The Topos of Music, 2002 Geometric Logic of Concepts, Theory, and Performance

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.- 24.2.1 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.

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