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The Topos of Music Geometric Logic of Concepts, Theory, and Performance

Bibliographic Details
Main Author:	Mazzola, Guerino
Format:	eBook
Language:	English
Published:	Basel Birkhäuser 2002, 2002
Edition:	1st ed. 2002
Subjects:	Algebraic Geometry Information Visualization Geometry Science / Philosophy Applications Of Mathematics Data And Information Visualization Mathematics Philosophy Of Science
Online Access:	https://doi.org/10.1007/978-3-0348-8141-8?nosfx=y
Collection:	Springer Book Archives -2004 - Collection details see MPG.ReNa

Table of Contents:

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
2.3.2 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
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
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
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
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
2
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 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
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
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
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”
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
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

The Topos of Music Geometric Logic of Concepts, Theory, and Performance

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