The Topos of Music

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 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.- 35.3Hierarchies.- 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 ofTemporal 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 Amplificationof 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.

Topos of Music is an extensive and elaborate body of mathematical investigations into music and involves several and ontologically different levels of musical description. Albeit the author Guerino Mazzola lists 17 contributors and 2 collaborators, the book should be characterized as a monograph. Large portions of the content represent original research of Mazzola himself, and the material from other work is exposed from Mazzola's point of view and is well referenced. The preface preintimates an intended double meaning of the term topos in the title. On the one hand, it provides a mathematical anchor, which is programmatic for the entire approach: the concept of a cartesian closed category with a subobject classifier. (...)

Zentralblatt MATH