The topos of music: 1 Theory : geometric logic, classification, harmony, counterpoint, motives, rhythm
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Cham, Switzerland
Springer
[2017]
|
| Schriftenreihe: | Computional music science
Computional music science |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031533292&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | xlix, 543, 110 Seiten Illustrationen, Diagramme |
| ISBN: | 9783319643632 9783030097172 |
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Datensatz im Suchindex
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| adam_text | Volume I Contents Part I Introduction and Orientation 1 2 What Is Music About? ................................................................................................................................ 3 1.1 Fundamental Activities.................................................................................................................. 1.2 Fundamental Scientific Domains................................................................................................... 3 5 Topography.......................................................................................................................................................... 2.1 10 10 11 11 11 12 13 13 14 15 15 15 15 17 19 Musical Ontology............................................................................................................................................. 21 3.1 Where Is Music? ............................................................................................................................ 3.2 Depth and Complexity.................................................................................................................. 21 23 Models and Experiments in Musicology............................................................................................... 27 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 30 31 32 2.2 2.3 2.4 2.5 3 4 9 Layers of Reality............................................................................................................................ 2.1.1 Physical Reality.................................................................................................................. 2.1.2 Mental Reality.................................................................................................................... 2.1.3 Psychological Reality........................................................................................................ 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............................................................................................... Semiosis........................................................................................................................................... 2.3.1 Expressions..........................................................................................................................
2.3.2 Content............................................................................................................................... 2.3.3 The Process of Signification............................................................................................... 2.3.4 A Short Overview of Music Semiotics.............................................................................. The Cube of Local Topography ................................................................................................... Topographical Navigation ............................................................................................................ xiii
XIV Volume I Contents Part II Navigation on Concept Spaces 5 Navigation.................................................................................................................................................. 5.1 Music in the EncycloSpace................................................................................................................ 5.2 Receptive Navigation ........................................................................................................................ 5.3 Productive Navigation...................................................................................................................... 35 36 39 39 6 Denotators ................................................................................................................................................ 41 6.1 Universal Concept Formats.............................................................................................................. 42 6.1.1 First Naive Approach to Denotators.................................................................................. 43 6.1.2 Interpretations and Comments............................................................................................ 48 6.1.3 Ordering Denotators and ‘Concept Leafing’....................................................................... 50 6.2 Forms.................................................................................................................................................... 52 6.2.1 Variable Addresses
................................................................................................................ 53 6.2.2 Formal Definition .................................................................................................................. 54 6.2.3 Discussion of the Form Typology........................................................................................ 56 6.3 Denotators .......................................................................................................................................... 57 6.3.1 Formal Definition of a Denotator......................................................................................... 57 6.4 Anchoring Forms in Modules............................................................................................................ 59 6.4.1 First Examples and Comments on Modules in Music....................................................... 60 6.5 Regular and Circular Forms ............................................................................................................ 64 6.6 Regular Denotators............................................................................................................................ 66 6.7 Circular Denotators............................................................................................................................ 72 6.8 Ordering on Forms and Denotators................................................................................................ 75 6.8.1 Concretizations and
Applications........................................................................................ 78 6.9 Concept Surgery and Denotator Semantics.................................................................................... 83 Part III Local Theory 7 Local Compositions....................................................................................................................... 7.1 The Objects of Local Theory............................................................................................................ 7.2 First Local Music Objects................................................................................................................ 7.2.1 Chords and Scales.................................................................................................................. 7.2.2 Local Meters and Local Rhythms........................................................................................ 7.2.3 Motives.................................................................................................................................... 7.3 Functorial Local Compositions........................................................................................................ 7.4 First Elements of Local Theory ...................................................................................................... 7.5 Alterations Are Tangents.................................................................................................................. 7.5.1 The Theorem of Mason-Mazzola
......................................................................................... 89 89 92 92 96 99 101 103 107 108 8 Symmetries and Morphisms.............................................................................................................. 8.1 Symmetries in Music.......................................................................................................................... 8.1.1 Elementary Examples............................................................................................................ 8.2 Morphisms of Local Compositions................................................................................................... 8.3 Categories of Local Compositions.................................................................................................... 8.3.1 Commenting on the Concatenation Principle..................................................................... 8.3.2 Embedding and Addressed Adjointness............................................................................... 8.3.3 Universal Constructions on Local Compositions............................................................... 8.3.4 The Address Question .......................................................................................................... 8.3.5 Categories of Commutative Local Compositions............................................................... 113 114 116 128 132 134 136 138 140 142
Volume I Contents 9 XV 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...................................................................... 445 147 150 152 153 153 155 10 Paradigmatic Classification............................................................................................................. 10.1 Paradigmata in Musicology, Linguistics, and Mathematics....................................................... 10.2 Transformation................................................................................................................................ 10.3 Similarity ........................................................................................................................................ 10.4 Fuzzy Concepts in the
Humanities............................................................................................... 157 158 162 163 164 11 Orbits..................................................................................................................................................... 167 11.1 Gestalt and Symmetry Groups..................................................................................................... 11.2 The Framework for Local Classification ...................................................................................... 11.3 Orbits of Elementary Structures................................................................................................... 11.3.1 Classification Techniques................................................................................................... 11.3.2 The Local Classification Theorem .................................................................................... 11.3.3 The Finite Case.................................................................................................................. 11.3.4 Dimension............................................................................................................................ 11.3.5 Chords.................................................................................................................................. 11.3.6 Empirical Harmonic Vocabularies..................................................................................... 11.3.7 Self-addressed
Chords........................................................................................................ 11.3.8 Motives................................................................................................................................ 11.4 Enumeration Theory...................................................................................................................... 11.4.1 Pólya and de Bruijn Theory............................................................................................. 11.4.2 Big Science for Big Numbers............................................................................................. 11.5 Group-Theoretical Methods in Composition and Theory........................................................... 11.5.1 Aspects of Serialism.......................................................................................................... 11.5.2 The American Tradition.................................................................................................... 11.6 Esthetic Implications of Classification......................................................................................... 11.6.1 Jakobson’s Poetic Function............................................................................................... 11.6.2 Motivic Analysis: Schubert/Stolberg “Lied auf dem Wasser zu singen...”..................... 11.6.3 Composition: Mazzola/Baudelaire “La mort des artistes”............................................. 11.7 Mathematical Reflections on Historicity in
Music...................................................................... 11.7.1 Jean-Jacques Nattiez’ Paradigmatic Theme.................................................................... 11.7.2 Groups as a Parameter of Historicity................................................................................ 167 168 168 169 170 177 178 180 181 185 187 190 190 196 198 199 202 211 212 214 218 220 221 223 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............................................................................................................ 225 225 226 228 230 232 232 238 240 241
xvi Volume I Contents Part IV Global Theory 13 Global Compositions ............................................................................................................................ 13.1 The Local-Global Dichotomy in Music ........................................................................................... 13.1.1 Musical and Mathematical Manifolds................................................................................. 13.2 What Are Global Compositions? ..................................................................................................... 13.2.1 The Nerve of an Objective Global Composition ............................................................... 13.3 Functorial Global Compositions....................................................................................................... 13.4 Interpretations and the Vocabulary of Global Concepts............................................................... 13.4.1 Iterated Interpretations......................................................................................................... 13.4.2 The Pitch Domain: Chains of Thirds, Ecclesiastical Modes, Triadic and Quaternary Degrees.................................................................................................................................... 13.4.3 Interpreting Time: Global Meters and Rhythms............................................................... 13.4.4 Motivic Interpretations: Melodies and Themes ................................................................. 245 246 251 252 253 256 258 258 259 266
270 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......................................................................... 273 273 274 280 281 283 285 15 Global Classification.............................................................................................................................. 15.1 Module Complexes ............................................................................................................................ 15.1.1 Global Affine Functions......................................................................................................... 15.1.2 Bilinear and Exterior
Forms................................................................................................. 15.1.3 Deviation: Compositions vs. “Molecules” ........................................................................... 15.2 The Resolution of a Global Composition ....................................................................................... 15.2.1 Global Standard Compositions............................................................................................. 15.2.2 Compositions from Module Complexes............................................................................... 15.3 Orbits of Module Complexes Are Classifying................................................................................. 15.3.1 Combinatorial Group Actions............................................................................................... 15.3.2 Classifying Spaces.................................................................................................................. 287 287 288 290 291 292 293 294 298 299 300 16 Classifying Interpretations.................................................................................................................. 16.1 Characterization of Interpretable Compositions............................................................................. 16.1.1 Automorphism Groups of InterpretableCompositions...................................................... 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” ................................................................................................................ 303 303 306 307 309 309 310 312 314 316 17 Esthetics and Classification................................................................................................................ 319 17.1 Understanding by Resolution: An IllustrativeExample................................................................ 319 17.2 Varese’s Program and Yoneda’s Lemma........................................................................................ 323
Volume I Contents xvii 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 ..................................................................................................................................... 327 327 328 329 330 337 331 334 339 345 349 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................................................................................................... 35! 351 354 356 357 20 Visualization Principles......................................................................................................................... 20.1 Problems............................................................................................................................................... 20.2 Folding Dimensions............................................................................................................................. 20.2.1 R2 — R..................................................................................................................................... 20.2.2 R” — Ж..................................................................................................................................... 20.2.3 An Explicit Construction of μ with Special Values............................................................ 20.3 Folding
Denotators............................................................................................................................. 20.3.1 Folding Limits......................................................................................................................... 20.3.2 Folding Colimits ..................................................................................................................... 20.3.3 Folding Powersets................................................................................................................... 20.3.4 Folding Circular Denotators................................................................................................. 20.4 Compound Parametrized Objects..................................................................................................... 20.5 Examples.............................................................................................................................................. 361 361 363 363 364 365 366 366 367 368 369 370 371 Part V Topologies for Rhythm and Motives 21 Metrics and Rhythmics......................................................................................................................... 21.1 Review of Riemann and Jackendoff-Lerdahl Theories................................................................... 21.1.1 Riemann’s Weights................................................................................................................. 21.1.2 Jackendoff-Lerdahl: Intrinsic Versus Extrinsic Time Structures...................................... 21.2
Topologies of Global Meters and Associated Weights................................................................... 21.3 Macro-events in the Time Domain.................................................................................................. 375 375 375 376 378 380 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....................................................................................................................... 383 384 385 386 388 389 390 391 393 394
xviii Volume I Contents 22.6.1 The Inheritance Property....................................................................................................... 22.6.2 Cognitive Aspects of Inheritance......................................................................................... 22.6.3 Epsilon Topologies................................................................................................................... 22.7 First Properties of the EpsilonTopologies ....................................................................................... 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.................................................................................................................................. 395 396 397 399 401 404 404 406 408 Part VI Harmony 23 Critical Preliminaries............................................................................................................................ 23.1 Hugo
Riemann.................................................................................................................................... 23.2 Paul Hindemith.................................................................................................................................. 23.3 Heinrich Schenker and Friedrich Salzer........................................................................................... 413 414 414 414 24 Harmonic Topology................................................................................................................................ 24.1 Chord Perspectives............................................................................................................................ 24.1.1 Euler Perspectives................................................................................................................... 24.1.2 12-Tempered Perspectives..................................................................................................... 24.1.3 Enharmonic Projection........................................................................................................... 24.2 Chord Topologies................................................................................................................................ 24.2.1 Extension and Intension......................................................................................................... 24.2.2 Extension and Intension Topologies..................................................................................... 24.2.3 Faithful
Addresses................................................................................................................... 24.2.4 The Saturation Sheaf............................................................................................................. 417 418 418 422 425 427 427 429 431 434 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....................................................................... 435 436 437 437 439 442 442 444 447 447 449 26 Cadence...................................................................................................................................................... 26.1 Making the Concept Precise............................................................................................................ 26.2 Classical Cadences Relating to 12-Tempered Intonation ............................................................. 26.2.1 Cadences in Triadic Interpretations of Diatonic Scales..................................................... 26.2.2 Cadences in More General Interpretations......................................................................... 26.3 Cadences in Self-addressed Tonalities of Morphology................................................................... 26.4 Self-addressed Cadences by Symmetriesand Morphisms.............................................................. 26.5 Cadences for Just Intonation............................................................................................................ 26.5.1 Tonalities in Third-Fifth Intonation..................................................................................... 26.5.2 Tonalities in Pythagorean Intonation................................................................................... 453 454 454 455
456 457 459 460 460 461
Volume I Contents xix 27 Modulation................................................................................................................................................ 27.1 Modeling Modulation by Particle Interaction................................................................................. 27.1.1 Models and the Anthropic Principle ................................................................................... 27.1.2 Classical Motivation and Heuristics..................................................................................... 27.1.3 The General Background....................................................................................................... 27.1.4 The Well-Tempered Case....................................................................................................... 27.1.5 Reconstructing the Diatonic Scale from Modulation......................................................... 27.1.6 The Case of Just Tuning...................................................................................................... 27.1.7 Quantized Modulations and Modulation Domains for Selected Scales........................... 27.2 Harmonic Tension............................................................................................................................... 27.2.1 The Riemann Algebra .......................................................................................................... 27.2.2 Weights on the Riemann Algebra......................................................................................... 27.2.3
Harmonic Tensions from Classical Harmony?..................................................................... 27.2.4 Optimizing Harmonic Paths ................................................................................................ 453 464 464 465 467 469 471 473 477 481 481 482 484 485 28 Applications.......................................................... 28.1 First Examples.................................................................................................................................... 28.1.1 Johann Sebastian Bach: Choral from “Himmelfahrtsoratorium” ................................... 28.1.2 Wolfgang Amadeus Mozart: “Zauberflöte”, Choir of Priests........................................... 28.1.3 Claude Debussy: “Préludes”, Livre 1, No.4 ....................................................................... 28.2 Modulation in Beethoven’s Sonata op.106, 1st Movement........................................................... 28.2.1 Introduction........................................................................................................................ 28.2.2 The Fundamental Theses of Erwin Ratz and Jürgen Uhde......................................... 28.2.3 Overview of the Modulation Structure........................................................................... 28.2.4 Modulation B , ՝~՝·* G via e՜3 in W................................................................................. 28.2.5 Modulation G E , via Ug in W.................................................................................. 28.2.6 Modulation £į ՝~՝-*·
D/b from W to W* ......................................................................... 28.2.7 Modulation D/b ՝~՝- В via = Ugi/a within W* ................................................... 28.2.8 Modulation В ՝~՝-»· Д, from W* to W............................................................................. 28.2.9 Modulation Д, Gi, via I4b within W......................................................................... 28.2.10 Modulation Gi, ՝~՝- G via Uab/a within W....................................................................... 28.3 Rhythmical Modulation in “Synthesis”......................................................................................... 28.3.1 Rhythmic Modes................................................................................................................ 28.3.2 Composition for Percussion Ensemble............................................................................. 487 488 488 490 492 495 495 497 498 499 499 499 500 500 501 501 501 502 503 Part VII Counterpoint 29 Melodic Variation by Arrows............................................................................................................ 29.1 Arrows and Alterations .................................................................................................................. 29.2 The Contrapuntal Interval Concept.............................................................................................. 29.3 The Algebra of Intervals.................................................................................................................. 29.3.1
The Third Torus................................................................................................................ 29.4 Musical Interpretation of the Interval Ring .................................................................................. 29.5 Self-addressed Arrows...................................................................................................................... 29.6 Change of Orientation .................................................................................................................... 507 507 508 509 510 511 514 515 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........................................................... 517 517 520 521
XX Volume I Contents 30.2.2 Induced Polarities............................................................................................................... 523 30.2.3 Empirical Evidence for the Polarity Function ............................................................... 523 30.2.4 Music and the Hippocampal Gate Function................................................................... 527 31 Modeling Counterpoint by Local Symmetries.............................. ............................................. 31.1 Deformations of the Strong Dichotomies....................................................................................... 31.2 Contrapuntal Symmetries Are Local............................................................................................. 31.3 The Counterpoint Theorem............................................................................................................. 31.3.1 Some Preliminary Calculations......................................................................................... 31.3.2 Two Lemmata on Cardinalities of Intersections............................................................. 31.3.3 An Algorithm for Exhibiting the Contrapuntal Symmetries....................................... 31.3.4 Transfer of the Counterpoint Rules to General Representatives of Strong Dichotomies... 31.4 The Classical Case: Consonances and Dissonances..................................................................... 31.4.1 Discussion of the Counterpoint Theorem in the Light ofReduced Strict Style......... 31.4.2 The Major Dichotomy—A
Cultural Antipode?............................................................. 31.4.3 Software for Counterpoint and Theoretical Extentions ............................................... 531 531 533 534 534 536 536 540 540 541 542 543 Part XXIV References and Index References.......................................................................................................................................................... R.l Index R.33
Book Set Contents Part I Introduction and Orientation 1 2 What Is Music About?................................................................................................................................ 3 1.1 1.2 3 5 Fundamental Activities....................................................................................................................... Fundamental Scientific Domains....................................................................................................... Topography......................................................................................................................................................... 2.1 Layers of Reality................................................................................................................................. 2.1.1 Physical Reality...................................................................................................................... 2.1.2 Mental Reality........................................................................................................................ 2.1.3 Psychological Reality............................................................................................................ 2.2 Molino’s Communication Stream.................................................................................................... 2.2.1 Creator and Poietic Level .................................................................................................... 2.2.2 Work and Neutral
Level........................................................................................................ 2.2.3 Listener and Esthesic Level................................................................................................... 2.3 Semiosis................................................................................................................................................ 2.3.1 Expressions............................................................................................................................... 2.3.2 Content.................................................................................................................................... 2.3.3 The Process of Signification................................................................................................... 2.3.4 A Short Overview of Music Semiotics................................................................................. 2.4 The Cube of Local Topography ....................................................................................................... 2.5 Topographical Navigation ................................................................................................................ 3 4 9 10 10 11 11 11 12 13 13 14 15 15 15 15 1? 19 Musical Ontology............................................................................................................................................. 21 3.1 3.2 Where Is Music? ................................................................................................................................ Depth
and Complexity....................................................................................................................... 21 23 Models and Experiments in Musicology.............................................................................................. 27 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 30 31 32 XXI
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| owner | DE-12 |
| owner_facet | DE-12 |
| physical | xlix, 543, 110 Seiten Illustrationen, Diagramme |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | Springer |
| record_format | marc |
| series2 | Computional music science |
| spellingShingle | Mazzola, Guerino 1947- The topos of music |
| title | The topos of music |
| title_auth | The topos of music |
| title_exact_search | The topos of music |
| title_full | The topos of music 1 Theory : geometric logic, classification, harmony, counterpoint, motives, rhythm Guerino Mazzola |
| title_fullStr | The topos of music 1 Theory : geometric logic, classification, harmony, counterpoint, motives, rhythm Guerino Mazzola |
| title_full_unstemmed | The topos of music 1 Theory : geometric logic, classification, harmony, counterpoint, motives, rhythm Guerino Mazzola |
| title_short | The topos of music |
| title_sort | the topos of music theory geometric logic classification harmony counterpoint motives rhythm |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031533292&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV046153199 |
| work_keys_str_mv | AT mazzolaguerino thetoposofmusic1 |