.3 Omission of Harmonic Analysis
42.4 Stemma Constructions
42.4.1 Performance Setup
42.4.2 Instrumental Setup
42.4.3 Global Discussion
XI Statistics of Analysis and Performance
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
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
44.2 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
44.3.4 Overview of Statistical Results
XII Inverse Performance Theory
45 Principles of Music Critique
45.1 Boiling down Infinity-Is Feuilletonism Inevitable?
45.2 "Political Correctness" in Performance-Reviewing Gould
45.3 Transversal Ethnomusicology
46 Critical Fibers
46.1 The Stemma Model of Critique
46.2 Fibers for Locally Linear Grammars
46.3 Algorithmic Extraction of Performance Fields
46.3.1 The Infinitesimal View on Expression
46.3.2 Real-time Processing of Expressive Performance
46.3.3 Score-Performance Matching
46.3.4 Performance Field Calculation
46.3.5 Visualization
46.3.6 The EspressoRUBETTE®: An Interactive Tool for Expression Extraction
46.4 Local Sections
46.4.1 Comparing Argerich and Horowitz
XIII Operationalization of Poiesis
47 Unfolding Geometry and Logic in Time
47.1 Performance of Logic and Geometry
47.2 Constructing Time from Geometry
47.3 Discourse and Insight
48 Local and Global Strategies in Composition
48.1 Local Paradigmatic Instances
48.1.1 Transformations
48.1.2 Variations
48.2 Global Poetical Syntax
48.2.1 Roman Jakobson's Horizontal Function
48.2.2 Roland Posner's Vertical Function
48.3 Structure and Process
49 The Paradigmatic Discourse on presto®
49.1 The presto®Functional Scheme
49.2 Modular Affine Transformations
49.3 Ornaments and Variations
49.4 Problems of Abstraction
50 Case Study I:"Synthesis" by Guerino Mazzola
50.1 The Overall Organization
50.1.1 The Material: 26 Classes of Three-Element Motives
50.1.2 Principles of the Four Movements and Instrumentation
50.2 1st Movement: Sonata Form
50.3 2nd Movement: Variations
50.4 3rd Movement: Scherzo
50.5 4th Movement: Fractal Syntax
51 Object-Oriented Programming in OpenMusic
51.1 Object-Oriented Language
51.1.1 Patches
51.1.2 Objects
51.1.3 Classes
51.1.4 Methods
51.1.5 Generic Functions
51.1.6 Message Passing
51.1.7 Inheritance
51.1.8 Boxes and Evaluation
51.1.9 Instantiation
51.2 Musical Object Framework
51.2.1 Internal Representation
51.2.2 Interface
51.3 Maquettes: Objects in Time
51.4 Meta-object Protocol
51.4.1 Reification of Temporal Boxes
51.5 A Musical Example
XIV String Quartet Theory
52 Historical and Theoretical Prerequisites
52.1 History
52.2 Theory of the String Quartet Following Ludwig Finscher
52.2.1 Four Part Texture
52.2.2 The Topos of Conversation Among Four Humanists
52.2.3 The Family of Violins
53 Estimation of Resolution Parameters
53.1 Parameter Spaces for Violins
53.2 Estimation
54 The Case of Counterpoint and Harmony
54.1 Counterpoint
54.2 Harmony
54.3 Effective Selection
XV Appendix: Sound
- A Common Parameter Spaces
- A.1 Physical Spaces
- A.1.1 Neutral Data
- A.1.2 Sound Analysis and Synthesis
- 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 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
XVI 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 p-groups
- C.3.4 Classification of 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.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.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.3.7 Complements on the Module of a Local Composition
- 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.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
- 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
XVII Appendix: Tables
- J Euler's Gradus Function
- K Just and Well-Tempered Tuning
- L Chord and Third Chain Classes
- L.1 Chord Classes
- L.2 Third Chain Classes
- M Two, Three, and Four Tone Motif Classes
- M.1 Two Tone Motifs in OnPiMod12,12
- M.2 Two Tone Motifs in OnPiMod5,12
- M.3 Three Tone Motifs in OnPiMod12,12
- M.4 Four Tone Motifs in OnPiMod12,12
- M.5 Three Tone Motifs in OnPiMod5,12
- N Well-Tempered and Just Modulation Steps
- N.1 12-Tempered Modulation Steps
- N.1.1 Scale Orbits and Number of Quantized Modulations
- N.1.2 Quanta and Pivots for the Modulations Between Diatonic Major Scales (No.38.1)
- N.1.3 Quanta and Pivots for the Modulations Between Melodic Minor Scales (No.47.1)
- N.1.4 Quanta and Pivots for the Modulations Between Harmonic Minor Scales (No.54.1)
- N.1.5 Examples of 12-Tempered Modulations for all Fourth Relations
- N.2 2-3-5-Just Modulation Steps
- N.2.1 Modulation Steps between Just Major Scales
- N.2.2 Modulation Steps between Natural Minor Scales
- N.2.3 Modulation Steps From Natural Minor to Major Scales
- N.2.4 Modulation Steps From Major to Natural Minor Scales
- N.2.5 Modulation Steps Between Harmonic Minor Scales
- N.2.6 Modulation Steps Between Melodic Minor Scales
- N.2.7 General Modulation Behaviour for 32 Alterated Scales
- O Counterpoint Steps
- O.1 Contrapuntal Symmetries
- O.1.1 Class Nr. 64
- O.1.2 Class Nr. 68
- O.1.3 Class Nr. 71
- O.1.4 Class Nr. 75
- O.1.5 Class Nr. 78
- O.1.6 Class Nr. 82
- O.2 Permitted Successors for the Major Scale
XVIII References
... weniger
Bibliographische Angaben
- Autor: Guerino Mazzola
- 2002, 1344 Seiten, Maße: 18,9 x 25,6 cm, Gebunden, Englisch
- Mitarbeit: Göller,Stefan; Müller, Stefan
- Verlag: Springer
- ISBN-10: 3764357312
- ISBN-13: 9783764357313
Sprache:
Englisch
Kommentar zu "The Topos of Music, w. CD-ROM"
0 Gebrauchte Artikel zu „The Topos of Music, w. CD-ROM“
| Zustand | Preis | Porto | Zahlung | Verkäufer | Rating |
|---|
Schreiben Sie einen Kommentar zu "The Topos of Music, w. CD-ROM".
Kommentar verfassen