Classification of Musical Objects for Analysis and Composition

Preface
Contents
Part I Initial Orientation
	1 The Basic Problem of Classification
		1.1 Structures
		1.2 Relationships
		1.3 The Overall Challenge
Part II General Formal Concepts
	2 Ontology, Oniontology, and Creativity
		2.1 Ontology and Oniontology
		2.2 Ontology: Where, Why, and How
		2.3 Oniontology: Facts, Processes, and Gestures
		2.4 A Short Characterization
		2.5 Oniontology for Classification
		2.6 Creativity
	3 Formal Representation of Musical Structures
		3.1 Scientific vs. Creative Value of Classification
		3.2 Software-oriented Perspectives
	4 Denotators over General Categories
		4.1 Formal Definition of Forms
		4.2 Formal Definition of Denotators
		4.3 The Category of Denotators
		4.4 First Examples of Denotators in Theory and Practice
			4.4.1 First Examples of Denotators
				4.4.1.1 Denotators over the Category Mod
				4.4.1.2 Denotators over the Category Digraph
			4.4.2 Hyperdenotators
				4.4.2.1 Hyperdenotators over the Category Mod
				4.4.2.2 Hyperdenotators over the Category Digraph
			4.4.3 Gestures over Topological Categories
	5 Composition Denotators and Classification
		5.1 Some Classical Composition Denotators
		5.2 The Role of Classification in Tonal Modulation Theory
		5.3 Modulations in Beethoven’s op. 106
	6 Gestural Denotators: A First Overview
		6.1 The Symmetric Concept Architecture between Compositions and Gestures
		6.2 The Role of Yoneda’s Lemma
		6.3 Software Perspectives
	7 The Escher Theorem for Compositions and Gestures
		7.1 The Escher Theorem and Escher Categories
			7.1.1 First Examples
Part III Local Classification
	8 Local Composition and Gesture Classification
		8.1 Classification of Local Compositions
			8.1.1 Compositions as Sequences of Module Elements
			8.1.2 Eliminating Diagonal Elements
			8.1.3 Transforming Sequences to Linear Maps
			8.1.4 The Grassmann Scheme for Local Compositions
		8.2 Classification Problems for Local Gestures
	9 Classification of Chords
	10 Motif Classes
		10.1 Three Element Motives in Analysis
			10.1.1 An Example from Classical Music
		10.2 Three Element Motives and an Example of a Melody from this Repertory in the Jazz Composition Synthesis
		10.3 A General Algorithm for the Construction of Generic Melodies Composed by Motives of n Elements Each
		10.4 Calculation of the Classes of n-element Motives in Z2 12
	11 Third Chain Classes
	12 Harmony through Third Chains
	13 Counterpoint Worlds
	14 Strong Interval Dichotomies
	15 Microtonal Contrapuntal Theories
		15.1 The Category of Strong Dichotomies
		15.2 Towers of Strong Dichotomies
	16 Dodecaphonic Rows
Part IV Global Classification
	17 Global Composition and Gesture Classification
		17.1 Why Global Compositions and Gestures
			17.1.1 Global Compositions
				17.1.1.1 Boulez and Webern
				17.1.1.2 Uhde/Wieland and Marek
				17.1.1.3 Hoffmann and Kaiser
				17.1.1.4 Graeser, Ruwet, and Nattiez
				17.1.1.5 Jackendoff and Lerdahl
				17.1.1.6 Schaeffer and Cage
				17.1.1.7 Musical and Mathematical Manifolds
				17.1.1.8 Global Gestures
				17.1.1.9 The Definition of Global Compositions and Global Gestures
				17.1.1.10 Examples of Global Compositions and Gestures
			17.1.2 The Nerve of a Global Composition or Gesture
		17.2 Classification through Modules of Affine Functions
			17.2.1 Local Compositions and Functions
			17.2.2 Morphisms and Function Spaces
			17.2.3 Global Standard Structures and Resolutions for Compositions
	18 The Classification Theorem for Global Compositions
		18.1 Preliminary Remarks
			18.1.1 Characterization of Interpretations
			18.1.2 An Example of a Non-interpretable Global Composition
		18.2 The Musical Meaning of Non-Interpretable Compositions: Varèse’s Program and Yoneda’s Lemma
			18.2.1 Global Dodecaphonic Classes
		18.3 Classification of Global Compositions
			18.3.1 The Resolution of a Global Composition
	19 The Classification Problem of Global Gestures
		19.1 Local Gesture and Functions
			19.1.1 Restriction of Gesture Types
			19.1.2 Global Standard Structures and Resolutions for Gestures
		19.2 A Conjectured Classification Theorem
			19.2.1 A Hypergesture for Human Bodies
	20 Singular Homology of Hypergestures
		20.1 Homology via Hypergestures
		20.2 Homology and Counterpoint
	21 Local Gestures, Structures of Knots, and Local Gestures as Local Compositions
		21.1 Local Gestures and Knot Theory
		21.2 Local Gestures as Local Compositions
			21.2.1 Characterstic Differences
			21.2.2 Conclusion and Future Topics
Part V Classification and Creativity
	22 Gestural Interpretation of Harmonic Dynamics in Tonal Modulation and Future Developments
		22.1 Gestural Modulation and Creativity
		22.2 Classification Problems for Performance Gestures
		22.3 Mirror Logic
			22.3.1 Introduction
			22.3.2 The Logic of Spatial Mirrors
			22.3.3 Temporal Specification of Chords
			22.3.4 Construction of First Species Counterpoint
			22.3.4.1 Hypergestures for Counterpoint Summary.
			22.3.5 Lead Sheet in Jazz
			22.3.6 Dance Gestures Associated with Musical Structures
			22.3.7 Conclusion and Future Topics
		22.4 Perspectives of Future Theory
Part VI References, Index
	23 Classification Lists
		23.1 List of Local Denotators in Z12, Chord Classes
		23.2 Third Chain Classes
		23.3 List of Local Denotators of Cardinality Two and Three in Z2 12
			23.3.1 Two Tone Motifs in Z2 12
			23.3.2 Two Tone Motifs in Z5 × Z2 12
			23.3.3 Three Tone Motifs in Z2 12
		23.4 List of Local Denotators of Cardinality Four in Z2 12
		23.5 List of Modulation Chords (Pivots)
References
Index