The Topos of Music: Geometric Logic of Concepts, Theory, and Performance, Volume 1

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Springer Science & Business Media, Sep 23, 2002 - Mathematics - 1335 pages
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The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Töne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of Geometrie der Töne in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a classification theory of musical objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies. The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical music theory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances. The book's formal language and models are currently being used by leading researchers in Europe and Northern America and have become a foundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples.
 

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User Review  - ztutz - LibraryThing

A pompous and confused application of category theory to music. The math is advanced, and so is the obscurity of the encyclopedic and voluminous prose. This is a pity, since Mazzola makes a number of ... Read full review

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A beautiful book. Mazzola has generated a majestic work on the theory of music. Music which stills the savage beast is a resonant phenomenon throughout human history. Why? Because music is much closer to cognition than the author recognizes. This book is more properly a first step to a physically accurate model of cognition.
Res ipsa loquitur
 

Contents

Topography
9
TOM_CD
12
Musical Ontology
23
Models and Experiments in Musicology
29
Navigation
39
Denotators
49
legal GPL
71
Local Compositions
105
Performance Grammars
747
Stemma Theory
755
Operator Theory
773
Architecture
807
The RUBETTE Family
813
Performance Experiments
833
Statistics of Analysis and Performance
853
Differential Operators and Regression
871

Symmetries and Morphisms
135
Yoneda Perspectives
175
Paradigmatic Classification
191
Orbits
203
software presto documentation
268
Topological Specialization
275
Global Compositions
299
Global Perspectives
333
Global Classification
349
Classifying Interpretations
369
Esthetics and Classification
387
Predicates
397
Topoi of Music
427
Visualization Principles
439
Topologies for Rhythm and Motives
453
Motif Gestalts
465
Critical Preliminaries
501
Harmonic Semantics
529
Cadence
551
Modulation
563
Applications
593
Melodic Variation by Arrows
617
Interval Dichotomies as a Contrast
629
Modeling Counterpoint by Local Symmetries
645
Local and Global Performance Transformations
663
Performance Fields
681
Initial Sets and Initial Performances
695
Hierarchies and Performance Scores
711
Taxonomy of Expressive Performance
733
Principles of Music Critique
905
Critical Fibers
911
Unfolding Geometry and Logic in Time
933
Local and Global Strategies in Composition
939
The Paradigmatic Discourse on presto
945
Synthesis by Guerino Mazzola
955
ObjectOriented Programming in OpenMusic
967
Historical and Theoretical Prerequisites
993
Estimation of Resolution Parameters
999
The Case of Counterpoint and Harmony
1007
A Common Parameter Spaces
1013
sources
1014
B Auditory Physiology and Psychology
1035
Sets Relations Monoids Groups
1057
Rings and Algebras
1075
E Modules Linear and Affine Transformations
1083
F Algebraic Geometry
1107
G Categories Topoi and Logic
1115
H Complements on General and Algebraic Topology
1145
Complements on Calculus
1153
J Eulers Gradus Function
1165
Two Three and Four Tone Motif Classes
1183
N WellTempered and Just Modulation Steps
1197
O Counterpoint Steps
1211
Bibliography
1221
nextrubato
1241
Index
1253
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