## The Topos of Music: Geometric Logic of Concepts, Theory, and Performance, Volume 1Man kann einen jeden BegrifJ, einen jeden Titel, darunter viele Erkenntnisse gehoren, einen logischen Ort nennen. Immanuel Kant [258, p. B 324] This book's title subject, The Topos of Music, has been chosen to communicate a double message: First, the Greek word "topos" (r01rex; = location, site) alludes to the logical and transcendental location of the concept of music in the sense of Aristotle's [20, 592] and Kant's [258, p. B 324] topic. This view deals with the question of where music is situated as a concept and hence with the underlying ontological problem: What is the type of being and existence of music? The second message is a more technical understanding insofar as the system of musical signs can be associated with the mathematical theory of topoi, which realizes a powerful synthesis of geometric and logical theories. It laid the foundation of a thorough geometrization of logic and has been successful in central issues of algebraic geometry (Grothendieck, Deligne), independence proofs and intuitionistic logic (Cohen, Lawvere, Kripke). But this second message is intimately entwined with the first since the present concept framework of the musical sign system is technically based on topos theory, so the topos of music receives its top os-theoretic foundation. In this perspective, the double message of the book's title in fact condenses to a unified intention: to unite philosophical insight with mathematical explicitness. |

### What people are saying - Write a review

#### 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

User Review - Flag as inappropriate

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

What is Music About? | 3 |

Topography | 9 |

Musical Ontology | 23 |

Models and Experiments in Musicology | 29 |

programs | 35 |

Navigation | 39 |

Denotators | 47 |

Local Compositions | 105 |

Taxonomy of Expressive Performance 733 | 736 |

Performance Grammars | 747 |

Stemma Theory | 755 |

Operator Theory | 773 |

Architecture | 807 |

The RUBETTE Family | 813 |

contrapunctus_III | 817 |

Performance Experiments | 833 |

Symmetries and Morphisms | 135 |

Yoneda Perspectives | 175 |

Paradigmatic Classification | 191 |

Orbits | 203 |

sources | 258 |

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 |

1 rubato | 546 |

Cadence | 551 |

Modulation | 563 |

program | 573 |

Applications | 593 |

Melodic Variation by Arrows | 617 |

Interval Dichotomies as a Contrast | 630 |

Modeling Counterpoint by Local Symmetries | 645 |

Local and Global Performance Transformations | 663 |

performances | 665 |

Performance Fields | 681 |

Initial Sets and Initial Performances | 695 |

Hierarchies and Performance Scores | 711 |

Statistics of Analysis and Performance | 853 |

Differential Operators and Regression | 871 |

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 |

kuriose_geschichte | 972 |

Historical and Theoretical Prerequisites | 993 |

Estimation of Resolution Parameters | 999 |

The Case of Counterpoint and Harmony | 1007 |

A Common Parameter Spaces | 1013 |

B Auditory Physiology and Psychology | 1035 |

Sets Relations Monoids Groups | 1057 |

Rings and Algebras | 1075 |

E Modules Linear and AffineTransformations | 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 |

1227 | |

1253 | |

### Common terms and phrases

abstract affine affine functions ambient space analysis analytical appendix approach arrows automorphism bijection cadence canonical cantus firmus chapter charts chord classification colimit common commutative complex concept construction context contrapuntal coordinates corresponding defined definition deformation denotators diagram dichotomy discussion dodecaphonic duration dynamics endomorphisms example fact Figure finite function functor gestalt given global compositions harmonic hierarchy interval inversion isomorphism lemma linear local compositions mathematical means melodic metrical minor scale module monoid morphism motif motives musicology notation objective local compositions onset operator orbit paradigm paradigmatic parameters performance field perspective physical pitch classes pitch space poetical precise predicates projection recursive relation Riemann RUBATO RUBETTE scale score semantic semiosis semiotic sequence shape type simplexes stemma stemmatic structure subfunctor Summary surjective symbolic symmetry tempo tempo curve theorem theory tonality topology transformation values vector weight zero-addressed