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

I Introduction and Orientation......Page 27
What is Music About?......Page 29
Fundamental Activities......Page 30
Fundamental Scientific Domains......Page 32
Topography......Page 35
Layers of Reality......Page 36
Physical Reality......Page 37
Molino\'s Communication Stream......Page 38
Creator and Poietic Level......Page 39
Listener and Esthesic Level......Page 40
Expressions......Page 42
A Short Overview of Music Semiotics......Page 43
The Cube of Local Topography......Page 45
Topographical Navigation......Page 47
Musical Ontology......Page 49
Where is Music?......Page 50
Depth and Complexity......Page 51
Models and Experiments in Musicology......Page 55
Interior and Exterior Nature......Page 58
What Is a Musicological Experiment?......Page 59
Questions---Experiments of the Mind......Page 60
New Scientific Paradigms and Collaboratories......Page 61
II Navigation on Concept Spaces......Page 63
Navigation......Page 65
Music in the EncycloSpace......Page 66
Receptive Navigation......Page 70
Productive Navigation......Page 71
Denotators......Page 73
Universal Concept Formats......Page 74
First Naive Approach To Denotators......Page 76
Interpretations and Comments......Page 81
Ordering Denotators and `Concept Leafing\'......Page 84
Variable Addresses......Page 87
Formal Definition......Page 89
Discussion of the Form Typology......Page 92
Formal Definition of a Denotator......Page 93
Anchoring Forms in Modules......Page 95
First Examples and Comments on Modules in Music......Page 96
Regular and Circular Forms......Page 102
Regular Denotators......Page 105
Circular Denotators......Page 111
Ordering on Forms and Denotators......Page 115
Concretizations and Applications......Page 119
Concept Surgery and Denotator Semantics......Page 125
III Local Theory......Page 129
Local Compositions......Page 131
The Objects of Local Theory......Page 132
First Local Music Objects......Page 134
Chords and Scales......Page 135
Local Meters and Local Rhythms......Page 140
Motives......Page 144
Functorial Local Compositions......Page 147
First Elements of Local Theory......Page 148
Alterations Are Tangents......Page 153
The Theorem of Mason--Mazzola......Page 155
Symmetries and Morphisms......Page 161
Symmetries in Music......Page 163
Elementary Examples......Page 165
Morphisms of Local Compositions......Page 180
Categories of Local Compositions......Page 184
Commenting the Concatenation Principle......Page 187
Embedding and Addressed Adjointness......Page 189
Universal Constructions on Local Compositions......Page 192
The Address Question......Page 195
Categories of Commutative Local Compositions......Page 197
Yoneda Perspectives......Page 201
Morphisms Are Points......Page 204
Yoneda\'s Fundamental Lemma......Page 207
The Yoneda Philosophy......Page 210
Painting and Music......Page 211
The Art of Object-Oriented Programming......Page 214
Paradigmatic Classification......Page 217
Paradigmata in Musicology, Linguistics, and Mathematics......Page 218
Transformation......Page 222
Similarity......Page 224
Fuzzy Concepts in the Humanities......Page 226
Gestalt and Symmetry Groups......Page 229
The Framework for Local Classification......Page 230
Classification Techniques......Page 231
The Local Classification Theorem......Page 233
The Finite Case......Page 242
Dimension......Page 243
Chords......Page 245
Empirical Harmonic Vocabularies......Page 247
Self-addressed Chords......Page 251
Motives......Page 254
Enumeration Theory......Page 257
Pólya and de Bruijn Theory......Page 258
Big Science for Big Numbers......Page 264
Group-theoretical Methods in Composition and Theory......Page 267
Aspects of Serialism......Page 269
The American Tradition......Page 273
Esthetic Implications of Classification......Page 284
Jakobson\'s Poetic Function......Page 285
Motivic Analysis: Schubert/Stolberg ``Lied auf dem Wasser zu singen...\'\'......Page 288
Composition: Mazzola/Baudelaire ``La mort des artistes\'\'......Page 294
Mathematical Reflections on Historicity in Music......Page 297
Groups as a Parameter of Historicity......Page 298
Topological Specialization......Page 301
What Ehrenfels Neglected......Page 302
Topology......Page 303
Metrical Comparison......Page 305
Specialization Morphisms of Local Compositions......Page 307
Topographic Determinants of Sound Descriptions......Page 310
Varieties of Sounds......Page 317
Semiotics of Sound Classification......Page 320
Making the Vague Precise......Page 321
IV Global Theory......Page 323
Global Compositions......Page 325
The Local-Global Dichotomy in Music......Page 326
Musical and Mathematical Manifolds......Page 333
What Are Global Compositions?......Page 334
The Nerve of an Objective Global Composition......Page 336
Functorial Global Compositions......Page 340
Interpretations and the Vocabulary of Global Concepts......Page 342
Iterated Interpretations......Page 343
The Pitch Domain: Chains of Thirds, Ecclesiastical Modes, Triadic and Quaternary Degrees......Page 344
Interpreting Time: Global Meters and Rhythms......Page 352
Motivic Interpretations: Melodies and Themes......Page 357
Musical Motivation......Page 359
Global Morphisms......Page 360
Local Domains......Page 367
Nerves......Page 369
Simplicial Weights......Page 371
Categories of Commutative Global Compositions......Page 373
Global Classification......Page 375
Global Affine Functions......Page 376
Bilinear and Exterior Forms......Page 379
Deviation: Compositions vs. ``Molecules\'\'......Page 381
Global Standard Compositions......Page 382
Compositions from Module Complexes......Page 384
Orbits of Module Complexes Are Classifying......Page 389
Combinatorial Group Actions......Page 390
Classifying Spaces......Page 392
Classifying Interpretations......Page 395
Characterization of Interpretable Compositions......Page 396
Automorphism Groups of Interpretable Compositions......Page 398
A Cohomological Criterion......Page 400
Tesselation......Page 402
Mosaics......Page 404
Classifying Rational Rhythms and Canons......Page 406
Global American Set Theory......Page 408
Interpretable ``Molecules\'\'......Page 411
Understanding by Resolution: An Illustrative Example......Page 413
Varèse\'s Program and Yoneda\'s Lemma......Page 418
What Is the Case: The Existence Problem......Page 423
Merging Systematic and Historical Musicology......Page 424
Textual and Paratextual Semiosis......Page 426
Textual and Paratextual Signification......Page 427
The Category of Denotators......Page 428
Textual Semiosis......Page 432
Atomic Predicates......Page 438
Logical and Geometric Motivation......Page 445
Paratextuality......Page 450
The Grothendieck Topology......Page 453
Cohomology......Page 456
Marginalia on Presheaves......Page 460
The Topos of Music: An Overview......Page 461
Problems......Page 465
R2 R......Page 468
Rn R......Page 469
An Explicit Construction of  with Special Values.......Page 470
Folding Denotators......Page 471
Folding Colimits......Page 472
Folding Circular Denotators......Page 474
Compound Parametrized Objects......Page 475
Examples......Page 477
V Topologies for Rhythm and Motives......Page 479
Review of Riemann and Jackendoff--Lerdahl Theories......Page 481
Riemann\'s Weights......Page 482
Jackendoff--Lerdahl: Intrinsic Versus Extrinsic Time Structures......Page 483
Topologies of Global Meters and Associated Weights......Page 485
Macro-Events in the Time Domain......Page 487
Motif Gestalts......Page 491
Motivic Interpretation......Page 492
Shape Types......Page 494
Examples of Shape Types......Page 495
Examples of Distance Functions......Page 498
Paradigmatic Groups......Page 499
Examples of Paradigmatic Groups......Page 501
Pseudo-metrics on Orbits......Page 503
The Inheritance Property......Page 505
Cognitive Aspects of Inheritance......Page 507
Epsilon Topologies......Page 508
First Properties of the Epsilon Topologies......Page 510
Toroidal Topologies......Page 513
Rudolph Reti\'s Motivic Analysis Revisited......Page 516
Review of Concepts......Page 517
Reconstruction......Page 519
Motivic Weights......Page 522
VI Harmony......Page 525
Critical Preliminaries......Page 527
Hugo Riemann......Page 528
Heinrich Schenker and Friedrich Salzer......Page 529
Harmonic Topology......Page 531
Euler Perspectives......Page 532
12-tempered Perspectives......Page 538
Enharmonic Projection......Page 540
Extension and Intension......Page 544
Extension and Intension Topologies......Page 546
Faithful Addresses......Page 549
The Saturation Sheaf......Page 552
Harmonic Semantics......Page 555
Harmonic Signs---Overview......Page 556
Chains of Thirds......Page 558
American Jazz Theory......Page 560
Hans Straub: General Degrees in General Scales......Page 563
Function Theory......Page 564
Canonical Morphemes for European Harmony......Page 566
Riemann Matrices......Page 569
Chains of Thirds......Page 571
Tonal Functions from Absorbing Addresses......Page 572
Cadence......Page 577
Making the Concept Precise......Page 578
Cadences in Triadic Interpretations of Diatonic Scales......Page 579
Cadences in More General Interpretations......Page 581
Cadences in Self-addressed Tonalities of Morphology......Page 582
Self-addressed Cadences by Symmetries and Morphisms......Page 584
Tonalities in Third-Fifth Intonation......Page 586
Tonalities in Pythagorean Intonation......Page 587
Modulation......Page 589
Modeling Modulation by Particle Interaction......Page 590
Classical Motivation and Heuristics......Page 591
The General Background......Page 594
The Well-Tempered Case......Page 597
Reconstructing the Diatonic Scale from Modulation......Page 600
The Case of Just Tuning......Page 602
Quantized Modulations and Modulation Domains for Selected Scales......Page 607
The Riemann Algebra......Page 612
Weights on the Riemann Algebra......Page 613
Harmonic Tensions from Classical Harmony?......Page 616
Optimizing Harmonic Paths......Page 617
Applications......Page 619
First Examples......Page 620
Johann Sebastian Bach: Choral from ``Himmelfahrtsoratorium\'\'......Page 621
Wolfgang Amadeus Mozart: ``Zauberflöte\'\', Choir of Priests......Page 624
Claude Debussy: ``Préludes\'\', Livre 1, No.4......Page 626
Introduction......Page 629
The Fundamental Theses of Erwin Ratz and Jrgen Uhde......Page 631
Overview of the Modulation Structure......Page 633
Modulation E D/b from W to W*......Page 634
Modulation B B from W* to W......Page 635
Rhythmical Modulation in ``Synthesis\'\'......Page 636
Rhythmic Modes......Page 637
Composition for Percussion Ensemble......Page 639
VII Counterpoint......Page 641
Arrows and Alterations......Page 643
The Contrapuntal Interval Concept......Page 645
The Third Torus......Page 646
Musical Interpretation of the Interval Ring......Page 648
Self-addressed Arrows......Page 651
Change of Orientation......Page 652
Interval Dichotomies as a Contrast......Page 655
Dichotomies and Polarity......Page 656
The Consonance and Dissonance Dichotomy......Page 660
Fux and Riemann Consonances Are Isomorphic......Page 661
Empirical Evidence for the Polarity Function......Page 663
Music and the Hippocampal Gate Function......Page 667
Deformations of the Strong Dichotomies......Page 671
Contrapuntal Symmetries Are Local......Page 673
Some Preliminary Calculations......Page 675
An Algorithm for Exhibiting the Contrapuntal Symmetries......Page 677
The Classical Case: Consonances and Dissonances......Page 681
Discussion of the Counterpoint Theorem in the Light of Reduced Strict Style......Page 682
The Major Dichotomy---A Cultural Antipode?......Page 683
VIII Structure Theory of Performance......Page 687
Local and Global Performance Transformations......Page 689
Performance as a Reality Switch......Page 691
Why Do We Need Infinite Performance of the Same Piece?......Page 692
The Coherence of Local Performance Transformations......Page 693
Differential Morphisms of Local Compositions......Page 694
Global Structure......Page 698
Modeling Performance Syntax......Page 700
The Formal Setup......Page 701
Performance qua Interpretation of Interpretation......Page 705
Tempo......Page 707
Intonation......Page 709
Dynamics......Page 711
Genesis of the General Formalism......Page 712
The Question of Articulation......Page 713
The Formalism of Performance Fields......Page 715
What Performance Fields Signify......Page 716
Th.W. Adorno, W. Benjamin, and D. Raffman......Page 717
Towards Composition of Performance......Page 719
Initial Sets and Initial Performances......Page 721
Taking off with a Shifter......Page 722
Anchoring Onset......Page 723
The Concert Pitch......Page 725
Initializing Articulation......Page 727
Hit Point Theory......Page 729
Distances......Page 730
Flow Interpolation......Page 732
Performance Cells......Page 737
The Category of Performance Cells......Page 739
Hierarchies......Page 740
Classification Issues......Page 744
Example: The Piano and Violin Hierarchies......Page 748
Local Performance Scores......Page 749
Instrumental Fibers......Page 754
IX Expressive Semantics......Page 757
Taxonomy of Expressive Performance......Page 759
Feelings: Emotional Semantics......Page 760
Motion: Gestural Semantics......Page 763
Understanding: Rational Semantics......Page 767
Cross-semantical Relations......Page 771
Performance Grammars......Page 773
Rule-based Grammars......Page 774
The KTH School......Page 775
Neil P. McAgnus Todd......Page 777
The Zurich School......Page 778
Remarks on Learning Grammars......Page 779
Stemma Theory......Page 781
Motivation from Practising and Rehearsing......Page 782
Does Reproducibility of Performances Help Understanding?......Page 783
Tempo Curves Are Inadequate......Page 784
The Stemma Concept......Page 788
The General Setup of Matrilineal Sexual Propagation......Page 789
The Primary Mother---Taking Off......Page 791
Mono- and Polygamy---Local and Global Actions......Page 795
Family Life---Cross-Correlations......Page 797
Operator Theory......Page 799
Why Weights?......Page 800
Discrete and Continuous Weights......Page 801
Weight Recombination......Page 802
Dynamics......Page 803
Agogics......Page 806
Tuning and Intonation......Page 808
Ornaments......Page 809
Analytical Weights......Page 811
Taxonomy of Operators......Page 813
Splitting Operators......Page 814
Symbolic Operators......Page 815
Physical Operators......Page 817
Field Operators......Page 818
Tempo Operator......Page 819
Scalar Operator......Page 820
The Theory of Basis-Pianola Operators......Page 821
Basis Specialization......Page 823
Locally Linear Grammars......Page 827
X RUBATO\"472......Page 831
Architecture......Page 833
The Overall Modularity......Page 834
Frame and Modules......Page 835
The RUBETTE\"472 Family......Page 839
MetroRUBETTE\"472......Page 840
MeloRUBETTE\"472......Page 842
HarmoRUBETTE\"472......Page 845
PerformanceRUBETTE\"472......Page 850
PrimavistaRUBETTE\"472......Page 857
A Preliminary Experiment: Robert Schumann\'s ``Kuriose Geschichte\'\'......Page 859
Full Experiment: J.S. Bach\'s ``Kunst der Fuge\'\'......Page 860
Metric Analysis......Page 861
Motif Analysis......Page 865
Stemma Constructions......Page 867
Performance Setup......Page 868
Instrumental Setup......Page 875
Global Discussion......Page 876
XI Statistics of Analysis and Performance......Page 879
General Motivation......Page 881
Hierarchical Smoothing......Page 883
Hierarchical Decomposition......Page 884
Comparing Analyses of Bach, Schumann, and Webern......Page 886
Differential Operators and Regression......Page 897
Analytical Data......Page 899
The Concept......Page 900
The Formalism......Page 903
The Full Model......Page 906
The Results of Regression Analysis......Page 907
Relations between Tempo and Analysis......Page 908
Complex Relationships......Page 909
Commonalities and Diversities......Page 910
Overview of Statistical Results......Page 923
XII Inverse Performance Theory......Page 929
Boiling down Infinity---Is Feuilletonism Inevitable?......Page 931
``Political Correctness\'\' in Performance---Reviewing Gould......Page 932
Transversal Ethnomusicology......Page 935
The Stemma Model of Critique......Page 937
Fibers for Locally Linear Grammars......Page 938
The Infinitesimal View on Expression......Page 942
Real-time Processing of Expressive Performance......Page 943
Score--Performance Matching......Page 944
Performance Field Calculation......Page 945
Visualization......Page 947
The EspressoRUBETTE\"472: An Interactive Tool for Expression Extraction......Page 948
Local Sections......Page 951
Comparing Argerich and Horowitz......Page 953
XIII Operationalization of Poiesis......Page 957
Unfolding Geometry and Logic in Time......Page 959
Performance of Logic and Geometry......Page 960
Constructing Time from Geometry......Page 961
Discourse and Insight......Page 963
Local and Global Strategies in Composition......Page 965
Transformations......Page 966
Global Poetical Syntax......Page 967
Roland Posner\'s Vertical Function......Page 968
Structure and Process......Page 969
The presto\"472 Functional Scheme......Page 971
Modular Affine Transformations......Page 974
Ornaments and Variations......Page 975
Problems of Abstraction......Page 978
Case Study I:``Synthesis\'\' by Guerino Mazzola......Page 981
Principles of the Four Movements and Instrumentation......Page 982
1st Movement: Sonata Form......Page 984
2nd Movement: Variations......Page 985
3rd Movement: Scherzo......Page 989
4th Movement: Fractal Syntax......Page 990
Object-Oriented Programming in OpenMusic......Page 993
Object-Oriented Language......Page 994
Objects......Page 995
Methods......Page 996
Inheritance......Page 997
Boxes and Evaluation......Page 998
Internal Representation......Page 999
Interface......Page 1001
Maquettes: Objects in Time......Page 1004
Meta-object Protocol......Page 1008
Reification of Temporal Boxes......Page 1010
A Musical Example......Page 1012
XIV String Quartet Theory......Page 1017
Historical and Theoretical Prerequisites......Page 1019
Theory of the String Quartet Following Ludwig Finscher......Page 1020
Four Part Texture......Page 1021
The Topos of Conversation Among Four Humanists......Page 1022
The Family of Violins......Page 1023
Estimation of Resolution Parameters......Page 1025
Parameter Spaces for Violins......Page 1026
Estimation......Page 1029
Counterpoint......Page 1033
Harmony......Page 1034
Effective Selection......Page 1035
XV Appendix: Sound......Page 1037
Physical Spaces......Page 1039
Neutral Data......Page 1040
Sound Analysis and Synthesis......Page 1044
Onset and Duration......Page 1054
Amplitude and Crescendo......Page 1055
Frequency and Glissando......Page 1057
Auditory Physiology and Psychology......Page 1061
Outer Ear......Page 1062
Inner Ear (Cochlea)......Page 1063
Cochlear Hydrodynamics: The Travelling Wave......Page 1067
Active Amplification of the Traveling Wave Motion......Page 1068
Neural Processing......Page 1070
Discriminating Tones: Werner Meyer-Eppler\'s Valence Theory......Page 1072
Euler\'s Gradus Function......Page 1075
von Helmholtz\' Beat Model......Page 1077
Counterpoint......Page 1078
Consonance and Dissonance: A Conceptual Field......Page 1079
XVI Appendix: Mathematical Basics......Page 1081
Sets......Page 1083
Relations......Page 1084
Graphs and Quivers......Page 1088
Monoids......Page 1089
Homomorphisms of Groups......Page 1092
Direct, Semi-direct, and Wreath Products......Page 1094
Classification of Groups......Page 1095
General Affine Groups......Page 1096
Permutation Groups......Page 1097
Basic Definitions and Constructions......Page 1101
Universal Constructions......Page 1103
Approximation of Real Numbers by Fractions......Page 1106
Integers, Rationals, and Real Numbers......Page 1107
Modules and Linear Transformations......Page 1109
Examples......Page 1110
Dimension......Page 1111
Semi-Simple Modules......Page 1113
Jacobson Radical and Socle......Page 1114
Categories of Modules and Affine Transformations......Page 1116
Affine Forms and Tensors......Page 1117
Biaffine Maps......Page 1119
Symmetries of the Affine Plane......Page 1122
Symmetries on Z2......Page 1123
Symmetries on Zn......Page 1124
Fiber Products and Fiber Sums in Mod......Page 1125
Localization......Page 1127
Projective Modules......Page 1128
Injective Modules......Page 1129
Lie Algebras......Page 1130
Locally Ringed Spaces......Page 1133
Spectra of Commutative Rings......Page 1134
Sober Spaces......Page 1136
Schemes and Functors......Page 1137
The Zariski Tangent Space......Page 1138
Grassmannians......Page 1139
Quotients......Page 1140
Categories Instead of Sets......Page 1141
Examples......Page 1142
Functors......Page 1143
Natural Transformations......Page 1144
Universal Constructions: Adjoints, Limits, and Colimits......Page 1146
Limit and Colimit Characterizations......Page 1148
Topoi......Page 1151
Subobject Classifiers......Page 1152
Definition of Topoi......Page 1153
Grothendieck Topologies......Page 1155
Sheaves......Page 1156
Propositional Calculus......Page 1157
Predicate Logic......Page 1161
A Formal Setup for Consistent Domains of Forms......Page 1163
General......Page 1171
The Category of Topological Spaces......Page 1172
Special Issues......Page 1173
Geometric Realization of a Simplicial Complex......Page 1174
Cohomology......Page 1176
Norms and Metrics......Page 1179
Completeness......Page 1180
Differentiation......Page 1181
The Fundamental Theorem: Local Case......Page 1182
The Fundamental Theorem: Global Case......Page 1184
Vector Fields and Derivations......Page 1186
Partial Differential Equations......Page 1187
XVII Appendix: Tables......Page 1189
Euler\'s Gradus Function......Page 1191
Just and Well-Tempered Tuning......Page 1193
Chord Classes......Page 1195
Third Chain Classes......Page 1201
Two Tone Motifs in  OnPiMod12,12......Page 1209
Two Tone Motifs in  OnPiMod5,12......Page 1210
Three Tone Motifs in  OnPiMod12,12......Page 1211
Four Tone Motifs in  OnPiMod12,12......Page 1214
Three Tone Motifs in  OnPiMod5,12......Page 1221
Scale Orbits and Number of Quantized Modulations......Page 1223
Quanta and Pivots for the Modulations Between Diatonic Major Scales (No.38.1)......Page 1225
Quanta and Pivots for the Modulations Between Melodic Minor Scales (No.47.1)......Page 1226
Quanta and Pivots for the Modulations Between Harmonic Minor Scales (No.54.1)......Page 1228
Modulation Steps between Just Major Scales......Page 1229
Modulation Steps between Natural Minor Scales......Page 1230
Modulation Steps From Natural Minor to Major Scales......Page 1231
Modulation Steps Between Harmonic Minor Scales......Page 1232
Modulation Steps Between Melodic Minor Scales......Page 1233
General Modulation Behaviour for 32 Alterated Scales......Page 1234
Class Nr. 64......Page 1237
Class Nr. 68......Page 1238
Class Nr. 71......Page 1239
Class Nr. 75......Page 1240
Class Nr. 78......Page 1241
Class Nr. 82......Page 1243
Permitted Successors for the Major Scale......Page 1244
XVIII References......Page 1247
Index......Page 1249