Category theory & music?

Mazzola, Guerino. The Topos of Music I: Theory Geometric Logic, Classification, Harmony, Counterpoint, Motives, Rhythm. Springer, 2017. Xxx, 1337 pp.

Adorno is referenced as an important thinker for this book, but I imagine he would consider it an abomination. As interesting as I find category theory and topos theory and their possible applications, the issue here is not the intellectual exercise but the threat posed by the implementation of artificial intelligence, to which category theory is now contributing. This author purports to translate the ineffable into topoi, something I do not want to see done, if there is in fact a possibility of making this happen. Here are some interesting quotes from the book. (RD)

1 September 2024

…. It was Theodor Wiesengrund Adorno who recapitulated the question of identity of a musical work with respect to performance. According to Adorno [6] we have to recognize this fact: “Die Idee der Interpretation gehört zur Musik selber und ist ihr nicht adkzidenziell.” Performance is a substantial part of a composition’s identity. This means that the work of art is not fully existential before or without performance. [p. 186]

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It seems that Adorno and Valéry have uniquely stressed the exterior performance activity in music and poetry, but their concern is much deeper. We should keep in mind that performance is strongly tied to its rhetoric function as a means to express understanding, and in this respect, performance is not only a perspective of action but instantiation of understanding, of interpretation of given structures. In other words, the Adorno and Valéry approach is classificatory on the level of functorial representation of works of art. [p. 187]

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We shall vaporize these mystifications [classification in musicology] in the following chapters, the second yet seems already partly settled from our previous discussion on the Yoneda philosophy: classification in the sense of a determination of isomorphism classes is nothing else than the overall task of understanding an object as we have identified it in the functorial approach in the spirit of Adorno, Valéry, and Bätschmann in chapter 9. [p. 192]

*      *     *

Adorno and Benjamin [110] have associated performative adequacy with an activity of “infinitesimal precision”. We make plausible that their language suggests the language of vector fields—though not explicitly stated by these authors. Diana Raffman’s argument for ineffability of musical nuances in performance [432] is discussed. We relate this admittance of ineffability to the search for a powerful language as an extension of the powerless common language. [p. 691]

*      *      *

As the wording is chosen, micrologic is a logic in the smallest dimensions of a composition and its performance. The text suggests that this procedure could be misunderstood as opposed to artistic fantasy. Adorno evokes Benjamin’s observation that fantasy is involved in the infinitely small (“im unendlich Kleinen”), more precisely in the interpolation towards the infinitely small. For Adorno, this is a revealing insight into the ultimate, true performance. Adorno asks for the discovery and inspection of the innermost interspaces (“sind die minimalen Hohlräume zu entdecken”). Infinite interpolation is the tool to do so. And this is not contrary to artistic fantasy, it is, so to speak, the strongest microscopic instrument we have, and should use. Artistic fantasy is not the pseudo-romantic blurredness, but a maximum of precision, of intensity and interplay of minimal movements and forces.

This absolutely central insight of Adorno and Benjamin is not only astonishing in the musicological environment (though not as a category of Adorno’s and Benjamin’s discourse), it is also a very problematic approach insofar as the humanities—where their text belongs—do not have any means of making such allusions precise. The text is a kind of schizophrenic claim of non-mathematical experts in the words of mathematical concept frameworks: Interpolation, infinitely small, etc.

To the mathematically trained, the allusion to calculus is straightforward. No doubt, the language of the infinitely small is calculus. Is it this kind of language which Adorno and Benjamin were aiming at? What is intriguing is that they are talking of infinite interpolation. Between what? The score is a radically discrete symbolism. The infinite interpolation is not a priori inscripted into the score structure. And, what are those cavities (“Hohlräume”) in terms of music parameters and processes? In between the discrete score events, there must be some infinitely divisible space which encompasses the cavities, Adorno and Benjamin are zooming in and penetrating. A solution of this conceptual approach could in fact be the continuous and differentiable interpolation suggested in the previous considerations of extensions of local compositions and maps. We claim that our theory is the mathematically adequate concept framework to the Adorno-Benjamin approach. [p. 692]

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These perspectives should make clear that we scarcely understand the genuine concepts of performative coherence in their musical phenomenology and accordingly cannot construct mathematical models on the basis of such a blurred phenomenology. Adorno and Benjamin have given us the catchwords for a deeper investigation of performance, catchwords which we could very well transform into adequate mathematical concepts. But these theorists did not elaborate their conceptual germs to a degree of differentiation that could help describing and understanding the concrete artistic shaping of performance. We can hardly understand and even less forgive the tremendous lack of musicological conceptualization and knowledge about performance in view of the overly fluffy and too often ridiculously blasé music criticism in the feuilletons of our newspapers. [p. 772]

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[6] Adorno Th W: Fragment über Musik und Sprache. Stuttgart, Jahresring 1956

[7] Adorno Th W: Dergetreue Korrepetitor (1963). Gesammelte Schriften, Bd. 15, Suhrkamp, Frankfurt am Main 1976 [p. 1223]

Contents

[Preface v-ix]

[Diagrams xxix-xxx]

I Introduction and Orientation 1

1 What is Music About? 3

1.1 Fundamental Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Fundamental Scientific Domains . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Topography 9

2.1 Layers of Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Physical Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  11
2.1.2 Mental Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  12
2.1.3 Psychological Reality . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Molino’s Communication Stream . . . . . . . . . . . . . . . . . . . . . . .12

2.2.1 Creator and Poietic Level . . . . . . . . . . . . . . . . . . . . . .  13
2.2.2 Work and Neutral Level . . . . . . . . . . . . . . . . . . . . . . .  14
2.2.3 Listener and Esthesic Level . . . . . . . . . . . . . . . . . . . .  14

2.3 Semiosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  17
2.3.3 The Process of Signification . . . . . . . . . . . . . . . . . . .  17
2.3.4 A Short Overview of Music Semiotics . . . . . . . . . . . 17

2.4 The Cube of Local Topography . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Topographical Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21

3 Musical Ontology 23

3.1 Where is Music? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Depth and Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  25

4 Models and Experiments in Musicology 29

4.1 Interior and Exterior Nature . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 What Is a Musicological Experiment? . . . . . . . . . . . . . . . . . . . 33
4.3 Questions—Experiments of the Mind . . . . . . . . . . . . . . . . . . . 34
4.4 New Scientific Paradigms and Collaboratories . . . . . . . . . . . . 35

II Navigation on Concept Spaces 37

5 Navigation 39

5.1 Music in the EncycloSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Receptive Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 Productive Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  45

6 Denotators 47

6.1 Universal Concept Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.1.1 First Naive Approach To Denotators . . . . . . . . . . . .  50
6.1.2 Interpretations and Comments . . . . . . . . . . . . . . . . . 55
6.1.3 Ordering Denotators and ‘Concept Leafing’ . . . . . . . 58

6.2 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.2.1 Variable Addresses . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.2 Formal Definition . . . . . . . . . . . . . . . . . . . . . . .. . . 63
6.2.3 Discussion of the Form Typology . . . . . . . . .  . . . 66

6.3 Denotators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.3.1 Formal Definition of a Denotator . . . . . . . . . . . .  67

6.4 Anchoring Forms in Modules . . . . . . . . . . . . . . . . . . . . . . 69

6.4.1 First Examples and Comments on Modules in Music . . .  70

6.5 Regular and Circular Forms . . . . . . . . . . . . . . . . . . . . . . .  76
6.6 Regular Denotators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.7 Circular Denotators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.8 Ordering on Forms and Denotators . . . . . . . . . . . . . . . . .  89

6.8.1 Concretizations and Applications . . . . . . . . . . . . .  93

6.9 Concept Surgery and Denotator Semantics . . . . . . . . . . . 99

III Local Theory 103

7 Local Compositions 105

7.1 The Objects of Local Theory . . . . . . . . . . . . . . . . . . . . . . . . 106

7.2 First Local Music Objects . . . . . . . . . . . . . . . . . . . . . . . . . . .108

7.2.1 Chords and Scales . . . . . . . . . . . . . . . . . . . . . . . . . . .109
7.2.2 Local Meters and Local Rhythms . . . . . . . . . . . . .  114
7.2.3 Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.3 Functorial Local Compositions . . . . . . . . . . . . . . . . . . . . . . .121
7.4 First Elements of Local Theory . . . . . . . . . . . . . . . . . . . . . .122

7.5 Alterations Are Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . .127

7.5.1 The Theorem of Mason–Mazzola . . . . . . . . . . . . . .129

8 Symmetries and Morphisms 135

8.1 Symmetries in Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  137

8.1.1 Elementary Examples . . . . . . . . . . . . . . . . . . . . . . .  139

8.2 Morphisms of Local Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.3 Categories of Local Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.3.1 Commenting the Concatenation Principle . . . . . . . . . . . . .. 161
8.3.2 Embedding and Addressed Adjointness . . . . . . . . . . . . . . . 163
8.3.3 Universal Constructions on Local Compositions . . . . . . . . 166
8.3.4 The Address Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  169
8.3.5 Categories of Commutative Local Compositions . . . . . . . . 171

9 Yoneda Perspectives 175

9.1 Morphisms Are Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
9.2 Yoneda’s Fundamental Lemma . . . . . . . . . . . . . . . . . . . . . . . . .. 181
9.3 The Yoneda Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

9.4 Understanding Fine and Other Arts . . . . . . . . . . . . . . . . . . . . . . . . . 185

9.4.1 Painting and Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.4.2 The Art of Object-Oriented Programming . . . . . . . . 188

10 Paradigmatic Classification 191

10.1 Paradigmata in Musicology, Linguistics, and Mathematics . . . . 192
10.2 Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10.3 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.4 Fuzzy Concepts in the Humanities . . . . . . . . . . . . . . . . . . . .  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 Pólya 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 Nattiez’ Paradigmatic Theme . . . . . . . . . 272
11.7.2 Groups as a Parameter of Historicity . . . . . . . . . . . . . . . 272

12 Topological Specialization 275

12.1 What Ehrenfels Neglected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

12.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

12.2.1 Metrical Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
12.2.2 Specialization Morphisms of Local Compositions . . . . .. . 281

12.3 The Problem of Sound Classification . . . . . . . . . . . . . . . . . . . . . . . 284

12.3.1 Topographic Determinants of Sound Descriptions . . . . . 284
12.3.2 Varieties of Sounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
12.3.3 Semiotics of Sound Classification . . . . . . . . . . . . . . . . . . . . 294

12.4 Making the Vague Precise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

IV Global Theory 297

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 333

14.1 Musical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
14.2 Global Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
14.3 Local Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
14.4 Nerves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
14.5 Simplicial Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
14.6 Categories of Commutative Global Compositions . . . . . . . . . 347

15 Global Classification 349

15.1 Module Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

15.1.1 Global Affine Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
15.1.2 Bilinear and Exterior Forms . . . . . . . . . . . . . . . . . . . . . . . 353
15.1.3 Deviation: Compositions vs. “Molecules” . . . . . . . . . . . . . 355

15.2 The Resolution of a Global Composition . . . . . . . . . . . . . . . .. . . . 356

15.2.1 Global Standard Compositions . . . . . . . . . . . . . . . . . . . . . 356
15.2.2 Compositions from Module Complexes . . . . . . . . . . . . .  358

15.3 Orbits of Module Complexes Are Classifying . . . . . . . . . . . . . . . 363

15.3.1 Combinatorial Group Actions . . . . . . . . . . . . . . . . . . . . .  364
15.3.2 Classifying Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

16 Classifying Interpretations 369

16.1 Characterization of Interpretable Compositions . . . . . . . . . . . . 370

16.1.1 Automorphism Groups of Interpretable Compositions . . . . . 372
16.1.2 A Cohomological Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 374

16.2 Global Enumeration Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .  376

16.2.1 Tesselation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
16.2.2 Mosaics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
16.2.3 Classifying Rational Rhythms and Canons . . . . . . . . . . 380

16.3 Global American Set Theory . . . . . . 382
16.4 Interpretable “Molecules” . . . . . . . . . . . . . . . . . . . . . . . 385

17 Esthetics and Classification 387

17.1 Understanding by Resolution: An Illustrative Example . . . . . .  387
17.2 Varèse’s Program and Yoneda’s Lemma . . . . . . . . . . . . . . . . . . 392

18 Predicates 397

18.1 What Is the Case: The Existence Problem . . . . . . . . . . . . . . . . . . 397

18.1.1 Merging Systematic and Historical Musicology . . . .  . . . 398

18.2 Textual and Paratextual Semiosis . . . . . . . . . . . . . . . . . . . . . . . 400

18.2.1 Textual and Paratextual Signification . . . . . . . . . . .. . . 401

18.3 Textuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

18.3.1 The Category of Denotators . . . . . . . . . . . . . . . . . . . . . . . 402
18.3.2 Textual Semiosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
18.3.3 Atomic Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
18.3.4 Logical and Geometric Motivation . . . . . . . . . . . . . . . . . 419

18.4 Paratextuality . . . . . . . . . . . . . . . . . . . . . 424

19 Topoi of Music 427

19.1 The Grothendieck Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

19.1.1 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
19.1.2 Marginalia on Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

19.2 The Topos of Music: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 435

20 Visualization Principles 439

20.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

20.2 Folding Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

20.2.1 R2 ! R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
20.2.2 Rn ! R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
20.2.3 An Explicit Construction of μ with Special Values. . . . . 444

20.3 Folding Denotators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

20.3.1 Folding Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
20.3.2 Folding Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
20.3.3 Folding Powersets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
20.3.4 Folding Circular Denotators . . . . . . . . . . . . . . . . . . . . . . 448

20.4 Compound Parametrized Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
20.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

V Topologies for Rhythm and Motives 453

21 Metrics and Rhythmics 455

21.1 Review of Riemann and Jackendoff–Lerdahl Theories . . . . . . . . . . . . . . . . 455

21.1.1 Riemann’s Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
21.1.2 Jackendoff–Lerdahl: Intrinsic Versus Extrinsic Time Structures . . . . . . 457

21.2 Topologies of Global Meters and Associated Weights . . . . . . . . . . . . . . . . 459
21.3 Macro-Events in the Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 461

22 Motif Gestalts 465

22.1 Motivic Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

22.2 Shape Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

22.2.1 Examples of Shape Types . . . . . . . . . . . . . . . . . . . . 469

22.3 Metrical Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

22.3.1 Examples of Distance Functions . . . . . . . . . . . . . . . 472

22.4 Paradigmatic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

22.4.1 Examples of Paradigmatic Groups . . . . . . . . . . . . . . . . 475

22.5 Pseudo-metrics on Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

22.6 Topologies on Gestalts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

22.6.1 The Inheritance Property . . . . . . . . . . . . . . . . . . . . . . . 479
22.6.2 Cognitive Aspects of Inheritance . . . . . . . . . . . . . . . . . 481
22.6.3 Epsilon Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

22.7.1 Toroidal Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

22.8 Rudolph Reti’s Motivic Analysis Revisited . . . . . . . . . . . . . . . . . 490

22.8.1 Review of Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
22.8.2 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

22.9 Motivic Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

VI Harmony 499

23 Critical Preliminaries 501

23.1 Hugo Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
23.2 Paul Hindemith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
23.3 Heinrich Schenker and Friedrich Salzer . . . . . . . . . . . . . . . . . . . . . . . . 503

24 Harmonic Topology 505

24.1 Chord Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

24.1.1 Euler Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
24.1.2 12-tempered Perspectives . . . . . . . . . . . . . . . . . . . . . . . .  512
24.1.3 Enharmonic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

24.2 Chord Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

24.2.1 Extension and Intension . . . . . . . . . . . . . . . . . . . . . . . . . . 518
24.2.2 Extension and Intension Topologies . . . . . . . . . . . . . . . . 520
24.2.3 Faithful Addresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
24.2.4 The Saturation Sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

25 Harmonic Semantics 529

25.1 Harmonic Signs—Overview . . . . . . . . . . 530

25.2 Degree Theory . . . . . . . . . . . . . . . . . . . . . . . . 532

25.2.1 Chains of Thirds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
25.2.2 American Jazz Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
25.2.3 Hans Straub: General Degrees in General Scales . . . . . .  537

25.3 Function Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

25.3.1 Canonical Morphemes for European Harmony . . . . . . 540
25.3.2 Riemann Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
25.3.3 Chains of Thirds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
25.3.4 Tonal Functions from Absorbing Addresses . . . . . . . . 546

26 Cadence 551

26.1 Making the Concept Precise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

26.2 Classical Cadences Relating to 12-tempered Intonation . . . . . . . 553

26.2.1 Cadences in Triadic Interpretations of Diatonic Scales . . . . 553
26.2.2 Cadences in More General Interpretations . . . . . . . . . . . . . 555

26.3 Cadences in Self-addressed Tonalities of Morphology . . . . . . . . . . 556
26.4 Self-addressed Cadences by Symmetries and Morphisms . . . . . . . 558

26.5 Cadences for Just Intonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

26.5.1 Tonalities in Third-Fifth Intonation . . . . . . . . . . . . . . . . . . 560
26.5.2 Tonalities in Pythagorean Intonation . . . . . . . . . . . . . . . . . 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

28 Applications 593

28.1 First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594

28.1.1 Johann Sebastian Bach: Choral from “Himmelfahrtsoratorium” . . . . . . 595
28.1.2 Wolfgang Amadeus Mozart: “Zauberfl¨ote”, Choir of Priests . . . . . . . . 598
28.1.3 Claude Debussy: “Pr´eludes”, Livre 1, No.4 . . . . . . . . . . . . . . . . . 600

28.2 Modulation in Beethoven’s Sonata op.106, 1st Movement . . . . . . . . 603

28.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
28.2.2 The Fundamental Theses of Erwin Ratz and Jrgen Uhde . . . . . 605
28.2.3 Overview of the Modulation Structure . . . . . . . . . . . . . . . . . . . . 607
28.2.4 Modulation B[ G via e−3 in W . . . . . . . . . . . . . . . . . . . . . . 608
28.2.5 Modulation G E[ via Ug in W . . . . . . . . . . . . . . . . . . . . . . . 608
28.2.6 Modulation E[ D/b from W to W_ . . . . . . . . . . . . . . . . . . . . 608
28.2.7 Modulation D/b B via Ud/d] = Ug]/a within W_ . . . . . . . . . 609
28.2.8 Modulation B B[ from W_ to W . . . . . . . . . . . . . . . . . . . . . 609
28.2.9 Modulation B[ G[ via Ub[ within W . . . . . . . . . . . . . . . . . . . 610
28.2.10Modulation G[ G via Ua[/a within W . . . . . . . . . . . . . . . . . . . 610
28.2.11Modulation G B[ via e3 within W . . . . . . . . . . . . . . . . . . . . . 610

28.3 Rhythmical Modulation in “Synthesis” . . . . . . . . . . . . . . . . . . . . 610

28.3.1 Rhythmic Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
28.3.2 Composition for Percussion Ensemble . . . . . . . . . . . . . . 613

VII Counterpoint 615

29 Melodic Variation by Arrows 617

29.1 Arrows and Alterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
29.2 The Contrapuntal Interval Concept . . . . . . . . . . . . . . . . . . . . . . . . . . 619

29.3 The Algebra of Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

29.3.1 The Third Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

29.4 Musical Interpretation of the Interval Ring . . . . . . . . . . . . . . . . . . . . . 622
29.5 Self-addressed Arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
29.6 Change of Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626

30 Interval Dichotomies as a Contrast 629

30.1 Dichotomies and Polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630

30.2 The Consonance and Dissonance Dichotomy . . . . . . . . . . . . . . . 634

30.2.1 Fux and Riemann Consonances Are Isomorphic . . . . . 635
30.2.2 Induced Polarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
30.2.3 Empirical Evidence for the Polarity Function . . . . . . . . 637
30.2.4 Music and the Hippocampal Gate Function . . . . . . . . . 641

31 Modeling Counterpoint by Local Symmetries 645

31.1 Deformations of the Strong Dichotomies . . . . . . . . . . . . . . . . . . . . . . . 645

31.2 Contrapuntal Symmetries Are Local . . . . . . . . . . . . . . . . . . . . . . . . . 647

31.3 The Counterpoint Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

31.3.1 Some Preliminary Calculations . . . . . . . . . . . . . . . . . . . . . . . . 649
31.3.2 Two Lemmata on Cardinalities of Intersections . . . . . . . . . . . . . . . 651
31.3.3 An Algorithm for Exhibiting the Contrapuntal Symmetries . . . . . . . . 651
31.3.4 Transfer of the Counterpoint Rules to General Representatives of Strong Dichotomies . . . . . . . . . . 655

31.4 The Classical Case: Consonances and Dissonances . . . . . . . . . . . . . . . . . 655

31.4.1 Discussion of the Counterpoint Theorem in the Light of Reduced Strict Style . . . . . . . . . . . . . . . 656
31.4.2 The Major Dichotomy—A Cultural Antipode? . . . . . . . . . . . . . . . 657

VIII Structure Theory of Performance 661

32 Local and Global Performance Transformations 663

32.1 Performance as a Reality Switch . . . . . . . . . . . . . . . . . . . . . . . . . . . 665
32.2 Why Do We Need Infinite Performance of the Same Piece? . . . . 666

32.3 Local Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667

32.3.1 The Coherence of Local Performance Transformations . . . . . 667
32.3.2 Differential Morphisms of Local Compositions . . . . . . . . . . . . 668

32.4 Global Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672

32.4.1 Modeling Performance Syntax . . . . . . . . . . . . . . . . . . . . . . . . . 674
32.4.2 The Formal Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
32.4.3 Performance qua Interpretation of Interpretation . . . . . 679

33 Performance Fields 681

33.1 Classics: Tempo, Intonation, and Dynamics . . . . . . . . . . . . . . . 681

33.1.1 Tempo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

33.1.2 Intonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

33.1.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

33.2 Genesis of the General Formalism . . . . . . . . . . . . . . . . . . . . . 686

33.2.1 The Question of Articulation . . . . . . . . . . . . . . . . . . . . 687
33.2.2 The Formalism of Performance Fields . . . . . . . . . . . 689

33.3 What Performance Fields Signify . . . . . . . . . . . . . . . . . . . . . . . . . . . 690

33.3.1 Th.W. Adorno, W. Benjamin, and D. Raffman . . . . . . . . . . . . . . . 691
33.3.2 Towards Composition of Performance . . . . . . . . . . . . . . . . . . . . 693

34 Initial Sets and Initial Performances 695

34.1 Taking off with a Shifter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696
34.2 Anchoring Onset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
34.3 The Concert Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
34.4 Dynamical Anchors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701
34.5 Initializing Articulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701

34.6 Hit Point Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

34.6.1 Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704
34.6.2 Flow Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706

35 Hierarchies and Performance Scores 711

35.1 Performance Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
35.2 The Category of Performance Cells . . . . . . . . . . . . . . . . . . . . . . . . . . 713

35.3 Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714

35.3.1 Operations on Hierarchies . . . . . . . . . . . . . . . . . . . . . . 718
35.3.2 Classification Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 718
35.3.3 Example: The Piano and Violin Hierarchies . . . . . . . 722

35.4 Local Performance Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

35.5 Global Performance Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . 728

35.5.1 Instrumental Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . 728

IX Expressive Semantics 731

36 Taxonomy of Expressive Performance 733

36.1 Feelings: Emotional Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
36.2 Motion: Gestural Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737
36.3 Understanding: Rational Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . 741
36.4 Cross-semantical Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745

37 Performance Grammars 747

37.1 Rule-based Grammars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748

37.1.1 The KTH School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749
37.1.2 Neil P. McAgnus Todd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751
37.1.3 The Zurich School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752

37.2 Remarks on Learning Grammars . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

38 Stemma Theory 755

38.1 Motivation from Practising and Rehearsing . . . . . . . . . . . . . . . . . . . . . 756

38.1.1 Does Reproducibility of Performances Help Understanding? . . . . . . . . 757

38.2 Tempo Curves Are Inadequate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758

38.3 The Stemma Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762

38.3.1 The General Setup of Matrilineal Sexual Propagation . . . . . . . . . . . 763
38.3.2 The Primary Mother—Taking Off . . . . . . . . . . . . . . . . . . . . . . 765
38.3.3 Mono- and Polygamy—Local and Global Actions . . . . . . . . . . . . . . 769
38.3.4 Family Life—Cross-Correlations . . . . . . . . . . . . . . . . . . . . . . . 771

39 Operator Theory 773

39.1 Why Weights? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774

39.1.1 Discrete and Continuous Weights . . . . . . . . . . . . . . . . . . . . . . . 775
39.1.2 Weight Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776

39.2 Primavista Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777

39.2.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
39.2.2 Agogics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780
39.2.3 Tuning and Intonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782
39.2.4 Articulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783
39.2.5 Ornaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783

39.3 Analytical Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785

39.4 Taxonomy of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787

39.4.1 Splitting Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788
39.4.2 Symbolic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789
39.4.3 Physical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791
39.4.4 Field Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792

39.5 Tempo Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793

39.6 Scalar Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794

39.7 The Theory of Basis-Pianola Operators . . . . . . . . . . . . . . . . . . . . . . . 795

39.7.1 Basis Specialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797
39.7.2 Pianola Specialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801

39.8 Locally Linear Grammars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801

X RUBATOr 805

40 Architecture 807

40.1 The Overall Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808
40.2 Frame and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809

41 The RUBETTEr Family 813

41.1 MetroRUBETTEr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814
41.2 MeloRUBETTEr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816
41.3 HarmoRUBETTEr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819
41.4 PerformanceRUBETTEr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824
41.5 PrimavistaRUBETTEr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831

42 Performance Experiments 833

42.1 A Preliminary Experiment: Robert Schumann’s “Kuriose Geschichte” . . . . . . 833
42.2 Full Experiment: J.S. Bach’s “Kunst der Fuge” . . . . . . . . . . . . . . . . . . . 834

42.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835

42.3.1 Metric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835
42.3.2 Motif Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839
42.3.3 Omission of Harmonic Analysis . . . . . . . . . . . . . . 841

42.4 Stemma Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841

42.4.1 Performance Setup . . . . . . . . . . . . . . . . . . . . . . . . 842
42.4.2 Instrumental Setup . . . . . . . . . . . . . . . . . . . . . . . . 849
42.4.3 Global Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 850

XI Statistics of Analysis and Performance 853

43 Analysis of Analysis 855

43.1 Hierarchical Decomposition . . . . . . . . . . . . . . . . . . . . . . . 855

43.1.1 General Motivation . . . . . . . . . . . . . . . . . . . . . . . 855

43.1.2 Hierarchical Smoothing . . . . . . . . . . . . . . . . . . . 857

43.1.3 Hierarchical Decomposition . . . . . . . . . . . . . . . . 858

43.2 Comparing Analyses of Bach, Schumann, and Webern . . . . . 860

44 Differential Operators and Regression 871

44.0.1 Analytical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873

44.1 The Beran Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874

44.1.1 The Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874
44.1.2 The Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877

44.2 The Method of Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 880

44.2.1 The Full Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880
44.2.2 Step Forward Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881

44.3 The Results of Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 881

44.3.1 Relations between Tempo and Analysis . . . . . . . . . . . . . . . . . . . 882
44.3.2 Complex Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883
44.3.3 Commonalities and Diversities . . . . . . . . . . . . . . . . . . . . . . . . 884
44.3.4 Overview of Statistical Results . . . . . . . . . . . . . . . . . . . . . . . . 897

XII Inverse Performance Theory 903

45 Principles of Music Critique 905

45.1 Boiling down Infinity—Is Feuilletonism Inevitable? . . . . . . . . . . . . . . . . . 905
45.2 “Political Correctness” in Performance—Reviewing Gould . . . . . . . . . . . . . 906
45.3 Transversal Ethnomusicology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 EspressoRUBETTEr: An Interactive Tool for Expression Extraction 922

46.4 Local Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925

46.4.1 Comparing Argerich and Horowitz . . . . . . . . . . . . . . . . . . . . . . 927

XIII Operationalization of Poiesis 931

47 Unfolding Geometry and Logic in Time 933

47.1 Performance of Logic and Geometry . . . . . . . . . . . . . . . . . . . . 934
47.2 Constructing Time from Geometry . . . . . . . . . . . . . . . . . . . . . 935
47.3 Discourse and Insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937

48 Local and Global Strategies in Composition 939

48.1 Local Paradigmatic Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940

48.1.1 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940
48.1.2 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941

48.2 Global Poetical Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941

48.2.1 Roman Jakobson’s Horizontal Function . . . . . . . . . . . . . . . . . . . 942
48.2.2 Roland Posner’s Vertical Function . . . . . . . . . . . . . . . . . . . . . . 942

48.3 Structure and Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943

49 The Paradigmatic Discourse on prestor 945

49.1 The prestor Functional Scheme . . . . . . . . . . . . . . . . . . . . . . . . . .  945
49.2 Modular Affine Transformations . . . . . . . . . . . . . . . . . . . . . . . . .  948
49.3 Ornaments and Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949
49.4 Problems of Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952

50 Case Study I:“Synthesis” by Guerino Mazzola 955

50.1 The Overall Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956

50.1.1 The Material: 26 Classes of Three-Element Motives . . . . . . 956
50.1.2 Principles of the Four Movements and Instrumentation . . . 956

50.2 1st Movement: Sonata Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958
50.3 2nd Movement: Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959
50.4 3rd Movement: Scherzo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963
50.5 4th Movement: Fractal Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964

51 Object-Oriented Programming in OpenMusic 967

51.1 Object-Oriented Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968

51.1.1 Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
51.1.2 Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
51.1.3 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970
51.1.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970
51.1.5 Generic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971
51.1.6 Message Passing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971
51.1.7 Inheritance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971
51.1.8 Boxes and Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 972
51.1.9 Instantiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973

51.2 Musical Object Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973

51.2.1 Internal Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973
51.2.2 Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975

51.3 Maquettes: Objects in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978

51.4 Meta-object Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982

51.4.1 Reification of Temporal Boxes . . . . . . . . . . . . . . . . . . . . . . . . . 984

51.5 A Musical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986

XIV String Quartet Theory 991

52 Historical and Theoretical Prerequisites 993

52.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994

52.2 Theory of the String Quartet Following Ludwig Finscher . . . .  994

52.2.1 Four Part Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995
52.2.2 The Topos of Conversation Among Four Humanists . . . 996
52.2.3 The Family of Violins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997

53 Estimation of Resolution Parameters 999

53.1 Parameter Spaces for Violins . . . . . . . . . . . . . . . . . . . . . . . 1000
53.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003

54 The Case of Counterpoint and Harmony 1007

54.1 Counterpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007
54.2 Harmony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008
54.3 Effective Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1009

XV Appendix: Sound 1011

A Common Parameter Spaces 1013

A.1 Physical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013

A.1.1 Neutral Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014
A.1.2 Sound Analysis and Synthesis . . . . . . . . . . . . . . . . 1018

A.2 Mathematical and Symbolic Spaces . . . . . . . . . . . . . . . . . . 1028

A.2.1 Onset and Duration . . . . . . . . . . . . . . . . . . . . . . . . . 1028
A.2.2 Amplitude and Crescendo . . . . . . . . . . . . . . . . . .  . 1029
A.2.3 Frequency and Glissando . . . . . . . . . . . . . . . . . . . . 1031

B Auditory Physiology and Psychology 1035

B.1 Physiology: From the Auricle to Heschl’s Gyri . . . . . . . . . . . 1036

B.1.1 Outer Ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036
B.1.2 Middle Ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037
B.1.3 Inner Ear (Cochlea) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037
B.1.4 Cochlear Hydrodynamics: The Travelling Wave . . . . . . . . . . . . . 1041
B.1.5 Active Amplification of the Traveling Wave Motion . . . . . . . . . .  1042
B.1.6 Neural Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044

B.2 Discriminating Tones: Werner Meyer-Eppler’s Valence Theory . . . . 1046

B.3 Aspects of Consonance and Dissonance . . . . . . . . . . . . . . 1049

B.3.1 Euler’s Gradus Function . . . . . . . . . . . . . . . . . . . . 1049
B.3.2 von Helmholtz’ Beat Model . . . . . . . . . . . . . . . . . . 1051
B.3.3 Psychometric Investigations by Plomp and Levelt . . . 1052
B.3.4 Counterpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052
B.3.5 Consonance and Dissonance: A Conceptual Field . . . . . 1053

XVI Appendix: Mathematical Basics 1055

C Sets, Relations, Monoids, Groups 1057

C.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057

C.1.1 Examples of Sets . . . . . . . . . . . . . . . . . . . . . . . 1058

C.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1058

C.2.1 Universal Constructions . . . . . . . . . . . . . . . . . 1062
C.2.2 Graphs and Quivers . . . . . . . . . . . . . . . . . . . . . 1062
C.2.3 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

C.3 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066

C.3.1 Homomorphisms of Groups . . . . . . . . . . . . . . . . . 1066
C.3.2 Direct, Semi-direct, and Wreath Products . . . . . 1068
C.3.3 Sylow Theorems on p-groups . . . . . . . . . . . . . .  . 1069
C.3.4 Classification of Groups . . . . . . . . . . . . . . . . . . . . 1069
C.3.5 General Affine Groups . . . . . . . . . . . . . . . . . . . . . 1070
C.3.6 Permutation Groups . . . . . . . . . . . . . . . . . . . . . .  1071

D Rings and Algebras 1075

D.1 Basic Definitions and Constructions . . . . . . . . . . . . . . . . 1075

D.1.1 Universal Constructions . . . . . . . . . . . . . . . . . . 1077

D.2 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080
D.3 Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080
D.4 Approximation of Real Numbers by Fractions . . . . . . 1080

D.5 Some Special Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

D.5.1 Integers, Rationals, and Real Numbers . . . . . 1081

E Modules, Linear, and Affine Transformations 1083

E.1 Modules and Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 1083

E.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084

E.2 Module Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085

E.2.1 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085
E.2.2 Endomorphisms on Dual Numbers . . . . . . . . . . . . . . . 1087
E.2.3 Semi-Simple Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
E.2.4 Jacobson Radical and Socle . . . . . . . . . . . . . . . . .. . . . . 1088
E.2.5 Theorem of Krull–Remak–Schmidt . . . . . . . . . . . . . . 1090

E.3 Categories of Modules and Affine Transformations . . . .  . . . 1090

E.3.1 Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
E.3.2 Affine Forms and Tensors . . . . . . . . . . . . . . . . . . . . . . 1091
E.3.3 Biaffine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
E.3.4 Symmetries of the Affine Plane . . . . . . . . . . . . .  . . . . . 1096
E.3.5 Symmetries on Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
E.3.6 Symmetries on Zn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
E.3.7 Complements on the Module of a Local Composition . . . . . 1099
E.3.8 Fiber Products and Fiber Sums in Mod . . . . . . . . . . . . 1099

E.4 Complements of Commutative Algebra . . . . . . . . . . . . . . . . . . 1101

E.4.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
E.4.2 Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
E.4.3 Injective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
E.4.4 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104

F Algebraic Geometry 1107

F.1 Locally Ringed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107

F.2 Spectra of Commutative Rings . . . . . . . . . . . . . . . . . . . . . . . . . 1108

F.2.1 Sober Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110

F.3 Schemes and Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111

F.4 Algebraic and Geometric Structures on Schemes . . . . . . . . . . 1112

F.4.1 The Zariski Tangent Space . . . . . . . . . . . . . . . . . . . . . . 1112

F.5 Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113
F.6 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114

G Categories, Topoi, and Logic 1115

G.1 Categories Instead of Sets . . . . . . . . . . . . . . . . . . . . . . . 1115

G.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116
G.1.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117
G.1.3 Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118

G.2 The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1120

G.2.1 Universal Constructions: Adjoints, Limits, and Colimits . . . . . . . . . . 1120
G.2.2 Limit and Colimit Characterizations . . . . . . . . . . . . . . . . . . . . . 1122

G.3 Topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125

G.3.1 Subobject Classifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126
G.3.2 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127
G.3.3 Definition of Topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127

G.4 Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129

G.4.1 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1130

G.5 Formal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131

G.5.1 Propositional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131
G.5.2 Predicate Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135
G.5.3 A Formal Setup for Consistent Domains of Forms . . . . . . . . . . . . . . 1137

H Complements on General and Algebraic Topology 1145

H.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145

H.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145
H.1.2 The Category of Topological Spaces . . . . . . . . . . 1146
H.1.3 Uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147
H.1.4 Special Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147

H.2 Algebraic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148

H.2.1 Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148
H.2.2 Geometric Realization of a Simplicial Complex . . . . . 1148
H.2.3 Contiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150

H.3 Simplicial Coefficient Systems . . . . . . . . . . . . . . . . . . . . 1150

H.3.1 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150

I Complements on Calculus 1153

I.1 Abstract on Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153

I.1.1 Norms and Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153

I.2 Ordinary Differential Equations (ODEs) . . . . . . . . . . . . . . . . . . . . . . . 1156

I.2.1 The Fundamental Theorem: Local Case . . . . . . . . . . . . . . . . . . . 1156

I.3 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161

XVII Appendix: Tables 1163

J Euler’s Gradus Function 1165

K Just and Well-Tempered Tuning 1167

L Chord and Third Chain Classes 1169

L.1 Chord Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169
L.2 Third Chain Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175

M Two, Three, and Four Tone Motif Classes 1183

M.1 Two Tone Motifs in OnPiMod12,12 . . . . . . . . . . . . . . . . . . . . . . . . . 1183
M.2 Two Tone Motifs in OnPiMod5,12 . . . . . . . . . . . . . . . . . . . . . . . . . . 1184
M.3 Three Tone Motifs in OnPiMod12,12 . . . . . . . . . . . . . . . . . . . . . . .  1185
M.4 Four Tone Motifs in OnPiMod12,12 . . . . . . . . . . . . . . . . . . . . . . . . . 1188
M.5 Three Tone Motifs in OnPiMod5,12 . . . . . . . . . . . . . . . . . . . . . . . . . 1195

N Well-Tempered and Just Modulation Steps 1197

N.1 12-Tempered Modulation Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197

N.1.1 Scale Orbits and Number of Quantized Modulations . . . . . . . . . . . . 1197
N.1.2 Quanta and Pivots for the Modulations Between Diatonic Major Scales
(No.38.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199
N.1.3 Quanta and Pivots for the Modulations Between Melodic Minor Scales
(No.47.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1200
N.1.4 Quanta and Pivots for the Modulations Between Harmonic Minor Scales
(No.54.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202
N.1.5 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
N.2.4 Modulation Steps From Major to Natural Minor Scales . . . . . . . . . . 1206
N.2.5 Modulation Steps Between Harmonic Minor Scales . . . . . . . . . . . . . 1206
N.2.6 Modulation Steps Between Melodic Minor Scales . . . . . . . . . . . . . . 1207
N.2.7 General Modulation Behaviour for 32 Alterated Scales . . . . . . . . . . . 1208

O Counterpoint Steps 1211

O.1 Contrapuntal Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1211

O.1.1 Class Nr. 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1211
O.1.2 Class Nr. 68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212
O.1.3 Class Nr. 71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213
O.1.4 Class Nr. 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214
O.1.5 Class Nr. 78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216
O.1.6 Class Nr. 82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217

O.2 Permitted Successors for the Major Scale . . . . . . . . . . . . . . . . 1218

XVIII References 1221

Bibliography 1223

Index 1255

Category Theory — History & Philosophy: An Introductory Bibliography

Cybernetics & Artificial Intelligence: Ideology Critique

Theodor W. Adorno & Critical Theory Study Guide