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AuthorMazzola, Guerino. author
TitleThe Topos of Music [electronic resource] : Geometric Logic of Concepts, Theory, and Performance / by Guerino Mazzola
ImprintBasel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2002
Connect tohttp://dx.doi.org/10.1007/978-3-0348-8141-8
Descript XCVI, 1344 p. online resource

CONTENT

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 Molinoโ{128}{153}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 Questionsโ{128}{148}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 โ{128}{152}Concept Leafingโ{128}{153} -- 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 Masonโ{128}{148}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 Yonedaโ{128}{153}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 Pรณlya 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 Jakobsonโ{128}{153}s Poetic Function -- 11.6.2 Motivic Analysis: Schubert/Stolberg โ{128}{156}Lied auf dem Wasser zu singen...โ{128}{157} -- 11.6.3 Composition: Mazzola/Baudelaire โ{128}{156}La mort des artistesโ{128}{157} -- 11.7 Mathematical Reflections on Historicity in Music -- 11.7.1 Jean-Jacques Nattiezโ{128}{153} 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. โ{128}{156}Moleculesโ{128}{157} -- 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 โ{128}{156}Moleculesโ{128}{157} -- 17 Esthetics and Classification -- 17.1 Understanding by Resolution: An Illustrative Example -- 17.2 Vareseโ{128}{153}s Program and Yonedaโ{128}{153}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 Jackendoffโ{128}{148}Lerdahl Theories -- 21.1.1 Riemannโ{128}{153}s Weights -- 21.1.2 Jackendoffโ{128}{148}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 Retiโ{128}{153}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 Signsโ{128}{148}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 โ{128}{156}Himmelfahrtsoratoriumโ{128}{157} -- 28.1.2 Wolfgang Amadeus Mozart: โ{128}{156}Zauberflรถteโ{128}{157}, Choir of Priests -- 28.1.3 Claude Debussy: โ{128}{156}Prรฉludesโ{128}{157}, Livre 1, No.4 -- 28.2 Modulation in Beethovenโ{128}{153}s Sonata op.106, 1stMovement --


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