Classification of Musical Objects for Analysis and Composition 9783031301827, 9783031301834

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Computational Music Science

Linshujie Zheng Guerino Mazzola

Classification of Musical Objects for Analysis and Composition

Computational Music Science Series Editors Guerino Mazzola, School of Music, University of Minnesota, Minneapolis, MN, USA Moreno Andreatta, Music Representation Team, IRCAM - CNRS, Paris, France Advisory Editors Emmanuel Amiot, Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, Perpignan, France Christina Anagnostopoulou, Department of Music Studies, National and Kapodistrian University of Athens, Athens, Greece Yves André, CNRS, Sorbonne Université - Université de Paris, Paris, France Gerard Assayag, Music Representation Team, IRCAM - CNRS, Paris, France Elaine Chew, King's College London, London, UK Johanna Devaney, Brooklyn College, Brooklyn, NY, USA Andrée C. Ehresmann, Faculte des Sciences Mathematiques, Université de Picardie Jules Verne, Amiens, France Thomas M. Fiore, Department of Mathematics and Statistics, University of Michigan–Dearborn, Dearborn, MI, USA Harald Fripertinger, Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität, NAWI-Graz, Graz, Austria Emilio Lluis-Puebla, Faculdad de Ciencias, Universidad Nacional Autónoma de México, México City, Mexico Mariana Montiel, Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, USA Thomas Noll, Escuela Superior de Música de Cataluña (ESMUC), Barcelona, Spain John Rahn, School of Music, Music Bldg, University of Washington, Seattle, WA, USA Anja Volk, Department of Information and Computing Sciences, Urtrecht University, Utrecht, The Netherlands

About this series - The CMS series covers all topics dealing with essential usage of mathematics for the formal conceptualization, modeling, theory, computation, and technology in music. The series publishes peer-reviewed only works. Comprehensiveness - The series comprises symbolic, physical, and psychological reality, including areas such as mathematical music theory, musical acoustics, performance theory, sound engineering, music information retrieval, AI in music, programming, soft- and hardware for musical analysis, composition, performance, and gesture. The CMS series also includes mathematically oriented or computational aspects of music semiotics, philosophy, and psychology. Quality - All volumes in the CMS series are published according to rigorous peer review, based on the editors' preview and selection and adequate refereeing by independent experts. Collaboration - The editors of this series act in strong collaboration with the Society for Mathematics and Computation in Music and other professional societies and institutions. Should an author wish to submit a manuscript, please note that this can be done by directly contacting the series Editorial Board, which is in charge of the peer-review process. THE SERIES IS INDEXED IN SCOPUS

Linshujie Zheng • Guerino Mazzola

Classification of Musical Objects for Analysis and Composition

Linshujie Zheng School of Music University of Minnesota Minneapolis, MN, USA

Guerino Mazzola School of Music University of Minnesota Minneapolis, MN, USA

ISSN 1868-0305 ISSN 1868-0313 (electronic) Computational Music Science ISBN 978-3-031-30182-7 ISBN 978-3-031-30183-4 (eBook) https://doi.org/10.1007/978-3-031-30183-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to Sina Asiedu Mazzola

Photo and  by Ida Mazzola

Preface

This book deals with the classification of musical objects. This means that we thematize the project of exhibiting all “types” of such objects up to equivalence following structural similarity. Of course, it is critical to make precise which musical objects we are addressing. Nobody can at the time being pretend to include all possible objects. For example, our attention does not include psychological aspects. Also are we focusing on objects that can be described in frameworks of mathematical conceptualization. More precisely, we shall deal with two conceptual approaches: compositions and gestures. Both concepts were introduced in Mazzola’s work, also in view of the software implementation of musical objects. Local and global compositions are essentially implemented in Mazzola’s prestor and RUBATOr software. The gestural classification topic is new here, and it deals with two topics: To begin with, local and global gestures are introduced in a quite parallel procedure to the structures of compositions. Local and global compositions have been classified in a precise sense, namely by exhibiting algebraic spaces, so-called schemes, a.k. as algebraic varieties in a more traditional language. Classes of local and global compositions can be represented as points in such schemes. But the remarkable difference between compositions and gestures is that the classification of gestures is far from settled. This is due to the purely mathematical framework of gestures. Their topological aspects cannot—at the time being—be classified yet. We describe these problems and will establish as much as possible—from our perspective—to specify what is missing. The classification topic will in both cases, compositions and gestures, be introduced in view of the musical signification of this endeavor. Classification is in fact essential for the creative work of composers, most of the classical musical techniques are intimately related to classification, be it in harmony, counterpoint, and motivic work. We shall present these correlations in a discussion of musical creativity. The prerequisites we ask the reader to comply with are introductory courses in algebra [2], category theory [22], and topology [21]. vii

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We are pleased to acknowledge the support from Springer’s science editor Francesca Bonadei for writing such a demanding treatise. Minneapolis, December 2022

Guerino Mazzola and Linshujie Zheng

Contents

Part I Initial Orientation 1

The Basic Problem of Classification . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Overall Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 4

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 7 8 9 9 9

3

Formal Representation of Musical Structures . . . . . . . . . . . . . . . 13 3.1 Scientific vs. Creative Value of Classification . . . . . . . . . . . . . . . . . 13 3.2 Software-oriented Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 First Examples of Denotators . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Hyperdenotators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Gestures over Topological Categories . . . . . . . . . . . . . . . . . .

15 15 16 18 19 19 24 25

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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 . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 30

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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33 34 34

The Escher Theorem for Compositions and Gestures . . . . . . . 37 7.1 The Escher Theorem and Escher Categories . . . . . . . . . . . . . . . . . . 37 7.1.1 First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

10 Motif Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Three Element Motives in Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 An Example from Classical Music . . . . . . . . . . . . . . . . . . . . 10.2 Three Element Motives and a Jazz Example . . . . . . . . . . . . . . . . . 10.3 A General Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 A RUBETTEr for Generic Melodies . . . . . . . . . . . . . . . . . . 10.4 Calculation of the Classes of n-element Motives in Z212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 42 43 44 45

49 49 50 51 52 53 53

11 Third Chain Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 12 Harmony through Third Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 13 Counterpoint Worlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 14 Strong Interval Dichotomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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15 Microtonal Contrapuntal Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 65 15.1 The Category of Strong Dichotomies . . . . . . . . . . . . . . . . . . . . . . . . 65 15.2 Towers of Strong Dichotomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 16 Dodecaphonic Rows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Part IV Global Classification 17 Global Composition and Gesture Classification . . . . . . . . . . . . . 17.1 Why Global Compositions and Gestures . . . . . . . . . . . . . . . . . . . . . 17.1.1 Global Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 71 81 81 81 82

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 Non-Interpretable Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 Global Dodecaphonic Classes . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Classification of Global Compositions . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 The Resolution of a Global Composition . . . . . . . . . . . . . .

85 85 85 85 87 91 91 91

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 . . . . . . . . . . . . . . . . . . . .

95 95 96 97 98 98

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20 Singular Homology of Hypergestures . . . . . . . . . . . . . . . . . . . . . . . 103 20.1 Homology via Hypergestures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 20.2 Homology and Counterpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 21 Local Gestures, Structures of Knots, and Local Gestures as Local Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 21.1 Local Gestures and Knot Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 21.2 Local Gestures as Local Compositions . . . . . . . . . . . . . . . . . . . . . . . 108 21.2.1 Characterstic Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 21.2.2 Conclusion and Future Topics . . . . . . . . . . . . . . . . . . . . . . . . 111

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Part V Classification and Creativity 22 Gestural Interpretation and Future Developments . . . . . . . . . . 115 22.1 Gestural Modulation and Creativity . . . . . . . . . . . . . . . . . . . . . . . . 115 22.2 Classification Problems for Performance Gestures . . . . . . . . . . . . . 117 22.3 Mirror Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 22.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 22.3.2 The Logic of Spatial Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . 119 22.3.3 Temporal Specification of Chords . . . . . . . . . . . . . . . . . . . . . 119 22.3.4 Construction of First Species Counterpoint . . . . . . . . . . . . 120 22.3.5 Lead Sheet in Jazz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 22.3.6 Dance Gestures Associated with Musical Structures . . . . . 121 22.3.7 Conclusion and Future Topics . . . . . . . . . . . . . . . . . . . . . . . . 121 22.4 Perspectives of Future Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Part VI References, Index 23 Classification Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 23.1 List of Local Denotators in Z12 , Chord Classes . . . . . . . . . . . . . . . 127 23.2 Third Chain Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 23.3 List of Local Denotators of Cardinality Two and Three in Z212 . . 139 23.3.1 Two Tone Motifs in Z212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 23.3.2 Two Tone Motifs in Z5 × Z12 . . . . . . . . . . . . . . . . . . . . . . . . 139 23.3.3 Three Tone Motifs in Z212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 23.4 List of Local Denotators of Cardinality Four in Z212 . . . . . . . . . . . 142 23.5 List of Modulation Chords (Pivots) . . . . . . . . . . . . . . . . . . . . . . . . . 150 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Part I

Initial Orientation

1 The Basic Problem of Classification

Summary. This initial chapter deals with the very definition of classification. Such a definition must specify the structures to be dealt with and also the relationships among these structures. –Σ– Classification of a determined domain of object means that one should specify the structures, together with relationships among such structures. It is evident that not any type of musical object can at present be classified, we will have to determine the object types that have the chance to be controlled by precise conceptualization. In our approach in this book, we shall exclude a significant class of objects that escape a precision as needed by our formal methodologies. In the next chapter, we shall expose the ontological limits of our approach.

1.1 Structures The type of structures we can deal with will be defined by a mathematical description, which was successful in mathematical music theory and its software reification. This description is given by what we call denotators, which are objects or points in spaces that are called forms. This approach is a restatement of the classical conceptualization in objectoriented programming, as given by classes and their instantiations, called objects. Denotators and forms are canonically restated in terms of objects in specific mathematical categories. The remarkable advantage of our approach will not only include structures, such as local compositions, e.g., chords or rhythms, structures that were dealt with our book [27]. It will also include local and global gestures, a new type of structures that were not present there. The significant point of this extension is that the conceptual approach of denotators and forms allows for a natural transfer to global gestures, by a covering local gestures without any © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_1

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major structural restatement. Gestures are a canonical extension of classical denotators and forms.

1.2 Relationships The conceptual setup of denotators enables a natural conceptualization of relationships among denotators. Classification deals with the understanding of how denotators can be compared to each other. This feature is a consequence of the fact that denotators can be viewed as objects in determined categories. Whenever one works in a category, its objects admit morphisms between any two of its objects. Classification of objects within a given category then means to determine isomorphism classes in such a category. This means that one determines classes of objects, which are isomorphic to each other. Isomorphism classes define a partition of the object class of the category into classes of isomorphic objects. In mathematics, this task can be an easy task, but it can also present unsolvable problems. Many important categories have not been understood with regard to their classification objective. For example, in the category of vector spaces over a determined field, the dimension of a vector space is the defining invariant of a vector space, i.e., two vector spaces are isomorphic iff their dimensions coincides. The classification of finite simples groups (groups without non-trivial normal subgroups) is also achieved, but this work only results from a huge collaboration of leading group theorists [50]. But, for example, the category of knots (closed continuous curves in topological spaces) has not been classified yet [16]. In our context, there are many subcategories of denotators, whose classification has not yet been achieved. Nevertheless, some important subcategories, namely zero-addressed local composition were classified. But local gesture denotators are not classified at the present.

1.3 The Overall Challenge Whatever might be the status of classification of denotators, the semantics of classification is not only a mathematical objective, but plays a crucial role for the creative challenge of realizing musical works. We shall learn that most systematic approaches to musical composition, analysis, and performance are intimately connected to classification objectives. For example, tonal modulation presupposes the classification of tonalities, and counterpoint is built upon the classification of sets of consonant and dissonant intervals, but also more modern approaches, such as dodecaphonic and serial composition methods, are derived from classification of melodic units, called rows in this case. We shall claim and prove that classification is a driving force backing musical creativity.

Part II

General Formal Concepts

2 Ontology, Oniontology, and Creativity

Summary. This short chapter introduces first the global architecture of ontology of music, which this book is going to use extensively as an initial classifying perspective. –Σ–

2.1 Ontology and Oniontology This section is about ontology of music, including three dimensions: realities, semiotics, and communication. It also includes the extension of ontology to the fourth dimension of embodiment. We call this extension “oniontology” for reasons that will become evident soon.

2.2 Ontology: Where, Why, and How Ontology is the science of being. We are therefore discussing the ways of being that are shared by music. As shown in Figure 2.1, we view musical being as spanned by three ‘dimensions’, i.e., fundamental ways of being. The first one is the dimension of realities. Music has a threefold articulated reality: physics, psychology, and mentality. Mentality means that music has a symbolic reality, which it shares with mathematics. This answers the question of “where” music exists. The second dimension, semiotics, specifies that musical being is also one of meaningful expression. Music is also an expressive entity. This answers the question of “why” music is so important: it creates meaningful expressions, the signs that point to contents. The third dimension, communication, stresses the fact that music exists also as a shared being between a sender (usually the composer or musician), the message (typically the composition), and the receiver (the audience). Musical communication answers the question of “how” music exists. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_2

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2 Ontology, Oniontology, and Creativity

Fig. 2.1: The three-dimensional cube of musical ontology. Guerino Mazzola

Fig. 2.2: The hypercube of musical oniontology. Guerino Mazzola

2.3 Oniontology: Facts, Processes, and Gestures Beyond the three dimensions of ontology, we have to be aware that music is not only a being that is built from facts and finished results. Music is strongly also processual, creative, and living in the very making of sounds. Musical performance is a typical essence of music that lives, especially in the realm of improvisation, while being created. The fourth dimension, embodiment, deals with this aspect; it answers the question “how to come into being?” It is ar-

2.6 Creativity

9

ticulated in three values: facts, processes, and gestures. This fourth dimension of embodiment gives the cube of the three ontological dimensions a threefold aspect: ontology of facts, of processes, and of gestures. This four-dimensional display can be visualized as a threefold imbrication of the ontological cube, and this, as shown in Figure 2.2, turns out to be a threefold layering, similar to an onion. This is the reason why we coined this structure “oniontology”—it sounds funny, but it is an adequate terminology.

2.4 A Short Characterization The four dimensions of musical oniontology can be put together to present a short characerization of music: Music embodies meaningful communication and mediates physically between its emotional and symbolic layers.

2.5 Oniontology for Classification Our classification business will not deal with all of these oniontological dimensions. In particular, the psychological dimension will not be addressed. This means in particular that, for example, psychological arguments for the selection of consonances are not dealt with. We shall only address mathematical constructions of consonance-dissonance concepts. Also, modulations will not be addresses as psychological processes, but only in their shape of mathematical models, despite the fact that such models may be motivated by psychological rationales. And denotators may also represent physical entities, such as Fourier or FM constructions, but we will not address these genuinely physical phenomena, we shall only deal with their mathematical models using denotators. Finally, gestures as typical entities of embodiment will only be represented in their mathematical form. In other words, the oniontological dimensions will only be addressed through their formal representation via denotators. Nevertheless, the original oniontology of denotators will be referred to when we discuss their impact on the creative implications within musical constructions in composition, analysis, and performance.

2.6 Creativity We want to describe here the essentials of our approach to creativity [36] in practical terms to be used in the book.

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2 Ontology, Oniontology, and Creativity

Our approach to creativity is not a psychological trick nor does it incite a state of ecstasy, possibly boosted by drugs and similar devices. It is a process of discovery and invention that begins with an open question and continues with a run through a sequence of well-defined operational steps. It might not be successful in finding a solution to the question, but we can start over and repeat the processual run until we find a solution. Creativity cannot be guaranteed, but there are good reasons for approaching it based upon a clear strategy. Again, we discuss many examples of successful creativity in the theory part and hope that the reader will accept our approach here as a basis for our practical work. The process of creativity is composed of the following steps: 1. 2. 3. 4. 5. 6.

Exhibiting the open question Identifying the semiotic context Finding the question’s critical sign or concept in the semiotic context Identifying the concept’s walls Opening the walls and displaying its new perspectives Evaluating the extended walls

Let us briefly describe them, together with a simple example: the invention of 3M’s Post-It (see Figure 2.3). Step 1, exhibiting an open question. Creativity should start with the will to find something new that will answer an open question. It has a target. Example: In 1968, Dr. Spencer Silver, a chemist Fig. 2.3: The Post-It at Minnesota’s 3M company, developed an adhesive that did not really glue. He did not know what notes. to do with it. What would be the usage of such a substance? Step 2 consists of identifying the context of this question. This is a natural requirement: We want to know what is the overall position of the question, since for one individual the question might be open but trivial, while for another it might be a big challenge. Also, we stress that the context is a semiotic one. What does this mean? It means that the context should be one of things, thoughts, and signs that mean something. The open question then would be one that creates new contents when answered. It would extend the given meaningful context. So creation in our understanding is about creating new contents, not just forms. Example: The context is the chemical industry. Including a variety of more than 50,000 3M products. Step 3 is the moment of focusing on a specific location in the context, a critical concept or thing where we guess that the open question could be made more precise. Example: The critical concept here was “adhesive.” Dr. Silver had invented an adhesive that was critical—it did not glue as required, but only “half of it.”

2.6 Creativity

11

Step 4 is very important. It asks us to identify the concept’s walls. Walls are a metaphor for properties, characteristics, and specificities of the concept that circumscribe the concept in a more or less explicit form. It is a delicate task here, since some properties might be so subtle that one is barely capable of recognizing them. Example: The walls of the concept “adhesive” was that one would expect that an adhesive has to glue, meaning 100 percent not just somewhat. It was a wall because it was so evident that an adhesive had to glue, by its very definition! Step 5 asks us to consider these walls and to try to ‘soften’ and to ‘open’ them. This means that we ask to what extent these walls are necessary for the critical concept, and whether we could possibly find ways to open them and to recognize new perspectives on the other side of the given walls. Example: The softening of the “glue” wall meant that one had to put into question this strong requirement of 100 percent gluing. It was Arthur Fry, who attended one of Silver’s seminars, who successfully opened that wall. He sang in a church choir and was frustrated with the paper bookmarks he used to mark the songs in his hymnal because they would not stay put. Fry realized that Silver’s adhesive met exactly his needs. Step 6 is the terminal step. It consists of the evaluation of wall extensions we may have found in step 5, and then the judgment of this evaluation’s result. Is it a successful extension of the original critical concept or didn’t we find an answer to the open question? If we are successful, everything is OK; otherwise, we have to go back to step 4 and find new walls and new extensions, or even to step 3 and look for new critical concepts. Example: In 1980 Post-It Notes were introduced nationwide in the U.S. and soon became a big success. Remark 1 It is, of course, easy to give an example of a creative strategy that very probably would not be successful. Suppose that an open question in number theory is given. The critical concept could be “number.” And the wall one could discover could be “the number’s color” when you write it down on paper, on a blackboard, or on a computer screen. You could have the ‘ingenious’ idea that the number’s color could be extended from the usual black, blue, or red to any fancy color. But it is quite clear that such an extension would not solve the original number-theoretical question. Remark 2 We should not conclude this informal introduction to our creativity process scheme without pointing out the deep impact of representing ideas and methods in a sensual way (touch, smell, taste, hearing, sight). We have already established the sight- and touch-oriented metaphor of a wall described in steps 4, 5, and 6. When unfolding creative processes, we should always embed our activity in an object-oriented environment, where we feel at home and comfortable. Humans cannot think in abstract categories without using sensual metaphors. Even the most abstract mathematical thoughts are always embodied in objects that can be manipulated by human gestures. This is one of

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2 Ontology, Oniontology, and Creativity

the secrets of successful mathematicians. Einstein, for example, admitted that his thoughts were always performed with an intense chewing activity. He was literally chewing ideas! Last but not least, we should be aware that probably the most creative force is love for life. After all, this is the strongest motor of human propagation. For a successful creation, the atmosphere should be a warm and loving one 1 ; creation out of hate or indifference is never for life. A creation is a mental baby; it needs a ‘mother’ and her carrying body.

1

Sidney Lanier says, “Music is love in search of a word.”

3 Formal Representation of Musical Structures

Summary. We discuss what classification (the determination of isomorphism classes) means for understanding and creating music. We analyze the scientific versus artistic perspective of classification, in particular its often subconscious presence while musical creativity is operated. A basis of any serious investigation of musical objects/structures is the project of their reliable and precise conceptual description. And also being useful for the software management of music. Such a representation should include not only the objects, but also their possible relationships. –Σ–

3.1 Scientific vs. Creative Value of Classification The scientific classification using formal representation of musical structures seems to contradict the creative process when constructing musical works. The reason is that creativity is frequently viewed as being an activity full of vagueness and a fuzzy multitude of possibilities and options. No doubt that such categories intervene. But the transition from phantasies to musical objects must be driven by tools that realize those options qua musical objects. To recur to our definition of creativity, the point is about an efficient and concrete recognition and transgression of the walls of the previous framework. The softening and opening of walls must comply with a concrete trajectory to learn about the potential success of such an extension. This approach must rely upon an optimal representation of musical concepts or objects to be instantiated in the present creative movement. There is a moment within the creative procedure, where vagueness has to yield to concrete precision to generate an intended musical extension. For this reason, a formal representation of critical concepts, together with their classification, becomes mandatory to reach the extension of the box’s walls.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_3

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3.2 Software-oriented Perspectives It becomes increasingly relevant to apply a technological methodology to classified structures to reach the intended creative objective. More concretely speaking, creative activity needs to face the software-driven implementation of musical concepts. In view of what was said in the preceding section, it seems important to understand the necessity of a formal representation of musical concepts or objects as a basic style of realistic creativity. Let us look at a representative example, the famous Structures pour deux pianos, 1952 and 1961, by Pierre Boulez [5, 6]. The creative work of this composer was mainly focused on formal manipulations, but without technological support, Boulez had to perform lengthy calculations by hand without a hope to redo those calculations in reasonable time. We have reconstructed this composition, now for twelve instruments, using our RUBATOr software [27], see also Figure 3.1. This re-creative work could realize Boulez’s idea in seconds (after having programmed the software tools). This gave us a much more general perspective upon this composer’s creative methodology of serial composition.

Fig. 3.1: The Boulez network in the Rubato software.

4 Denotators over General Categories

Summary. We recapitulate the successful concept architecture of forms and denotators, which was also essential in software developments, in particular for the software RUBATOr . However, with regard to the newer concepts of musical gestures, we now extend forms and denotators from module categories to include directed graph topoi as well as topological categories. –Σ–

4.1 Formal Definition of Forms Summary. The definition of a form is given, including name, type, coordinator, and the space functor identifier. In our new approach, the basic category Mod of modules will be enriched by a general category Cat, especially with regard to the category Digraph of digraphs that is crucial for the formalism of gestures. –Σ– Recall that Cat denotes the topos of contravariant functors F u : Cat → Ens from the category Cat to the category Ens of sets. Recall also that for functors F u in Cat@ , an object X which is an argument of F u is called an address of F u, and a morphism f : X → Y in Cat is called an address change. Recall the notation X@F u for the value set of F u at address X. Recall finally the subobject classifier Ω in the topos Cat@ [23]. @

Definition 1 A form F is a quadruple F = (N F, T F, CF, IF ) where (i) NF is a string of Unicode characters; it is called the name of F and denoted by N (F ). (ii) TF is one of the symbols1 1

They turn out to symbolize five types of operators, but in the definition, we just need the symbols, i.e., the character strings.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_4

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1. Simple, 2. Syn,2 3. Power, 4. Limit, 5. Colimit; it is called the type of F and denoted by T (F ). (iii) CF is one of the following objects according to the previous symbols: A. For Simple, CF is an object M of Cat, B. for Syn, and Power, CF is a form, C. for Limit and Colimit, CF is a diagram D of forms; it is called the coordinator of F and denoted by C(F ). The diagram D is a diagram of functors F un(Fi ), as defined in (iv), for a family (Fi )i of forms. (iv) IF is a monomorphism of functors IF : F u  X in Cat@ , with this data: 1. For Simple, X = @M , 2. for Syn, X = F un(CF ), 3. for Power, X = Ω F un(CF ) , 4. for Limit, X = lim(D), 5. for Colimit, X = colim(D); it is called the identifier of F and denoted by I(F ), whereas its domain F u is called the space (functor) of F and denoted by F un(F ). The codomain of the identifier is called the frame space of the form. To denote a form F , we inherit the notation of the setup established in naive context [27, Ch. 6] and add the identifier below the arrow: N ame

−→

Identif ier

T ype(Coordinator).

(4.1)

A more formulaic representation of a form can be written by N ame : Identif ier.T ype(Coordinator).

4.2 Formal Definition of Denotators Summary. Based on the definition of a form (Subsection 4.1), we define denotators. –Σ– Definition 2 Let M be an address in Cat. A M -addressed denotator is a triple D = (N D, F D, CD) where 2

In the generic definition of a form, the synonym form type is superfluous. We however maintain this type for semiotic reasons: synonymy is a proper type of reference which is meant to be different from any other reference mode.

4.2 Formal Definition of Denotators

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(i) ND is a string of Unicode characters; it is called the name of D and denoted by N (D). (ii) FD is a form; it is called the form of D and denoted by F (D). (iii) CD is an element of M @F un(F (D)); it is called the coordinates3 of D and denoted by CT (D). According to the naive setup, a denotator is denoted by N ame : Address F orm(Coordinates)

(4.2)

So, in the full notation, a denotator D with form F is symbolized as follows: N (D) : Address N (F )

−→

Identif ier

T ype(Coordinator)(Coordinates),

(4.3)

but this clumsy writing is only used if absolutely unavoidable. Again, as already described for the naive setup, we shall write myN ame : myAddress@AF orm(myCoordinates)

(4.4)

for a denotator of form AF orm and coordinates myCoordinates, or even myN ame : @AF orm(myCoordinates)

(4.5)

if myAddress is also clear. Naming. The naming policy is identical with the naive situation. Recall that we can have the empty denotator name. Address. The address object M is an important generalization, however, we shall not overstress it since in the everyday language we speak of denotators independently of their address, as if there were just one ambient space instead of an entire space functor. Form. The form of a denotator englobes the whole recursion information as well as the form’s functor. It is the latter which contains the coordinates. Coordinates. The coordinates are one “point” 4 or element of the form’s functor at the given address M . Let us look at the shape of coordinates as a function of the particular form: 1. For Simple, the coordinator is an object N of he basic category Cat, and the coordinates CT (D) are identified (via the identifier!) with an element of M @N , i.e. a morphism CT (D) : M → N of Cat objects. 2. For Syn, the coordinates CT (D) identify with an element of M @F un(C(F (D))). So the coordinates are described by recursion to the coordinator C(F (D)). 3 4

This is a plural in singular mode. In algebraic geometry, an element of M @F for a functor F ∈ Mod@ is called an “M -valued point of F”.

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3. For Power, the coordinates CT (D) identify with an element of ∼

M @Ω F un(C(F )) → Sub(@M × F un(C(F ))), i.e. a subfunctor of @M × F un(C(F )). 4. For Limit, suppose that the diagram D of forms has vertex forms Fi and vertex functors F ui := F un(Fi ) at vertexes i, as well as natural transformations m(f ) : F utail(f ) → F uhead(f ) for arrows f in D. Recall ([27, Appendix G.2.2.1]) that at a given address M , we have a natural isomorphism ∼

M @lim(D) → Y M @F ui | m(f )(xtail(f ) ) = xhead(f ) , all arrows f of D} {x ∈ i

so that the coordinates of Limit denotators are special tuples in the product of all M @F ui . In other words, the limit denotators are canonically related to the product denotators from the naive setup. Therefore the general setup does resemble the naive one as being focused on a general, but fixed address. 5. For Colimit, let us inherit the notation of the preceding situation. Recall ([27, Appendix G.2.2.1]) that for a given ` address M , we have the equivalence relation ∼ on the coproduct i M @F ui , generated by the relation x ∼ y, x ∈ M @F ui and y ∈ M @F uj , iff there is an arrow f in D with i = tail(f ), j = head(f ), and m(f )(x) = y. We then have a natural isomorphism a ∼ M @colim(D) → M @F ui / ∼ i

and recognize that the naive coproduct is just the basic space of the Colimit denotator space before dividing through the equivalence relation ∼, and again, selecting a general, but fixed address M . So—up to general addressing—the coordinates of a Colimit denotator are those of the coproduct modulo an equivalence relation defined by the diagram’s arrows.

4.3 The Category of Denotators Denotators can be understood as objects of the category Deno(Cat) of denotators over Cat as follows. Let D = DN ame : DAddress@DF orm(DCoordinates),

4.4 First Examples of Denotators

19

E = EN ame : EAddress@EF orm(ECoordinates) be two denotators over Cat. This includes two elements (the coordinates) DCoordinates : DAddress → F un(DF orm), ECoordinates : EAddress → F un(EF orm). A morphism f : D → E is a pair f = (Add(f ), F un(f )), where Add(f ) : DAddress → EAddress is a morphism in Cat and F un(f ) : DF orm → EF orm is a natural transformation (a morphism of functors) such that the following square diagram commutes: DCoordinates

@DAddress −−−−−−−−−→ DF orm     @Add(f )y yF un(f ) ECoordinates

@EAddress −−−−−−−−−→ EF orm To understand this setup, recall that the Yoneda Lemma identifies N @F for the evaluation of a presheaf F at N with the set N at(@N, F ) = @N @F of natural transformations from the representable presheaf @N to F. The horizontal arrows in the above square diagram are the Yoneda-associated natural transformations for the two evaluations DCoordinates, ECoordinates.

4.4 First Examples of Denotators in Theory and Practice With the definition of morphisms among denotators over Cat we may start investigating the task of classification in Deno(Cat), i.e., the determination of classes of isomorphic denotators. Before we investigate this major task, it is useful to present a variety of denotators, which are significant in theory and practice. 4.4.1 First Examples of Denotators We want to present first examples of denotators, starting with the category Cat = Mod of modules, and then addressing gestural denotators over the category Digraph of digraphs. 4.4.1.1 Denotators over the Category Mod The first example is for form Loudness : Id.Simple(ZST RG). where Id denotes the identity identifier, with module ZST RG, the Z monoid algebra over the monoid ST RG of strings (words) over an alphabet ST RG that

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4 Denotators over General Categories

could be the ASCII alphabet or the extended alphabet Unicode. A denotator could be mezzof orte : 0@Loudness(mf ) denoting the mezzoforte symbol mf. The address 0 (zero module) means that we are given an affine map x : 0 → ZST RG, which defines its image point x(0) = mf . For the module Z2 we get a form HiHat : Id.Simple(Z2 ), and we may consider the denotator HihatOpen : 0@HiHat(1) to denote the open state 1 of a HiHat, whereas the value 0 would denote a closed HiHat. For module Z, we get the pitch form P itch : Id.Simple(Z), and we may consider the denotator myP itch : 0@P itch(p) to denote a pitch of value p. For module R, we get the onset form Onset : Id.Simple(R), and we may consider the denotator myOnset : 0@Onset(o) to denote an onset of value o. With these forms, we may define a simple version of a note by the form N ote : Id.Limit(Onset, P itch) where a denotator myN ote : 0@N ote(thisOnset, thisP itch) would denote a note with two zero-addressed coordinates thisOset, thisP itch in the simple forms Onset, P itch, which are a diagram without arrows and two Forms. We may use a pause form of limit type over the forms Onset and Duration Duration : Id.Simple(R).

4.4 First Examples of Denotators

21

We then define P ause : Id.Limit(Onset, Duration), and then a form GeneralN ote would be GeneralN ote : Id.Colimit(N ote, P ause) such that a general note would be either a note or a pause. For a score (of a very simple type), we would define SimpleScore : Id.P ower(GeneralN ote) whose zero-addressed denotators would be myScore : 0@SimpleScore{n1 , n2 , . . . nk } a collection of k notes n1 , n2 , . . . nk . Here Id is the canonical embedding of the functor 2X in the topos-theoretically given functor Ω X , see [27, Ch. 6.2.3]. Forms may also be defined in a circular way, as illustrated by the following example, which formalizes a hierarchical score variant, also used implicitly in Schenker analysis, called M acroScore: M acroScore : Id.P ower(N ode), with N ode : Id.Limit(N ote, M acroScore) A macroscore denotator is a collection of nodes, which are each defined by an “anchor note” and a “satellite” collection. Trills are typical denotators of such a form. The circularity boils down to a finite information when satellites become empty sets. To represent a Fourier decomposition of a periodic function of time t, X f (t) = A0 + Ai sin(2πif + P hi ) i>0

we define the form F ourier : Id.Limit(Basis, F requencey, Spectrum) with the basic value A0 Basis : Id.Simple(R), the frequency f in F requency : Id.Simple(R), and Spectrum : Id.Limit(Amplitude, P hase, Spectrum),

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with Amplitude : Id.Simple(R), and P hase : Id.Simple(R). Here the circularity is strictly infinite as required for the Fourier formula. One may also represent the Frequency Modulation formula X f (t) = Ai sin(2πfi + P hi + M odulatori ) i

by the representation of this sum via F M Object : Id.P ower(Carrier) with Carrier : Id.Limit(Support, F M Object), where Support : Id.Limit(Amplitude, F requency, P hase, M odulator). The form M odulator is synonymous to FMObject: M odualator : Id.Syn(F M Object). Because of the powerset in this form setup, finite solutions are feasible, as usually realized for FM soft- and hardware, typically realized by Yamaha’s DX7 synthesizer. Denotators over non-zero addresses are useful to denote important constructions in modern compositional methodologies, such as the serial method that is built upon a sequence of rows in a number of musical dimensions, such as pitchclass, duration, loudness, and attack. Pierre Boulez’s (in)famous composition Structures pour deux pianos (1952 and 1961) is a good example [5]. He considers these four forms P itchClass : Id.Simple(Z12 ) DurationClass : Id.Simple(Z12 ) LoudnessClass : Id.Simple(Z12 ) AttackClass : Id.Simple(Z12 ) A serial composition refers to the form SerialComp : Id.Limit(P itchClass, DurationClass, LoudnessClass, AttackClass), and the denotators in a serial composition are quadruples of denotators in SerialComp:

4.4 First Examples of Denotators

23

mySerialComp : Z11 @SerialComp(p, d, l, a) which means that each of the four coordinates is Z11 -addressed. In other words, each coordinate is given by the images of the twelve affine basis vectors e0 = 0, e1 = (1, 0 . . . 0), . . . e11 = (0, . . . 0, 1). This approach allows Boulez to consider address changes c : Z11 → Z11 in order to generate derived rows by address changes, i.e., without any transformation on the four parameter spaces. Boulez with this approach avoids the problem of performing transformations on duration, loudness, and attack, which would make no musical sense! See [20] for a thorough discussion of Boulez’s serial composition. 4.4.1.2 Denotators over the Category Digraph We present examples of integral curves of performance, gestures over topological categories, sets of gestures, gestures in topological vector spaces, glissandi, crescendi, tempo curves (fermata etc.) Simple forms over the category Digraph refer to the coordinators, which are digraphs. For classical gestures, these digraphs are derived from topological → − spaces X as follows. We consider the digraph X , whose vertices x are the elements x of X, and whose arrows a are the continuous functions a : I → X, I = [0, 1] ⊂ R, with the head h(a) = a(1) and tail t(a) = a(0). A simple form over X is the form → − Xgestures : Id.Simple( X ) with Id being the identity identifier. A ∆-addressed gesture in this form is aGesture : ∆@Xgestures(g) → − with coordinate g : ∆ → X a digraph morphism. A concrete example is an integral curve i : I → REHLD of a performance −−−−−→ field, represented as a gesture g :↑→ REHLD : a 7→ i in the four-dimensional real vector space REHLD of onset E, pitch H, loudness L, and duration D. A second example is the triply articulated hand movement of a pianist: go down to the keyboard, stay there for the duration of a sound, go up away from the keys. Here we take the representation of the hand by five finger tips and one carpus, each in the spacetime R4 , adding up to a movement in R24 . −−→ This defines a gesture g : ∆ → R24 of address ∆ = • → • → •, one curve for every arrow. For a non-simple form, take −−−−−→ CurveSets : Id.P ower(REHLD ) whose denotators aCurveSet :↑ @CurveSets{c1 , c2 , . . .}

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denote sets {c1 , c2 , . . .} of integral curves ci : I → REHLD as represented by −−−−−→ ↑-addressed gestures ci :↑→ REHLD . Here, the identifier Id is not the identity, but the embedding of the subfunctor 2X within the full powerset functor Ω X , see [27, Ch. 6.2.3]. 4.4.2 Hyperdenotators This section presents the construction of hyperdenotators for both categories, Mod and Digraph. 4.4.2.1 Hyperdenotators over the Category Mod For a module M , the set of M -addressed denotators in a simply form Spl : Id.Simple(N ) over module N identifies with the set M @N = Hom(M, N ) of homomorphisms f : M → N (when neglecting their names). But M @N is again a module, and we may consider denotators at address R in the simple form SplM : Id.Simple(M @N ). These denotators correspond to elements of the set R@M @N , which is isomorphic5 to the set (RM )@N . Such denotators are called hyperdentators since they are denotators whose coordinates are again denotators, denotators of denotators. For example, if R = Zd , M = Z11 , N = Z12 , we may interprete denotators in T welveRows : Id.Simple(Z11 @Z12 ) with address Zd as d-tuples of twelvetone rows, i.e. as hyperdenotators in Z12 . 4.4.2.2 Hyperdenotators over the Category Digraph → − To define hypergestures, take the simple form Gst : Id.Simple( X ) over a topological space X. For a digraph ∆, the ∆-addressed gestures → − N g : ∆@Gst(g : ∆ → X ) → − are elements of the morphism set ∆@ X . But since ∆ is the colimit of its arrows and vertexes, we have → − ∼ → − ∆@ X → lim ↑ @ X . ∆

→ − ∼ But ↑ @ X → I@Top X, the set of I-addressed curves in X. Giving this set the → − compact-open topology, we may define a topology on ∆@ X by the limit of the → − → − compact-open topologies on the ↑ @ X for the ∆’s arrows. This turns ∆@ X into → − a topological space, which we denote by ∆ @ X, and we may consider gestures → − → − in Γ @ ∆ @ X, these are gesture of gestures, i.e., hypergestures. 5

 denotes the tensor product in the category of modules with affine transformations.

4.4 First Examples of Denotators

25

4.4.3 Gestures over Topological Categories In this section, we deal not only with topological spaces for gestures, but moreover with topological categories. A topological category is a category which is internal in the category of topological spaces, i.e., the set of morphisms is a topological space, the domain and codomain maps are continuous, and the composition of morphisms is continuous. Continuous maps in gesture theory will be replaced by continuous functors. We first need to turn the unit interval into a topological category ∇. As a topological space it is the set ∇ = {(x, y)|x, y ∈ I, x ≤ y}, with the topology inherited from R × R, and domain d(x, y) = (x, x), codomain c(x, y) = (y, y). For a topological space X we define the graph category, again denoted by X, as the category, whose objects are the elements of X, whereas the morphisms are the elements of X × X, the topology being the product topology. Clearly, continuous curves f : I → X correspond to continuous functors f : ∇ → X. The category TopCat of topological categories with continuous functors has morphism sets for pairs of topological categories K, L, denoted by K@L, which are subsets of the sets of general functors f : K → L. The functor Top → TopCat identifying topological spaces and their continuous functions with corresponding graph categories is fully faithful. For a topological category X and a digraph Γ , we define the space digraph → − X of X with the arrow set being the topological functors ∇ → X with the vertices as the restrictions of arrows to the initial and final arguments in ∇. → − The gestures of skeleton Γ are then the set Γ @ X . To turn this latter set into → − ∼ a topological category, observe that ↑ @ X → ∇@X. This latter space becomes a topological category as follows: The objects are the continuous functors f : ∇ → X, the morphisms are continuous natural transformations between such functors, which are defined by continuous maps n : I → X. The topology on ∇@X is essentially the compact-open topology, see [27, Ch. 62.1] for details. Now, since Γ = colim ↑, the colimit of its arrows, we may define the limit → − ∼ → − topology colim ↑ @ X → lim ∇@X. This defines the topological category Γ @ X . With this construction, we may step over to hypergestures in topological categories.

5 Composition Denotators and Classification

Summary. In this chapter we discuss some denotators over Mod, traditionally called (local) composition denotators. We also give examples of their classification in music theory of tonal modulation. –Σ–

5.1 Some Classical Composition Denotators We consider denotators in forms F N ame : Id.P ower(M ) with the canonical Id = 2@M ⊂ Ω @M whose denotators DN ame : A@F N ame{x1 , x2 , . . . xk } are finite sets of A-addressed elements xi : A → M . Such denotators are called A-addressed local compositions or simply compositions in M . Such denotators were the first objects in mathematical music theory since its initial development (see [25]) in the 80th of the 20th century. Morphisms between local compositions K : A@KF orm{x1 , . . . xk } in KF orm : Id.P ower(M ) and

L : A@LF orm{y1 , . . . yl } in LF orm : Id.P ower(N )

were defined by commutative squares for a fixed address A and two carrier modules M, N : K −−−−→ A@M     fy yA@h L −−−−→ A@N where f is a set map that is a restriction of the evaluation A@h of an affine module morphism h : M → N at A, and where the horizontal arrows are set inclusions. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_5

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5 Composition Denotators and Classification

This morphism concept is slightly more general than what we had defined for denotator morphisms. The common information is the part A@h, but the difference here is that f needn’t be surjective, f is only landing in L, not onto L. However, when we later deal with isomorphism, this difference disappears.

5.2 The Role of Classification in Tonal Modulation Theory Let us now discuss the classification role in tonal modulation. This theory usually deals with so-called tonalities and the mechanism of constructing modulations between such tonalities. To begin with, modulations deal with tonalities that qua denotators are isomorphic to each other. One starts with scales, such as the major scales, and then develops mechanisms that involve isomorphisms between such scales. A scale is defined to be a zero-addressed denotator S@0Chroma{s1 , s2 , . . . s7 } with form of Chroma : Id.P ower(Z12 ). The C-major scale would be Cmaj : 0@Chroma{0, 2, 4, 5, 7, 9, 11} with the standard interpretation of 0 (the Cmajor tonic) for C, 2 for D etc., 11 for B. The isomorphisms would be the affine inversion isomorphisms T t .(−1) and the affine transpositions T t . The group of these isomorphisms on Chroma is denoted by T ±, ± standing for ±1 where T t ± (x) = t ± x. By definition, a major scale is the image T t ± (Cmaj). For example, T 8 (Cmaj) = Emaj, the major scale over E. The scale Cmaj has a unique automorphism T 4 .(−1). Every major scale has a unique inversion automorphism. More generally, the image T t (−1)(Xmaj) is the major scale whose third is the image of the tonic of Xmaj. Modulation deals with scales in the orbit of Cmajor under the group T ±. This is a simple first background of music theory that starts from classification. A tonality is a scale together with a covering of the scale by seven standard triads, so-called degrees, i.e., X ⊂ Xmaj that for Cmaj consists of these three-element degree sets: ICmaj = {0, 4, 7}, IICmaj = {2, 5, 9}, IIICmaj = {4, 7, 11}, IVCmaj = {5, 9, 0}, VCmaj = {7, 11, 2}, V ICmaj = {9, 0, 4}, V IICmaj = {11, 2, 5}. Observe that this setup does not identify a tonic with special functions. If we had to specify a tonic, i.e., an element of a major scale, this would define a mode, but this is not addressed in the present setup. What is significant for the definition of a tonality is that it exceeds the simple scale and adds seven parts in the sense of the covering of a geographic atlas by charts. This is the origin of a concept of global compositions (and gestures) that will be dealt with Chapter 18. The configuration of the seven degree chords that is defined by their intersections is geometrically represented by the following construction: Represent each degree of Xmaj by a point in three-space. Connect two degrees if they

5.2 The Role of Classification in Tonal Modulation Theory

29

have a non-empty intersection, and connect three degrees by a triangular surface having these degree points as its vertices. Then we get what is called the harmonic band Xmaj 3 . This band turns out to be a Möbius strip, see Figure 5.1.

Fig. 5.1: The harmonic band of triads.

Observe that every degree in a tonality is mapped to a degree of another tonality under a connecting isomorphism T t ±. More precisely, for T t , every degree XY maj is mapped to the degree of same name in the image tonality. For T t (−1), we have these maps: I 7→ II, II 7→ V, III 7→ IV, IV 7→ III, V 7→ II, V I 7→ III, V II 7→ V II. This means that under T t ± we not only have isomorphisms of scales, but also of tonalities. Arnold Schönberg in his harmony book [44] described the configuration of the lines connecting intersecting triadic degrees, but he did not make evident the present geometric shape issued by the triangles of three intersection degrees. It can be shown that the non-orientability of the harmonic band is a geometric reason for the failure of Hugo Riemann’s attempted harmony theory, see [27, CH. 13.4.2.1]. After these preliminary thoughts, we need to explicate the concept of a tonal modulation m : Xmaj 3 → Y maj 3 using the classification, i.e., the fact that these two tonalities are isomorphic to each other under the two isomorphism in the group T ±. The modulation should “materialize” such a connecting isomorphism since symmetries are not collections of notes, the latter being a conditio sine qua non for a concrete musical object. So the question arises about how to generate sets of notes that represent connecting isomorphisms. This problem is similar to the theory of physical forces, where one aims at materializing such forces by quanta, e.g., the electromagnetic force being represented by photons. We want to follow the modulation model as proposed by Schönberg in his harmony book. He divides the modulatory process into three stages: (A) neu-

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tralization of the tonality of departure, by playing neutralization degrees, (B) playing the pivotal degrees of the targeted new tonality, (C) playing cadential degrees of the new tonality. To this end we have to define what is a cadence in our context. Definition 3 A cadence in a tonality Xmaj 3 is a set C of degrees such that no other scale contains all of these degrees. A minimal cadence is a cadence with a minimal number of degrees. There are five minimal cadences in Xmaj 3 , namely C1 = {II, III}, C2 = {II, V }, C3 = {III, IV }, C4 = {IV, V }, C5 = {V II}. In stage (A), we may play any combination of degrees that does not contain a cadence, for example I, V I. The open problem here is the calculation of the set of pivotal degrees in stage (B). The main structure here is dealing with the pivotal degrees of a modulation. Formally speaking a modulation m : Xmaj I 3 → Y maj 3 is a pair m = (s, C) where s ∈ T ± such that s(Xmaj 3 ) = Y maj 3 , s being called the modulator and C is a minimal cadence of Y maj 3 . For such a modulation m, a modulation quantum M is a set of pitch classes with the following properties: 1. We have s(M ) = M , i.e., the modulator s is an “inner symmetry” of M ; 2. The cadence C of Y maj 3 is a set of degrees, all of which are contained in M , i.e., the quantum M is large enough to contain C; 3. the intersection of T ±∩Sym(Y maj ∩M ) is trivial and Y maj ∩M is covered by degrees of Y maj 3 ; 4. the quantum M is a minimal set with properties (1) and (2). Observe that the last property does not guarantee property (3). Therefore the existence of M is not automatic. With such a quantum, if it exists, the Y maj 3 degrees in M are by definition the pivotal degrees of modulation m. We call m quantized if it admits a modulation quantum M . Theorem 1 For every modulation m : Xmaj 3 → Y maj 3 , there exists a modulation quantum M , i.e., every modulation is quantized. The list of all possible quantized modulations is shown in Section 23.5.

5.3 Modulations in Beethoven’s op. 106 Beethoven’s “Hammerklavier” Sonata op. 106 is a challenge for modulation theory. We find some modulations that are executed very fast, with a minimum of notes, although they connect tonalities that are far from each other in terms of fourth distance. For example, in the Allegro movement, there is a modulation B[ → G[ between distant tonalities in measures 238-239. But there are other modulations that have a huge anatomy that is difficult to understand without a deeper understanding of the modulatory process.

5.3 Modulations in Beethoven’s op. 106

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Fig. 5.2: A complex modulation Gmaj → E[maj.

Let us look at an example of such a complex modulation. It is an excellent example not only of Beethoven’s virtuosity, but also an illustration of the presence of the modulator within the composed score, i.e., of the presence of the classificatory mechanism of isomorphisms between categorical objects, tonalities in our case. Consider the modulation Gmaj → E[maj in measures 124-129 of the Allegro movement, see Figure 5.2. All modulations in the Allegro movement have been analyzed and completely understood using our modulation theory, see [27, Section 28.2]. Let us look at the modulation Gmaj → Emaj[ in measures 124-129 of the Allegro movement. This modulation is bipartite (first part: measures 124-127, second part: measures 128-129). Before we encounter the fundamental degrees V II − V − V II in E[ in part two, according to our modulation table in Section 23.5, we hear note g as an octave interval: pedal and stationary voice in the first part. The pitches of the first part, when transposed into the octave spanned by the two g notes, show a regular melodic structure, see Figure 5.2, bottom. This structure has two parts: the first in measures 124-125, and the second in measures 126-127. They are related to each other by the inversion at d[, which is the same as the inversion at g in pitch classes. This first part of the

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modulation makes evident the inversion Id[ before we see the fundamental degrees in the second part of the modulation. But why this preliminary inversion? It is the modulator for the modulation in our model, Id[ (G(3) ) = E[(3) . This strategy is a beautiful compositional realization of what our model specifies. The model does predict fundamental degrees, and it does so on the basis of modulation forces that are provided by modulator symmetries. Beethoven not only writes down the fundamental degrees, but also makes evident the modulator symmetry in the first part of the modulation. Our interpretation in this analysis does not assume that Beethoven has performed his construction using the ideas of our model. But he might have done so instinctively; one cannot know such hidden layers of creativity. This situation is parallel to what happens in physics. We discover physical laws, but we cannot know whether a divine creator (if this is the underlying cosmological hypothesis) has constructed the universe according to these laws, which are our way to understand nature. Nevertheless, the laws hold, and so does our modulation model for the critical system of modulations in Beethoven’s composition. Concluding this chapter, we should add that our model also holds for other compositions by Beethoven, for example for modulations in the Cavatina movement of String Quartet op. 130.

6 Gestural Denotators: A First Overview

Summary. We discuss the formalism of denotators in digraph categories to represent gestures generalizing the already successful denotator approach. This is also a first step to a software-oriented implementation of gestural perspectives. This generalization generates a powerful conceptual symmetry between compositions and gestures. Both contexts start with simple forms, be it over the category of modules or the category of digraphs. Presheaves over these categories then introduce higher compound concepts by one and the same topostheoretical technique starting from Yoneda’s Lemma. –Σ–

6.1 The Symmetric Concept Architecture between Compositions and Gestures It is remarkable that the two conceptual threads, starting from the category Mod of modules for local compositions or from the category Digraph of digraphs, show a symmetric shape, namely the fact that compound forms in both cases are built from limit, colimit, and powerset constructions. Only simple forms are different here, compound forms are deduced from the same topostheoretical tools. This symmetry does however not imply that the classification of local compositions and gestures shares the same properties. We will see in Chapter 19 that classification of gestures is highly problematic, not because of philosophical aspects, but because the mathematical categories of gestures are far from being understood, in opposition to the categories of local compositions, where important classification theorems are available.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_6

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6.2 The Role of Yoneda’s Lemma In both categories the Yoneda Lemma plays a crucial role. In both cases the Lemma’s statement that for a presheaf F : Cat → Set and an argument X in Cat, the evaluation X@F of a F at X is identified with the set N at(@X, F ) = @X@F of natural transformations from the representable presheaf @X to F . This enables the definition of the categories of denotators for local compositions and for gestures. This fact unites the two types of denotators, but it does not guarantee an equally demanding classification procedure. The difference in classification is focused upon simple forms, which are much more problematic for gestures than for local compositions. Let us look at two concrete examples, where gestural classification be→ − comes problematic. The first example deals with the space ↑ @ REHLD of one−−−−−→ arrow gestures g :↑→ REHLD in the classical topological space of symbolic onset E, pitch H, loudness L, and duration D. Such gestures intervene for performance theory, which describes the performance functions ℘ : REHLD → Rehld with codomain Rehld and physical parameters e, h, l, d corresponding to the symbolic E, H, L, D. The performance maps ℘ are diffeomorphisms1 , and the images ℘ ◦ g of g describe the performance of determined points (notes) in ∼ the symbolic space REHLD . We therefore obtain an isomorphism g → ℘ ◦ g of gestures, both with domain digraph ↑, that are generated by the performance map ℘. Such isomorphisms of gestures are difficult to classify since the classification of curves modulo homeomorphisms of underlying topological spaces is not accomplished yet. In performance theory, the diffeomorphisms ℘ are not arbitrary homeomorphisms, they are vice versa given to define isomorphisms, which is not the proper classification objective. A second, simpler, but even more dramatic, example is the classification → − of simple loop gestures g : → X . Their classification is equivalent to the classification of knots, which is far from settled. Loops in topological spaces are critical structures, which we shall discuss in Chapter 21.1. We should observe that the classification of zero-addressed local composition powerset denotators in a module is much simpler than the same problem of zero-addressed gestures within a topological space. The former have the linear algebra structures, which we shall discuss later, but for gestures, the classification of collections of points in a topological space is far from understood.

6.3 Software Perspectives Software for local compositions has been developed in the RUBATOr environment, programmed in Java [26]. The formal software representation of simple gestures would be the starting point of an implementation of gestures since 1

A diffeomorphism is an isomorphism in the category, whose morphisms are differentiable maps.

6.3 Software Perspectives

35

compound forms are implemented literally by the same methods and objects as for local compositions. The critical question here is to represent a simple → − gesture g : ∆ → X for digraph ∆ and topological space X. The software representation of a digraph is not difficult, one has to define a vertex set plus a set of arrows. But the topological space X is more problematic. More precisely, how would one define continuous curves c : I → X? The map c must be given by a formula that defines the map’s evaluation at arguments in I. This amounts to implementing a set of possible formulas, polynomial, sinusoidal, exponential, say. But the evaluation of such formulas must also be feasible, yielding function values in X. This presupposes that X is given terms that may describe values of available functions. In other words, the implementation of simple gestures presupposes a collection of space constructions, together with a corresponding collection of continuous functions. The software construction should provide the user with flexible methods of constructing continuous functions c : I → X. For local compositions in RUBATOr construction methods rely on standard linear algebra constructions of modules via direct sums, projections, and the like. For gestures, we need to construct very different objects.

7 The Escher Theorem for Compositions and Gestures

Summary. The Escher Theorem establishes an isomorphism (classification) among hyperdenotator (denotators of denotators) sets for a sequence of addresses, i.e., modules or digraphs. It establishes an isomorphism between different permutations of address sequences when applied to simple forms. –Σ–

7.1 The Escher Theorem and Escher Categories A hyergesture is a gesture of gestures. This is reasonable since the set of ges→ − tures Γ @ X is a topological space/category [35], and therefore, one may con→ − sider gestures with values in Γ @ X. The case of modules, leading to hyperde∼ notators, follows from the adjunction (with affine morphisms) M @(R@N ) → (M  R)@N , where M  R = M ⊗ R ⊕ M ⊕ R is the affine tensor product. The two hyperstructural constructions share a commutativity property, which we coined Escher Theorem in [33], a proof for hypergestures can be found in [35]. Theorem 2 If R, M, N are three modules then there is a canonical isomorphism of hyperdenotator sets ∼

R@M @N → M @R@N. If Γ, ∆ are two digraphs, and X a topological category, then there is a canonical isomorphism of topological categories of hypergestures → − → − → − → − ∼ Γ @∆@X → ∆@Γ @X This may be generalized to arbitrary permutations of addresses for hyperdenotators and hypergestures: Let π be a permutation of the set {1, 2, . . . k} of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_7

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natural numbers, and take a sequence (Mi )i=1...k of modules as well as a sequence (∆i )i=1...k of digraphs. Further, let M be a module and X a topological category. Then we have canonical isomorphisms of hyperdenotator modules or topological categories, respectively: ∼

M1 @M2 @ . . . Mk @M → Mp1 @Mp2 @ . . . Mpk @M and

→ → − − → − → − → − → − ∼ ∆1 @ ∆2 @ . . . ∆k @ X → ∆p1 @ ∆p2 @ . . . ∆pk @ X.

The Escher Theorem unites local composition denotator theory with gesture theory. We might also call more intuitively the categories Mod and Digraph Escher categories. It is an open problem to set up an axiomatic formalism unifying these categories in a wider context, i.e., making precise what are the general consequences of an Escher Theorem for such categories. This problem is comparable to the situation for general homology theories. 7.1.1 First Examples For modules and their hyperdenotators, refer to the dodecaphonic example in ∼ Section 4.4.2.1. We have Zd @Z11 @Z12 → Z11 @Zd @Z12 , which means that d+1tuples of dodecaphonic rows may be interpreted as 12-tuples of d + 1-phonic rows. This generates a reinterpretation of rows in serial music: For example with d = 3, we get four rows in Boulez’s Structures pour deux pianos, which now may be interpreted as a sequence of 12 rows, each for the four parameter rows in pitch, duration, attack, and loudness. For digraphs ∆, Γ and topological categories X, the Escher isomorphism − → → − → → − − ∼ ∆@Γ @X → Γ @∆@X → − means that a ∆ hypergesture in Γ @ X can be viewed as a Γ hypergesture − → in ∆ @ X. For example, if ∆ = Γ =↑ and X = REHLD , the exchange of → − → − the two copies of ↑ in ↑ @ ↑ @ REHLD means to interpret a deformation of an performance (qua integral curve as described in Section 4.4.1.2) as a performance of a deformation, which is a remarkable reinterpretation of given hypergstures. In practice the Escher Theorem enables an important application for collaborative performance. Suppose that two musicians are interacting through → − → − morphisms f between their hypergestures: f : ∆ @ . . . X → Γ @ . . . Y . The codomain of f is a hypergestural structure with initial digraph Γ . If the Escher Theorem is applied to the codomain, one might get another initial digraph, and the morphism f would change into a morphism whose digraph map differs from f ’s digraph map. Musically, this means to change the perspective of the codomain of morphisms. So the musician who represents the domain of f views his/her action under a different interpretation of the involved codomain and its musician’s role. In jazz improvisation this is a well known procedure, focusing on different aspects of the addressed musician’s behavior.

Part III

Local Classification

8 Local Composition and Gesture Classification

Summary. Local compositions and gestures are the basic structures in classical musical concept spaces. They are essentially subsets in modules, such as chords, motifs, and rhythms, or else digraph maps from a digraph to a topological space/category. They are local since they are not built as configurations of smaller subsets. We describe in detail the theorem of classification of local compositions, including its proof. –Σ–

8.1 Classification of Local Compositions We present and discuss the theory of classification of local compositions, which we first discussed 1985 in our first book [25]. In this approach, we dealt with what now are called orbifolds, and what is essential, though not mathematically explicated, in a more recent approach by Dmitri Tymoczko [46], who did not know my German book. The classification in this section deals with simple musical objects, namely local compositions, which are denotators d : 0@M odulF orm(M ) for simple forms M oduleF orm : Id.P ower(M ) for modules M and Id the identity. There are approaches to the classification of such local compositions for non-zero addresses and for general modules, we refer to [27, Ch. 11.2] for those approaches. But here we want to present a context, which for special modules shows the general methods without being mathematically too demanding. The modules in our approach will be modules over a semi-simple com∼ mutative ring R, which means that R → k1 × . . . kn is a direct product of a finite sequence of commutative fields ki . We shall not prove those facts for such modules, which we use here, but simply inform the reader about what can be assumed for such modules. So we consider modules M , where we have (1) the usual structure of an additive commutative group, yielding sums m + n © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_8

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of module elements m, n, and (2) the scalar multiplication r.m of a module element m with a ring element r, with the usual distributive property (r + s).(m + n) = r.m + s.m + r.n + s.n. We classify local, zero-addressed compositions (K, M ), i.e. sets K ⊂ M of elements of the M . Morphisms between such objects are set maps f : K → L which are the restriction of an affine module morphism1 h : M → N , i.e., f (k) = h(k) for all k ∈ K. Isomorphisms are isomorphisms in this category, i.e., morphisms f which have right and left inverses g with f ◦ g = IdL and g ◦ f = IdK . For a local composition (K, M ), we have its associated module RK ⊂ M , which is generated by all differences k − k0 , k ∈ K with respect to a fixed element k0 ∈ K. The module RK is independent of the chosen fixed element k0 since for another fixed element k1 , we have k − k1 = (k − k0 ) − (k1 − k0 ). This association (K, M ) 7→ RK is in fact a functor from the category of our local compositions to the category Mod. For the present semi-simple rings R, it can be shown that every local composition (K, M ) is isomorphic to a generating local composition (L, N ), which means that RL = N . We shall therefore restrict our discussion to generating local compositions. For example, the major scales are (Xmaj, Z12 ) are generating local compositions (attention: the ring Z12 is not semi-simple, it is the product Z3 × Z4 , where the second factor is not a field). The classification of these local compositions splits into a sequence of intermediate bijections of classifying sets as follows: 8.1.1 Compositions as Sequences of Module Elements We start with a local composition (K, M ) having t + 1 elements. Consider the symmetric group St+1 of permutations of natural numbers 1, 2, . . . t + 1. Then K may be interpreted as a sequence K. = (k1 , . . . kt+1 ) of its elements, i.e., an element of M t+1 modulo the action of St+1 , the σ ∈ St+1 taking k. = (k1 , . . . kt+1 ) to σ.k. = (kσ(1) , . . . kσ(t+1) ). The elements of the sequence K. must be pairwise different. The subset of M t+1 of pairwise different element sequences is denoted by M t+1 . The local compositions with t + 1 elements are the orbits in M t+1 /St+1 of the St+1 action. 8.1.2 Eliminating Diagonal Elements Here we reduce the sequence K. to the set of its differences. We first consider the short exact sequence of R-modules 0 → ∆M → M t+1 →d M t → 0, 1

An affine module morphism h : M → N is the composition of a linear morphism h0 : M → N with a translation T t : N → N on the codomain N of h0 .

8.1 Classification of Local Compositions

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where ∆M is the diagonal submodule and d(x1 , . . . xt+1 ) = (x2 − x1 , . . . xt+1 − x1 ). The group operation of St+1 on M t+1 leaves ∆M invariant, therefore St+1 ∼ operates on M t → M t+1 /∆M . On the other hand, the linear group GL(M ) operates on M t , and these two operations are compatible with each other. This entails the Lemma 1 Let M t be the subset of M t such that its elements are (1) different from zero, and (2) the sequences (y1 , . . . yt ) generate M . Then the set of isomorphism classes of t + 1-element generating local compositions in M is in bijection with the orbit set GL(M )\M t /St+1 . Proof. If (K, M ) is a t + 1-element local composition, it defines an orbit by taking its sequence (x2 − x1 , . . . xt+1 − x1 ) in M t and then stepping over to the orbit by permutation of the sequence and the GL(M )-orbit. Every such local composition can be found in the orbit space. If two local compositions are associated with the same orbit, this means that their original element sequences are transformed by an element of GL(M ) and possibly by a permutation of their sequences of differences. That we have to take linear maps in GL(M ) is due to the fact that transpositions cancel out for differences of K elements. So they are isomorphic, QED. 8.1.3 Transforming Sequences to Linear Maps This third step deals with the restatement of sequences of module elements as linear maps between modules. This will be used to introduce subspaces of modules instead of sequences, and thereby representing isomorphism classes in geometric spaces (Grassmanians) whose points are such subspaces of modules, a technique that is standard in modern algebraic geometry. ∼ We use the basic fact that M t → Hom(Rt , M ) sending a t-tuple (mi ) ∈ t M to the linear map which evaluates at ei = (0, . . . 0, 1, 0 . . . 0) with the 1 at position i to mi . The above operation of St+1 on M t caries over to an operation on2 Hom(Rt , M ) and thereby in particular to Rt . Now consider the sets Xt (R) = {V ⊂ Rt , V a submodule with ei , ei − ej 6∈ V all i 6= j} and

∼

XtM (R) = {V ∈ Xt (R), Rt /V → M }.

Evidently, St+1 operates on both, Xt (R) and XtM (R), and we have Lemma 2 The orbit space XtM (R)/St+1 is in bijection to the space LgM,t+1 of generating t + 1-element local compositions on a module isomorphic to M . 2

The R-linear maps.

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Proof. This bijection is induced by the map f 7→ Ker(f ). Let us verify that this map generates a bijection. To begin with, by Lemma 1, such a local composition is identified with an orbit in GL(M )\M t /St+1 , which means that it is defined by the orbit of a surejctive linear map f : Rt → M . Then Ker(f ) is an element of XtM (R), and the action of the permutation group identifies the orbits in both spaces. The action of GL(M ) identifies the GL(M ) orbits by definition of LgM,t+1 . This association injects the space LgM,t+1 via kernels into XtM (R)/St+1 . But it is also surjective since a subspace V defines a surjection Rt → M with the defining properties in XtM (R)/St+1 . This identification means that the set of isomorphism classes of generating local compositions can be identified with the set a Xt (R)/St+1 = XtM (R)/St+1 M ∈M

where M is a set of representatives of module isomorphism classes. This may be stated as a theorem: Theorem 3 The orbit space Xt (R)/St+1 is in bijection with the set of isomorphism classes of generating t + 1-element local compositions over R. Since every such local composition is isomorphic to a generating local composition by simply taking the generated module RK and shifting K into RK if necessary, this classification also holds without the “generating” attribute. 8.1.4 The Grassmann Scheme for Local Compositions In this final section, we shall interpret the orbit space Xt (R)/St+1 as a set of points in an algebraic variety (also called scheme in modern algebraic geometry), the Grassmannian. This discussion presupposes some knowledge in algebraic geometry, but it means that isomorphism classes may be viewed as points in a geometric space. The scheme Grass(r,t) is the scheme, whose point set Grass(r,t) (R) for the ring R is defined as the set of direct summands V ⊂ Rt which have a quotient Rt /V that is of rank r. The latter means that such a quotient is free of rank r when taking the local modules S ⊗ Rt /V over localizations S of R at its prime ideals. The operation of St+1 on Grass(r,t) is evidently the restriction of its operation on Xt (R). This means that the conditions ei − ej 6= 0, ei 6= 0 define an open subscheme of Grass(r,t) . In other words Lemma 3 The subfunctor Xtr ⊂ Grass(r,t) is represented by an open, St+1 invariant subscheme. This open subscheme has its orbits which are finite. This implies in view of general theorems from algebraic gometry that there is a quotient scheme Xtr /St+1 whose orbits are the isomorphism classes.

8.2 Classification Problems for Local Gestures

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Theorem 4 The set of isomorphism classes of local, t+1-element compositions over R is represented as the set of R-valued points of a scheme of finite type (defined by a finite number of indeterminates over the base field).

8.2 Classification Problems for Local Gestures In opposition to local compositions, there is not (yet) a classification theo→ − rem for local gestures, i.e., digraph maps g : ∆ → X for a digraph ∆ and a topological space/category X. Classifying such objects in their given category turns out to be an open mathematical problem. Let us focus on a critical example: ∆ = . A local gesture over this digraph is essentially (equivalent to) a continuous map from the unit circle in R2 to X. Such a map is called a knot, and the classification of knots is an open problem in mathematics, but see Chapter 21.1 for more details. This means that there is no classification theorem for local gestures.

9 Classification of Chords

Summary. This chapter deals with chords, i.e., zero-addressed local composition denotators K = {c1 , . . . ck } ⊂ Z12 , i.e. K : 0@P itchClass{c1 . . . ck } for the form P itchClass : Id.Simple(Z12 ). Such denotators can also be interpreted as a 12-periodic rhythms, we shall occasionally consider this latter point of view. –Σ– This classification follows the general concepts, but uses some mathematical tools which we cannot develop here in our elementary approach. We may however state that two local compositions K : 0@P itchClass{c1 . . . ck }, L : 0@P itchClass{d1 . . . dl } are isomorphic if and only if there is an affine automorphism f = T t .r, r = 1, 5, 7, 11 such that f (K) = L. The list of all representatives of isomorphism classes is displayed in Section 23.1. Let us discuss some immediate consequences of this classification. It is a well known fact that when we step from C Major to F Major to B[ Major to E[ Major to A[ Major to D[ Major to G[ Major, the number of [s increases by one for each step, starting with no [ for C Major and ending with six [s on G[ Major. Such a regular increase would not be the case if we had to step from C harmonic minor to F harmonic minor, etc. Why is such and increase happening? The reason is that in our list of chord classes we recognize that the C Major scale is the complement of class 38.1, and the latter is isomorphic to class 38, i.e., the C Major scale is isomorphic to the complement of class 38. ∼ The isomorphism is given by f : {0, 2, 4, 5, 7, 9, 11} → {11, 0, 1, 2, 3, 4, 5} with f = f −1 = .7, yielding 7.11 = 5, 7.0 = 0, 7.1 = 7, 7.2 = 2, 7.3 = 9, 7.4 = 4, 7.5 = 11. Now, stepping from C Major to F Major etc., always is performed by a transposition T 5 by a fourth. Then F M ajor = T 5 (CM ajor) etc. until G[M ajor = T 5 (D[M ajor). Transforming these relationships we get f (F M ajor) = .7(5 + CM ajor) = 11 + {11, 0, . . . 5} = {10, 11, 0, 1, 2, 3, 4}. This is just a transposition of class 38 representative by one unit, i.e., the fourth transposition morphs to a minor second transposition. But hereby, we lose one point at one end of the chromatic point sequence and add another point at © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_9

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the other end. This operation is repeated for every fourth transposition of the original major scale, we always add a new note by a [ and take away a note from the previous scale. One could simply state that the increasing sequence of [s is due to the fact that a major scale is an uninterrupted sequence of fifths or, in the other direction, of fourths. A second, even more important, fact from this classification is given by ∼ class 87, which is autocomplementary, there is an isomorphism f : 87 → ˆ denotes the complement of 87. To understand this, observe that ˆ 87 87,where class representative 87 = {0, 1, 3, 4, 7, 8} is a transposition of {1, 2, 5, 6, 10, 11} by 2. But the latter represents the set of dissonances in classical Fuxian counterpoint, and the autocomplementarity function of the dissonances with the consonances {0, 3, 4, 7, 8, 9} is T 2 .5. It is also the unique function that connects consonances to dissonances, since the automorphisms of 87 are trivial. This result is the basis of a mathematical counterpoint model that derives important rules, e.g., forbidden parallels of fifths, and the dissonant qualification of the fourth, see [3].

10 Motif Classes

Summary. This chapter sets a perspective onto the classification of motives, ie., local compositions K ⊂ Z212 . The general case is not made explicit yet, but for card(K) ≤ 4, we have complete lists and some creative applications. –Σ– Motives are the two-dimensional elements of melodic structures. We shall deal with this gluing process that generates melodies and more general “global compositions” in Chapter 18. Here we look at the classification of motives with the aim of constructing global compositions, i.e., set-theoretically speaking, the union of local motives to build melodies of a certain character with respect to the classes of local motives.

10.1 Three Element Motives in Analysis To begin with, the isomorphism classes of three element motives K ⊂ Z212 are 26 in number, their list is shown in Section 23.3. We also show the list in a geometric, two-dimensional way in Figure 23.1. And we show the Hasse diagram of these classes in Figure 23.2, which means the following: There is a dominance arrow from class m to class n iff there is an affine map d : Z212 → Z212 such that d(m) = n, and d cannot be an isomorphism since the classes m, n are supposed to be different. For example, class 10 dominates class 18 via a multiplication by 2 of its coordinates. The Hasse diagram denotes all shortest dominance arrows, i.e., those which cannot be decomposed into a sequence of two or more arrows. For example, the dominance 10 → 18 decomposes into a dominance 10 → 12 and a dominance 12 → 18. It results that class 10 is a unique “generic” three-element motif, all three-element motives are dominated by class 10. It is remarkable that three-element classes are characterized by two items: the classes of their two-element submotives and the volume of a three-element class. The volume is the Z12 element derived from the second exterior power © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_10

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50

Λ2 (Z12 K) of the motif K. Unfortunately, this characterization holds no longer for four- and higher -element motif classes, see our list in Section 23.4.

Xj. b Xk Xj

j X XjXj

X Xk @ K @ XK

I

Mit-ten im 14

Schim-mer der 14

spie-geln-den 19

Wel-len

II

glei-tet wie 14

Schwä-ne der 14

wan-ken-de 6

Kahn

III1

Ach auf der 23

Freu-de sanft 10

schim-mern-den 11

Wel-len

III2

... 11

... 10

... 11

...

IV1

glei-tet die 3

See-le da10

hin wie der 11

Kahn

IV2

... 3

... 10

... 11

...

V

Denn von dem 14

Him-mel her10

ab auf die 14

Wel-len

VI1

tan-zet das 10

A-bend-rot 10

rund um den 10

Kahn

VI2

... 10

... 14

... 10

...

XJ . b X Xj K

Fig. 10.1: The highly symmetric arrangement of motive classes in the dactylus grid of the voice score in Schubert’s setting of Stolberg’s poem Lied auf dem Wasser zu singen, für meine Agnes.

10.1.1 An Example from Classical Music We consider the analysis for Schubert’s composition Auf dem Wasser zu singen, für meine Agnes. The composition shows a highly symmetric arrangement

10.2 Three Element Motives and a Jazz Example

51

of three-element melodic motives, see Figure 10.1, the numbers refer to the numbers in our classification list in Section 23.3.3.

6 & 8 X X X b X b X X X X b X b X b X X X X X X X X X b X X XJ b X b X X X Xj X 15

19

24 25

14

12

9

3

26

16

22

6

13

23

11

4

5 10

2 1

7

8 21

17

18

20

Fig. 10.2: The germinal melody of the Synthesis concert for piano, percussion, and bass is a patchwork of 26 three-element motives, each representing one motif class in OnP iM od12,12 . The melody is shown as a score and as local composition (red-colored points).

10.2 Three Element Motives and an Example of a Melody from this Repertory in the Jazz Composition Synthesis For the construction of the composition Synthesis, a germinal melody was constructed as a patchwork/union of representatives of all 26 classes, see Figure 10.2. This procedure guarantees the presence of all of 26 classes and thereby

52

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generates a melody that is as generic as possible. Here the classification is used for a melodic construction.

10.3 A General Algorithm for the Construction of Generic Melodies Composed by Motives of n Elements Each This section proposes an algorithm to construct a melody M that is the union of representatives of all classes of n-element motives. The idea is as follows. Such an algorithm is mandatory because there are hundreds of classes in general, for example for n = 4 we already have 215 classes, see the list in Section 23.4. We suppose that we have constructed a partial melody M ∗ that is the union of the first k representatives, and we then choose the first n-element motif that is not contained in M ∗ . We add this new motif to M ∗ , and we continue until representatives of all classes are added. This method should enable us to generate melodies that are generic in the sense that they are the union of representatives of all classes of n-element motives. Such a method can be realized in many ways, e.g., adding disjoint new representatives, but we want better solutions, i.e., solutions that add a minimal number of new notes. We first want to check all the classes that are not 2 ). Suppose we found a first such candidate N . contained in M ∗ (modulo Z12 Then, we want to look for the maximal part of N 0 ⊂ N that is contained in M ∗ . We then add the missing elements of N to M ∗ . We could do so by taking the first representatives and adding them to M ∗ . We should do so by taking the first representatives of these elements with the smallest values in Z × Z12 . The open question so far is to find the first missing motives N . This can be achieved by taking in the Hasse diagram of n-element motives the first missing N . What does this mean? If we look at the distance function on this Hasse diagram, defined by the minimal length sequence of arrows from the M ∗ n-motives. Within the missing motives, we take one with such a minimal distance to the M ∗ motives and in this selection taking the first for an a priori given linear ordering among the n-element motives. This selection method guarantees that we add a N that is nearest to the given set of n-motives in M ∗ . This method depends on the a priori given linear order and on the Hasse diagram, in particular, given the linear ordering, we may start the entire procedure with the first candidate to get off the ground. Let us write down this algorithm a bit more formally: Suppose we are 2 . given a linear ordering of the set Class(n) of n-element motive classes in Z12 1. Start the procedure with a representative M0 of the first class. 2. Suppose we have defined a partial melody M ∗ that is the union of k representatives of Class(n). Take the set C(M ∗ ) of classes that are not contained in M ∗ . Take in C(M ∗ ) the first candidate N with a maximal number of elements, i.e., a maximal submotif N 0 ⊂ N ∩ M ∗ and with minimal Hasse distance, and therein the first of the given order.

2 10.4 Calculation of the Classes of n-element Motives in Z12

53

3. Take elements representing missing points in N and add them to M ∗ . Thereby a representative of the class of N is added to M ∗ . 10.3.1 A RUBETTEr for Generic Melodies 2 For three-element motives in Z12 , we have the complete information: all 26 classes are calculated, the Hasse diagram is known, and for every class, the classes of its two-element submotives are given. Also, the isomorphism of twoelement subcompositions, together with the motives’ volumes, are sufficient for the determination of isomorphism classes. This implies that the Java programming of a RUBETTEr is feasible for the task of generating generic melodies from three-element motives on the RUBATOr platform. The linear ordering of the 26 classes is the only variable in this project, but 26! is nevertheless a big number, more than the number of possible 12! dodecaphonic rows. The musical idea behind the concept of a generic melody is to produce a melody that subsumes all possible classes of motives of a given size. This generates musical objects that present a maximal variability of its submotives of a given size. The construction of “interesting” melodies is a still unsolved problem: When is a melody interesting, interesting under what perspective? Of course, historical examples tend to be characterized by a property of becoming “ear worms”. Our approach here will very probably not comply with this psychological condition, but we believe that our construction yields melodies, whose anatomy is as rich as possible.

10.4 Calculation of the Classes of n-element Motives in Z212 Despite the general classification of local comopositions, one might also try to approach classification from a recursive point of view. Here is the method. Given a local composition K ⊂ M , one can define its simplicial weight as follows. On looks at al the proper subcompositions L ⊂ K and then exhibits all their nonempty intersections, called simplices. So a simplex isTa set σ = {L1 , . . . Lk } of proper subcompositions with non-empty intersection σ. This configuration is called the composition’s nerve n(K). For every σ, one exhibits its isomorphism class cl(σ). The map σ 7→ cl(σ) is called the simplicial weight, it defines the totality of isomorphism classes of proper subcompositions. ∼ It is an open question, whether two nerves being isomorphic n(K) → n(L) ∼ implies K → L. If this can be proved, one would get a recursive method to determine isomorphism classes of local compositions. The isomorphism of simplicial weights is necessary for an isomorphism of two local compositions, but it may be insufficient. The basic problem is the continuation of isomorphisms: If we have two isomorphic local compositions K, L with isomorphic subcompositions ∼ K 0 , L0 of K, L, respectively, can we extend a given isomorphism φ0 : K 0 → L0 ∼ to an isomorphism φ : K → L?

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We see that such extensions are not always possible. In particular, this means that isomorphisms of simplicial weights are necessary but not sufficient for isomorphisms of the underlying local compositions. An example is shown in Figure 10.3. Here intervals of length one are isomorphic, but no automorphism of the motif extends their isomorphisms in general, the only automorphism of the motif being the diagonal exchange.

Fig. 10.3: A motif which has only the diagonal exchange as an automorphism since its horizontal and vertical parts are rigid.

11 Third Chain Classes

Summary. Third chains are chords that are built from a concatenation of minor or major thirds. A major triad 0, 4, 7 is a third chain, concatenating a major third 0, 4 with a minor third 4, 7. Third chains are important in harmony, and we shall demonstrate how they are applied to the management of harmonic contents of general chords. This approach is important for the harmonic analysis of musical compositions. –Σ– In Section 23.2 all third chains are listed. They are denotators of the following type. Let Interval : id.Simple(Z12 ) be the simple form for intervals. Let Base : id.Simple(Z12 ) be a starting pitch form. Let T hirdChain.n : id.Limit(Base, Interval, . . . Interval) be the form with n intervals. Then a third chain with two intervals, such as (3, 4), and based upon starting pitch 0, is given by the denotator Cm : 0@T hirdChain(B, I1 , I2 ) with B : 0@Base(0), I1 : 0@Interval(3), I2 : 0@Interval(4). Third chains are a finer construction mode than local compositions. The construction of pitch classes via T 3 , T 4 and a variable initial pitch class follows the idea of David Lewin and the author, where sets are replaced by operations that generate sets, and objects plus such operations may be compared in a finer approach. For example, the minor triad third chain 2, 5, 9 is isomorphic to the major triad third chain 11, 3, 6, but as chains they are different. In the next chapter, we shall discuss a harmony in Hugo Riemann’s sense that uses third chains to define harmonic functions of local compositions.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_11

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12 Harmony through Third Chains

Summary. We discuss the application of third chain classification to Riemannian harmony for general chords. –Σ– Third chains are important for harmonic theory by the following method. We suppose that third chain chords are given a determined harmonic function. Then a general chord Ch may be embedded in third chains, e.g. Ch = {2, 9}, the fifth interval chord over d = 2. We may embed Ch in the third chains Dm = 2 + (0, 3, 7) or DM = 2 + (0, 4, 7). The harmonic function of Ch may then be determined by a combination of he harmonic functions of Dm and DM . Every chord can be embedded in at least one third chain, e.g. the third chain of the entire pitch class set, namely (0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8, 11). With this in mind, one may define the third chain spectrum T hirdChainSpectrum(Chord) as being the set of minimal third chains that contain Chord. For the above Ch, we have T hirdChainSpectrum(Ch) = {Dm, DM }. Therefore we should discuss potential harmonic functions of third chains. If we construct harmonic functions f (T hirdStream) that are numerical, we may define for a chord Chord its function by X 1 f (Chord) = f (Ch), N Ch∈T hirdChainSpectrum(Chord)

N being the number of third chains in T hirdChainSpectrum(Chord). The point now is to define f (X) for third chains X. To this end, we first define the tonality for which this functions should be defined. Let us suppose that we have two choice: Y major or Y minor. With such a choice, that would be symbolize by the Y + for major an Y − for minor. Then, we define a harmonic value of harm(x, Y ±) for pitch class x and tonality Y major (+) or Y minor (−). With these values, we define X harm(X, Y, ±) = harm(x, Y ±). x∈X

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_12

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12 Harmony through Third Chains

This defines the value f (Chord) with the values of the third chains in its spectrum and the tonality information (Y ±). One might also refine this information by defining harmonic values harm(x, Y ±, T onic), harm(x, Y ±, Dominant), harm(x, Y ±, Subdominant), and then calculating the chord’s value with respect to the tonality parameters Y, R, R = T onic, Dominant, Subdominant. All in all, this defines a so-called Riemann matrix Riemann(Chord)(Y, R) with twelve values for Y and six values for R, three in major and three in minor mode. The point of this approach is that a chord may have values for all 12 × 6 = 72 matrix entries, but not in the sense of “yes” or “no”, the values are a fuzzy logic representation of the harmonic value of a chord. This approach was implemented in RUBATOr ’s HarmoRubette, a software component for Riemannian harmonic analysis. Therefore the classification of third chains offers a tool for Riemannian harmony with fuzzy numerical values.

13 Counterpoint Worlds

Summary. The classification of strong dichotomies yields a basis for the rules of counterpoint. We discuss some “exotic” counterpoint worlds (five besides the Fux world) that are generated by this classification. –Σ– This chapter discusses some counterpoint topics from the perspective of classification. We shall however not penetrate the mathematical model of counterpoint, for which we refer to [27, Part VII] and to [3]. Our approach to counterpoint gets off the ground with a redefinition of consonant and dissonant intervals. Classically, interval qualities are derived from the historical approach as introduced by the Pythagorean school under the form of their tetractys, see Figure 13.1.

Fig. 13.1: The Greek tetractys symbol.

This form is a triangular display of 1,2,3, and 4 points, starting from the top. It is the expression of a “world formula” in this Greek cosmology. Music was then understood as an audible expression of this tetractys in the sense that successive fractions, 1/2, 2/3, 3/4 would represent the intervals of an octave, fifth, and fourth, where these fractions were meant to refer to relations of the length of a string of the monochord, i.e., 1/2 represents going to half the full length of the string, producing an octave higher, 2/3 represents the fraction of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_13

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the full length representing the fifth, and 3/4 represents the fraction of the full string representing the fourth. Equivalently, the inverse fractions 2/1, 3/2, 4/3 would represent the relative frequencies of the fractions of the monochord’s string. Later, with Zarlino’s approach in the 16th century, the number 5 was added, generating the pentactys and enabling the fraction of 4/5 that represents the major third. These fractions were understood to define consonant intervals via frequency ratios. This means that an interval would be classified as a consonant if it represented one of these fractions, which were also the basis of the just tuning. This classification however had the disadvantage that the fourth 3/4 was no longer understood as being consonant in the 16th century, when counterpoint theory had fully developed since its first appearance around 900 with the Gregorian chorale. After 600 years of contrapuntal experiments with variable concepts of consonances vs. dissonances, the final shape of counterpoint in composition and theory had produced an interval concept that was no longer congruent with the tetractys and pentactys. The point here is that the contrapuntal concept of a consonance, 0,3,4,7,8,9 semitones between the two notes of the interval, i.e., prime, two thirds, fifth, two sixths, was no longer a consequence of the Pythagorean approach. In particular, the fourth had become a dissonant interval. The simple fractions were no longer a valid criterion! There have been several attempts, also in the 20th century, to redefine consonances using fractions, but none could create the contrapuntal conceptualization, which had been set up with Palestrina and later, in 1725, canonized by Johann Joseph Fux in his famous Gradus ad parnassum. Moreover, the contrapuntal rules also forbid the “parallels of fifths”, i.e., the immediate succession of a fifth after a fifth. The prestigious German music theorist Carl Dahlhaus argued that this mysterious situation in counterpoint theory was probably due to an undiscovered argument from musical syntax in composition. Our approach to counterpoint therefore started with a fundamental critique of the traditional definition of a consonance. The idea was that in composition, a consonant interval is not consonant by an attribute of a single interval, but that such a quality is derived from the fact that in composition, consonances are defined as collections of intervals that are interesting when creating a musical work. In other words, we argued that being consonant is a property of being a member of a distinguished collection of intervals, and not a property of a single interval without considering other intervals. The solution of our approach was a criterion relating to the dichotomy (K = {0, 3, 4, 7, 8, 9}, D = {1, 2, 5, 6, 10, 11}) of consonant v. dissonant intervals. In fact, it turned out that the dichotomy (K, D) is strong, which by definition means that there is a unique automorphism p of Z12 , which exchanges K and D. This automorphism is called polarity, and its formula is p = T 2 .5. We have p(K) = D, and, by the way, also p(D) = K. For example, p(3) = 5.3 + 2 = 5.

13 Counterpoint Worlds

61

This polarity is used to derive all contrapuntal rules of Fux, in particular the forbidden parallels of fifths. See [27, Part VII] and [3]. This theory does not refer to any psychological argument, it is a purely mathematical model, a fact that opposes it to the usual psychology of contrapuntal arguments. The problem here is the question, why the dichotomy (K, D) would be distinguished among all possible strong dichotomies. This is a classification problem: Describe all possible strong dichotomies and find arguments for the choice of (K, D). Such arguments can be determined, when one considers the ∼ geometry of the interval space Z12 → Z3 × Z4 . In the geometry of this space, the Fux dichotomy has a maximal separation of consonances vs. dissonances.

14 Strong Interval Dichotomies

Summary. Classical Fuxian counterpoint is based upon a dichotomy of the set of 12 intervals within the set of pitch classes, specifying six consonances versus six dissonances. We characterize this dichotomy by its unique autocomplementarity symmetry that exchanges consonances and dissonances, and we calculate all possible dichotomies with this property. –Σ– Definition 4 In the module is Z2n , a dichotomy is an ordered pair (K, D) where each part K, D has cardinality n and they are disjoint. In other words, Z2n = K ∪ D, and K ∩ D = ∅, and card(K) = card(D) = n. The most famous example is the Fux dichotomy ({0, 3, 4, 7, 8, 9}, {1, 2, 5, 6, 10, 11}) from classical counterpoint, where the first half denotes the consonant intervals of a prime, two thirds, the fifth, and two sixths. To develop a counterpoint theory, one has to establish rules that define allowed and forbidden successive intervals. For example, parallels of fifths are forbidden, i.e., a fifth may not immediately succeed another fifth. To derive the set of allowed successions k1 → k2 of consonances, one needs to apply another property of dichotomies: Definition 5 A dichotomy (K, D) is called strong if its parts are isomorphic by a unique symmetry, which is called the dichotomy’s polarity. To exhibit all strong dichotomies, one inspects the list or chord classes 23.1 and searches for those classes C of six-element chords, which are isomorphic to their complement. Evidently, if a dichotomy (K, D) is strong, all its transformations (g(K), g(D) by isomorphisms of Z12 are also strong. These classes are six in number, namely classes C = 64, 71, 75, 78, 82, 87. The Fux dichotomy is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_14

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class 87. It is remarkable that some of these dichotomies arise in natural musical contexts. For example, class 64 is represented by the dichotomy {2, 4, 5, 7, 9, 11} of all proper intervals in the major scale when calculated from the major tonic. Scriabin’s mystical chord {0, 2, 4, 6, 9, 10} represents dichotomy class 78. And Boulez’s pitch class series in his Structures pour deux pianos has its first six pitch classes defining a strong dichotomy of class 71. Conceptually speaking, counterpoint theory only needs the specification of a strong dichotomy class, all technical methods work without additional hypotheses. This means that we have a counterpoint theory for each of the six strong classes. The Fux theory is just the case of class 87. It should be mentioned that for the Fux theory, the allowed transition k1 → k2 are exactly those described by Fux in his famous Gradus ad parnassum [9]. There is a mathematically defined geometric reason for selecting class 87. To understand this fact, one represents the first half of a strong dichotomy as a subset of the third torus Z3 × Z4 , see Figure 14.1.

Fig. 14.1: The third torus separates consonances (K) from dissonances (D) in an optimal way.

The Fux dichotomy shows a maximal separation of K and D on the geometry of the torus. To select the Fux dichotomy within the class 87, one observes, that the Fux selection is defined by a consonance part that is closed under multiplication, it is a multiplicative monoid. This implies that the Fux dichotomy is uniquely determined by a set of purely mathematical conditions that are derived from the classification of chords, no psychology is required.

15 Microtonal Contrapuntal Theories

Summary. We shortly present the extension of counterpoint worlds to microtonal environments, an extension that would be very problematic, if at all possible, without the theory exposed above. –Σ– After the classical environment of pitch classes in Z12 had been investigated relating to counterpoint, Octavio Alberto Agustín-Aquino, a Mexican mathematician, started investigating counterpoint theories in finer environments Z2n for any non-trivial n, especially quarter-tone space Z24 . It turned out that such an extension is feasible in a remarkably coherent way: Not only is it possible to perform counterpoint in such refined spaces, but it is also possible to build “towers” of contrapuntal theories, connecting theories in smaller spaces to theories in larger spaces. It may be possible with an increasing sophistication of music technology to write compositions that live in such towers. The computer scientist Julien Junod has written software in this direction. All these efforts were published in a joint book [3]. We should recall that all of this was only possible because of the new definition of consonances qua sets of intervals with properties that are generic from the mathematical point of view and relate to classification. In this chapter, we want to account for this revolutionary development of a counterpoint theory that transcends all traditional efforts to understand, what counterpoint is about.

15.1 The Category of Strong Dichotomies The category SD of strong dichotomies has as objects the strong dichotomies X = (C, D) on a given Zn , n an even number, together with their polarities pX , which are affine automorphisms of Zn exchanging C and D.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_15

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For two strong dichotomies X1 = (C1 , D1 ) on Z2k1 with polarity p1 , and X2 = (C2 , D2 ) on Z2k2 with polarity p2 , a morphism f : X1 → X2 is an affine morphism f : Z2k1 → Z2k2 such that the following diagram commutes: f

C1 −−−−→   p1 y

C2  p2 y

f

D1 −−−−→ D2 Morphisms of strong dichotomies are connecting counterpoint theories that live in different domains, and in particular are refinement of each other.

15.2 Towers of Strong Dichotomies The basic theorem (proved by Agustín-Aquino, see [3]) states that there exist arbitrary long sequences of such morphisms: Theorem 5 There is an infinite sequence of the canonical injective morphisms of strong dichotomies (a “tower”) X1  X2  . . . X m  . . . with Xm on Z3.2m for all m = 1, 2, 3 . . .. This means that we start on Z6 , the go to the traditional Z12 , then to microtones on Z24 , etc. We may in particular generate such a tower with the Fux dichotomy on Z12 .

16 Dodecaphonic Rows

Summary. This section concludes the discussion of local classification with a presentation of the variety of dodecaphonic series. –Σ– The dodecaphonic composition method was introduced by Arnold Schoönberg around 1921. It is a radical deviation from the classical structures of tonal compositions, such as consonant intervals, standard chord (triads etc.) and harmonic classification, for example via cadences and modulations. Schönberg proposes to base a dodecaphonic composition on the selection of a row, i.e., a determined sequence r = (p1 , p2 , . . . p12 ) of all pitch classes in Z12 . There are 12! = 479, 001, 600 such rows. Together with this selected row, −→∗ Schönberg allows the composer to include all rows f (r), where f ∈ GL , the group of all 48 affine automorphisms f of Z212 which are generated by inversions, retrogrades, retrograde inversions and transpositions. Such a composition would −→∗ consist of a patchwork of rows in GL (r), a set of 48 transformed rows starting from r. In general, due to possible symmetries of r, not all 48 derived rows will be different. In terms of classification, one considers the set R of all rows and −→∗ then takes one of the orbits GL (r). Schönberg also allows for successive row elements to have the same onset. The problem of this approach is that there is no rule that would define a global organization of such a dodecaphonic patchwork. When listening to such a composition, it often turns out to be very difficult—if not impossible—to recognize the basic row and its transformations. The method has no determined syntax, which would be required for tonal compositions. In terms of denotators, a dodecaphonic row is a denotator r : Z11 → Z12 with address Z11 , where the images r(ei ) are given for the affine basis e0 = (0, 0, . . . 0), e1 = (1, 0, 0, . . . 0), . . . e11 = (0, 0, . . . 0, 1). A sequence of d + 1 rows would be a denotator in Zd @Z11 @Z12 , a hyperdenotator of address Zd . In the generalized dodecaphonic composition method, the serial method, one has to define such a sequence of rows, each row mapping into a set of parameters, such © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_16

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pitch, duration, loudness, and attack in the case of a piano composition, with d = 3. Pierre Boulez has constructed such a composition Structures pour deux pianos around 1952. In the tradition of dodecaphonic composition, they have tried to exhibit rows, which have special properties. A first property is what is called an all interval row. Such a row is characterized by the fact that the intervals of successive row elements pi , pi+1 are all possible non-zero intervals, an example is r = (1, 4, 11, 5, 10, 0, 9, 8, 6, 2, 3, 7), defining the interval sequence 3, 7, 6, 5, 2, 9, 11, 10, 8, 1, 4. Composer and theorist Herbert Eimert [7] was the first to present the total collection of all 3, 856 all interval rows. Musically speaking, an all interval row is generic since it generates all intervals. Many rows exhibit a quasi-tonal infrastructure, e.g., r = (0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8, 11), where the diminished seventh chord (0, 3, 6, 9) is represented three times in succession. The idea to consider specific infrastructures in rows is a special case of the fundamental approach to music objects, which S deals with coverings of musical objects o by specific subobjects oi , i.e., o = i oi . Boulez in his seminal book Musidenken heute [4, vol. 1] discusses such global objects with respect to their musical signification. This perspective opens up a vast theoretical field of socalled global compositions, which we will discuss in the following chapters, and in particular addressing the classification topic for global compositions.

Part IV

Global Classification

17 Global Composition and Gesture Classification

Summary. This chapter opens up the concept of a global musical structure with a number of typical examples from theory and composition. It turns out that most situations in “musical structuralism” are essentially related to global phenomena. –Σ–

17.1 Why Global Compositions and Gestures 17.1.1 Global Compositions Here we summarize the rationales for introducing global compositions. Intuitively speaking, global compositions are patchworks of local compositions, i.e., compositions that are not local compositions but compositions that are covered by local compositions to generate structures that are only “locally” represented by local compositions. 17.1.1.1 Boulez and Webern A musical composition is never created as a local composition. In classical European literature, e.g., for sonatas, string quartets, or symphonies, which consist of 104 to 105 tone events, it would be impossible to start from such large local compositions. Rather does the composer start from ‘small’ local compositions, such as motives, themes, chords, rhythms, and similar elements as a basis of a ‘creative combinatorics’ and then merges these parts by use of various transformations to build a compound totality. An excellent example is Boulez’ example of a dodecaphonic series in [138, I]: He describes a series together with its internal structure, i.e., its composition from partial series and their transformations, see Figure 17.1. Boulez’ discussion refers to the local composition in ambient space of integer onset and common © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_17

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pitch classes. We see local subcompositions A;A1;B;G;H of this row and some transformations, drawn as subsets of points which are surrounded by rectangles, defining a total of 15 partial series. These subcompositions are not disjoint in general, they define a sophisticated covering of the given series.

a)

A1

( ).G 1 0 0 2

U.B1

b) U.A1 A G

K.G

U.A1 K.U.B1

( ).U.A1 1 0 0 7

B1

B

A1

C

H

( ).U.A1 = B1 1 0 0 8

Fig. 17.1: The covering of a dodecaphonic row by Boulez.

In his discussion of compositional principles of Schönberg, Berg, and Webern, the idea of hierarchies of local compositions is made explicit by Boulez [138, I, p. 86]: Bei Webern findet sich der Keim einer äusserst fruchtbaren Idee, die die Reihe als Einigungsfaktor von Untergruppen und Obergruppen betrachtet. Im Effekt sind alle isomorphen Figuren einer Grundstruktur von der Tatsache abhängig, dass sie sich immer innerhalb der gle-

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ichen Ordnung abwickeln, entsprechend den gegebenen Transpositionen und Umkehrungen; sie integrieren sich in die Conditio sine qua non der Zwölftonreihe: die chromatische Totalität. Diese isomorphen Figuren bilden die Basis von privilegierten Mengen, die ihrerseits auf einer höheren Ebene wiederum das vorstellen, was die isomorphen Figuren selbst innerhalb der Reihe bedeuten. Durch Verkettung fügen sich Reihenformen, welche festgelegte Privilegien besitzen, zu einer Ganzheit, einer in gewisser Weise “höheren Reihe” zusammen. Die Grundreihe kann dann als strukturelle Kraft der Vermittlung zwischen Unter- und Obergruppen betrachtet werden. Boulez had learned from the second Viennese school that there is a strong local-global principle in serial composition. 17.1.1.2 Uhde/Wieland and Marek Performance is essential in the constitution of a musical work; this was already pointed out by Adorno [1]: Musikalische Interpretation ist der Vollzug, der als Synthesis die Sprachähnlichkeit festhält und zugleich alles einzelne Sprach-ähnliche tilgt. Darum gehört die Idee der Interpretation zur Musik selber und ist ihr nicht akzidentiell. According to Uhde’s and Wieland’s comment in [47], the meaning of “alles einzelne Sprach-ähnliche” is a semantic moment the meaning of involved signs. Therefore, music need not be decoded, but requires “imitation” of itself. Semiotically speaking, Uhde seems to refer to Jakobson’s poetical function (see [14]) as a projection of the paradigmatic axis to the syntagmatic axis of the sign system. A semantic which doesn’t consider the poetical function is excluded. This view of musical performance as a poetical oriented activity is essentially syntactical articulation and paradigmatic intertwining. It establishes an articulated whole, built from local and elementary parts of the composition. In practical performance theory, e.g., in Ceslav Marek’s standard work “Lehre des Klavierspiels” [24], the basic insights of Adorno and Uhde/Wieland are realized in the artisanal details. Here, the metrical grouping of the melody creates successive parts which induce the shaping in performance by dynamical and phrasing (legato/staccato) prescriptions. This level of grouping into local units is also essential for mnemotechnical purposes and in order to give the fingering strategy a support. In fact, it would be bad piano playing to phrase against the fingering strategy. 17.1.1.3 Hoffmann and Kaiser After the foundation of music criticism by Mattheson, Rousseau, or Avison, to name some of the important contributors, it was the merit of Ernst Theodor

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Amadeus Hoffmann—above all in his famous review of Beethoven’s Fifth symphony in the Leipzig-based “Allgemeine Musikalische Zeitung” in 1810—of having given music criticism importance as a contribution to the aesthesic identification of the work. Since this achievement, music criticism was cultivated by profiled critics, such as Eduard Hanslick or Joachim Kaiser, and has contributed to the self-estimation of music and to the history of its reception. It happens only as nonprofessional side-effect that this criticism is eager to celebrate the “uniquely valid” interpretation and performance, or else to lament its vanishing. Although Kaiser may be right in his Beethoven book [17] that whereas it is a rare event in our days when a fundamentally new reading of the 32 Beethoven sonatas is presented, it is a never-ending objective in this field of classical literature to set forth new approaches to the spiritual torso of these sonatas, approaches which add to the given ones new perspectives, variations of explanatory power. In the media and concert business, the music critics should do exactly the job of commenting on this process and of comparing, questioning and evaluating the singular approaches in the spirit of a work in progress towards understanding an infinite evolution. If the poietic work of the artist is expressed by articulation and correlation of the composition’s local parts, then this a fortiori is the work of the critic— only on the level of aesthesis. The overall impression of a performance integrates knowledge, prejudice, and personal disposition, which may result in tiny local effects on agogics, dynamics, articulation, and tuning. This is a central effort to open the access of a broader public to the present work. With the technology of saving works on LP, CD, MC and other media, the articulated listening to music which was founded by music criticism has been enriched by a new aspect: multiply repeated perception. Listening repeatedly to a work on CD changes one’s articulation and grouping activity. Every new listening changes or questions the relevant local compositions and their mutual relations, some are eliminated, others are added. Successively, the listener accumulates a patchwork of elements of comprehension. 17.1.1.4 Graeser, Ruwet, and Nattiez Opposed to the perspectives of composers are positioned the musicologists whose efforts for an adequate analysis are characterized by the need to retrace the composer’s thoughts. It is not astonishing that in the analysis of musical works, organically composed hierarchies are common structures. They start from small local elements, such as chords, degrees, tonalities, tonal functions, voice leading, contrapuntal and harmonic progressions, and end up with large local compositions, such as exposition, development, recapitulation, and coda in the sonata form. This principle of hierarchical organisms was explicitly evidenced in 1924 by Wolfgang Graeser in his analysis of Bach’s Kunst der Fuge [10]. He describes a contrapuntal form as follows ([10, p.17]): Bezeichnen wir die Zusammenfassung irgendwelcher Dinge zu einem Ganzen als eine Menge dieser Dinge und die Dinge selber als

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Elemente der Menge, so bekommen wir etwa das folgende Bild einer kontrapunktischen Form: eine kontrapunktische Form ist eine Menge von Mengen von Mengen. Das klingt etwas abstrus, wir wollen aber gleich sehen, was wir uns darunter vorzustellen haben. Bauen wir einmal ein kontrapunktisches Werk auf. Da haben wir zunächst ein Thema. Dies ist eine Zusammenfassung gewisser Töne, also eine Menge, deren Elemente Töne sind. Aus diesem Thema bilden wir eine Durchführung in irgendeiner Form. Immer wird dies Durchführung die Zusammenfassung gewisser Themaeinsätze zu einem Ganzen sein, also eine Menge, deren Elemente Themen sind. Da die Themen selber Mengen von Tönen sind, so ist die Durchführung eine Menge von Mengen. Und eine kontrapunktische Form, ein kontrapunktisches Musikstück ist die Zusammenfassung gewisser Durchführungen zu einem Ganzen, also ein Menge, deren Elemente Mengen von Mengen sind, wir können also sagen: eine Menge von Mengen von Mengen. The explicit reference to set theory is historically interesting since set theory was a new language in mathematics in 1924. The text is somewhat misleading since it suggests that tones are abstract objects. This is however not the case: Graeser views tones as points in a geometric space, and he also recognizes the role of symmetries, such as rotations and reflections, in such a space, transformations which may be applied to alter sets of tones or to compare different tone-sets ([10, p.13]): Gegenstand der Untersuchungen sind aber nicht die Töne selbst, denn deren spezielle Beschaffenheit spielt gar keine Rolle, sondern die Verknüpfungen und Verbindungen der Töne untereinander. Es wird uns interessieren, ob wir gewisse Analogien zwischen den Gebilden, die man in der Geometrie aus Punkten aufbaut, und unseren aus Tönen hergestellten erkennen können. Das wichtigste Grundprinzip der festen Körper, und die Geometrie ist nichts anderes als das Studium der festen Körper, ist die Eigenschaft der Symmetrie. The Paris school of structuralist linguistics has applied results of semiology after Saussure and Jakobson to musical analysis, above all in the investigations of Nicolas Ruwet [42] and Jean-Jacques Nattiez [39]. The method of neutral analysis which was developed by these authors starts from a hierarchical ordering into units and subunits which are a function of the given work. These units are associated with each other by certain equivalence relations [39]: Les divers unités ont entre elles des rapports d’équivalence de toutes sortes, rapports qui peuvent unir, par exemple, des segments de longueur inégale — tel segment apparaîtra comme une expansion, ou comme une contraction, de tel autre — et aussi des segments empiétinant les uns sur les autres.

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The equivalence relations are not arbitrary but realized by specific transformations: Les unités paradigmatiquement associées sont équivalentes d’un point de vue donné (le thème paradigmatique), rarement identiques, et reliées entre elles par des transformations qui décrivent les variants par rapport a des invariants. This language resembles the one which Graeser seems to aim at. However the geometric aspect—embedding tones in spaces which admit symmetry transformations—is more radical with Graeser, though less flexible regarding the transformations which may be applied (Graeser limits his approach to transformations of the “rigid” geometry, i.e., isometries). 17.1.1.5 Jackendoff and Lerdahl Explicit grouping concepts are described by Ray Jackendoff and Fred Lerdahl in [15]. Grouping is described from an aesthesic point of view of music psychology. It deals with portions of notes which are heard as building an auditory unit. In this approach, a group is always defined by a time interval, i.e., a group consists of strictly time-adjacent notes and cannot be restricted to proper subsets within time-slices, and which would be defined by voice splitting or parametric splitting of pitch or loudness, for example. Also this grouping is a nearly perfect hierarchy under inclusion: Overlapping neighboring groups may only contain one common onset and are treated as very special situations in this theory. A more general “web of motivic associations” would be beyond the theory because it is not hierarchical [15, p.17]. This approach resembles Graeser’s concept of a contrapuntal structure which is also a hierarchical grouping of parts, but Graeser’s idea was more general insofar as it did not strictly ask for time slices. It is also less paradigmatic than Graeser’s, Ruwet’s, and Nattiez’ because no significant statements are made concerning the association of groups under symmetry transformations. From the remarks in [15, p.286] it follows that these authors have no concept of the transformation groups which may be adapted to specific contexts, as it is explicitly proposed by Ruwet and Nattiez under the flag of “paradigmatic theme” (see [39] for this concept). But it is precisely the psychological claim of grouping—even in its strictly hierarchical appearance of the Jackendoff-Lerdahl theory—as a cognitive basic which undermines the fact of global structures in music. 17.1.1.6 Schaeffer and Cage In the last years of the 1940s of the 20th century, the new technology of tape music became a paradigm for local-global constructs. The American music for tape movement was initiated by Otto Luening and Vladimir Ussachevsky and

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applied by John Cage in the early fifties1 since his first music for tape Imaginary landscape. At the same time, in Paris, Pierre Schaeffer and Pierre Henry initiated the “musique concrète” movement, also based on tape as a flexible medium of syntactical combination and recombination. The new tape medium became a tool for concretely cutting and merging time-slices of music. This was a starting point for an entire group of technological realizations of local-global patchworks. At present, it is largely extended and refined in various software for musical composition, notation, and postproduction as compared to hard disk recording. The local parts are termed tracks, parts, global and local scores, etc. But the tape music movement had not only consequences for music software, it also changed the concept of a score. In particular, Cage realized compositions, such as his concert for piano and orchestra (1957), where the modular structure of the composition, showing a number of relatively autonomous local parts, and an intended ambiguity of identifying such parts, became an explicit and primordial feature of his poiesis. In view of these rich traces of a local-global paradigm in music, Christian von Ehrenfels’ approach to gestalt which stresses super-summativity (in fact a warmed-up version of the Aristotelian principle that “the whole is more than the sum of its parts”) does not look very original. But it does provoke the question how much more the added value exceeds the sum of the parts. The above examples show that this may be a very complex question, in fact the only interesting point of super-summativity. How is the whole constructed from the collection of its parts? When are two wholes different, though having identical parts? How can we compare wholes if we suppose that their parts are comparable? Fact 1 As long as no precise structure theory of wholes qua constructs from local parts is available, no real understanding of music is possible. By the arsenal of the preceding examples, this is a problem which touches all levels of the communicative axis. And it is a problem which involves interpretative activity and its innate ambiguities. 17.1.1.7 Musical and Mathematical Manifolds In history of science, the mathematical and musicological concepts of global gestalts is situated around 1854 when Eduard Hanslick (Figure 17.3) published his famous treatise “Vom Musikalisch Schönen” [11]. Musical content was recognized as being “tönend bewegte Formen”. Hanslick added that these forms are by no means elementary but composed in an artistic way, and building a unity within the manifold, as restated by music theorist Hugo Riemann. 1

In the Project of Music for Magnetic Tape, together with Morton Feldman, David Tudor, and Christian Wolff.

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Fig. 17.2: Bernhard Riemann introduced global structures (manifolds) as compound mathematical spaces.

Fig. 17.3: Eduard Hanslick described music as a compound structure, built artistically from parts.

It is remarkable that in the same year, the mathematician Bernhard Riemann (Figure 17.2) conceived a far reaching generalization of the mathematical concept of space to so-called manifolds [40]. These are understood as being patchworks of locally cartesian charts, similar to geographic atlases. The shared conceptual basis of these approaches is that locally trivial structures can add up to aesthetically valid configurations if united in a nontrivial way. A simple and well-known example of such a global shape is the Möbius strip. Its fascination stems from gluing together the ends of an ordinary ‘belt’-shaped strip after a rotation of half the full circle of one end. For example, in musicology, the Möbius strip is realized by the harmonic strip of triadic degrees within a diatonic scale, we shall discuss this item below. The basic difference between musical and mathematical manifolds is that musical manifolds are defined with a fixed atlas, whereas mathematical manifolds are not tied to fixed atlases. Intuitively speaking, the sphere of our globe, viewed as a mathematical manifold, is the same if we add new small or large charts as long as they are compatible with the given ones. If I add a regional map, nobody will complain that the geographic identity of the globe has changed. Mathematically speaking, we may go to the colimit of all atlases. But in music, this is completely different: A given covering of a composition by a determined set of charts, such as chords, motives, or periods, is essential for the identification of the composition, two different coverings change the composition qua global structure or gestalt, to use Christian von Ehrenfels’ concept. Mathematically speaking, the colimit of charts (going to all possi-

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79

ble additional local perspectives) is not allowed, the individual interpretational activity is an integral part of the object’s identity. 17.1.1.8 Global Gestures In practice, gestures are not compactly defined, but reveal a union of local gestures, which coincide on determined subgestures. A typical example would be the dancing gesture, where two persons are represented by their local gesture, but their holding hands share a common subgestural configuration, see Figure 17.4.

Fig. 17.4: Two dancers holding each other by their hands.

17.1.1.9 The Definition of Global Compositions and Global Gestures Definition 6 A (global) objective composition is defined by the following data: (i) (ii) (iii) (iv) (v)

A set G and a finite, non-empty covering I of G by non-emty subsets, a family (Kt , Ft )t∈T of local objective compositions, a surjection I? : T → I : t 7→ It , ∼ a bijection φt : Kt → It for each t ∈ T , for each couple s, t ∈ T such that Is ∩ It 6= ∅, the induced bijection −1 −1 φs,t := φ−1 t ◦ φs : φs (Is ∩ It ) → φt (Is ∩ It )

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(restricted to the respective domains and codomains) defines an isomorphism ∼ −1 φs,t /1 : (φ−1 s (Is ∩ It ), Fs ) → (φt (Is ∩ It ), Ft ) of local compositions (we only discuss zero-addressed compositions here, but a general approach is feasible, see [28, vol. 1]). The data (iii) to (v) are called an A-addressed atlas Φ for the covering I of G. The bijections φt (or—by abuse of language—the local compositions Kt ) are called the charts of the atlas Φ. Two A-addressed atlases Φ, Ψ for the covering I of G are called equivalent iff their disjoint sum Φ q Ψ is an atlas for the covering I of G. An objective global composition is a covering I of G (often abbreviated by GI ) together with an equivalence class of atlases for GI . If no confusion is likely, we may abbreviate the entire data by saying that we are given an objective global composition G, i.e., by just naming the support set G of the objective global composition. The surjection I? refers to the option to have several charts for the same local composition part. Global gestures are defined in the same sprit as follow, similar to the definition of global compositions: Definition 7 A global gesture consists of these components: 1. a digraph Γ , the support, and a finite covering I of Γ by non-empty subdigraphs, 2. a surjection I? : T → I : t 7→ It , − → 3. a family (kt : It → Kt )T of local gestures, 4. for each couple s, t ∈ I such that Is ∩ It 6= ∅, the identity Ids,t of digraphs Is ∩ It extends to an isomorphism fs,t of local gestures, ∼

fs,t : ks |Is ∩ It → kt |Is ∩ It . of local gestures. The data in 2. and 3. are called an atlas of the global gesture, which is often denoted by Γ I . The local gestures kt are called the charts of the atlas. Two atlases are equivalent if their disjoint union is also an atlas for Γ I . The global gesture is defined by an equivalence class of atlases. 17.1.1.10 Examples of Global Compositions and Gestures Global compositions and gestures may be derived from corresponding local → − compositions K ⊂ M and gestures ∆ → X by a simple covering of the ambient spaces, i.e., of K or of ∆. Given such a covering K I or ∆J , for each subset Ii ∈ I, we get a restricted local composition Ki ⊂ M , and for each subset → − Jj ∈ J, we get a restricted local gesture ∆j → X . The non-empty intersections

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81

in both cases are connected by the identities of the ambient spaces. Such global structures are called interpretations of the given local structures. It is remarkable that not every global composition is isomorphic to the interpretation of a local composition. Here is such a “pathological” example. It means that this case cannot be played in a musical performance since there is not ambient module of musical parameters, where the performance could take place. Nevertheless, non-interpretable global compositions are conceptually important as “germs” of interpretable global compositions. We shall show in Section 17.2.3 how this can be achieved. 17.1.2 The Nerve of a Global Composition or Gesture If GI or Γ I is a global composition or gesture, its nerve n(GI ) or n(Γ I ) is a subset of the powerset of covering index set I defined by the following properties: 1. Every element i of I has the singleton {i} in the nerve. 2. If T a finite set of chartes σ = {i1 , i2 , . . . ik } ⊂ I has a non-empty intersection σ 6= ∅, then σ is in the nerve. The elements σ of the nerve with k elements are called k − 1-simplices. So the 0-simplices are the charts. The nerve is also called the simplicial complex of the global structure. One may represent the nerve by writing the 0-simplices as points in Rn , the 1-simplices as straight lines connecting their two 0-simplices, the 2-simplices as triangular surfaces limited by their 1-simplices lines, the 3simplices as full tetrahedra limited by their 2-simplices triangles. The construction goes on for higher dimensions, but these cannot be visualized in 3-space. See [27, Ch. H.2.1] for a complete mathematical definition. A classical example is the global composition of a triadic interpretation X (3) of a major scale X defined by its seven triadic degrees as charts. Its nerve is a Möbius strip, as shown in Figure 5.1.

17.2 Classification through Modules of Affine Functions The general classification of global compositions and gestures uses functions on such global structures to describe them in the context of function spaces and their transformations by natural group operations on standard structures. 17.2.1 Local Compositions and Functions We describe the classification method of global compositions using affine function modules. The idea is a development starting from the tools, which were used in classification of local compositions. Let us look first at the local situation. We may first consider the so-called standard local composition of dimension n, ∆n ⊂ Rn consisting of the n + 1 points e0 = 0, e1 = (1, 0 . . . 0), . . . en =

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(0, . . . , 0, 1). For every local n + 1-element composition K ⊂ M , we have a one-to-one morphism f : ∆n → K sending e0 to k0 , etc., and en to kn , for an enumeration K = {k0 , . . . kn } of K’s elements. Recall that we used such a function in local classification, when considering the module morphism Rn → M . This map f can be used to induce an injective linear map K@R → ∆n @R, sending an affine function g ∈ K@R to the composed function g ◦ f . Here, K@R denotes the R-module (addition and scalar multiplication element-wise) of local composition morphisms from K to the full local composition R ⊂ R. This identifies the module of affine functions K@R with a submodule of the module of affine functions ∆n @R. This affine function methods will be sufficient to create a tool for global classification. The idea is to classify the affine functions qua functions on the standard composition and to generate orbits of such function sets under the action of an adequate permutation group. If we take a function module K 0 ⊂ K@R ← M @R, we may create a local composition K 0∗ ⊂ K 0 @R by sending the base elements ei of ∆n to their image ei∗ in K 0 @R via e∗i (x) = x(ei ) = x(f (ei )), x ∈ K 0 . If for any two different ei , ej there is a function x that has different images, we call K 0 separating. In this case, the new local composition K 0∗ is the bijective image of the standard composition ∆n . This new K 0∗ has been constructed from the module K 0 ⊂ ∆n @R of affine functions, and if we can show that this new composition is also isomorphic to the initial K, we may think of classification via modules of affine functions over the standard local composition. Classification using this method can be achieved if the ambient module M of K ⊂ M is finitely generated projective, i.e., if we M is a direct summand of a free module Rm . In this case, the coordinate functions pi : Rm → R yield separating candidates. Such local compositions are separating, and K 0∗ is in bijection with the original K. The bijection K → K 0∗ is defined by sending k ∈ K to the evident k ∗ ∈ K 0 @R ⊂ M ∗ @R, M ∗ = Hom(M, R). 17.2.2 Morphisms and Function Spaces We should add to our previous constructions of local compositions and gestures from function spaces the behavior of morphisms with respect to this function space approach. Let us first consider a morphism f : K → L of local compositions (K, M ), (L, N ) that is (by definition) induced by an affine module morphism h : M → N . We have this commutative diagram of local compositions:

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83

∼

→

K

- K 0∗

f ∗

f

→

∼

? M

-

-

∼

L

→

- L0∗

? - M ∗∗

-

-

? ? ∼ → - N ∗∗ N with bijections K → K 0∗ , L → L0∗ , where K 0∗ is defined by the module of affine functions that are restrictions of linear functions M → R, where L0∗ is defined by the module of affine functions that are restrictions of linear functions N → R, and where the codomain local compositions are derived from the above affine function modules within the standard local compositions and ∆m , ∆n if m = card(K), n = card(L). In other words, the map K 7→ K 0∗ is a natural transformation, bijective for separating local compositions. In the special case, where M is projective and finitely generated, the bidual map M → M ∗∗ , M ∗∗ the linear bidual of M , is an isomorphism, and the morphism K → K 0∗ is an isomorphism. In this situation (ambient modules finitely generated projective) the natural transformation K 7→ K 0∗ can be used to generate an isomorphism of a global compositions between the given global composition and a global composition that is derived from affine function modules. If in a global composition GI , we have two charts with local compositions K, L, such that the intersections (subcompositions) are isomorphic, ∼ K|L → L|K, then the bidual of this configuration for finitely generated projective ambient modules produces intersections (K|L)0∗ ⊂ K 0∗ and (L|K)0∗ ⊂ L0∗ also being isomorphic. In other words, the global composition GI produces an isomorphic bidual global composition G∗I via the bidual natural transformation. And the bidual global composition is generated by affine function modules. The next step deals with the following global affine functions on a global standard composition. 17.2.3 Global Standard Structures and Resolutions for Compositions For a covering GI of a set G by non-empty subsets Gi , i ∈ I, we may create a global composition ∆GI together with an obvious bijective morphism

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17 Global Composition and Gesture Classification

res : ∆GI → GI by the following method. Take the vector space Rn , where G is supposed to consist of n+1 elements. Take the global composition ∆GI that consist of all n + 1 affine basis vectors e0 = 0, e1 = (1, 0, . . . 0), . . . en = (0, . . . 0, 1), take any bijection of this basis with G, and define the charts ∆i of ∆GI as being the inverse image of the charts Gi . This defines an evident bijective morphism res : ∆GI → GI , and we call this the resolution of GI .

18 The Classification Theorem for Global Compositions

Summary. This theorem is very involved conceptually and mathematically. We first develop the conceptual framework, which is based on the construction of the resolution of a global composition. Every global composition turns out to be a kind of projection of its resolution. In this context, we discuss systems of functions on resolutions, and we show, how these function systems are used to classify global compositions. From this theory we derive the general classification theorem of global compositions –Σ–

18.1 Preliminary Remarks 18.1.1 Characterization of Interpretations To find out whether a global composition is interpretable, i.e., isomorphic to an interpretation, one has to consider affine functions on global compositions. An affine function on a global composition GI is a function f : G → R, R being the underlying ring of its charts. Such a function is, by definition, such that all of its restrictions to the composition’s charts are affine functions. In general, it is not true that every affine function f : Gi → R on a chart Gi can be extended to an affine function on G. If every chart affine function can be extended to all of G, the global composition is called flasque. It can be proved that Theorem 6 A global composition G is interpretable, iff all its affine functions on charts can be extended to functions on all of G, i.e., iff it is flasque. See [27, Ch. 16] for a proof. 18.1.2 An Example of a Non-interpretable Global Composition The theory of global structures exhibits global structures that are not interpretable, i.e., they are not isomorphic to interpretations. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_18

85

86

18.2.

18 The Classification Theorem for Global Compositions

An example of a non-interpretable global composition is shown in Figure

Example 1 We discuss a non-interpretable global composition GI in the zero address, and over the field of real numbers, see Figure 18.2. It has six points, G = {x1 , x2 , x3 , x4 , x5 , x6 },

ˇ “*

and three charts for its covering, i.e., I = {U1 , U2 , U3 } with U1 = {x1 , x2 , x3 , x4 }, U2 = {x1 , x2 , x5 , x6 }, U3 = {x3 , x4 , x5 , x6 }.

Fig. 18.1: A non-interpretable composition together with its resolution.

The three covering charts are all in bijection with a ‘square vertex’ composition S = {a = (0, 0), b = (1, 0), c = (1, 1), d = (0, 1)} ⊂ 0@R R2 as follows:

18.2 Non-Interpretable Compositions

87

∼

7 b, x4 7→ c, U1 → S : x1 7→ a, x2 7→ d, x3 → ∼ U2 → S : x1 7→ a, x2 7→ d, x5 7→ c, x6 7→ b, ∼ U3 → S : x3 7→ b, x4 7→ c, x5 7→ a, x6 7→ d, so that the intersections of any two of these charts yield an isomorphism of a two-point composition in R2 . If we visualize this configuration by means of a subdivision of each chart as a union of two triangular surfaces, we obtain a Moebius strip, see Figure 18.2. The resolution of GI in the same surface representation is also shown. Here, the charts are no longer plane compositions but tetrahedra in three space. Let us show that GI is not interpretable. It is evidently sufficient to see that the global affine functions on this composition do not separate points. Let f : G → R be an affine function (we suppress the zero address and just work in the respective ambient spaces), set f (xi ) = fi for i = 1, 2, 3, 4, 5, 6 and fi,j = fi − fj . Since f is affine on each chart, we have on chart U1 : f1,2 = f4,3 , on chart U2 : f1,2 = f5,6 , on chart U3 : f4,3 = f6,5 . Therefore f5,6 = f6,5 , and f5,6 + f6,5 = 0, by definition, so 0 = f1,2 = f4,3 = f6,5 , and no one of the pairs (x1 , x2 ), (x3 , x4 ), (x5 , x6 ) can be separated by f .

18.2 The Musical Meaning of Non-Interpretable Compositions: Varèse’s Program and Yoneda’s Lemma Summary. A non-interpretable global composition cannot be “played” in a score since there are not enough affine functions to describe the composition’s coordinates, a condition that is essential for a composition to be represented as a set of points with coordinates in a “score space”. Implicitly, the part of Yoneda’s Lemma dealing with variations of perspectives is of primordial importance in compositional concepts. In Edgar Varèse’s programmatic writings, a thoroughly geometric approach to the Yoneda philosophy is sketched. It accomplishes the classical variational principle in composition; we give an overview to this central connection between modern mathematics and music. –Σ– Let us first recapitulate the impact of the resolution of a (commutative) global composition on the esthetics of music. The resolution ∆GI of a global composition GI generates points which are in general position in every chart of the atlas. The resolution is interpretable, and GI can be rebuilt from the retracted function module in the resolution. Since the resolution’s nerve is isomorphic to the original nerve, and since the resolution projects bijectively onto the original composition, no note event and no overlapping relation of charts is destroyed in the resolution. So we virtually have the same note set, but they are enriched by a collection of parameters

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18 The Classification Theorem for Global Compositions

Fig. 18.2: Edgar Varèse (1883 – 1965).

which allow us to place these events in optimal relative position. So the resolution can—in principle—be performed by physical instruments, and it can—in principle—be played such that the old idea may be heard since old parameters are preserved. The freedom of choice for an optimal realization of the resolution (which after all is only determined up to isomorphism) is also mandatory since a good physical performance has to meet a number of additional conditions of human cognition. The auditory system, the instrumental skill of an artist, the material possibilities, the time frame at disposition, etc., all these conditions impose serious value boundaries on the allowed parameter values which are acceptable in a good realization. Typically, these parameters must reflect the syntactical structure of the composition, and not only the resolution’s general position context. So time must be given a crucial role in the parametrization of events. And the distinction of events must also be optimized when the temporal unfolding of a performance represents a good communication stream. Nonetheless, the resolution classification technique yields very important necessary conditions for a comprehensible performative parameter setting. What could now be the program of classification? Its core objective is that it deals with understanding musical works. And we should stress that our concept of a musical work is not the narrow one which restricts to those individual opera which—especially in Europe—started to flourish in the Renaissance. It includes as well general musical corpora such as scales, systems, everything that can be represented by means of global compositions, and—in the limit—any denotator if we admit the most general topos of this theory.

18.2 Non-Interpretable Compositions

89

From the precise parametric description of a work and its ambiguities, this work appears as a point configuration in a more or less complex space (or form, if one prefers the denotator terminology). However this configuration is already a defined perspective which enables many relations among its ingredients. It is the composer’s perspective (now including an abstract ‘composer’ or creator of a general musical structure like a scale). For example, the choice of tonality, instrumentation, tempo, etc., are points of view which may or may not pertain to the composition, this is a question of the epoch of creation. But their character can undoubtedly be subject to variation. Among others, here we do address the question of historical instrumentation for early music. In order to understand the relations among different parts of a composition, and even to simply recognize them, a change of the given perspective is mandatory. If a never seen object must be inspected, what should we do? You walk around it. This is the most common version of Yoneda’s Lemma. The analogy to cartography is straightforward: The natural perspective of the landscape in which we live does not coincide with the perspective which best meets our need for orientation. To reach this end, we build maps which show the landscape from an infinitely far vertical viewpoint. The same is the case in music. You play a piece in slow motion ‘from very near’, in a zoomed perspective, a complex chord is arpeggiated, i.e., viewed from a skew angle, and so forth. This idea of variation of the perspective has also been integrated in the compositional thinking of the 20th century. We want to illustrate this remarkable fact by a citation from Edgar Varèse’s comments on his composition “Intégrales” [49, p.67]: Die Intégrales wurden für eine räumliche Projektion entworfen. (...) Während wir in unserem musikalischen System Klänge anordnen, deren Werte festgelegt sind, suchte ich eine Verwirklichung, bei der die Werte fortwährend im Verhältnis zu einer Konstanten verändert werden. (...) Um dies besser zu begreifen, übertragen wir, da das Auge viel schneller und geübter ist als das Ohr, diese Vorstellungen ins Optische und betrachten die wechselnde Projektion einer geometrischen Figur auf eine Fläche, wobei Figur und Fläche sich beide im Raum bewegen, aber jede nach ihren eigenen Geschwindigkeiten, die veränderlich und verschieden sind, die sich verschieben und rotieren. Die augenblickliche Form der Projektion ist durch die Relation zwischen Figur und Fläche in diesem Augenblick bestimmt. Aber wenn man erlaubt, daß die Figur und die Fläche ihre eigenen Bewegungen haben, ist es möglich, mit der Projektion ein äußerst komplexes und scheinbar unvorhersehbares Bild zu erhalten. Diese Qualitäten können noch vermehrt werden, wenn man die Form der geometrischen Figur ebenso wie ihre Geschwindigkeiten variiert. (...) Ich hoffe, innerhalb kurzer Zeit einen Apparat zur Verfügung zu haben, der es erlauben wird, ein räumliches Relief zu geben. Nur des Beweises wegen würde ich daran interessiert sein, die Intégra-

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18 The Classification Theorem for Global Compositions

les eimal so zu realisieren, wie sie ursprünglich konzipiert worden sind. In 1960 Maurizio Kagel transferred these principles on paper strips and discs of the score for Transición II for piano, percussion, and two tapes. Varèse’s idea basically is a remake of the classical variation principle. Bach’s Goldberg Variations (BWV988), Beethoven’s Diabelli Variations (op.120), or Webern’s Variationen für Klavier (op.27) are compositions that follow this spirit. The subject is always an artistically interwoven change of perspectives of a theme: the variation in the parts of the theme and their relations. Principle 1 Variation as a principle of musical shaping is nothing else than the identification of an idea—such as the theme—as a integral of its perspectives. For Webern a composition is a cellular organism, a connected manifold of transformations, of ever changing perspectives, of metamorphoses of a single cellular germ (in the sense of Goethe), which in fact is Schönberg’s dodecaphonic rows. In front of this historical background the classification problem of global compositions—together with its central process of resolution—appears as a canonical program. In particular, the nerve of a resolution, a concept related to that of a “cell complex” from algebraic topology, reminds us of the cellular organism alluded to by Webern. The projections described by Varèse in a visionary fashion show a surprisingly similar geometry to the projections of a resolution onto the original composition, projections which are distinguished in that they project a general position onto specializations. Finally, the variation of these projections corresponds to the variation of the modules of affine functions, i.e., the variation of the compositions which are distinguished from each other via their retracted function modules on one and the same resolution. The variational principle is not only a compositional strategy, it even more dramatically applies to the level of performance. Performance deals with a transformation from the mental score space to the physical space of an acoustic realization. But this transformation locally is a deformation of the “rigid” parameter values set out on the score. Why should the artist deform a perfect opus? Wouldn’t this be blasphemy or at least a lack of respect? No, the added value of such a deformation is not a destruction of given structure, it is a subtle change of parametric perspectives which let the auditory still recognize the written relations, but on top of that puts configurations into general position such that their generic, or better: resolved, structure becomes ‘visible’ on the auditory level—to restate it in the wording of Varèse. Principle 2 So the structural rationale of performance is a strategy of small changes of the composer’s perspective to make the resolution of the composition audible and thereby ease understanding of the underlying composition class (in the strict sense of classification).

18.3 Classification of Global Compositions

91

18.2.1 Global Dodecaphonic Classes Dodecaphonic rows are also interpreted by “Binnenstrukturen” (“inland structures”), especially in Boulez’ theory.

18.3 Classification of Global Compositions The classification of global compositions is based upon a special type of global compositions, global standard compositions. These objects are “free” global compositions, they are always interpretable. And every global composition is a kind of image of a global standard composition, which is called the global composition’s resolution. The information about a global composition may be “retracted” into its resolution, where it is represented as a module of affine functions. The representation can be used to calculate all isomorphism classes of global compositions qua points in adequate algebraic varieties (schemes). Technically speaking, this classification method is quite demanding, so we shall limit our presentation to some key points without delving into mathematical subtleties. We refer to [28, Vol.I, Ch. 15] for a thorough trajectory. 18.3.1 The Resolution of a Global Composition The resolution of a global composition GI runs as follows. We first introduce the standard global composition ∆GI . We suppose that G consists of n + 1 points, and that its charts Gi , i ∈ I have ni points. Then we look at the local composition ∆n ⊂ Rn that consists of the (affine) base points e0 = 0, e1 = (1, 0, . . . 0), . . . en = (0, . . . 0, 1) ∈ Rn . We now enumerate the points of GI and associate the chart points of GI with the n + 1 base points of ∆. This defines a covering of ∆ by corresponding charts ∆i , each being defined by the corresponding points of the GI charts. This global composition ∆GI evidently interpretable, and we get a bijective morphism p : ∆GI → GI , this is the resolution of GI . The idea now is to represent GI by its affine functions, which are represented on ∆GI as follows. We first consider the affine functions on GI . This means that on each chart Gi of GI , we take the R-module of affine functions a : Gi → R, i.e., the set Γi = Gi @R. We also look at the affine function modules Γσ = Gσ @R, where Gσ denotes the subcomposition for a non-empty intersection σ of a number of charts. We have evident restriction module homomorphisms Γσ → Γτ whenever you σ ⊂ τ . De note this system of functions by N (GI ), and call it a function complex. The bijection (not an isomorphism in general!) p induces an injection N (GI ) → N (∆GI ). It is this injection of functions that we will use to reconstruct GI . The complex N (GI ) is used to generate a new global composition. On the subompositions σ, we consider the liner dual modules N (GI )(σ)∗ and within

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18 The Classification Theorem for Global Compositions

these modules, we define points by taking base points ei in σ and then consider their duals ei∗ ∈ N (GI )(σ)∗ , sending f ∈ N (GI )(σ) to the e∗i (f ) = f (ei ), a construction, which we discussed in Section 17.2.1. To reconstruct GI , we must be assured that the ei∗ , e∗j are different if you i 6= j. This is guaranteed if there are ‘enough’ functions in every N (GI )(σ), i.e., if there is a function f such that ei (f ) 6= ej (f ). In this case, we call N (GI ) separating. It can be shown (referring to the discussion in Section 17.2.1) that the complex N (GI ) is separating whenever the carrier modules of GI are finitely generated projective. Under this latter hypothesis, we now constructed a global composition built by the ei∗ , which are defined over function modules N (GI )(σ). This global composition, which we denote by ∆/GI is isomorphic to the original GI , when sending its points to the corresponding e∗i under the given bijection p, we omit the very technical proof of this fact and refer to the technical details in [28, Vol.I, Ch. 15]. This construction may be embedded in a commutative triangle ∆GI p

(18.1)



-

∆/GI

- GI

∼ →

Moreover, if we are given a morphism F : GI → H J , this one extends to the above triangle: ∆ GI pG

-

∆/GI

∆F

- GI

∼

→

? ∆H J

F pH

-

? ∆/H J

∼

→

- ? - HJ

This result means that we may reconstruct separating global compositions from their complexes of affine functions, as symbolized by ∆/GI . Moreover,

18.3 Classification of Global Compositions

93

these complexes are defined on the standard compositions, and one can prove that isomorphic global compositions are defined by isomorphisms of function complexes that are derived from permutations of the standard composition’s points. In other words: Isomorphism classes of global compositions are defined by the permutation orbits (on standard global compositions) of separating function complexes. Similar to the classification theorem of local compositions, see Section 8.1.4, we have this classification theorem: Theorem 7 The isomorphism classes of global compositions over finitely generated projective modules can be parametrized as rational points on schemes of finite type. The technically precise content of this theorem is displayed in [28, Section 15.3.2, Theorem 18], but we cannot discuss this level because it requires some higher algebraic geometry. Nevertheless, our presentation captures the main ideas leading to this result.

19 The Classification Problem of Global Gestures

Summary. According to the description of local gestures using function spaces as described in Section 19.1, it seems reasonable to go parallel to the classification of global compositions. We may consider the system (a complex) of functions Im(|gi |)∗ ⊂ |∆i |∗ and then step over to the orbits of such function complexes according to the action of the finite permutation group of an underlying global digraph. However, as already shown for the local gesture situation, classification of gestures is not feasible yet. Our discussion will therefore focus on techniques that could help classify gestures in future developments. –Σ–

19.1 Local Gesture and Functions The method of functions for compositions can also be applied to local gestures → − g : ∆ → X . The point here is to find continuous functions f : Im(g) → D, where Im(g) is the topological subspace of X of the curves of g, with a codomain topological space D such that it separates points in Im(g). In this case, we may proceed for local gestures in a similar way as with local compositions: Take the associated continuous map |g| : |∆| → Im(g) ⊂ X and then the inverse image of Im(g)∗ = Im(g)@D in |∆|∗ = |∆|@D. We get a canonical continuous map Im(g) → Im(g)∗∗ and its extension to X → X ∗∗ . We now get a gesture g ∗∗ : |∆| → Im(g)∗∗ ⊂ X ∗∗ , the values x ∈ |∆| being sent to g ∗∗ (x) : h 7→ h(x). We get a commutative diagram for gestures as follows. Take a morphism of gestures: → − g ∆ −−−−→ X    → ty y−f → − h Γ −−−−→ Y which corresponds to a commutative diagram of topological spaces: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_19

95

96

19 The Classification Problem of Global Gestures |g|

|∆| −−−−→   |t|y

X   yf

|h|

|Γ | −−−−→ Y Taking the “bidual” |∆|∗∗ = (|∆|@T op D)@T op D etc., with a suitable codomain topological space D, we get a commutative diagram |g|∗∗

|∆|∗∗ −−−−→ X ∗∗    f ∗∗ |t|∗∗ y y |h|∗∗

|Γ |∗∗ −−−−→ Y ∗∗ This induces the following commutative diagram for gestures, analogous to the previous one for local compositions, and where Imageg , Imageh are the images in the biduals, respectively: |∆|

|g|

⊂

- Im(|g|)

- X

bi j.

f

? |Γ |

Imageg

f

? - Im(|h|)

⊂

⊂

|h|

-

-

|t|

- X ∗∗

? - Y

f ∗∗

j.

bi

⊂

-

-

? Imageh

? - Y ∗∗

The critical difference with respect to local compositions is that there is no analogy to finitely generated projective modules in topology, i.e., there is no chance to have the biduals being isomorphic to the original objects. 19.1.1 Restriction of Gesture Types We may try to choose a codomain D that would work for a good selection of gestures. The topological problem is that there are too many fundamentally different topologies to enable a generic approach. For example, one may consider topologies deduced from metrics, which are very different from topologies deduced from algebraic geometry, such as Zariski topologies, where points are

19.1 Local Gesture and Functions

97

not comparable, there are thick and thin points, a situation that would not occur for metrically defined topologies. We therefore decide to consider the following framework that is frequently encountered in gestural practice. We first suppose that the topological spaces, where gestures are projected, are compact sets within Rn , which means that they are closed and limited. We then consider the codomain D = [0, 1] = I, the real unit interval, which is also compact. Under these conditions, the space X@Top D is compact metric with the distance function d(ξ, η) = maxx∈X ξ(x) − η(x). The same construction yields a compact metric space X ∗∗ = (X@Top D)@Top D. Let us now look at the bidual map q : Im(|g|) → Im(|g|)∗∗ for a gesture g, and Im(|g|)∗∗ being the image of Im(|g|) in X ∗∗ . To begin with, this map is continuous. In fact for x, y ∈ Im(|g|), d(x∗∗ , y ∗∗ ) = maxf ∈X ∗ x∗∗ (f ) − y ∗∗ (f ) = maxf ∈X ∗ f (x) − f (y). If x and y are near to each other in the given metric, the distance d(x∗∗ , y ∗∗ ) is small since all f are continuous, and by compactness we may choose a finite number of such f . Conversely, the map is also open, i.e., the inverse is continuous. This follows from the observation that if d(x∗∗ , y ∗∗ ) is small, then all function distances f (x)−f (y) are small, especially the coordinate functions of x, y must have a small difference. This means that we have a ∼ homeomorphism q : Im(|g|) → Im(q) ⊂ Im(|g|)∗∗ . The function q can also be obtained from the embedding |g|∗ : Im(|g|)∗ → ∗ |∆| by stepping over to the dual of this map, and then composing it with the embedding |∆| → |∆|∗∗ according to the following commutative diagram. |g| surjective

|∆| −−−−−−−−→ Im(|g|)     yq y |∆|∗∗

|g|∗∗

−−−−→

Im(|g|∗∗

19.1.2 Global Standard Structures and Resolutions for Gestures To construct a global standard gesture ∆I for a global gesture DI , we consider the standard topological space |∆| deduced from the digraph ∆ of DI , its parts are the unit interval lines [0, 1] ⊂ R, one for each arrow of ∆, and glued together (the colimit) on the shared vertices. This topological space is the union of its subspaces that are defined by the global gesture’s charts and |∆i |, i ∈ I. For −−→ every chart ∆i , we have the map ∆i → |∆i | defining the ith chart of the global gesture ∆I . This is the global standard gesture for digraph’s ∆ covering I. For a global gesture DI on ∆ with a covering I, we obtain the evident morphism ∆I → DI sending the standard charts on ∆i to the charts Di of DI . We call this morphism to resDI : ∆I → DI the resolution of the global gesture DI . The continuous functions in Im(g)@D should have a codomain that is good enough as a separator for all gestures. To this end one could take a set D with cardinality larger than all relevant domains and take the indiscrete topology, further take as Im(g)@D all constant functions, which are automatically

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19 The Classification Problem of Global Gestures

continuous and thereby separate points of the domain. The problem here is that the bidual Im(g)∗∗ will not be homeomorphic to Im(g). This is a significant difference to the compositions’ situation, where the bidual for “good” modules is isomorphic to the original.

19.2 A Conjectured Classification Theorem So far the classification runs parallel to the business for global compositions. But the next step: viewing such function complexes as module complexes, fails. The function spaces |∆i |∗ are not algebraic structures, they are compact topological spaces without algebraic specifications. A fortiori, the Grassmann formalism does not apply. And we do not recognize an analog mechanism in the topological realm. This suggests/conjectures Theorem 8 (Conjecture) The global gestures over real numbers as described above can be classified by finite group orbits of topological spaces of continuous functions on standard spaces |∆| together with the orbits of colimit homeomorphism actions on the topological carrier spaces. These orbits are not recognized as being points on manifolds analog to the Grassmann schemes used in the classification of global compositions. 19.2.1 A Hypergesture for Human Bodies In this Section, we want to develop the global hypergesture for a human body. We construct it as a hypergesture of lines of loops. Figures 19.1 and 19.2 show the idea. Figures 19.3 and 19.4 show the namings of parts and the measured coordinate values from a superposition of the human body image with our system of parts. Figure 19.5 shows the line skeleton, and Figure 19.6 shows the Mathematicar file with these coordinates in form of sequences of line endpoints. Finally, Figure 19.2 shows the Mathematicar surface generated from the given data. Each line of loops is first generated as a model at the origin of the 3D coordinate system and then mapped to its position on the corresponding line via straightforward linear algebra operations. For the global gesture example, we suppose that two human bodies qua hypergestures have been constructed. They can be combined to a global hypergesture by identifying a hand of each body that would hold a hand of the other body, which would be an isomorphism of hands’ parts of the two (local) hypergestural bodies, see Figure 19.7

19.2 A Conjectured Classification Theorem

Fig. 19.1: The figure of a human body’s hypergesture.

99

Fig. 19.2: The human body’s computergraphical representation via Mathematicar software.

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19 The Classification Problem of Global Gestures

Fig. 19.3: The symbols of defined parts of the hypergestural skeleton.

Fig. 19.4: The measured coordinates of the parts from Figure 19.3.

19.2 A Conjectured Classification Theorem

Fig. 19.5: The line skeleton.

Fig. 19.6: The Mathematicar coordinates of the parts from Figure 19.3.

101

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19 The Classification Problem of Global Gestures

Fig. 19.7: Two connected human bodies in a computergraphical representation via Mathematicar software; the two connected hands would generate isomorphic parts of the two local hypergestures.

Fig. 19.8: Two connected human bodies in a computergraphical representation via Mathematicar software; the two connected hands are identified, when holding each other.

20 Singular Homology of Hypergestures

Summary. A preliminary investigation of hypergestural classification is the introduction of singular homology and its signification for counterpoint. –Σ– Homology is an algebraic theory that gives a powerful tool for classification of topological spaces. It does not provide us with classification, but produces a number of important “invariants”, i.e., mathematical objects that are only a function of the isomorphism class of an object in a category. Classification would mean to have a complete set of invariants. This is a very difficult task in the category Top. The idea of homology can be connected to Yoneda’s Lemma. This lemma, valid for every category C, states (among others) that two objects X, Y of C are isomorphic iff their presheaves (contravaiant functors) @X, @Y are so. This means that the Yoneda philosophy holds: To understand an object X, it is necessary and sufficient to understand its functor @X, i.e., to understand how X behaves when “viewed” from all objects S of the category via the morphisms f : S → X, the morphism set S@X is the set of perspectives when ‘looking’ at X from the ‘point of view’ S. Singular homology is the special Yoneda point of view in C = Top taken from the objects I n = [0, 1]n defined by the closed unit interval I ⊂ R of real numbers, i.e., one considers the set I n @X of so-called singular n-chains c : I n → X with values in X. Evidently, the choice of I stresses the topology of R and neglects topologies, such as the Zariski topology in algebraic geometry, which are radically different from R. The mathematician Alexander Grothendieck (among others) has extended homological theories to setups which are significant for algebraic geometry (e.g. étale cohomology) and which requires the introduction of new types of topologies, which are finer than Zariski’s topology, and eventually led to the solution of the famous Weil conjectures by Grothendieck’s student Pierre Deligne in 1974. The strong statement of the Yoneda Lemma is restricted to useful domains I n in singular homology, and we now want to interpret this homology as a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_20

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special case in hypergesture theory. So it is important to recall that homology deals with a preliminary approach to classification via Yoneda.

20.1 Homology via Hypergestures Singular homology can be defined by continuous functions with I n as domain or with n-dimensional simplices. Both approaches yield the same results, and we choose the “cubic” approach with I n since it harmonizes better with our hypergestural formalism. The singular n-chains are the continuous functions c : I n → X, elements of the set I n @X. This set is a topological space which is homeomorphic to the space I@I n−1 @X, and so forth, we have ∼

I n @X → I@I@ . . . I@X the n-fold iteration X 7→ I@X 7→ I@I@X 7→ . . . I@I . . . I@X up to the result that is homeomorphic to I n @X. This fact is trivial set-theoretically, and also easy to prove as a fact about topological spaces of continuous functions. This latter result was used in the Escher Theorem. To step over to hypergestures, we observe that I n @X identifies with the ∼ hypergesture space ↑ @ ↑ @ . . . ↑ @X =↑n @X since I@X →↑ @X. To define homology for a hypergesture space Γn @Γn−1 @ . . . Γ1 @X, we first choose a → − commutative ring R. We define the R-module RΓn @Γn−1 @ . . . Γ1 @ X as being the free R-module over the basis set Γn @Γn−1 @ . . . Γ1 @X. The elements of this module are called gestural n-chains. In homology, one considers boundary homomorphisms ∂n : RI n @X → n−1 RI @X. For our hypergestural situation, we consider an infinite sequence of digraphs Γ0 , Γ1 , . . . but we suppose that only a finite number of digraphs Γ0 , Γ1 , . . . Γd−1 define this sequence. This is of course the case if all digraphs are ↑, as in the classical case. With this condition, the sequence of digraphs can be described as a d-adic number X.c1 c2 c3 . . . with 0 ≤ ci < d with X as the topological space that defines our context. For example, if we set Γ0 = ∅, Γ1 =↑, then the classical approach would be described by X.111 . . . 1000 . . .. To define gestural n-chain modules, we take the direct sum Cn of all → − modules RΓi1 @Γi2 @ . . . Γin @ X, where i1 , . . . in is a partial (not necessarily contiguous) sequence of indices in the set 0.c1 c2 c3 . . .. For the classical case → − X.1111 . . . we would have Cn as the module R ↑ @ ↑ @ . . . ↑ @ X with n copies of ↑. In case two sequences yield the same sequence of digraphs, we only take the sequence once. In particular, C0 = RX, and for negative n the n-chain modules are defined to be the zero module. The next step consists in defining the boundary maps ∂n : Cn → Cn−1 . It is evidently sufficient to define ∂g for an n-chain g. to this end, let Γ be a digraph. Take an arrow a of Γ . Then we define Γ |a− to be the digraph obtained by removing the tail ta of a and all arrows connected to ta . We define by Γ |a+

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to be the digraph obtained by removing the head ha and all arrows connected to ha . → − Then we define g ◦ to be the following: Let g : Γ → X be a gesture. Then 1. if AΓ = ∅, then g◦ =

X

g(v).

v∈VΓ

2. The general case is defined recursively on the number of arrows: X (g|a− )◦ − (g|a+ )◦ . g◦ = a∈AΓ

For a general chain, we extend this definition by linearity. Now, if g ∈ Γ0 @Γ1 . . . Γn−1 @X, then call gi the hypergesture obtained by the Escher Theorem transforming g to the corresponding hypergesture in Γi @Γ0 . . . Γi−1 @Γi+1 @ . . . Γn−1 @X. Again, we extend this definition linearly to a linear combination of hypergestures. Finally, we define X ∂n g = (−1)i gi◦ , which we extend linearly to any linear combination of hypergestures. With these definitions, we have ∂n−1 ◦ ∂n = 0. A proof thereof can be found in [30, Ch. 63.3]. This implies that Im(∂n ) ⊂ Ker(∂n−1 ). This enables us to define the n-th homology group Hn = Ker(∂n )/Im(∂n+1 ). This invariant of a hypergesture is an important information, however it is not yet a complete characterization of the isomorphism class of that hypergesture.

20.2 Homology and Counterpoint It is remarkable that this hypergestural homology theory applies to our counterpoint theory. Recall that in our theory, we deal with maximizing the intersection K[] ∩ g(K[]) of the set of consonant intervals in Z12 [] = Z12 [x]/(x2 ) and its g-transformed version g(Z12 [] = Z12 [x]/(x2 )) for a symmetry g of the interval ring Z12 []. This ring can be given a topology, and we may consider the closure K[] ∩ g(K[]). Then we have this theorem [27, Ch. 79.4]: Theorem 9 with the above notations, we have dim(H1 (K[] ∩ g(K[])) = card(K[] ∩ g(K[])).

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This means that we may view that critical cardinality of the intersection as being the dimension of a module. The latter is no longer restricted to the finiteness of the intersection. We may now generalize this counterpoint condition to cases, where the cardinality is infinite, and only consider the dimension, which is an essential perspective when focusing on towers of dichotomies as describe in Section 15.2.

21 Local Gestures, Structures of Knots, and Local Gestures as Local Compositions

Summary. This section discusses the classification problem of gestures for the special case of circular gestures, which in mathematics are known as knots. Knots cannot be classified yet. We compare the category of local compositions to the category of gestures and discuss essential differences. For the special case of topological R-vector spaces, we present a method to envisage and classify gestures and local compositions on a common ground. –Σ–

21.1 Local Gestures and Knot Theory → − Local gestures are complex. A special example is the type of gestures g : → X with the loop digraph . A special case is a gesture, whose image Im(g) is a circular curve without self-intersection, except the first and last values being identical. It can be shown that the isomorphism classes of such circular gestures have isomorphic fundamental groups of their complement X − Im(g). Let us recall the concept of a fundamental group π1,x (X) of a topological space X at a point x ∈ X. It consists of equivalence classes of closed curves c : I → X that start and end in x. Two such curves c, d are equivalent if there is a continuous map h : I 2 → X such that h|I × {0} is c, and h|I × {1} is d and all values are h|{0} × I and all values are h|{1} × I are constant equal x. Such a map is called isotopy. This defines an equivalence relation among the closed curves in x. Two such equivalence classes can be composed with each other by connecting the end of the first with the start of the second, and the resulting structure is the group π1,x (X). In particular, the inverse of a curve is the curve obtained by traversing the values in reversed order. If the space is pathwise connected, this group is independent (up to isomorphism) of the base point x, it is denoted by π1 (X) and it is called the fundamental group of X. The knot theory theorem here states that two isomorphic gestures (without self-intersection in their images) have isomorphic fundamental groups of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_21

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their images’ complements. This however does not mean that the fundamental groups determine these knot isomorphism classes, but the groups may be isomorphic without isomorphic knots. In knot theory, a complete set of invariants is not known. Knots and their combinations (so-called links) have not been classified yet, only the first six billion classes have been described, see [16], for example. In particular, the conductor’s gestures, which are knots, are not classified yet. The dimension of the space X is crucial for the knot classification. Usually, knots are embedded in X = R3 . But if we view them as being embedded in R4 , the situation changes dramatically. In fact, in four dimensions, any nonintersecting closed loop of one-dimensional string is equivalent to an unknot, i.e., the knot without any non-trivial property, the closed circle knot.

21.2 Local Gestures as Local Compositions The classical type of musical objects, called local compositions, is defined in its simplest shape as an ordered pair (K, M ), where K ⊂ M is a subset of a module M over a commutative ring R. Local compositions may be combined as “charts of an atlas” generating a global composition. See [29, vol. I] and Chapter 8 for the technical details of this conceptualization. Local and global compositions can be connected by determined ‘functions’, so-called morphisms, and thereby define categories (Loc) of local and categories (Glob) of global compositions. For quite general system parameters, the isomorphism classes of local and global compositions have been determined, a fact that is also know as “classification” of such structures. The global classification theorem is [29, vol. I, 15.3.2, Theorem 18], see also Chapter 18. Local and global compositions are essentially algebraic concepts, the R-modules M that carry local/global compositions K have no topological specification, and the classification of local/global compositions refers to thoroughly algebraic affine module morphisms. The algebraic nature of local and global compositions was recognized around 2002, ironically the year of the first publication of [29], as a conceptual limitation that could not embrace the gestural and embodied ‘dancing’ phenomena of music, in particular of performance by human musicians. This topological aspect of music was also stressed in free jazz pianist Cecil Taylor’s statement that “I try to imitate on the piano the leaps in space a dancer makes.” [8]. It should be observed that classical performance theory, as developed by Johan Sundberg’s school in Stockholm and Guerino Mazzola’s school in Zürich, among others, did not focus on gestural dimensions, but only on the performance transformation ℘ : SymbolicReality → P hysicalReality of sound elements that are defined by scores (notes, chords, melodies, etc.). It abstracts from the concrete traditional genealogy of performed sound, namely the transformation of score symbols into gestures, which in turn would interact with an instrumental interface to generate the sounding output. Cecil Taylor’s dance

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approach would even avoid the symbolic start and get off the ground via gestures that do not represent ‘frozen’ score symbols. For these reasons of conceptual deficiency in music and performance theory, a mathematical theory of musical gestures and hypergestures (gestures of gestures) was initiated in [33] and elaborated in [30, vol. III], giving the theory of free jazz creativity [34] a powerful conceptualization. By an adjunction theorem [33], the set Γ @X of gestures from Γ to X is in a canonical bijection to the set Top(|Γ |, X) of continuous functions starting from the topological space |Γ |, which is the gluing (colimit) of copies of I, one for each arrow of Γ , and singletons {v}, one for each vertex of Γ , with projections I → {v} to heads and tails of Γ . Gestures are easily classified in an abstract setup (but not in a geometric model, see Chapter 19). In fact, there is a left group action of Aut(X) × Aut(Γ )opp , which acts upon the X values of a gesture and the digraph according to the definition of gesture morphisms. The isomorphism classes of gestures in Γ @X are this group action’s orbits, the elements of Γ @X/Aut(X) × Aut(Γ )opp . In this chapter, we want to investigate some relationships between the theories of local compositions and gestures. The main result will be that these two categories are fundamentally different, there is no reasonable functor that would relate isomorphism classes of local compositions to isomorphism classes of gestures. We can nevertheless construct canonical local compositions from gestures for distinguished body spaces. The only reasonable common ground of these two categories must be a type of spaces that are simultaneously modules and topological spaces. We want to investigate a natural mathematical type of such shared spaces, namely the category of topological vector spaces over the coefficient ring R of reals. A topological R-vector space is a R-vector space M whose addition and scalar multiplication are both continuous for the usual topology on R and a given topology on M . A typical space of this type is the 4-dimensional space REHLD parametrizing musical events with the four real coordinates E for onset, H for pitch, L for loudness, and D for duration. For a topological vector space M over R, the space Γ @M of gestures g : Γ → M becomes a topological R-vector space by its description as a limit of the following diagram of R-vector spaces: For every arrow a of Γ , we take Ma = Top(I, M ) with its canonical R-vector space structure. For every vertex v of Γ , we take the space Mv = M . For every head map ha : a 7→ v, we take the linear map Mha : Ma → Mv sending f ∈ Ma to Mha (f ) = f (1), and for the tail map ta : a 7→ v, we take Mta (f ) = f (0). The limit of this diagram of topological vector spaces defines the topological vector space Γ @R M . Thereby, the set of gestures is given the structure of a topological vector space, which we call the topological gesture space with skeleton Γ and body M . With this setup, a local gestural composition over Γ, M is a local composition K ⊂ Γ @R M .

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21.2.1 Characterstic Differences In this section we shall discuss some important differences between gesture and local composition theories. To begin with, suppose that gesture g ∈ Γ @R M is isomorphic to h ∈ ∼ Γ @R M by the homeomorphism q : M → M . Then there is a gesture t ∈ Γ @R M such that h = g + t. This means that isomorphism classes of gestures in Γ @R M are also translation classes, i.e., the algebraic translation classes are finer than the topological classes. The critical gesture h is defined as follows. Suppose ∼ that there is a homeomorphism f : M → M such that h = f.g. This means that for any parameter x ∈ |Γ |, we have h(x) = f (g(x)). Then the gesture t(x) = h(x) − g(x) answers our question.—However, the reverse direction is false, i.e., if h = g + t, it is not true in general that h and g are related by a homeomorphism of M . This is evident for the case where g(x) = g(y) for two different digraph parameters x, y. A homeomorphism would send this point to one single point, whereas the translation by t could send the point to two difference values. Also, if a local gestural composition K with more than one element is isomorphic by a single homeomorphism (topological isomorphism) to a local gestural composition L, they will not be translations of each other in general. Let us look at a special example for the musical standard space M = REHLD and Γ = • → • → • → •. Gestures of this data are the typical case as imagined by David Lewin [19]. Local gestural compositions K ⊂ Γ @R M might describe sets of gesturally interpreted seventh chords, e.g., in a jazz lead sheet. Local gestural compositions on REHLD can be transformed in a classical way: Adding a constant gesture g, i.e., a gesture with a single overall value, will generate a transformation of K by that single value, e.g., adding a pitch or onset, for example, to K’s coordinates. Also, retrograde or inversion can be applied by a scalar multiplication of K by −1 in its relevant coordinates. The general addition of a gesture may generate ‘deformations’ of K’s gestures, an operation that is not possible in the traditional setup of such gestures.—The digraph’s arrows may also be used to represent glissandi or crescendi. More generally, these gestures can be taken to represent functions of one variable (the digraph’s arguments), and one may use addition or scalar multiplication of gestures to represent these operations on such functions, e.g., the linear combination of sinusoidal functions in Fourier analysis. In any case, Theorem 10 For a digraph Γ and a topological R-vector space M , local gestural compositions in Γ @R M can be classified, i.e., their ‘linear’ isomorphism classes (as distinguished from the topological classes) are completely calculated following the classification theorem in Section 8.1.4. Observe that the action of Aut(Γ ) on a local gestural composition is absorbed by the corresponding affine automorphisms of Γ @R M .

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21.2.2 Conclusion and Future Topics The next theoretical step would be to investigate and hopefully classify global gestural compositions over topological R-vector spaces. Also, more extensive applications and examples in this context are to be expected, specifically to understand the structural and dynamic aspect of gesturally driven improvisation.

Part V

Classification and Creativity

22 Gestural Interpretation of Harmonic Dynamics in Tonal Modulation and Future Developments

Summary. We give a gestural interpretation of tonal modulations. We discuss some music software which uses classification results, especially the RUBATO r components for rhythmical and harmonic analysis. This chapter gives some perspectives relating to future directions in theory and software, especially regarding classification in performance. –Σ–

22.1 Gestural Modulation and Creativity

Fig. 22.1: The force field generated by the Lie bracket of two tonalities.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_22

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The classical approach to tonal modulation, as described in Section 5.2, involves classification of tonalities via symmetries of their triadic interpretations of X 3 . These symmetries are however not embodied as movements among musical objects. The symmetries are only present via modulation quanta, i.e., sets of notes that are invariant under these symmetries. The symmetries are not ‘visible’, but their movements only appear as abstract relationships connecting corresponding note objects. When introducing gestures, one would like to materialize these symmetries as gestural movements and thereby map the abstract setup into a gestural movement. This goal could be achieved by a new gestural model of tonal modulation, where notes are gesturally moved to describe the modulatory process. The gestural modulation model gets off the ground by the creation of a force field by a mathematical tool in differential geometry, called Lie bracket of vector fields. Figure 22.1 shows the field generated from tonality C major and tonality F major. The circle shows the pitch classes as immersed in the force field. This force field generates force curves that define gestures connecting notes of triads, as shown in Figure 22.2.

Fig. 22.2: The gestural connection moving a triad into another, thereby defining modulation.

Without going into details of this gestural modulation model, as developed in [30, Ch. 80], we can state that it provides us with the same results as the classical symmetry model. There is however an important advantage of the gestural approach, namely the fact that the gestural model does not a priori presuppose that the involved tonalities are in one and the same isomorphism class. This classical condition disappears in the gestural model. In particular,

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this new modulation model could be used to construct a theory, where modulation moves between tonalities of different classes, e.g., between major and harmonic minor tonalities, or even between exotic tonalities of general type. This shift from the category of compositions to the category of gestures generates an enrichment of theory, which should be applied to musical composition. The general principle of composition, namely the projection from a mathematical category into the realm of musical objects and relations, is now enriched by the switch from algebraic categories (compositions) to topological categories (gestures). The intuitive meaning thereof is a transformation from ‘rigid’ abstract concepts to more ‘elastic’ ones of gestural embodiment!

22.2 Classification Problems for Performance Gestures We discussed the gestural interpretation of performance integral curves of performance transformations ℘ : REHLD → Rehld , see Figure 22.3, in Section 6.2.

∆ X

x Ts

℘ X0

x0

Fig. 22.3: The performance transformation specifies integral curves, which are gestures in REHLD and their isomorphic images in Rehld .

Classification of such performance curve gestures goes beyond the abstract classification of these ↑-gestures. We have to recall that they are the result of a performance vector field qua integral curves thereof, together with the associated performance transformation ℘. If we can prove that two such gestures (curves) (ending at the same symbolic point of X in Figure 22.3) are isomorphic, we have to investigate the associated inverse problem, namely to determine the changes in generating parameters of the two corresponding performance fields. This is a semantic problem: How are field parameters related if their integral curves are isomorphic? We are far from a solution, but we now understand the deep consequences of

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gestural classification. This topic is evidently very important for understanding the geneology of performance, to understand, why performances are generated to produce the audible shape the listener may perceive.

22.3 Mirror Logic Summary. We present a logical approach to the construction of hypergestures in musical composition. This logic is built upon a “mirror” construction using fiber products and operations on the musical time domain. We discuss four examples: Temporality of harmonic structures, counterpoint, lead sheets, and dance hypergestures. –Σ– 22.3.1 Introduction The mathematical theory of musical gestures, as developed in [29, vol. III], provides us with an adequate description of the category of gestures and hypergestures, i.e., gestures of gestures. However, the creative process of a musical composition (including improvisation) has not been dealt with in Mazzola’s gesture theory. The musical logic that drives the construction of hypergestures is not embodied in the “static” categories of hypergestures. It is evidently impossible to include the limitless variety of creative musical procedures in a single theory, we understand this infinity from our own practice as a free jazz pianist (Guerino Mazzola). Nevertheless, we claim that it is feasible to describe some logical methods that are omnipresent in the basic construction mechanisms of complex musical hypergestures. The naive gestural context of a given composition is first of all a complex space of hypergestures Γ1 @Γ2 @ . . . Γn @X that are built starting from a basic topological space X (we don’t extend to topological categories in this chapter). But this hypergestural setup is often too simple. Instead of descending along a sequence of digraphs, the hypergestures are built from a ramification of spaces that is defined by cartesian products, e.g., Γ1 @(X1 ×X2 × . . . Xm ), where X1 = Γ1,1 @(X1,1 × . . . X1,m1 ) etc. Such cartesian products may occur, when the gestures involve a number of simultaneous musical parameters, such as pitch and time, for example. This means that the gestural codomain involves compound musical spaces. And the cartesian factors of such spaces are often also hypergestural constructions. In this case, we have the canonical ∼ isomorphism Γ1 @(X1 × X2 × . . . Xm ) → Γ1 @X1 × Γ1 @X2 × . . . Γ1 @Xm , and so on for its factors if they are cartesian products. Dually, codomains of gestures may also be coproducts, i.e., disjoint unions of spaces for alternatives of choice, among orchestral instruments, for example. But we shall not discuss this dual, but not fundamentally different structural situation here. Our claim means that in such compound hypergestural architectures, the factors (or cofactors) are not generated by a single stroke of creativity, but that

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there often might be a logical order that determines a hierarchy of construction, i.e., that some factors of hypergestures may be constructed before others are, and that the subordinate later factors are determined by the earlier and higher ranked factors in a functional logical dependence. 22.3.2 The Logic of Spatial Mirrors Our language of gestures has two adjoint setups: either a gesture is a digraph → − morphism g : Γ → X from a digraph Γ (the skeleton) to the body digraph → − X = Top(I, X), I = [0, 1] the real unit interval, or it is by adjunction of functors a continuous map |g| : |Γ | → X. We shall use the adjoint version here. The musical time [32] is defined by the topological space |Γ |, and by a limit of topological digraph spaces for a diagram of hypergestures. Let us focus on the simpler case |Γ |@X. A gesture f : |Γ | → X is described by knowing which symbolic time t goes to which point of X. This association can be deduced from the functor Y @X = Top(Y, X) → Top(Y ×X |Γ |, |Γ |) defined by the fiber product. In fact, if Y = {x} is a singleton, i.e., a point of X, then Y ×X |Γ | is the fiber of f over x, and these fibers identify f . We view this procedure as a kind of “mirror” at X, yielding the musical times associated with spatial objects over X. In the following sections, we describe such a mirroring procedure to understand a type of logic for the creation of hypergestures. Our examples focus on four situations: (1) temporal specification of chords, (2) construction of first species counterpoint, (3) lead sheets in jazz, and (4) dance gestures associated with musical structures. 22.3.3 Temporal Specification of Chords This situation starts from a gesture f : |Γ | → X1 × X2 , where X1 is a space of harmonic structures (chords) and X2 is a space of symbolic onset times, in quarter-note units, say. The logical challenge here would be to associate onset times with given harmonic structures, more concretely, to specify onset times for chords. The logic would work as follows: One specifies a number of harmonic structures, which are given as a set P = {p1 , p2 , . . . pn } of points in X1 . We now suppose that the gesture f1 : |Γ | → X1 is given, i.e. the musical time projecting to the harmonic structure space X1 . Then the fiber over P would be the set F (P ) = {t1,1 , t1,2 . . . t2,1 , . . . tn,1 , . . .} of pairwise different musical times, where ti,j is the musical time of index j over structure pi . In other words, we have a possibly multiple musical time over every structure pi . On this set, we may now define a set function f2 from |Γ | into the time space X2 , and, in order to avoid any topological restriction, we may suppose that X2 is given the indiscrete topology. The two maps f1 , f2 now define the desired gesture f = (f1 , f2 ) : |Γ | → X1 × X2 . Here, the structural space

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part with f1 is hierarchically superior to the temporal part, where the onsets of chords are defined using their musical time for f1 . Of course, the onset assignment may follow a musical reason (or logic) that the composer should specify in the concrete situation. For example, a cadence would determine the onset sequence of its chords of type I, IV, V, I, say. If desired, one could also specify the harmonic space X1 as being the space of gestures with values in a pitch space, following David Lewin’s idea of a gestural interpretation of chords [19]. 22.3.4 Construction of First Species Counterpoint Traditional Fuxian counterpoint [9] in its elementary first species is defined by a cantus firmus line CF = (c1 , c2 , . . . cm ) of pitch classes with increasing onset times plus some normalized durations, which we don’t consider here. The points ci are events with onset and pitch class coordinates, possibly within a given scale. This CF line is complemented by a discantus line D = (d1 , d2 , . . . dm ) of the same onset and pitch class parameter space, where the onsets of a ci and di coincide, and where the difference of pitch classes of ci and di should be consonances (pitch class differences 0,3,4,7,8,9), see also [3]. The terminology “punctus contra punctum” does not refer to the D set as an opposition (“contra”) to CF , it refers to the suspence-packed succession of intervals Ii defined by ci , di , see [43]. But the CF, D setup defines a simpler constructive description of first species counterpoint. This situation can be described by a gesture f : |Γ | → XCF × XD , where the space XCF consists of points with onset and pitch class coordinates, while XD consists of the six consonant pitch class numbers. Both spaces carry the indiscrete topology. We may suppose that CF is represented by a gesture f1 : |Γ | → XCF , where the elements of CF are generating the fiber set T in |Γ |. Observe that a time t projected to a CF point c needn’t be single, f1 may show a number of times sitting over c. This fiber set T may now be projected via gesture map f2 : |Γ | → XD to a set of consonances in XD , which means that for every t ∈ T , we have a discantus dt = f1 (t) + f2 (t). The hierarchy here is dominated by the cantus firmus line defined by f1 , whereas the consonances associated with musical times of CF values are chosen as subordinate data. 22.3.4.1 Hypergestures for Counterpoint Summary. We discuss the application of the Escher Theorem of hypergestures to counterpoint. –Σ– An application of the Escher Theorem deals with first species counterpoint. There are two well-known views of this contrapuntal situation: The first

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views such a counterpoint as a hypegestural movement from the gesture of cantus firmus to the gesture of the discantus. The second views the counterpoint as a hypergesture of the sequence of interval gestures. The Escher Theorem interprets this duality as the result of exchanging the two digraphs. This means that both perspectives are equivalent under that digraph permutation. These perspectives are isomorphic with each other, but one should recall that only the second, a hypergesture of interval gestures, is historically and theoretically valid. The “contra” in counterpoint is an opposition of successive interval gestures. 22.3.5 Lead Sheet in Jazz Lead sheets in jazz have to components: the sequence of chords (defining the chord changes) and the basic melodic line flying over the chord changes, see also [37]. This construction may be viewed as a hierarchy, where the chord sequence is the dominant factor, while the melodic line is derived in accordance with the harmonic path of chord changes. We may interpret this data as a gesture f = (f1 , f2 ) : |Γ | → XCh × XM el , where XCh is a space of Lewin-type gestures for chords, but enriched by a symbolic time coordinate, following the ideas of the first example. The space XM el should consist of melodic gestures with values in pitchonset-duration space. The mirror logic would then define a gesture of f2 : |Γ | → XM el with specific values on the fiber in |Γ | over the chord set defined by f1 . 22.3.6 Dance Gestures Associated with Musical Structures A dance standard setup has two components: dance gestures and musical structures that are related to these dance gestures. Accordingly, we have to consider gestures f = (fdance , fmusic ) : |Γ | → Xdance × Xmusic . The space Xdance is a space of hypergestures that eventually terminate in a space with spatial 3D and time coordinates. The space Xmusic should be a hypergesture space whose elements describe musical structures by hypergestures with values in typical onset, pitch, loudness, duration coordinates of a common score. The mirror logic here associates dance gestures with musical times taken from the fiber of fmusic . The hierarchy is generating dance information from the fibers of musical gesturality. Of course, as with the above examples, the details of the compositional logic deduced from the mirror logic are to be elaborated, but they all must be realized in the mirror logic’s framework. In modern dance, the logic is also reversed, meaning that the dance gestures are given first, and then musical structures are deduced from dance information. 22.3.7 Conclusion and Future Topics We have shown with four typical examples how musical composition in complex hypergestural configurations may be created following a mirror logic, where

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22 Gestural Interpretation and Future Developments

fiber products are key. Future topics should include an investigation of characteristic differences between composition of scores and improvisation. In the latter case, psychological logic may play a key role for the selection of hierarchically secondary layers of musical creativity.

22.4 Perspectives of Future Theory The role of classification for musical creativity should be a dominant topic in future theory and software. From our discussion in this book it appears that creativity should focus on the transformation of classification, which is a mathematical theme, into musical structures, compositions and gestures. The overall image is this: We are given a set of musical structures, i.e., compositions or gestures. Their classification generates a corresponding set of orbits, i.e., classes of equivalent structures. The underlying equivalence relations are mathematical objects, generated by transformations among members of such orbits. The musical creativity deals with the translation of these transformations into musical objects. As we have learned from the examples in this book, from tonal modulation to serial approaches and counterpoint, important compositional techniques are designed to “musicalize” classifying transformations, i.e., representing them within a creative compositional strategy. For tonal modulation, one musicalizes the transformational symmetry between two tonalities, which means to represent that symmetry in terms of a modulation quantum, whose triadic degrees materialize the symmetry as a set of notes. This model also includes a syntax of the modulatory architecture, which means that not only the materialization is given, but also its temporal unfolding. For dodecaphonic and serial composition, one materializes the orbit of a row by the display of all its 48 equivalent transformed rows. However, this brute method lacks any syntactical explication. For counterpoint, the orbit of a strong dichotomy is used to determine successive intervals, in fact, the deformations of a dichotomy generate successive interval options. In this spirit, musical/compositional creativity consists of a musically comprehensible projection of abstract logical structures into the material ontology of sensible (audible, gestural) musical objects. In the case of improvisation, the dialogue between its performers (including the dialogue with oneself) creates new logical expressions, materializes them and receives one’s collaborator’s reply on the flight. It is a compositional creation in the moment of it’s making. But it follows the same general methodology as composition. In terms of the concept of creativity, what is the “box” that has to be opened and transgressed in our situation? This box is the logically closed architecture of musical constructions in the language of mathematics (or just

22.4 Perspectives of Future Theory

123

the mental formalism of compositional strategies). Often, musical constructors believe this framework is sufficient for a valid composition. Their compositional work follows this setup without taking care of its transformation into musical objects. This is an often hidden box, where the difference between thinking and making/creating music is not recognized. Creativity here stems from the insight that transforming thoughts into music must build musical objects, sounds, notes, gestures, etc., that are a sufficiently faithful image of initial thoughts. This is what “making music” means. Figure 22.4 shows this projection from mathematical categories to musical realm with its musical parts and transformations. Classification is involved since the most prominent examples deal with morphisms that are isomorphisms, i.e., those concepts, which define classification.

Fig. 22.4: Musical creativity is a projection from mathematical categories (and in particular isomorphisms) to the realm of music with its musical parts and transformation qua musical entities.

Part VI

References, Index

23 Classification Lists

Summary. This chapter displays lists of classified objects. –Σ–

23.1 List of Local Denotators in Z12 , Chord Classes This section contains the list of all isomorphism classes of zero-addressed chords in P iM od12 . Here we give a short definition. • Class Nr. is the number of the isomorphism class, numbers with extension “.1” indicate the class number for classification under symmetries from Z (no fifth or fourth transformations). Autocomplementary classes have a star after the number. • Representative of Nr. without hat is the number’s representative in full circles, the one with hat is the complementary chord. • Group of symmetries is Sym(N r.). To keep notation readable, we use the notation with linear factor to the left. • Conj. Class denotes the conjugacy class symbol of Sym(N r.) and refers to the numbering 1, 2, . . . 19 from [38]. d • Card. End. Cl. N r.|N r. is the pair of numbers of conjugacy classes of end domorphisms in Nr. and in its complement N r., respectively.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4_23

127

128

23 Classification Lists

Chord Classes Class Representative Group of d Nr. N r. = •, N r. = ◦ Symmetries −→ 1 • • • • • • • • • • •• GL(Z12 )

Conj. ] End. d Class N r.|N r. 19

28|28

8

1|31

3

3|23

8

3|25

8

3|19

8

3|31

13

3|28

2

4|14

1

4|30

1

8|36

4

4|20

6

5|29

One/Eleven Element

2

• ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ Z× 12 Two/Ten Elements

3

• • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ h−1e−1 i

3.1

• ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦◦

4

• ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ {1, 7, −1e−2 , 5e−2 }

5 6

• ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ {1, 5, 7e

−3

, −1e

8

−3

8

• ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦◦ {1, 7, 5e , −1e } Z× 12

7

• ◦ ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦◦

8

• • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ h−1e−2 i

8.1

• ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ ◦◦

9

• • ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ {1}

9.1

• ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦

10

• • ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦◦ {1}

ne

6Z12

}

Three/Nine Elements

10.1 • ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦◦ 11 12

• • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦◦ h5i 6

• • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦◦ h7e i

13

• ◦ • ◦ • ◦ ◦ ◦ ◦ ◦ ◦◦ {1, 7, −1e , 5e }

8

4|18

14

• ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦◦ h7i

6

8|31

8

5|32

15

4|20

3

4|8

1

5|19

1

5|19

1

7|23

15 16

8

8

6

6

• ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ ◦◦ {1, 5, −1e , 7e } • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦◦

Z× 12

ne

4Z12

Four/Eight Elements

17

• • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ h−1e−3 i

17.1 • ◦ • ◦ ◦ • ◦ • ◦ ◦ ◦◦ 18

• • • ◦ • ◦ ◦ ◦ ◦ ◦ ◦◦ {1}

18.1 • ◦ • ◦ • ◦ ◦ • ◦ ◦ ◦◦ 19

• • • ◦ ◦ • ◦ ◦ ◦ ◦ ◦◦ {1}

19.1 • • ◦ • ◦ ◦ ◦ ◦ • ◦ ◦◦ 20

• • • ◦ ◦ ◦ • ◦ ◦ ◦ ◦◦ {1}

23.1 List of Local Denotators in Z12 , Chord Classes

Chord Classes—Continued Class Representative Group of Conj. ] End. d d Nr. N r. = •, N r. = ◦ Symmetries Class N r.|N r. 20.1 • • ◦ ◦ ◦ • ◦ • ◦ ◦ ◦◦ 21 22

• • • ◦ ◦ ◦ ◦ • ◦ ◦ ◦◦ {1, 7, −1e−2 , 5e−2 } • • ◦ • • ◦ ◦ ◦ ◦ ◦ ◦◦ h−1e

−4

i

9

7|9

2

6|20

4

5|13

22.1 • ◦ • ◦ ◦ • ◦ ◦ ◦ • ◦◦ 23

• • ◦ • ◦ • ◦ ◦ ◦ ◦ ◦◦ h5i

24

• • ◦ • ◦ ◦ • ◦ ◦ ◦ ◦◦ h7e i

6

6|17

25

• • ◦ • ◦ ◦ ◦ • ◦ ◦ ◦◦ {1}

1

0|3

1

12|31

6

25.1 • • ◦ • ◦ • ◦ ◦ ◦ ◦ ◦◦ 26

• • ◦ • ◦ ◦ ◦ ◦ ◦ • ◦◦ {1}

26.1 • ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦◦ 27 28

• • ◦ • ◦ ◦ ◦ ◦ ◦ ◦ •◦ {1, 7e−3 , 5e2 , −1e−1 } 11 7

• • ◦ ◦ • • ◦ ◦ ◦ ◦ ◦◦ h−1e i

5|13

3

6|14

6

10|23

4

11|23

28.1 • • ◦ ◦ ◦ • ◦ ◦ • ◦ ◦◦ 29 30

• • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦◦ h7i 4

• • ◦ ◦ • ◦ ◦ ◦ • ◦ ◦◦ h5e i

31

• • ◦ ◦ • ◦ ◦ ◦ ◦ • ◦◦ {1, −1e

32

• • ◦ ◦ ◦ • • ◦ ◦ ◦ ◦◦ {1, 5, −1e6 , 7e6 }

−1

, 5e

−4

, 7e } 10 3

9|19

8

7|15

33

• • ◦ ◦ ◦ ◦ • • ◦ ◦ ◦◦ {1, 7, −1e , 5e , e6 , 7e6 , 5e5 , −1e5 }

14

7|14

34

• ◦ • ◦ • ◦ • ◦ ◦ ◦ ◦◦ {1, 7, −1e6 , 5e6 }

9

6|17

8

11|19

35

−1

4

−1

4

• ◦ • ◦ • ◦ ◦ ◦ • ◦ ◦◦ {1, 7, −1e , 5e }

36

• ◦ • ◦ ◦ ◦ • ◦ • ◦ ◦◦ {1, 7, −1e , 5e , e6 , 7e6 , 5e4 , −1e4 }

13

9|28

37

3Z12 • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦◦ Z× 12 n e

17

7|21

2

5|7

1

6|10

1

8|12

1

8|12

−2

−2

Five/Seven Elements

38

• • • • • ◦ ◦ ◦ ◦ ◦ ◦◦ h−1e−4 i

38.1 • ◦ • ◦ • ◦ ◦ • ◦ • ◦◦ 39

• • • • ◦ • ◦ ◦ ◦ ◦ ◦◦ {1}

39.1 • • ◦ • ◦ • ◦ ◦ ◦ ◦ •◦ 40

• • • • ◦ ◦ • ◦ ◦ ◦ ◦◦ {1}

40.1 • • ◦ • ◦ ◦ • ◦ • ◦ ◦◦ 41

• • • • ◦ ◦ ◦ • ◦ ◦ ◦◦ {1}

129

130

23 Classification Lists

Chord Classes—Continued Class Representative Group of Conj. ] End. d d Nr. N r. = •, N r. = ◦ Symmetries Class N r.|N r. 41.1 • • • ◦ ◦ • ◦ • ◦ ◦ ◦◦ 42

• • • ◦ • • ◦ ◦ ◦ ◦ ◦◦ {1}

1

6|16

1

8|20

42.1 • • ◦ • ◦ • ◦ ◦ • ◦ ◦◦ 43

• • • ◦ • ◦ • ◦ ◦ ◦ ◦◦ {1}

43.1 • • ◦ • ◦ • ◦ • ◦ ◦ ◦◦ 44

• • • ◦ • ◦ ◦ • ◦ ◦ ◦◦ h7i

6

7|9

45

• • • ◦ • ◦ ◦ ◦ • ◦ ◦◦ {1}

1

16|22

4

5|12

2

8|14

1

8|18

1

10|18

2

9|13

6

9|11

, 7e , 5e } 8

7|17

45.1 • • ◦ ◦ • ◦ • ◦ • ◦ ◦◦ 46 47

• • • ◦ • ◦ ◦ ◦ ◦ • ◦◦ h5e−4 i • • • ◦ • ◦ ◦ ◦ ◦ ◦ •◦ h−1e

−2

i

47.1 • ◦ • ◦ • ◦ • ◦ ◦ • ◦◦ 48

• • • ◦ ◦ • • ◦ ◦ ◦ ◦◦ {1}

48.1 • • ◦ • ◦ ◦ ◦ • • ◦ ◦◦ 49

• • • ◦ ◦ • ◦ ◦ • ◦ ◦◦ {1}

49.1 • • ◦ • ◦ ◦ ◦ ◦ • • ◦◦ 50

• • • ◦ ◦ • ◦ ◦ ◦ • ◦◦ h−1e−2 i

50.1 • • ◦ • • ◦ ◦ ◦ • ◦ ◦◦ 51

• • • ◦ ◦ ◦ • • ◦ ◦ ◦◦ h7i

52

• • • ◦ ◦ ◦ • ◦ • ◦ ◦◦ {1, −1e

53

• • ◦ • • ◦ • ◦ ◦ ◦ ◦◦ {1}

−2

6

4

1

10|20

1

14|26

53.1 • • ◦ • ◦ ◦ • ◦ ◦ ◦ •◦ 54

• • ◦ • • ◦ ◦ • ◦ ◦ ◦◦ {1}

54.1 • • ◦ ◦ • ◦ • ◦ ◦ • ◦◦ 55

• • ◦ • ◦ • • ◦ ◦ ◦ ◦◦ {1, 5, −1e6 , 7e6 }

8

8|8

56

• • ◦ • ◦ • ◦ ◦ ◦ • ◦◦ h5i

57

4

16|16

6

6

12|16

6

• • ◦ • ◦ ◦ • • ◦ ◦ ◦◦ h7e i

58

• • ◦ • ◦ ◦ • ◦ ◦ • ◦◦ h7e i

6

18|23

59

• • ◦ • ◦ ◦ ◦ • ◦ • ◦◦ h7i

6

13|29

60

• • ◦ ◦ • • ◦ ◦ • ◦ ◦◦ h5i

4

11|19

61

• • ◦ ◦ • ◦ ◦ • • ◦ ◦◦ {1, 7, −1e4 , 5e4 }

8

14|14

8

11|19

62

4

4

• ◦ • ◦ • ◦ • ◦ • ◦ ◦◦ {1, 7, −1e , 7e }

23.1 List of Local Denotators in Z12 , Chord Classes

Chord Classes—Continued Class Representative Group of Conj. ] End. d d Nr. N r. = •, N r. = ◦ Symmetries Class N r.|N r. Six/Six Elements

63*

• • • • • • ◦ ◦ ◦ ◦ ◦◦ h−1e−5 i

3

5|5

1

9|9

1

9|9

2

12|6

63.1* • • ◦ • ◦ • ◦ ◦ • ◦ •◦ 64*

• • • • • ◦ • ◦ ◦ ◦ ◦◦ {1}

64.1* • • ◦ • ◦ • ◦ • ◦ ◦ •◦ 65

• • • • • ◦ ◦ • ◦ ◦ ◦◦ {1}

65.1 • • • ◦ • ◦ ◦ • ◦ • ◦◦ 66

• • • • • ◦ ◦ ◦ • ◦ ◦◦ h−1e−4 i

66.1 • • • ◦ ◦ • ◦ • ◦ • ◦◦ 67*

• • • • ◦ • ◦ • ◦ ◦ ◦◦ h5e−2 i

5

6|6

68*

• • • • ◦ • ◦ ◦ • ◦ ◦◦ {1}

1

9|9

69

• • • • ◦ • ◦ ◦ ◦ • ◦◦ {1}

1

15|11

69.1 • • ◦ • • ◦ • ◦ • ◦ ◦◦ 70*

• • • • ◦ • ◦ ◦ ◦ ◦ •◦ {1, 5, −1e−3 , 7e−3 }

10

6|6

71*

• • • • ◦ ◦ • • ◦ ◦ ◦◦ {1}

1

11|11

6

8|10

i

3

13|9

74*

• • • • ◦ ◦ ◦ • • ◦ ◦◦ h−1e−3 i

3

7|7

75*

• • • ◦ • • ◦ ◦ • ◦ ◦◦ {1}

1

17|17

71.1* • • • ◦ ◦ • ◦ • • ◦ ◦◦ 72 73

• • • • ◦ ◦ • ◦ • ◦ ◦◦ h7e6 i • • • • ◦ ◦ • ◦ ◦ • ◦◦ h−1e

−3

73.1 • • ◦ • ◦ ◦ • ◦ • • ◦◦

75.1* • • ◦ ◦ • • ◦ ◦ • ◦ •◦ 76*

• • • ◦ • • ◦ ◦ ◦ • ◦◦ h5e−5 i

4

10|10

77*

• • • ◦ • ◦ • ◦ • ◦ ◦◦ h5e4 i

4

14|14

78*

• • • ◦ • ◦ • ◦ ◦ ◦ •◦ {1}

1

23|23

6

18|10

9

15|11

4

11|11

1

17|17

13

12|12

78.1* • • ◦ • ◦ • ◦ • ◦ • ◦◦ 79 80

• • • ◦ • ◦ • • ◦ ◦ ◦◦ h7i • • • ◦ • ◦ ◦ • ◦ ◦ •◦ {1, 7, 5e

81*

• • • ◦ • ◦ ◦ ◦ • • ◦◦ h5e

82*

• • • ◦ ◦ • • ◦ ◦ • ◦◦ {1}

−4

−2

, −1e

−2

}

i

82.1* • • ◦ • • ◦ ◦ • • ◦ ◦◦ 83*

• • • ◦ ◦ ◦ • • • ◦ ◦◦ {1, −1e−2 , 5e−2 , 7}

131

132

23 Classification Lists

Chord Classes—Continued Class Representative Group of Conj. ] End. d d Nr. N r. = •, N r. = ◦ Symmetries Class N r.|N r. ne6Z12 84* 85

• • ◦ • • ◦ • ◦ ◦ • ◦◦ h7e3 i • • ◦ • • ◦ ◦ • ◦ • ◦◦ {1, −1e

86*

• • ◦ • ◦ ◦ • • ◦ • ◦◦ h7i n e

87*

• • ◦ • • ◦ ◦ • • ◦ ◦◦ {1}

88*

Z× 12

• ◦ • ◦ • ◦ • ◦ • ◦ •◦

−4

, 5e

6Z12

ne

Z12

−4

, 7}

7

14|14

8

15|23

12

20|20

16

12|12

18

12|12

23.2 Third Chain Classes

133

23.2 Third Chain Classes The following list of third chain translation classes shows the class number in the first column, where equivalence (∼) means that the same pc set is generated. The second column shows the pitch classes in the order of appearance along the third chain. The third column shows the third chain, the fourth column shows the chord class of the pc set, and the fifth column shows lead-sheet symbols. Third Chains Chain Nr. Pitch Classes

Third

Chord Lead-Sheet

∼ equiv.

Chain

Class Symbols

from 0

Two Pitch Classes 1

0,3

3

5

trd

2

0,4

4

6

T rd

3

0,3,6

33

15

C0, Cm5-

4

0,3,7

34

10.1

Cm

5

0,4,7

43

10.1

C

6

0,4,8

44

16

C+, C5+

Three Pitch Classes

Four Pitch Classes 7

0,3,6,9

333

37

C07-

8

0,3,6,10

334

26.1

C07

9

0,3,7,10

343

22.1

Cm7

10

0,3,7,11

344

30

Cm7+

11

0,4,7,10

433

26.1

C7

12

0,4,7,11

434

28.1

C7+

13

0,4,8,11

443

30

C+7+

14

0,3,6,9,1

3334

58

C07-/9-

15

0,3,6,10,1

3343

53.1

C09-

16

0,3,6,10,2

3344

56

C09

17

0,3,7,10,1

3433

53.1

Cm9-

18

0,3,7,10,2

3434

42.1

Cm9

19

0,3,7,11,2

3443

59

Cm7+/9, Cmmaj7/9

20

0,4,7,10,1

4333

58

C9-

21

0,4,7,10,2

4334

47.1

C9

22

0,4,7,11,2

4343

42.1

C7+/9, Cmaj7/9

23

0,4,7,11,3

4344

60

C7+/9+

24

0,4,8,11,2

4433

56

C+7+/9

Five Pitch Classes

134

23 Classification Lists Third Chains—Continued

Chain Nr. Pitch Classes

Third

Chord Lead-Sheet

∼ equiv.

from 0

Chain

Class Symbols

25

0,4,8,11,3

4434

60

26

0,3,6,9,1,4

33343

84*

C07-/9-/11-

27

0,3,6,9,1,5

33344

85b

C07-/9-/11

28

0,3,6,10,1,4

33433

79

C09-/11-

29

0,3,6,10,1,5

33434

65.1b C09-/11

30

0,3,6,10,2,5

33443

69.1

C011

31

0,3,7,10,1,4

34333

84*

Cm9-/11-

32

0,3,7,10,1,5

34334

64.1* Cm9-/11

33

0,3,7,10,2,5

34343

63.1* Cm11

34

0,3,7,10,2,6

34344

75.1* Cm11+

35

0,3,7,11,2,5

34433

69.1

Cm7+/11

36

0,3,7,11,2,6

34434

82*

Cm7+/11+

37

0,4,7,10,1,5

43334

73.1

C9-/11

38

0,4,7,10,2,5

43343

64.1* C11

39

0,4,7,10,2,6

43344

78.1* C11+

40

0,4,7,11,2,5

43433

65.1b C7+/11

41

0,4,7,11,2,6

43434

66.1b C7+/11+

42

0,4,7,11,3,6

43443

82*

C7+/9+/11+

43

0,4,8,11,2,5

44333

85b

C+7+/11

44

0,4,8,11,2,6

44334

78.1* C+7+/11+

45

0,4,8,11,3,6

44343

75.1* C+7+/9+/11+

46

0,4,8,11,3,7

44344

87*

47

0,3,6,9,1,4,7

333433

58b

48

0,3,6,9,1,4,8

333434

54.1b C07-/9-/11-/13

49

0,3,6,9,1,5,8

333443

54.1b C07-/9-/13

50

0,3,6,10,1,4,7

334333

58b

C09-/11-/13-

51

0,3,6,10,1,4,8

334334

47.1

C09-/11-/13

52

0,3,6,10,1,5,8

334343

38.1

C09-/13

53

0,3,6,10,1,5,9

334344

54.1b C09-/13+

54

0,3,6,10,2,5,8

334433

47.1b C013

55

0,3,6,10,2,5,9

334434

54.1b C013+

56

0,3,7,10,1,4,8

343334

54.1b Cm9-/11-/13

57

0,3,7,10,1,5,8

343343

38.1b Cm9-/13

C+7+/9+

Six Pitch Classes

C+7+/9+/(11)/13-

Seven Pitch Classes C07-/9-/11-/13-

23.2 Third Chain Classes Third Chains—Continued Chain Nr. Pitch Classes

Third

Chord Lead-Sheet

∼ equiv.

from 0

Chain

Class Symbols

58

0,3,7,10,1,5,9

343344

47.1b Cm9-/13+

59

0,3,7,10,2,5,8

343433

38.1b Cm13

60

0,3,7,10,2,5,9

343434

38.1b Cm13+

61

0,3,7,10,2,6,9

343443

54.1b Cm11+/13+

62

0,3,7,11,2,5,8

344333

54.1b Cm7+/13

63

0,3,7,11,2,5,9

344334

47.1b Cm7+/13+

64

0,3,7,11,2,6,9

344343

54.1b Cm7+/11+/13+

65

0,3,7,11,2,6,10

344344

60b

66

0,4,7,10,1,5,8

433343

54.1b C9-/13

67

0,4,7,10,1,5,9

433344

54.1b C9-/13+

68

0,4,7,10,2,5,8

433433

47.1b C13

69

0,4,7,10,2,5,9

433434

38.1b C13+

70

0,4,7,10,2,6,9

433443

47.1b C11+/13+

71

0,4,7,11,2,5,8

434333

54.1b C7+/13

72

0,4,7,11,2,5,9

434334

38.1b C7+/13+

73

0,4,7,11,2,6,9

434343

38.1b C7+/11+/13+

74

0,4,7,11,2,6,10

434344

45.1b C7+/11+/(13)/15-

75

0,4,7,11,3,6,9

434433

54.1b C7+/9+/11+/13+

76

0,4,7,11,3,6,10

434434

55b

77

0,4,8,11,2,5,9

443334

54.1b C+7+/13+

78

0,4,8,11,2,6,9

443343

47.1b C+7+/11+/13+

79

0,4,8,11,2,6,10

443344

62b

80

0,4,8,11,3,6,9

443433

54.1b C+7+/9+/11+/13+

81

0,4,8,11,3,6,10

443434

45.1b C+7+/9+/11+/(13)/15-

82

0,4,8,11,3,7,10

443443

60b

Cm7+/11+/(13)/15-

C7+/9+/11+/(13)/15-

C+7+/11+/(13)/15-

C+7+/9+/(11)/13-/15-

Eight Pitch Classes C07-/9-/11-/13- . . .

83

0,3,6,9,1,4,7,10

3334333

37b

84

0,3,6,9,1,4,7,11

3334334

26.1b C07-/9-/11-/13- . . .

85

0,3,6,9,1,4,8,11

3334343

22.1b C07-/9-/11-/13 . . .

86

0,3,6,9,1,5,8,11

3334433

26.1b C07-/9-/13 . . .

87

0,3,6,10,1,4,7,11

3343334

29b

88

0,3,6,10,1,4,8,11

3343343

18.1b C09-/11-/13 . . .

89

0,3,6,10,1,5,8,11

3343433

17.1b C09-/13 . . .

90

0,3,6,10,2,5,8,11

3344333

26.1b C013 . . .

91

0,3,6,10,2,5,9,1

3344344

31b

C09-/11-/13- . . .

C013+ . . .

135

136

23 Classification Lists Third Chains—Continued

Chain Nr. Pitch Classes

Third

Chord Lead-Sheet

∼ equiv.

from 0

Chain

Class Symbols

92

0,3,7,10,1,4,8,11

3433343

31b

93

0,3,7,10,1,5,8,11

3433433

18.1b Cm9-/13 . . .

94

0,3,7,10,2,5,8,11

3434333

22.1b Cm13 . . .

95

0,3,7,10,2,5,9,1

3434344

18.1b Cm13+ . . .

96

0,3,7,10,2,6,9,1

3434434

29b

Cm11+/13+ . . .

97

0,3,7,11,2,5,9,1

3443344

34b

Cm7+/13+ . . .

98

0,3,7,11,2,6,9,1

3443434

29b

Cm7+/11+/13+ . . .

99

0,3,7,11,2,6,10,1

3443443

28b

Cm7+/11+/(13)/15- . . .

100

0,4,7,10,1,5,8,11

4333433

29b

C9-/13 . . .

101

0,4,7,10,2,5,8,11

4334333

18.1b C15

102

0,4,7,10,2,5,9,1

4334344

22.1b C13+ . . .

103

0,4,7,10,2,6,9,1

4334434

26.1b C11+/13+ . . .

104

0,4,7,11,2,5,9,1

4343344

18.1b C7+/13+ . . .

105

0,4,7,11,2,6,9,1

4343434

17.1b C7+/11+/13+ . . .

106

0,4,7,11,2,6,10,1

4343443

25.1b C7+/11+/(13)/15- . . .

107 ∼ 84 0,4,7,11,3,6,9,1

4344334

26.1b C7+/9+/11+/13+ . . .

108 ∼ 87 0,4,7,11,3,6,10,1

4344343

29b

C7+/9+/11+/(13)/15- . . .

109

0,4,7,11,3,6,10,2

4344344

30b

C7+/9+/11+/(13)/15- . . .

110

0,4,8,11,2,5,9,1

4433344

31b

C+7+/13+ . . .

111

0,4,8,11,2,6,9,1

4433434

18.1b C+7+/11+/13+ . . .

112

0,4,8,11,2,6,10,1

Cm9-/11-/13 . . .

C+7+/11+/(13)/15- . . .

4433443

34b

113 ∼ 85 0,4,8,11,3,6,9,1

4434334

22.1b C+7+/9+/11+/13+ . . .

114 ∼ 88 0,4,8,11,3,6,10,1

4434343

18.1b C+7+/9+/11+/(13)/15- . . .

115

0,4,8,11,3,6,10,2

4434344

35b

C+7+/9+/11+/(13)/15- . . .

116 ∼ 92 0,4,8,11,3,7,10,1

4434433

31b

C+7+/9+/(11)/13-/15- . . .

117

4434434

30b

C+7+/9+/(11)/13-/15- . . .

0,4,8,11,3,7,10,2

Nine Pitch Classes 118

0,3,6,9,1,4,7,10,2

33343334

15b

C07-/9-/11-/13- . . .

119

0,3,6,9,1,4,7,11,2

33343343

9.1b

C07-/9-/11-/13- . . .

120

0,3,6,9,1,4,8,11,2

33343433

9.1b

C07-/9-/11-/13 . . .

121

0,3,6,9,1,5,8,11,2

33344333

15b

C07-/9-/13 . . .

122

0,3,6,10,1,4,7,11,2

33433343

10b

C09-/11-/13- . . .

123

0,3,6,10,1,4,8,11,2

33433433

13b

C09-/11-/13 . . .

124

0,3,6,10,1,5,8,11,2

33434333

9.1b

C09-/13 . . .

125

0,3,6,10,2,5,9,1,4

33443443

10b

C013+ . . .

23.2 Third Chain Classes

137

Third Chains—Continued Chain Nr. Pitch Classes

Third

Chord Lead-Sheet

∼ equiv.

from 0

Chain

Class Symbols

126

0,3,7,10,1,4,8,11,2

34333433

10b

Cm9-/11-/13 . . .

127

0,3,7,10,1,5,8,11,2

34334333

9.1b

Cm9-/13 . . .

128

0,3,7,10,2,5,9,1,4

34343443

9.1b

Cm13+ . . .

129

0,3,7,10,2,6,9,1,4

34344343

15b

Cm11+/13+ . . .

130

0,3,7,11,2,5,9,1,4

34433443

13b

Cm7+/13+ . . .

131 ∼ 119 0,3,7,11,2,6,9,1,4

34434343

9.1b

Cm7+/11+/13+ . . .

132

34434344

14b

Cm7+/11+/13+ . . .

133 ∼ 122 0,3,7,11,2,6,10,1,4

34434433

10b

Cm7+/11+/(13)/15- . . .

134

0,3,7,11,2,6,10,1,5

34434434

11b

Cm7+/11+/(13)/15- . . .

135

0,4,7,10,1,5,8,11,2

43334333

15b

C9-/13 . . .

136

0,4,7,10,1,5,8,11,3

43334334

10.1b C9-/13 . . .

137

0,4,7,10,2,5,8,11,3

43343334

10.1b C17

138

0,4,7,10,2,6,9,1,5

43344344

10.1b C11+/13+ . . .

139

0,4,7,11,2,6,9,1,5

43434344

8.1b

C7+/11+/13+ . . .

140

0,4,7,11,2,6,10,1,5

43434434

12b

C7+/11+/(13)/15- . . .

141

0,4,7,11,3,6,9,1,5

43443344

14b

C7+/9+/11+/13+ . . .

142

0,4,7,11,3,6,10,1,5

43443434

12b

C7+/9+/11+/(13)/15- . . .

143

0,4,7,11,3,6,10,2,5

43443443

11b

C7+/9+/11+/(13)/15- . . .

144

0,4,8,11,2,6,9,1,5

44334344

10.1b C+7+/11+/13+ . . .

145

0,4,8,11,2,6,10,1,5

44334434

14b

146

0,4,8,11,3,6,9,1,5

44343344

10.1b C+7+/9+/11+/13+ . . .

147

0,4,8,11,3,6,10,1,5

44343434

8.1b

C+7+/9+/11+/(13)/15- . . .

148

C+7+/9+/11+/(13)/15- . . .

0,3,7,11,2,6,9,1,5

C+7+/11+/(13)/15- . . .

0,4,8,11,3,6,10,2,5

44343443

14b

149 ∼ 136 0,4,8,11,3,7,10,1,5

44344334

10.1b C+7+/9+/(11)/13-/15- . . .

150 ∼ 137 0,4,8,11,3,7,10,2,5

44344343

10.1b C+7+/9+/(11)/13-/15- . . .

151

44344344

16b

0,4,8,11,3,7,10,2,6

C+7+/9+/(11)/13-/15- . . .

Ten Pitch Classes 152

0,3,6,9,1,4,7,10,2,5

333433343

5b

C07-/9-/11-/13- . . .

153

0,3,6,9,1,4,7,11,2,5

333433433

4b

C07-/9-/11-/13- . . .

154

0,3,6,9,1,4,8,11,2,5

333434333

5b

C07-/9-/11-/13 . . .

155

0,3,6,10,1,4,7,11,2,5

334333433

3b

C09-/11-/13- . . .

156

0,3,6,10,1,4,8,11,2,5

334334333

4b

C09-/11-/13 . . .

157 ∼ 152 0,3,6,10,2,5,9,1,4,7

334434433

5b

C013+ . . .

158

0,3,6,10,2,5,9,1,4,8

334434434

6b

C013+ . . .

159

0,3,7,10,1,4,8,11,2,5

343334333

5b

Cm9-/11-/13 . . .

138

23 Classification Lists Third Chains—Continued

Chain Nr. Pitch Classes

Third

Chord Lead-Sheet

∼ equiv.

from 0

Chain

Class Symbols

160

0,3,7,10,1,4,8,11,2,6

343334334

6b

Cm9-/11-/13 . . .

161

0,3,7,10,1,5,8,11,2,6

343343334

3.1b

Cm9-/13 . . .

162

0,3,7,10,2,5,9,1,4,8

343434434

3.1b

Cm13+ . . .

163

0,3,7,10,2,6,9,1,4,8

343443434

7b

Cm11+/13+ . . .

164

0,3,7,11,2,5,9,1,4,8

344334434

6b

Cm7+/13+ . . .

165

0,3,7,11,2,6,9,1,4,8

344343434

3.1b

Cm7+/11+/13+ . . .

166

0,3,7,11,2,6,9,1,5,8

344343443

7b

Cm7+/11+/13+ . . .

167 ∼ 160 0,3,7,11,2,6,10,1,4,8

344344334

6b

Cm7+/11+/(13)/15- . . .

168 ∼ 161 0,3,7,11,2,6,10,1,5,8

344344343

3.1b

Cm7+/11+/(13)/15- . . .

169

0,3,7,11,2,6,10,1,5,9

344344344

6b

Cm7+/11+/(13)/15- . . .

170

0,4,7,10,1,5,8,11,2,6

433343334

7b

C9-/13 . . .

171

0,4,7,10,1,5,8,11,3,6

433343343

3.1b

C9-/13 . . .

172

0,4,7,10,2,5,8,11,3,6

433433343

6b

C19

173

0,4,7,10,2,6,9,1,5,8

433443443

6b

C11+/13+ . . .

174

0,4,7,11,2,6,9,1,5,8

434343443

3.1b

C7+/11+/13+ . . .

175 ∼ 170 0,4,7,11,2,6,10,1,5,8

434344343

7b

C7+/11+/(13)/15- . . .

176

434344344

3.1b

C7+/11+/(13)/15- . . .

177 ∼ 171 0,4,7,11,3,6,9,1,5,8

434433443

3.1b

C7+/9+/11+/13+ . . .

178

0,4,7,11,3,6,10,1,5,8

434434343

6b

C7+/9+/11+/(13)/15- . . .

179

0,4,7,11,3,6,10,1,5,9

434434344

7b

C7+/9+/11+/(13)/15- . . .

180 ∼ 172 0,4,7,11,3,6,10,2,5,8

434434433

6b

C7+/9+/11+/(13)/15- . . .

181

0,4,7,11,3,6,10,2,5,9

434434434

3.1b

C7+/9+/11+/(13)/15- . . .

182

0,4,8,11,2,6,10,1,5,9

443344344

6b

C+7+/11+/(13)/15- . . .

183

0,4,8,11,3,6,10,1,5,9

443434344

3.1b

C+7+/9+/11+/(13)/15- . . .

184

0,4,8,11,3,6,10,2,5,9

443434434

7b

C+7+/9+/(11)/13-/15- . . .

185

0,4,8,11,3,7,10,1,5,9

443443344

6b

C+7+/9+/(11)/13-/15- . . .

186

0,4,8,11,3,7,10,2,5,9

443443434

3.1b

C+7+/9+/(11)/13-/15- . . .

187

0,4,8,11,3,7,10,2,6,9

443443443

6b

C+7+/9+/(11)/13-/15- . . .

0,4,7,11,2,6,10,1,5,9

Eleven Pitch Classes 188

0,3,6,9,1,4,7,10,2,5,8

3334333433 2b

C07-/9-/11-/13- . . .

189

0,3,6,9,1,4,7,11,2,5,8

3334334333 2b

C07-/9-/11-/13- . . .

190

0,3,6,10,1,4,7,11,2,5,8

3343334333 2b

C09-/11-/13- . . .

191

0,3,6,10,1,4,7,11,2,5,9

3343334334 2b

C09-/11-/13- . . .

192

0,3,6,10,1,4,8,11,2,5,9

3343343334 2b

C09-/11-/13 . . .

193 ∼ 191 0,3,6,10,2,5,9,1,4,7,11

3344344334 2b

C013+ . . .

2 23.3 List of Local Denotators of Cardinality Two and Three in Z12

139

Third Chains—Continued Chain Nr. Pitch Classes

Third

Chord Lead-Sheet

∼ equiv.

Chain

Class Symbols

from 0

194 ∼ 192 0,3,6,10,2,5,9,1,4,8,11

3344344343 2b

C013+ . . .

195

0,3,7,10,1,4,8,11,2,5,9

3433343334 2b

Cm9-/11-/13 . . .

196

0,3,7,10,1,4,8,11,2,6,9

3433343343 2b

Cm9-/11-/13 . . .

197

0,3,7,10,1,5,8,11,2,6,9

3433433343 2b

Cm9-/13 . . .

198 ∼ 195 0,3,7,10,2,5,9,1,4,8,11

3434344343 2b

Cm13+ . . .

199

0,4,7,10,1,5,8,11,2,6,9

4333433343 2b

C9-/13 . . .

200

0,4,7,10,1,5,8,11,3,6,9

4333433433 2b

C9-/13 . . .

201

0,4,7,10,2,5,8,11,3,6,9

4334333433 2b

C21

202 ∼ 199 0,4,7,10,2,6,9,1,5,8,11

4334434433 2b

C11+/13+ . . .

203 ∼ 191 0,4,7,11,3,6,10,2,5,9,1

4344344344 2b

C7+/9+/11+/(13)/15- . . .

204 ∼ 192 0,4,8,11,3,6,10,2,5,9,1

4434344344 2b

C+7+/9+/(11)/13-/15- . . .

205 ∼ 195 0,4,8,11,3,7,10,2,5,9,1

4434434344 2b

C+7+/9+/(11)/13-/15- . . .

206 ∼ 196 0,4,8,11,3,7,10,2,6,9,1

4434434434 2b

C+7+/9+/(11)/13-/15- . . .

Twelve Pitch Classes 207

0,3,6,9,1,4,7,10,2,5,8,11 33343334333 1b

C07-/9-/11-/13- . . .

208 ∼ 207 0,4,7,10,2,5,8,11,3,6,9,1 43343334334 1b

C23

209 ∼ 207 0,4,7,10,2,6,9,1,5,8,11,3 43344344334 1b

C11+/13+ . . .

210 ∼ 207 0,4,8,11,3,7,10,2,6,9,1,5 44344344344 1b

C+7+/9+/(11)/13-/15- . . .

23.3 List of Local Denotators of Cardinality Two and Three in Z212 23.3.1 Two Tone Motifs in Z212 ClassN r. Representative 1

(0, 0), (0, 1)

2

(0, 0), (0, 2)

3

(0, 0), (0, 3)

4

(0, 0), (0, 4)

5

(0, 0), (0, 6)

23.3.2 Two Tone Motifs in Z5 × Z12

140

23 Classification Lists

ClassN r. Representative 1

(0, 0), (0, 1)

2

(0, 0), (0, 2)

3

(0, 0), (0, 3)

4

(0, 0), (0, 4)

5

(0, 0), (0, 6)

6

(0, 0), (1, 0)

7

(0, 0), (1, 1)

8

(0, 0), (1, 2)

9

(0, 0), (1, 2)

10

(0, 0), (1, 4)

11

(0, 0), (1, 6)

2 23.3 List of Local Denotators of Cardinality Two and Three in Z12

141

23.3.3 Three Tone Motifs in Z212 The order of these representatives is a historical one. After this table, the representatives are also visualized on a 12 × 12 square in list 23.1. Three-Element Motif Classes in OnP iM od12,12 Class Nr. Representative Kernel Class Weight Volume 1

(0, 0), (1, 0), (2, 0) Z.(1, 2) × Z.(1, 1)

(1, 1, 2)

0

2

(0, 0), (1, 0), (3, 0) Z.(1, 2) × Z.(0, 1)

(1, 2, 3)

0

3

(0, 0), (1, 0), (4, 0) Z.(1, 0) × Z.(0, 1)

(1, 3, 4)

0

4

(0, 0), (1, 0), (5, 0) Z.(1, 2) × Z.(1, 1)

(1, 1, 4)

0

5

(0, 0), (1, 0), (6, 0) Z.(1, 2) × Z.(1, 0)

(1, 1, 6)

0

6

(0, 0), (2, 0), (4, 0) (Z4 × 2Z4 ) × Z.(1, 1) (2, 2, 4)

0

7

(0, 0), (2, 0), (6, 0) (Z4 × 2Z4 ) × Z.(0, 1) (2, 4, 6)

0

8

(0, 0), (3, 0), (6, 0) Z.(1, 2) × (Z24 )

Z23

× Z.(1, 1)

(3, 3, 6)

0

(4, 4, 4)

0

9

(0, 0), (4, 0), (8, 0)

10

(0, 0), (1, 0), (0, 1) 0 × 0

(1, 1, 1)

1

11

(0, 0), (2, 0), (0, 1) Z.(2, 0) × 0

(1, 1, 2)

2

12

(0, 0), (3, 0), (0, 1) 0 × Z.(1, 0)

(1, 1, 3)

3

13

(0, 1), (0, 2), (3, 0) 0 × Z.(1, 1)

(1, 1, 1)

3

14

(0, 0), (0, 1), (4, 0) Z.(1, 0) × 0

(1, 1, 4)

4

15

(0, 0), (1, 2), (2, 0) Z.(1, 2) × 0

(1, 1, 2)

4

2Z24

×0

16

(0, 0), (2, 0), (0, 2)

(2, 2, 2)

4

17

(0, 0), (6, 0), (0, 1) Z.(2, 0) × Z.(1, 0)

(1, 1, 6)

6

18

(0, 0), (3, 0), (0, 2) Z.(2, 0) × Z.(0, 1)

(1, 2, 3)

6

19

(0, 0), (0, 2), (3, 1) Z.(2, 0) × Z.(1, 1)

(1, 1, 2)

6

20

(0, 0), (4, 0), (0, 2) (Z4 × 2Z4 ) × 0

(2, 2, 4)

4

(3, 3, 3)

3

(2, 2, 6)

0

(2, 2, 2)

0

(4, 4, 4)

4

(3, 3, 6)

6

(6, 6, 6)

0

21

(0, 0), (4, 0), (0, 4) 0 ×

22

(0, 0), (6, 0), (0, 2)

23

(0, 2), (0, 4), (6, 0)

24

(0, 0), (4, 0), (0, 4)

25

(0, 0), (6, 0), (0, 3)

26

(0, 0), (6, 0), (0, 6)

Z23

2Z24 × Z.(1, 0) 2Z24 × Z.(1, 1) Z24 × 0 Z.(2, 0) × Z23 2Z24 × Z23

142

23 Classification Lists

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

Fig. 23.1: Three Tone Motifs in Z212

23.4 List of Local Denotators of Cardinality Four in Z212 This list was calculated by Straub in [45]. The list’s numbering follows Straub’s algorithm; * denotes classes which are not determined by volume and class weight.

23.4 List of Local Denotators of Cardinality Four in Z212

143

10 11

12

13 21

15

19

14

17

18 25

16

3

20 24

5

2

23 22 1

4 6

26

8

7

9

Fig. 23.2: Hasse diagram of dominance/specialization among the 26 isomorphism classes of motives in Z212 .

Four-Element Motif Classes Class Nr. Representative Class Weight Volume 0 1 2 3 4 5 6 7 8 9 10

(0,0),(0,1),(0,2),(0,7) (0,0),(0,1),(0,2),(0,3) (0,0),(0,1),(0,2),(0,6) (0,0),(0,1),(0,2),(0,5) (0,0),(0,1),(0,2),(0,4) (0,0),(1,0),(0,5),(0,6) (0,0),(0,1),(0,4),(0,5) (0,0),(0,1),(0,3),(0,5) (0,0),(0,1),(0,4),(0,8) (0,0),(0,1),(0,6),(0,7) (0,0),(0,1),(0,3),(0,6)

(1,1,5,5) (1,1,2,2) (1,4,5,7) (1,4,2,3) (1,2,3,6) (4,4,5,5) (4,4,3,3) (4,2,2,6) (4,3,3,9) (5,5,5,5) (5,2,2,8)

0 0 0 0 0 0 0 0 0 0 0

144

23 Classification Lists

Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34* 35* 36* 37* 38* 39* 40* 41* 42* 43 44*

(0,0),(0,1),(0,3),(0,7) (0,0),(0,1),(0,4),(0,7) (0,0),(0,1),(0,3),(0,10) (0,0),(0,1),(0,3),(0,4) (0,0),(0,1),(0,3),(0,9) (0,0),(0,1),(0,4),(0,9) (0,0),(0,2),(6,0),(6,10) (0,0),(0,2),(0,4),(6,0) (0,0),(0,2),(0,4),(6,2) (0,0),(0,2),(0,4),(0,6) (0,0),(0,2),0,4),(0,8) (0,0),(0,2),(6,0),(6,2) (0,0),(0,2),(6,0),(6,6) (0,0),(0,2),(0,6),(6,2) (0,0),(0,2),(0,6),(6,0) (0,0),(0,2),(0,6),(0,8) (0,0),(0,3),(0,6),(0,9) (0,0),(0,6),(6,0),(6,6) (0,0),(0,1),(0,2),(1,0) (0,0),(0,1),(0,5),(1,0) (0,0),(0,1),(0,6),(1,0) (0,0),(0,1),(0,3),(1,0) (0,0),(0,1),(0,4),(1,0) (0,0),(0,1),(1,0),(1,5) (0,0),(0,1),(1,0),(7,7) (0,0),(0,1),(1,0),(1,1) (0,0),(0,1),(1,0),(3,5) (0,0),(0,1),(1,0),(3,11) (0,0),(0,1),(1,0),(1,2) (0,0),(0,1),(1,0),(5,10) (0,0),(0,1),(1,0),(4,10) (0,0),(0,1),(1,0),(2,4) (0,0),(0,1),(1,0),(2,5) (0,0),(0,1),(1,0),(1,3)

(5,2,3,7) (5,3,3,8) (2,2,2,2) (2,2,3,3) (2,3,7,8) (3,3,3,3) (23,23,22,22) (23,6,22,7) (6,6,22,22) (6,6,7,7) (6,7,7,9) (22,22,22,22) (22,22,22,26) (22,22,7,7) (22,7,7,26) (7,7,7,7) (8,8,8,8) (26,26,26,26) (1,10,10,11) (4,10,10,14) (5,10,10,17) (2,10,11,12) (3,10,12,14) (10,10,10,10) (10,10,10,10) (10,10,10,10) (10,10,10,13) (10,10,10,13) (10,10,11,11) (10,10,11,11) (10,10,11,15) (10,10,11,15) (10,10,11,19) (10,10,12,12)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

23.4 List of Local Denotators of Cardinality Four in Z212

Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 45* 46* 47 48* 49* 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73* 74* 75* 76* 77* 78*

(0,0),(0,1),(1,0),(7,9) (0,0),(0,1),(1,0),(3,3) (0,0),(0,1),(1,0),(6,8) (0,0),(0,1),(1,0),(1,4) (0,0),(0,1),(1,0),(4,4) (0,0),(0,1),(1,0),(1,6) (0,0),(0,1),(1,0),(2,2) (0,0),(0,1),(1,0),(2,3) (0,0),(0,1),(1,0),(6,9) (0,0),(0,1),(1,0),(8,8) (0,0),(0,1),(1,0),(3,4) (0,0),(0,1),(2,0),(3,1) (0,0),(0,1),(2,0),(3,4) (0,0),(0,1),(3,0),(4,1) (0,0),(0,1),(0,2),(2,1) (0,0),(0,1),(0,2),(2,0) (0,0),(0,1),(0,5),(2,0) (0,0),(0,1),(0,5),(2,1) (0,0),(0,1),(0,6),(2,1) (0,0),(0,1),(0,6),(2,0) (0,0),(0,1),(0,3),(2,0) (0,0),(0,1),(0,3),(2,1) (0,0),(0,1),(0,4),(2,1) (0,0),(0,1),(0,4),(2,0) (0,0),(0,1),(2,0),(4,6) (0,0),(0,1),(2,0),(4,0) (0,0),(0,1),(2,0),(6,6) (0,0),(0,1),(2,0),(6,0) (0,0),(0,1),(2,0),(2,1) (0,0),(0,1),(2,0),(2,5) (0,0),(0,1),(2,0),(2,7) (0,0),(0,1),(2,0),(2,11) (0,0),(0,1),(2,0),(6,5) (0,0),(0,1),(2,0),(6,11)

(10,10,12,12) (10,10,12,12) (10,10,15,19) (10,10,14,14) (10,10,14,14) (10,10,17,17) (10,11,11,13) (10,11,13,15) (10,11,12,18) (10,13,14,14) (10,12,15,18) (11,11,12,12) (11,12,12,15) (12,12,14,14) (1,11,11,15) (1,11,11,16) (4,11,11,14) (4,11,11,20) (5,5,11,11) (5,22,11,11) (2,11,15,18) (2,11,16,18) (3,11,14,18) (3,11,20,18) (23,11,11,15) (6,11,11,14) (22,11,15,17) (7,11,14,17) (11,11,11,11) (11,11,11,11) (11,11,11,11) (11,11,11,11) (11,11,11,19) (11,11,11,19)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

145

146

23 Classification Lists

Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 79* 80* 81* 82* 83* 84* 85* 86* 87* 88* 89* 90* 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112

(0,0),(0,1),(2,0),(4,7) (0,0),(0,1),(2,0),(8,11) (0,0),(0,1),(2,0),(2,2) (0,0),(0,1),(2,0),(8,10) (0,0),(0,1),(2,0),(4,1) (0,0),(0,1),(2,0),(8,5) (0,0),(0,1),(2,0),(8,4) (0,0),(0,1),(2,0),(2,4) (0,0),(0,1),(2,0),(6,7) (0,0),(0,1),(2,0),(6,1) (0,0),(0,1),(2,0),(2,9) (0,0),(0,1),(2,0),(2,3) (0,0),(0,1),(2,0),(4,3) (0,0),(0,1),(2,0),(4,2) (0,0),(0,1),(2,0),(4,9) (0,0),(0,1),(2,0),(4,8) (0,0),(0,1),(4,2),(6,1) (0,0),(0,1),(4,2),(6,4) (0,0),(0,1),(4,0),(6,1) (0,0),(0,1),(4,0),(6,4) (0,0),(0,1),(0,2),(3,0) (0,0),(0,1),(0,2),(3,1) (0,0),(0,1),(0,5),(3,0) (0,0),(0,1),(0,4),(3,2) (0,0),(0,1),(0,6),(3,2) (0,0),(0,1),(0,6),(3,1) (0,0),(0,1),(0,6),(3,0) (0,0),(0,1),(0,3),(3,2) (0,0),(0,1),(0,3),(3,1) (0,0),(0,1),(0,3),(3,0) (0,0),(0,1),(0,4),(3,0) (0,0),(0,1),(0,4),(3,1) (0,0),(0,1),(3,0),(6,0) (0,0),(0,3),(0,6),(3,0)

(11,11,15,15) (11,11,15,15) (11,11,15,16) (11,11,15,16) (11,11,14,14) (11,11,14,14) (11,11,14,20) (11,11,14,20) (11,11,17,17) (11,11,17,17) (11,11,18,18) (11,11,18,18) (11,15,15,19) (11,15,16,19) (11,14,14,19) (11,14,20,19) (15,15,17,17) (15,16,18,18) (14,14,17,17) (14,20,18,18) (1,13,12,18) (1,12,12,19) (4,4,12,12) (4,3,13,12) (5,13,13,17) (5,12,12,17) (5,12,12,25) (2,13,12,19) (2,12,12,18) (2,12,21,18) (3,3,12,12) (3,3,12,21) (8,12,12,17) (8,21,21,25)

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3

23.4 List of Local Denotators of Cardinality Four in Z212

Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 113* 114* 115* 116 117 118* 119* 120* 121* 122* 123 124 125 126 127 128* 129* 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146

(0,0),(0,1),(3,0),(3,5) (0,0),(0,1),(3,0),(3,11) (0,0),(0,1),(3,0),(9,11) (0,0),(0,1),(3,2),(3,8) (0,0),(0,1),(3,0),(3,2) (0,0),(0,1),(3,0),(9,7) (0,0),(0,1),(3,0),(3,7) (0,0),(0,1),(3,0),(3,1) (0,0),(0,1),(3,0),(9,3) (0,0),(0,1),(3,0),(3,3) (0,0),(0,1),(3,0),(6,5) (0,0),(0,1),(3,0),(6,1) (0,0),(0,1),(3,0),(3,6) (0,0),(0,1),(3,0),(3,10) (0,0),(0,1),(3,0),(6,9) (0,0),(0,3),(3,0),(3,3) (0,0),(0,3),(3,0),(9,9) (0,0),(0,3),(3,0),(3,6) (0,0),(0,1),(0,2),(4,3) (0,0),(0,1),(0,2),(4,1) (0,0),(0,1),(0,2),(4,0) (0,0),(0,1),(0,5),(4,2) (0,0),(0,1),(0,5),(4,3) (0,0),(0,1),(0,5),(4,0) (0,0),(0,1),(0,5),(4,1) (0,0),(0,1),(0,6),(4,3) (0,0),(0,1),(0,6),(4,1) (0,0),(0,1),(0,6),(4,0) (0,0),(0,1),(0,3),(4,2) (0,0),(0,1),(0,3),(4,1) (0,0),(0,1),(0,3),(4,0) (0,0),(0,1),(0,3),(4,3) (0,0),(0,1),(0,4),(4,1) (0,0),(0,1),(0,4),(4,0)

(13,13,12,12) (13,13,12,12) (13,13,12,12) (13,13,17,17) (13,12,19,18) (12,12,12,12) (12,12,12,12) (12,12,12,12) (12,12,12,21) (12,12,12,21) (12,12,19,19) (12,12,17,17) (12,12,17,25) (12,12,18,18) (12,21,18,18) (21,21,21,21) (21,21,21,21) (21,21,25,25) (1,15,15,15) (1,15,14,14) (1,15,14,20) (4,15,15,14) (4,15,15,20) (4,14,14,14) (4,14,14,24) (5,5,15,15) (5,5,14,14) (5,7,15,14) (2,2,15,15) (2,2,14,20) (2,3,15,14) (2,3,15,20) (3,3,14,14) (3,3,14,24)

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

147

148

23 Classification Lists

Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161* 162* 163* 164* 165* 166* 167* 168* 169 170 171* 172* 173* 174* 175 176 177 178 179 180

(0,0),(0,2),(2,0),(6,10) (0,0),(0,2),(2,0),(4,4) (0,0),(0,1),(4,0),(8,6) (0,0),(0,2),(0,4),(2,0) (0,0),(0,2),(0,4),(4,2) (0,0),(0,2),(0,4),(4,0) (0,0),(0,2),(2,0),(2,6) (0,0),(0,2),(4,0),(6,2) (0,0),(0,2),(0,6),(2,0) (0,0),(0,2),(0,6),(4,0) (0,0),(0,2),(0,6),(4,2) (0,0),(0,1),(4,0),(8,0) (0,0),(0,2),(4,0),(8,0) (0,0),(0,4),(0,8),(4,0) (0,0),(0,1),(4,2),(4,7) (0,0),(0,1),(4,2),(4,3) (0,0),(0,1),(4,0),(4,7) (0,0),(0,1),(4,0),(4,11) (0,0),(0,1),(4,0),(4,2) (0,0),(0,1),(4,0),(4,10) (0,0),(0,1),(4,0),(4,1) (0,0),(0,1),(4,0),(4,5) (0,0),(0,1),(4,0),(4,4) (0,0),(0,2),(2,0),(2,2) (0,0),(0,2),(2,0),(8,8) (0,0),(0,2),(2,0),(2,4) (0,0),(0,2),(4,0),(4,2) (0,0),(0,2),(4,0),(4,10) (0,0),(0,2),(4,0),(4,4) (0,0),(0,4),(4,0),(4,4) (0,0),(0,1),(0,2),(6,1) (0,0),(0,1),(0,2),(6,3) (0,0),(0,1),(0,2),(6,4) (0,0),(0,1),(0,2),(6,0)

(23,16,16,16) (23,16,20,20) (6,15,15,14) (6,16,16,20) (6,20,20,20) (6,20,20,24) (22,22,16,16) (22,22,20,20) (22,7,16,20) (7,7,20,20) (7,7,20,24) (9,14,14,14) (9,20,20,20) (9,24,24,24) (15,15,15,15) (15,15,15,15) (15,15,14,14) (15,15,14,14) (15,15,14,20) (15,15,14,20) (14,14,14,14) (14,14,14,14) (14,14,14,24) (16,16,16,16) (16,16,20,20) (16,16,20,20) (20,20,20,20) (20,20,20,20) (20,20,20,24) (24,24,24,24) (1,1,17,17) (1,2,19,18) (1,23,18,18) (1,22,19,17)

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6

23.4 List of Local Denotators of Cardinality Four in Z212

Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204* 205* 206* 207* 208* 209* 210* 211* 212* 213* 214*

(0,0),(0,1),(0,5),(6,0) (0,0),(0,1),(0,4),(6,5) (0,0),(0,1),(0,5),(6,3) (0,0),(0,1),(0,5),(6,1) (0,0),(0,1),(0,6),(6,5) (0,0),(0,1),(0,6),(6,1) (0,0),(0,1),(0,6),(6,2) (0,0),(0,1),(0,6),(6,4) (0,0),(0,1),(0,6),(6,3) (0,0),(0,1),(0,6),(6,0) (0,0),(0,1),(0,3),(6,0) (0,0),(0,1),(0,3),(6,4) (0,0),(0,1),(0,3),(6,5) (0,0),(0,1),(0,3),(6,1) (0,0),(0,1),(0,3),(6,3) (0,0),(0,1),(0,4),(6,1) (0,0),(0,1),(0,4),(6,3) (0,0),(0,1),(0,4),(6,2) (0,0),(0,1),(0,4),(6,0) (0,0),(0,1),(0,4),(6,4) (0,0),(0,1),(6,3),(6,9) (0,0),(0,3),(0,6),(6,3) (0,0),(0,3),(0,6),(6,0) (0,0),(0,1),(6,0),(6,5) (0,0),(0,1),(6,0),(6,11) (0,0),(0,1),(6,2),(6,3) (0,0),(0,1),(6,2),(6,5) (0,0),(0,1),(6,0),(6,1) (0,0),(0,1),(6,0),(6,7) (0,0),(0,1),(6,0),(6,3) (0,0),(0,1),(6,0),(6,9) (0,0),(0,1),(6,3),(6,4) (0,0),(0,1),(6,3),(6,10) (0,0),(0,3),(6,0),(6,3)

(4,4,17,17) (4,3,19,18) (4,6,18,18) (4,7,19,17) (5,5,19,19) (5,5,17,17) (5,22,19,19) (5,22,18,18) (5,8,18,18) (5,26,17,17) (2,2,17,25) (2,2,18,18) (2,23,19,18) (2,22,17,18) (2,22,18,25) (3,3,17,25) (3,3,18,18) (3,6,19,18) (3,7,17,18) (3,7,18,25) (22,8,18,18) (8,8,25,25) (8,26,25,25) (19,19,17,17) (19,19,17,17) (19,19,18,18) (19,19,18,18) (17,17,17,17) (17,17,17,17) (17,18,18,25) (17,18,18,25) (18,18,18,18) (18,18,18,18) (25,25,25,25)

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

149

150

23 Classification Lists

Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 215*

(0,0),(0,3),(6,0),(6,9) (25,25,25,25) 6

23.5 List of Modulation Chords (Pivots) Quanta and Pivots for the Modulations Between Diatonic Major Scales Transl. p Cadence Quantum Modulator Pivots 1

{II, V }

• ◦ • • ◦ • • • • • •• e5 11

{II, III, V, V II}

1

{II, III} • ◦ • • ◦ • • • • • •• e5 11

{II, III, V, V II}

2

{V II}

◦ • • ◦ • • ◦ • ◦ ◦ ◦• e6 11

{II, IV, V II}

2

{II, V }

◦ • • ◦ • • ◦ • ◦ • ◦• e 11

{II, IV, V, V II}

2

{IV, V } ◦ • • ◦ • • ◦ • ◦ • ◦• e6 11

{II, IV, V, V II}

3

{II, V }

3 4 4 4

6

• ◦ • ◦ ◦ • ◦ • • • •• e7 11

{II, III, V, V II}

8

{II, IV, V, V II}

8

{II, III, V, V II}

8

{V, V II}

9

{II, IV, V II}

{II, III} • ◦ • ◦ ◦ • ◦ • • • •• e 11 {V II}

{II, III, V, V II}

7

◦ ◦ • • ◦ • • ◦ ◦ • ◦• e 11

{IV, V } ◦ • • • • • • • ◦ • ◦• e 11 {II, III} • • • • ◦ • • • • • ◦• e 11

5

{V II}

6

{II, III} ◦ • • • • • ◦ • • • •• e6

◦ ◦ • ◦ • • ◦ • ◦ ◦ •• e 11

{II, III, V, V II}

6

{IV, V } ◦ • • • • • • • • • ◦• e 11

{II, IV, V, V II}

6

{IV, V } • • • • ◦ • • • • • ◦• e6

{II, IV, V, V II}

10

6

{II, III} • • • • ◦ • ◦ • • • •• e 11

7

{V II}

8 8 8 9

10

{V II}

• ◦ • ◦ ◦ • • ◦ ◦ • ◦• e11 11

{II, V II}

0

{II, IV, V, V II}

0

{II, III, V, V II}

1

{II, IV, V, V II}

1

{II, IV, V, V II}

◦ • • ◦ ◦ • ◦ • ◦ ◦ •• e 11

{II, III} • • • • ◦ • ◦ • ◦ • •• e 11 ◦ ◦ • ◦ • • • • • • ◦• e 11

9

{IV, V } ◦ ◦ • ◦ • • • • • • ◦• e 11

10

{V II}

10 10

{II, V }

{III, V, V II}

0

{IV, V } ◦ • • • • • ◦ • • • •• e 11 {II, V }

{II, III, V, V II}

• ◦ • • ◦ • ◦ ◦ ◦ • ◦• e2 11

{III, V, V II}

2

{II, III, V, V II}

2

{II, III, V, V II}

• ◦ • • ◦ • ◦ • ◦ • ◦• e 11

{II, III} • ◦ • • ◦ • ◦ • ◦ • ◦• e 11

23.5 List of Modulation Chords (Pivots)

Quanta and Pivots for Diatonic Major Scales—Continued Class Transl. p Cadence Quantum Modulator Pivots 11 11

{II, V }

◦ • • ◦ • • • • • • •• e3 11 3

{IV, V } ◦ • • ◦ • • • • • • •• e 11

{II, IV, V, V II} {II, IV, V, V II}

151

References

1. Adorno Th W: Fragment über Musik und Sprache. Stuttgart, Jahresring 1956 2. Lang S: Algebra. Springer 2002 3. Agustín-Aquino O A, J Junod, G Mazzola: Computational Counterpoint Worlds. Springer Series Computational Music Science, Heidelberg 2015 4. Boulez P: Musikdenken heute I,II; Darmstädter Beiträge V, VI. Schott, Mainz 1963, 1985 5. Boulez P: structures, premier livre. UE, London 1953 6. Boulez P: structures, deuxiéme livre. UE, London 1967 7. Eimert H: Grundlagen der musikalischen Reihentechnik. Universal Edition, Wien 1964 8. Funkhouser Ch: being matter ignited...an interview with Cecil Taylor. Hambone 1994 9. Fux J J: Gradus ad Parnassum (1725). Dt. und kommentiert von L. Mitzler, Leipzig 1742; English edition: The Study of Counterpoint. Translated and edited by A Mann. Norton & Company, New York, London 1971 https://www.thegreatcoursesdaily.com/kurt-godel-wanted-to-reviseour-concept-of-time 10. Graeser, W: Bachs “Kunst der Fuge”. In: Bach-Jahrbuch 1924 11. Hanslick E: Vom Musikalisch-Schönen. Breitkopf und Härtel (1854), Wiesbaden 1980 12. Hecquet S and A Prokhoris: Fabriques de la danse. PUF, Paris 2007 13. Hjelmslev L: Prolégomènes á une théorie du langage. Minuit, Paris 1966 14. Jakobson R: Linguistics and Poetics. In: Seboek, TA (ed.): Style in Language. Wiley, New York 1960 15. Jackendoff R and Lerdahl F: A Generative Theory of Tonal Music. MIT Press, Cambridge MA, 1983 16. Johnson I and Henrich A K: Knot Theory. Aurora 2017 17. Kaiser J: Beethovens 32 Klaviersonaten und ihre Interpreten. Fischer, Frankfurt/Main 1979 18. Kant I: Kritik der reinen Vernunft. Meiner, Hamburg 1956 19. Lewin D: Generalized Musical Intervals and Transformations (1987). Cambridge University Press 1987 20. Ligeti G: Pierre Boulez: Entscheidung und Automatik in der Structure Ia. Die Reihe IV, UE, Wien 1958 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4

153

154 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

35.

36. 37. 38.

39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

References Kelley J L: General Topology. Van Nostrand 1955 Mac Lane S: Categories for the Working Mathematician. Springer 1998 Mac Lane S and Moerdjik I: Sheaves in Geometry and Logic. Springer 1992 Marek C: Lehre des Klavierspiels. Atlantis 1977 Mazzola G: Gruppen und Kategorien in der Musik. Heldermann, Berlin 1985 Mazzola G: www.rubato.org Mazzola G et al.: The Topos of Music—Geometric Logic of Concepts, Theory, and Performance. Birkhäuser, Basel et al. 2002 Mazzola G et al.: The Topos of Music I: Theory. (First volume of second edition of [27]) Springer, Heidelberg 2018 Mazzola G et al.: The Topos of Music II: Performance. (Second volume of second edition of [27]) Springer, Heidelberg 2018 Mazzola G et al.: The Topos of Music III: Gestures. (Third volume of second edition of [27]) Springer, Heidelberg 2018 Mazzola G et al.: The Topos of Music IV: Roots. (Fourth volume of second edition of [27]) Springer, Heidelberg 2018 Mazzola G et. al.: Making Musical Time. Springer 2021 Mazzola G and M Andreatta: Diagrams, Gestures, and Formulas in Music. Journal of Mathematics and Music 2007, Vol. 1, no. 1, 2007 Mazzola G and P B Cherlin: Flow, Gesture, and Spaces in Free Jazz—Towards a Theory of Collaboration. Springer Series Computational Music Science, Heidelberg 2009 Mazzola G: Categorical Gestures, the Diamond Conjecture, Lewin’s Question, and the Hammerklavier Sonata. Journal of Mathematics and Music Vol. 3, no. 1, 2009 Mazzola G, J Park, F Thalmann: Musical Creativity. Springer, Heidelberg 2011 Mazzola G, M Mannone, Y Pang: Cool Math for Hot Music. Springer, Heidelberg 2016 Noll Th: http://www.cs.tu-berlin.de/ noll/ChordDictionary.sea.hqx, TU Berlin 1996 social and cultural context (pp. 33–56). Gainesville: University Presses of Florida, 1981 Nattiez J-J: Fondements d’une Sémiologie de la Musique. Edition 10/18 Paris 1975 Riemann B (1867): Über die Hypothesen, welche der Geometrie zugrunde liegen (Habilitationsreferat 1854). Gott. Abh. No.13, 1867 Riemann H: System der musikalischen Rhythmik und Metrik. Breitkopf und Härtel, Leipzig 1903 Ruwet N: Langage, Musique, Poesie. Seuil, Paris 1972 Sachs K-J: Der Contrapunctus im 14. und 15. Jahrhundert. AMW, Franz Steiner, Wiesbaden 1974 Schönberg A: Harmonielehre (1911). Universal Edition, Wien 1966 Straub H: Beiträge zur modultheoretischen Klassifikation musikalischer Motive. Diplomarbeit ETH-Zürich, Zürich 1989 Tymoczko D: A Geometry of Music. Oxford University Press, 2011 Uhde J and Wieland R: Denken und Spielen. Bärenreiter, Kassel et al. 1988 Uhde J and R Wieland: Forschendes Üben. Bärenreiter 2002 Varèse E: Erinnerungen und Gedanken. In: Darmstädter Beiträge III. Schott, Mainz 1960

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Index

C(F ), 16 CT (D), 17 Deno(Cat), 18 F (D), 17 GI , 80 I(F ), 16 N (D), 17 N (F ), 15 T (F ), 16 T ±, 28 Cat@ , 15 Mod, 15 TopCat, 25 ∇, 25 Colimit, 18 Digraph, 15, 23 Limit, 18 Power, 18 Ens, 15 Simple, 17 STRG, 19 Syn, 17 RUBATOr software, 34 Colimit, 16 Limit, 16 Power, 16 Simple, 16 Syn, 16 3M, 10 A activity, interpretative -, 77, 79 address change, 15, 23 Adorno, Theodor Wiesengrund, 73

agogics, 74 ambiguity, 77 analysis, neutral -, 75 articulated, listening, 74 articulation, 74 ASCII, 20 aspect, psychological -, vii atlas, 78 A-addressed -, 80 Avison, Charles, 73 B Bach, Johann Sebastian, 74, 90 Beethoven, Ludwig van, 30, 74, 90 Boulez, Pierre, 14, 22 C cadence, 30 Cage, John, 76 canonical program, 90 category topological -, 25 graph -, 25 cell complex, 90 cellular organism, 90 change of perspective, 89 chart, 78, 80 chord, 74 classification creative value of -, 13 definition of -, 3 of gestures, 34 scientific value of -, 13 semantics of -, 4

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Zheng, G. Mazzola, Classification of Musical Objects for Analysis and Composition, Computational Music Science, https://doi.org/10.1007/978-3-031-30183-4

157

158

Index

coda, 74 colimit, 78 communication, 7 complex, cell -, 90 composition global -, 28 resolution of a -, 87 global objective -, 79 local -, 27 modular -, 77 concept grouping -, 76 score -, 77 concert for piano and orchestra, 77 context, semiotic -, 10 contrapuntal form, 74 coordinator, form -, 16 covering, 78 equivalent -, 80 creativity, 9 D dactylus grid, 50 degree, 29, 74 pivotal -, 30 denotator, 3, 16 development, 74 Diabelli Variations, 90 diffeomorphism, 34 digraph, 23 dodecaphonic series, 90 row, 38 DX7, 22 dynamics, 74 E Einstein, Albert, 12 embodiment, 7, 8 equivalence relation, 75 equivalent covering, 80 Escher category, 38 Theorem, 37 aesthesic identification, 74 aesthetics of music, 87 exposition, 74 F fact, 8

Feldman, Morton, 77 Fifth symphony, 74 fingering, 73 form, 3, 15 circular -, 21 contrapuntal -, 74 coordinator, 16 functor, 16 identifier, 16 sonata -, 74 space, 16 type, 16 coordinator of a -, 15 identifier of a -, 15 name of a -, 15 type of a -, 15 Fourier decomposition, 21 frame space, 16 frequency modulation, 22 Fry, Arthur, 11 function poetical -, 73 tonal -, 74 functor, form -, 16 G general note, 21 germinal melody, 51 gestalt, global -, 77 gesture, 4, 8, 23, 33 global -, 28, 79, 80 global gestalt, 77 objective composition, 79 score, 77 Goethe, Johann Wolfgang von, 90 Goldberg Variations, 90 Graeser, Wolfgang, 74 grid, dactylus -, 50 group, finite simple -, 4 grouping concept, 76 metrical -, 73 H Hammerklavier Sonata, 30 Hanslick, Eduard, 74, 77 hierarchical organism, 74 hierarchy, 76

Index HiHat, 20 historical instrumentation, 89 Hofmann, Ernst Theodor Amadeus, 73 homology theories, 38 hyperdenotator, 24, 37 hypergesture, 37 I identification, esthesic -, 74 identifier, form -, 16 instrumentation, historical -, 89 Intégrales, 89 integral of perspectives, 90 interpretative activity, 77, 79 isometry, 76 isomorphism class, 4 J Jackendoff, Ray, 76 Jakobson, Roman, 75 Java, 34 jazz improvisation, 38 K Kagel, Maurizio, 90 Kaiser, Joachim, 73, 74 knot, 4, 34 Kunst der Fuge, 74 L Lanier, Sidney, 12 Lerdahl, Fred, 76 limits, ontological -, 3 linguistics, structuralist -, 75 listening, articulated -, 74 local score, 77 local-global patchwork, 77 locally trivial structure, 78 loop, 34 loudness, 19 Luening, Otto, 76 M Möbius strip, 29, 78 macroscore, 21 manifold, 78 Marek, Ceslav, 73 Mattheson, Johann, 73 melody, germinal -, 51 metrical grouping, 73

mode, 28 modular composition, 77 modulation model, 29 quantized -, 30 quantum, 30 tonal -, 28 modulator, 30 music aesthetics of -, 87 characterization of -, 9 critic, 74 criticism, 74 ontology, 7 psychology, 76 software, 77 tape -, 76 musique concrète, 77 N naming policy, 17 Nattiez, Jean-Jacques, 74 neutral analysis, 75 note, 20 O object, 3 objective global composition, 79 oniontology, 7, 9 onset, 20 ontology, music -, 7 organism cellular, 90 hierarchical -, 74 P part, 77 patchwork, local-global -, 77 pause, 20 performance structural rationale of -, 90 performance theory, 34 perspective, change of -, 89 perspectives, integral of -, 90 pitch, 20 poetical function, 73 Post-It, 10 presheaf, 34

159

160

Index

principle, variation -, 90 process, 8 program, canonical -, 90 progression harmonic -, 74 contrapuntal -, 74 Project of Music for Magnetic Tape, 77 psychology, music -, 76 Q question, open -, 10 R realities, 7 recapitulation, 74 relation, equivalence -, 75 representation, formal -, 13 resolution of a global composition, 87 Riemann, Bernhard, 78 Riemann, Hugo, 29, 77 Rousseau, Jean-Jacques, 73 Rubato software, 14 Ruwet, Nicolas, 74 S Saussure, Ferdinand de, 75 scale, 28 major -, 28 Schönberg, Arnold, 29, 90 Schaeffer, Pierre, 76 Schenker analysis, 21 score concept, 77 global -, 77 local -, 77 semiotics, 7 serial composition, 22 series, dodecaphonic -, 90 set support -, 80 theory, 75 sign, critical -, 10 Silver, Spencer, 10 software, 34 music -, 77 sonata form, 74

Hammerklavier -, 30 space, form -, 16 Straub, Hans, 142 structural rationale of performance, 90 structuralist linguistics, 75 structure, locally trivial -, 78 Structures pour deux pianos, 14, 22, 38 support set, 80 symmetry transformation, 75, 76 T tönend bewegte Formen, 77 tape music, 76 theory, set -, 75 time-slice, 77 tonal function, 74 tonality, 28, 74 track, 77 transformation, symmetry -, 75, 76 Transición II, 90 Tudor, David, 77 tuning, 74 type, form -, 16 U understanding, 90 musical works, 88 Unicode, 20 Ussachevsky, Vladimir, 76 V Varèse, Edgar, 87, 89 variation principle, 90 Variationen für Klavier, 90 voice leading, 74 von Ehrenfels, Christian, 78 W walls, 10 extended -, 10 Webern, Anton von, 90 Wolff, Christian, 77 Y Yoneda Lemma, 19, 34, 89 Yoneda, Nobuo, 87