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

User Review - ztutz - LibraryThingA 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 |

1241 | |

1253 | |

### Other editions - View all

The Topos of Music: Geometric Logic of Concepts, Theory, and Performance Guerino Mazzola Limited preview - 2012 |

The Topos of Music: Geometric Logic of Concepts, Theory, and ..., Volume 1 Guerino Mazzola No preview available - 2002 |

The Topos of Music: Geometric Logic of Concepts, Theory, and Performance Guerino Mazzola No preview available - 2013 |

### Common terms and phrases

abstract according action analysis appendix approach associated calculated called canonical chapter charts chord classes classification common complex concept consider construction coordinates corresponding curve deal defined definition denotators described discussion domain duration dynamics elements evident example expression extension fact field Figure finite formal function functor give given global compositions harmonic hierarchy initial interpretation interval isomorphism linear local compositions look mathematical means metrical module morphism motif motives natural notes objects onset operator parameters particular performance physical pitch position possible precise predicates present problem projection question reference relation representative respect scale score selection sense sequence shape shows situation sound space specific structure Summary Suppose symbolic symmetry tempo theory third tonality topology transformation understanding values weight whereas