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

The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der TC6ne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of Geometrie der TC6ne 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 afoundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples.

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.- 24.2.1 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