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2023 | Buch

The Basic Problem of Classification Kapitel lesen Erstes Kapitel lesen

Classification of Musical Objects for Analysis and Composition

verfasst von: Linshujie Zheng, Guerino Mazzola

Verlag: Springer International Publishing

Buchreihe : Computational Music Science

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents and discusses the fundamental topic of classification of musical objects, such as chords, motifs, and gestures. Their classification deals with the exhibition of isomorphism classes. Our structure types include local and global constructions, the latter being similar to global structures in geometry, such as differentiable manifolds.

The discussion extends to the role, which classification plays for the creative construction of musical compositions. Our examples include references to classical compositions, such as Beethoven’s sonatas, and some of the author’s own compositions of classical and jazz styles.

We also discuss software that enables the application of classification to musical creativity.

The volume is addressed to an audience that would apply classification to programming and creative musical construction.

Inhaltsverzeichnis

Frontmatter

Initial Orientation

Frontmatter
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.
Linshujie Zheng, Guerino Mazzola

General Formal Concepts

Frontmatter
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.
Linshujie Zheng, Guerino Mazzola
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.
Linshujie Zheng, Guerino Mazzola
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 RUBATO®. 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.
Linshujie Zheng, Guerino Mazzola
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.
Linshujie Zheng, Guerino Mazzola
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.
Linshujie Zheng, Guerino Mazzola
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.
Linshujie Zheng, Guerino Mazzola

Local Classification

Frontmatter
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.
Linshujie Zheng, Guerino Mazzola
9. Classification of Chords
Summary
This chapter deals with chords, i.e., zero-addressed local composition denotators \( \textit{K} = \{c_{1},...c_{k}\} \subset \mathbb{Z}_{12} \), i.e. K : 0@PitchClass{c1 . . . ck} for the form PitchClass : Id.Simple(\( \mathbb{Z}_{12} \)). Such denotators can also be interpreted as a 12-periodic rhythms, we shall occasionally consider this latter point of view.
Linshujie Zheng, Guerino Mazzola
10. Motif Classes
Summary
This chapter sets a perspective onto the classification of motives, ie., local compositions \( \textit{K} \subset \mathbb{Z}^{2}_{12} \). The general case is not made explicit yet, but for card(K) ≤ 4, we have complete lists and some creative applications.
Linshujie Zheng, Guerino Mazzola
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.
Linshujie Zheng, Guerino Mazzola
12. Harmony through Third Chains
Summary
We discuss the application of third chain classification to Riemannian harmony for general chords.
Linshujie Zheng, Guerino Mazzola
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.
Linshujie Zheng, Guerino Mazzola
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.
Linshujie Zheng, Guerino Mazzola
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.
Linshujie Zheng, Guerino Mazzola
16. Dodecaphonic Rows
Summary
This section concludes the discussion of local classification with a presentation of the variety of dodecaphonic series.
Linshujie Zheng, Guerino Mazzola

Global Classification

Frontmatter
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.
Linshujie Zheng, Guerino Mazzola
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
Linshujie Zheng, Guerino Mazzola
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 \( \textit{Im} (|g_{i}|)^{\ast} \subset |\vartriangle_{i}|\ast \) 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.
Linshujie Zheng, Guerino Mazzola
20. Singular Homology of Hypergestures
Summary
A preliminary investigation of hypergestural classification is the introduction of singular homology and its signification for counterpoint.
Linshujie Zheng, Guerino Mazzola
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 \( \mathbb{R} \)-vector spaces, we present a method to envisage and classify gestures and local compositions on a common ground.
Linshujie Zheng, Guerino Mazzola

Classification and Creativity

Frontmatter
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® components for rhythmical and harmonic analysis. This chapter gives some perspectives relating to future directions in theory and software, especially regarding classification in performance.
Linshujie Zheng, Guerino Mazzola

References, Index

Frontmatter
23. Classification Lists
Summary
This chapter displays lists of classified objects.
Linshujie Zheng, Guerino Mazzola
Backmatter
Metadaten
Titel
Classification of Musical Objects for Analysis and Composition
verfasst von
Linshujie Zheng
Guerino Mazzola
Copyright-Jahr
2023
Electronic ISBN
978-3-031-30183-4
Print ISBN
978-3-031-30182-7
DOI
https://doi.org/10.1007/978-3-031-30183-4

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