The Topos of Music

Guerino Mazzola

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

In Collaboration with Stefan Goller and Stefan Miiller Contributions by Carlos Agon, Moreno Andreatta, Gerard Assayag, Jan Beran, Chantal Buteau, Roberto Ferretti, Anja Fleischer, Harald Fripertinger, Jorg Garbers, Werner Hemmert, Michael Leyton, Emilio LIuis Puebia, Mariana Montiel Hernandez, Thomas Noll, Joachim Stange-Elbe, Oliver Zahorka

Birkhauser Verlag Basel • Boston • Berlin

Contents I

Introduction and Orientation

1

1 What is Music About? 1.1 Fundamental Activities 1.2 Fundamental Scientific Domains

3 4 6

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 Seniiosis 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

9 10 11 12 12 12 13 14 14 16 16 17 17 17 19 21

3 Musical Ontology 3.1 Where is Music? 3.2 Depth and Complexity

23 24 25

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

29 32 33 34 35

xi

xii

II

CONTENTS

Navigation on Concept Spaces

37

5 Navigation 5.1 Music in the EncycloSpace 5.2 Receptive Navigation 5.3 Productive Navigation

39 40 44 45

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

47 48 50 55 58 61 61 63 66 67 67 69 70 76 79 85 89 93 99

III

Local Theory

103

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

105 106 108 109 114 118 121 122 127 129

8

135 137 139 154 158

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

CONTENTS 8.3.1 8.3.2 8.3.3 8.3.4 8.3.5

xiii Commenting the Concatenation Principle Embedding and Addressed Adjointness Universal Constructions on Local Compositions The Address Question Categories of Commutative Local Compositions

161 163 166 169 171

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

175 178 181 184 185 185 188

10 Paradigmatic Classification 10.1 Paradigmata in Musicology, Linguistics, and Mathematics 10.2 Transformation 10.3 Similarity 10.4 Fuzzy Concepts in the Humanities

191 192 196 198 200

11 Orbits 203 11.1 Gestalt and Symmetry Groups 203 11.2 The Framework for Local Classification 204 11.3 Orbits of Elementary Structures 205 11.3.1 Classification Techniques 205 11.3.2 The Local Classification Theorem 207 11.3.3 The Finite Case 216 11.3.4 Dimension 217 11.3.5 Chords 219 11.3.6 Empirical Harmonic Vocabularies 221 11.3.7 Self-addressed Chords 225 11.3.8 Motives 228 11.4 Enumeration Theory 231 11.4.1 Polya and de Bruijn Theory 232 11.4.2 Big Science for Big Numbers 238 11.5 Group-theoretical Methods in Composition and Theory 241 11.5.1 Aspects of Serialism 243 11.5.2 The American Tradition 247 11.6 Esthetic Implications of Classification 258 11.6.1 Jakobson's Poetic Function 259 11.6.2 Motivic Analysis: Schubert/Stolberg "Lied auf dem Wasser zu singen..." . 262 11.6.3 Composition: Mazzola/Baudelaire "La mort des artistes" 268 11.7 Mathematical Reflections on Historicity in Music 271 11.7.1 Jean-Jacques Natticz' Paradigmatic Theme 272 11.7.2 Groups as a Parameter of Historicity 272

xiv

CONTENTS

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

275 276 277 279 281 284 284 291 294 295

IV

297

Global Theory

13 Global Compositions 299 13.1 The Local-Global Dichotomy in Music 300 13.1.1 Musical and Mathematical Manifolds 307 13.2 What Are Global Compositions? 308 13.2.1 The Nerve of an Objective Global Composition 310 13.3 Functorial Global Compositions 314 13.4 Interpretations and the Vocabulary of Global Concepts 316 13.4.1 Iterated Interpretations 317 13.4.2 The Pitch Domain: Chains of Thirds, Ecclesiastical Modes, Triadic and Quaternary Degrees 318 13.4.3 Interpreting Time: Global Meters and Rhythms 326 13.4.4 Motivic Interpretations: Melodies and Themes 331 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

333 333 334 341 343 345 347

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

349 350 350 353 355 356 356 358 363 364

CONTENTS 15.3.2 Classifying Spaces

xv 366

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"

369 370 372 374 376 376 378 380 382 385

17 Esthetics and Classification 17.1 Understanding by Resolution: An Illustrative Example 17.2 Varesc's Program and Yoneda's Lemma

387 387 392

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

397 397 398 400 401 402 402 406 412 419 424

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

427 427 430 434 435

20 Visualization Principles 20.1 Problems 20.2 Folding Dimensions 20.2.1 R 2 ->• M 20.2.2 R™ -> R 20.2.3 An Explicit Construction of fi 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

439 439 442 442 443 444 445 446 446 448 448

xvi

CONTENTS 20.4 Compound Parametrized Objects 20.5 Examples

V

Topologies for Rhythm and Motives

449 451

453

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

455 455 456 457 459 461

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

465 466 468 469 472 472 473 475 477 479 479 481 482 484 487 490 491 493 496

VI

Harmony

499

23 Critical Preliminaries 23.1 Hugo Riemann 23.2 Paul Hindemith 23.3 Heinrich Schenker and Friedrich Salzer

501 502 503 503

24 Harmonic 24.1 Chord 24.1.1 24.1.2 24.1.3

505 506 506 512 514

Topology Perspectives Euler Perspectives 12-tcmpered Perspectives Enharmonic Projection

CONTENTS 24.2 Chord 24.2.1 24.2.2 24.2.3 24.2.4

xvii Topologies Extension and Intension Extension and Intension Topologies Faithful Addresses The Saturation Sheaf

518 518 520 523 526

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

529 530 532 532 534 537 538 540 543 545 546

26 Cadence 26.1 Making the Concept Precise 26.2 Classical Cadences Relating to 12-tempercd 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

551 552 553 553 555 556 558 560 560 561

27 Modulation 563 27.1 Modeling Modulation by Particle Interaction 564 27.1.1 Models and the Anthropic Principle 565 27.1.2 Classical Motivation and Heuristics 565 27.1.3 The General Background 568 27.1.4 The Well-Tempered Case 571 27.1.5 Reconstructing the Diatonic Scale from Modulation 574 27.1.6 The Case of Just Tuning 576 27.1.7 Quantized Modulations and Modulation Domains for Selected Scales . . . 581 27.2 Harmonic Tension 586 27.2.1 The Riemann Algebra 586 27.2.2 Weights on the Riemann Algebra 587 27.2.3 Harmonic Tensions from Classical Harmony? 590 27.2.4 Optimizing Harmonic Paths 591

xviii

CONTENTS

28 Applications 28.1 First Examples 28.1.1 Johann Sebastian Bach: Choral from "Himmelfahrtsoratorium" 28.1.2 Wolfgang Amadeus Mozart: "Zauberflote", Choir of Priests 28.1.3 Claude Debussy: "Preludes", Livre 1, No.4 28.2 Modulation in Beethoven's Sonata op. 106, Is* Movement 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 Bb ~» G via e" 3 in W 28.2.5 Modulation G -» Eb via Ug in W 28.2.6 Modulation Eb ~~> D/b from W to W* 28.2.7 Modulation D/b •* B via Ud/dt = Ugt/a within W* 28.2.8 Modulation B ~» B b from W* to W 28.2.9 Modulation Bb -^ Gb via Ubk within W 28.2.10 Modulation G\, ~» G via Uaja within W 28.2.11 Modulation G ~* B b via e3 within W 28.3 Rhythmical Modulation in "Synthesis" 28.3.1 Rhythmic Modes 28.3.2 Composition for Percussion Ensemble

VII

Counterpoint

593 594 595 598 600 603 603 605 607 608 608 608 609 609 610 610 610 610 611 613

615

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

617 617 619 620 620 622 625 626

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

629 630 634 635 637 637 641

31 Modeling Counterpoint by Local Symmetries 31.1 Deformations of the Strong Dichotomies 31.2 Contrapuntal Symmetries Are Local 31.3 The Counterpoint Theorem

645 645 647 649

CONTENTS 31.3.1 31.3.2 31.3.3 31.3.4

Some Preliminary Calculations Two Lemmata on Cardinalities of Intersections An Algorithm for Exhibiting the Contrapuntal Symmetries 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

xix 649 651 651 655 655 656 657

661

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

663 665 666 667 667 668 672 674 675 679

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

681 681 681 683 685 686 687 689 690 691 693

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

695 696 697 699 701 701 703 704 706

xx

CONTENTS

35 Hierarchies and Performance Scores 35.1 Performance Cells 35.2 The Category of Performance Cells 35.3 Hierarchies 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

711 711 713 714 718 718 722 723 728 728

IX

731

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

733 734 737 741 745

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

747 748 749 751 752 753

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

755 756 757 758 762 763 765 769 771

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

773 774 775 776 777 777 780 782 783

CONTENTS 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®

xxi 783 785 787 788 789 791 792 793 794 795 797 801 801

805

40 Architecture 40.1 The Overall Modularity 40.2 Frame and Modules

807 808 809

41 The 41.1 41.2 41.3 41.4 41.5

813 814 816 819 824 831

RUBETTE® Family MetroRUBETTE® MeloRUBETTE® HarmoRUBETTE® PcrformanceRUBETTE® 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

833 833 834 835 835 839 841 841 842 849 850

853 855 855 855

xxii

CONTENTS

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 Feuillctonism Inevitable? 45.2 "Political Correctness" in Performance—Reviewing Gould 45.3 Transversal Ethnomusicology

857 858 860 871 873 874 874 877 880 880 881 881 882 883 884 897

903 905 905 906 909

46 Critical Fibers 911 46.1 The Stemma Model of Critique 911 46.2 Fibers for Locally Linear Grammars 912 46.3 Algorithmic Extraction of Performance Fields 916 46.3.1 The Infinitesimal View on Expression 916 46.3.2 Real-time Processing of Expressive Performance 917 46.3.3 Score Performance Matching 918 46.3.4 Performance Field Calculation 919 46.3.5 Visualization 921 46.3.6 The EspressoRUBETTE®: An Interactive Tool for Expression Extraction 922 46.4 Local Sections 925 46.4.1 Comparing Argerich and Horowitz 927

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

931 933 934 935 937

CONTENTS

xxiii

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

939 940 940 941 941 942 942 943

49 The 49.1 49.2 49.3 49.4

945 945 948 949 952

Paradigmatic Discourse on presto® The presto® Functional Scheme Modular Affine Transformations Ornaments and Variations 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 1 s t Movement: Sonata Form 50.3 2nd Movement: Variations 50.4 3 r d Movement: Scherzo 50.5 4th Movement: Fractal Syntax

955 956 956 956 958 959 963 964

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 of Temporal Boxes 51.5 A Musical Example

967 968 969 969 970 970 971 971 971 972 973 973 973 975 978 982 984 986

xxiv

XIV

CONTENTS

String Quartet Theory

991

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

993 994 994 995 996 997

53 Estimation of Resolution Parameters 53.1 Parameter Spaces for Violins 53.2 Estimation

999 1000 1003

54 The Case of Counterpoint and Harmony 54.1 Counterpoint 54.2 Harmony 54.3 Effective Selection

XV

Appendix: Sound

1007 1007 1008 1009

1011

A Common Parameter Spaces A.I 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

1013 1013 1014 1018 1028 1028 1029 1031

B Auditory Physiology and Psychology B.I Physiology: From the Auricle to Heschl's Gyri B.I.I Outer Ear B.1.2 Middle Ear B.1.3 Inner Ear (Cochlea) B.1.4 Cochlear Hydrodynamics: The Travelling Wave B.I.5 Active Amplification of 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 Hehnholtz' Beat Model B.3.3 Psychometric Investigations by Plomp and Levelt B.3.4 Counterpoint B.3.5 Consonance and Dissonance: A Conceptual Field

1035 1036 1036 1037 1037 1041 1042 1044 1046 1049 1049 1051 1052 1052 1053

CONTENTS

XVI

Appendix: Mathematical Basics

xxv

1055

C Sets, Relations, Monoids, Groups C.I Sets C.I.I 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

1057 1057 1058 1058 1062 1062 1063 1066 1066 1068 1069 1069 1070 1071

D Rings and Algebras D.I Basic Definitions and Constructions D.I.I 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

1075 1075 1077 1080 1080 1080 1081 1081

E Modules, Linear, and Affine Transformations E.I Modules and Linear Transformations E.I.I 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.3.5 Symmetries on Z 2 E.3.6 Symmetries on Z n E.3.7 Complements on the Module of a Local Composition E.3.8 Fiber Products and Fiber Sums in Mod E.4 Complements of Commutative Algebra

1083 1083 1084 1085 1085 1087 1087 1088 1090 1090 1091 1091 1093 1096 1097 1098 1099 1099 1101

xxvi

CONTENTS E.4.1 Localization E.4.2 Projective Modules E.4.3 Injective Modules E.4.4 Lie Algebras

F Algebraic Geometry F.I 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.I Categories Instead of Sets G.I.I Examples G.I.2 Functors G.I.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.I Topology H.I.I General H.I.2 The Category of Topological Spaces H.I.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

1101 1102 1103 1104

1107 1107 1108 1110 1111 1112 1112 1113 1114

1115 1115 1116 1117 1118 1120 1120 1122 1125 1126 1127 1127 1129 1130 1131 1131 1135 1137

1145 1145 1145 1146 1147 1147 1148 1148 1148 1150 1150

CONTENTS H.3.1 Cohomology I

Complements on Calculus 1.1 Abstract on Calculus 1.1.1 Norms and Metrics 1.1.2 Completeness 1.1.3 Differentiation 1.2 Ordinary Differential Equations (ODEs) 1.2.1 The Fundamental Theorem: Local Case 1.2.2 The Fundamental Theorem: Global Case 1.2.3 Flows and Differential Equations 1.2.4 Vector Fields and Derivations 1.3 Partial Differential Equations

XVII

Appendix: Tables

xxvii 1150 1153 1153 1153 1154 1155 1156 1156 1158 1160 1160 1161

1163

J Euler's Gradus Function

1165

K Just and Well-Tempered Tuning

1167

L Chord and Third Chain Classes L.I Chord Classes L.2 Third Chain Classes

1169 1169 1175

M Two, Three, and Four Tone Motif Classes M.I Two Tone Motifs in OnPiModl2.\2 M.2 Two Tone Motifs in OnPiModbA2 M.3 Three Tone Motifs in OnPiMod12,12 M.4 Four Tone Motifs in OnPiModl2,\2 M.5 Three Tone Motifs in OnPiMod5.12

1183 1183 1184 1185 1188 1195

N Well-Tempered and Just Modulation Steps 1197 N.I 12-Tempered Modulation Steps 1197 N.I.I Scale Orbits and Number of Quantized Modulations 1197 N.I.2 Quanta and Pivots for the Modulations Between Diatonic Major Scales (No.38.1) 1199 N.I.3 Quanta and Pivots for the Modulations Between Melodic Minor Scales (No.47.1) 1200 N.I.4 Quanta and Pivots for the Modulations Between Harmonic Minor Scales (No.54.1) 1202 N.I.5 Examples of 12-Tempered Modulations for all Fourth Relations 1203 N.2 2-3-5-Just Modulation Steps 1203 N.2.1 Modulation Steps between Just Major Scales 1203 N.2.2 Modulation Steps between Natural Minor Scales 1204 N.2.3 Modulation Steps From Natural Minor to Major Scales 1205

xxviii

CONTENTS N.2.4 N.2.5 N.2.6 N.2.7

Modulation Steps From Major to Natural Minor Scales Modulation Steps Between Harmonic Minor Scales Modulation Steps Between Melodic Minor Scales General Modulation Behaviour for 32 Alterated Scales

O Counterpoint Steps O.I Contrapuntal Symmetries 0.1.1 Class Nr. 64 0.1.2 Class Nr. 68 0.1.3 Class Nr. 71 0.1.4 Class Nr. 75 0.1.5 Class Nr. 78 0.1.6 Class Nr. 82 0.2 Permitted Successors for the Major Scale

XVIII

References

1206 1206 1207 1208 1211 1211 1211 1212 1213 1214 1215 1216 1217

1219

Bibliography

1221

Index

1253