With contributions by numerous experts

With contributions by numerous experts

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.- 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 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.…