Musical Variation. Toward a Transformational Perspective 9783031314506, 9783031314513

Table of contents :
Preface
Acknowledgements
References
Contents
List of Figures
List of Tables
Part I Decontextualized Variation
1 Basic Concepts
1.1 Derivative Work
1.2 Derivative Impact and Relations of Similarity
1.3 Derivative Space
1.4 Levels of Derivative Work
References
2 Decomposable Variation
2.1 Phases of Decomposable Variation
2.2 Classifications of Decomposable Variation
2.3 Attributes
2.3.1 Pitch
2.3.2 Time
2.3.3 Harmony
2.4 Matrix of Attributes
References
3 Measurement of Similarity (I)
3.1 General Information
3.2 Measurement of Similarity in the Pitch Domain
3.3 Measurement of Similarity in the Temporal Domain
3.4 Measurement of Similarity in the Harmonic Domain
3.5 Global Similarity Between Musical Ideas
3.6 Spatial Representation of Similarity Relations
References
4 Transformational Operations
4.1 A Set-Functional Conception
4.2 Classifications of Operations
4.2.1 Types
4.2.2 Domains of Application
4.2.3 Scope of the Argument
4.2.4 Complementary Symbology
4.2.5 Alteration of Cardinality
4.2.6 Collateral Effects
4.3 Descriptions
4.3.1 Addition
4.3.2 Augmentation
4.3.3 Change of Mode
4.3.4 Change of Register
4.3.5 Chordal Inversion
4.3.6 Chromatic Alteration
4.3.7 Chromatic Inversion
4.3.8 Chromatic Transposition
4.3.9 Contextual Dyadic Transformation
4.3.10 Deletion
4.3.11 Diatonic Inversion
4.3.12 Diatonic Transposition
4.3.13 Diminution
4.3.14 Extension
4.3.15 Interpolation
4.3.16 Merging of Durations
4.3.17 Metric Displacement
4.3.18 Permutation
4.3.19 Re-harmonization
4.3.20 Re-partition
4.3.21 Replication
4.3.22 Rest Substitution
4.3.23 Retrogradation
4.3.24 Rotation
4.3.25 Split Duration
4.3.26 Subtraction
4.3.27 Suppression
4.4 Composition of Operators
4.5 Summary
4.6 Transformational-Derivative Analysis
4.6.1 Case 1: Simple/Pure Decomposable Variation
4.6.2 Case 2: Compound/Pure Decomposable Variation
4.6.3 Case 3: Simple/Hybrid Decomposable Variation
4.6.4 Case 4: Compound/Hybrid Decomposable Variation
References
5 Measurement of Similarity (II)
5.1 Similarity Between Non-compatible Cardinalities
5.1.1 Example 1
5.1.2 Example 2
5.1.3 Example 3
5.2 Some Concluding Remarks
Part II Variation on Time
Reference
6 Grundgestalt
6.1 Grundgestalt-Components
6.2 Variables
6.3 Orders of Grundgestalten
6.4 Identification of a Grundgestalt
References
7 Developing Variation
7.1 A Transformational Typology of Developing Variation
7.2 Relative and Absolute Developing Variation
7.3 Genealogical Notation
7.4 Derivation of Variables
7.5 Quotient
7.6 Crossover
7.7 Inter-Attribute Equivalence
7.8 Thematic Transformation, Linkage, and Metric Displacement
7.9 Speciation
7.10 Teleology
7.11 Involution
References
Part III Analysis: Brahms—Intermezzo in A Major Op. 118/2
References
8 Formal, Harmonic, and Metric Structure
8.1 Form
8.2 High-Level Harmony (Tonal Relations)
8.3 Low-Level Harmony (Chord Progressions)
8.4 Metric Structure
8.4.1 Segmentation
8.4.2 Metric Organization of Passages I–IX
References
9 Derivative Analysis
9.1 Methodology
9.2 Grundgestalt, Variables, and Segmentation
9.3 Derivative Segment 1
9.3.1 Derivation of uA 02
9.3.2 Derivation of uA 03
9.3.3 Derivation of uA 05
9.3.4 Derivation of uA 06
9.3.5 Derivation of uA 07
9.3.6 Derivation of uA 08
9.3.7 Derivation of uA 10
9.3.8 Derivation of uA 13
9.3.9 Derivation of uA 14
9.3.10 Derivation of uA 15
9.3.11 Derivation of uA 16
9.3.12 Overview of Segment 1
9.4 Derivative Segment 2
9.4.1 Derivation of uA 17
9.4.2 Derivation of uA 18
9.4.3 Derivation of uA 19
9.4.4 Derivation of uA 21
9.4.5 Derivation of uA 22
9.4.6 Derivation of uA 23
9.4.7 Derivation of uA 24
9.4.8 Overview of Segment 2
9.5 Derivative Segment 3
9.5.1 Derivation of uA 25
9.5.2 Derivation of uA 26
9.5.3 Derivation of uA 28
9.5.4 Derivation of uA 29
9.5.5 Derivation of uAs 31 and 32
9.5.6 Derivation of uAs 33 and 34
9.5.7 Derivation of uAs 36
9.5.8 Overview of Segment 3
9.6 Derivative Segment 4
9.6.1 Derivation of uA 37
9.6.2 Derivation of uA 41
9.6.3 Derivation of uA 42
9.7 Derivative Segment 5
9.7.1 Derivation of uA 45
9.7.2 Derivation of uAs 46 and 48
9.7.3 Derivation of uAs 47, 49, and 50
9.7.4 Derivation of uA 51
9.7.5 Derivation of uAs 52 and 54
9.7.6 Derivation of uAs 53 and 55
9.7.7 Overview of Segment 5
9.8 Derivative Segment 6
9.8.1 Derivation of uA 56
9.8.2 Derivation of uAs 57, 58, and 59
9.8.3 Derivation of uA 60
9.8.4 Derivation of uAs 61 and 62
9.8.5 Overview of Segment 6
9.9 Derivative Segment 7
9.9.1 Derivation of uA 63
9.9.2 Derivation of uA 64
9.9.3 Derivation of uA 65
9.9.4 Derivation of uA 67
9.9.5 Derivation of uA 68
9.9.6 Derivation of uA 69
9.9.7 Derivation of uA 71
9.9.8 Overview of Segment 7
9.10 Segment 8
9.11 Discussion
9.11.1 A Qualitative Perspective of the Variants Production
9.11.2 High-Level Variants
9.11.3 Developing Variation
9.11.4 Low-Level Variants
9.11.5 The Derivative Role of the Kopfnote
References
Afterword
Afterword
A Variation in Non-tonal Contexts
B MDA*
Pitch Filtering
Rhythmic Filtering
Variation in MDA*
C Algorithms

Citation preview

Computational Music Science

Carlos de Lemos Almada

Musical Variation

Toward a Transformational Perspective

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, KarlFranzens-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 peerreview process. THE SERIES IS INDEXED IN SCOPUS

Carlos de Lemos Almada

Musical Variation Toward a Transformational Perspective

Carlos de Lemos Almada School of Music Federal University of Rio de Janeiro Rio de Janeiro, Brazil

ISSN 1868-0305 ISSN 1868-0313 (electronic) Computational Music Science ISBN 978-3-031-31450-6 ISBN 978-3-031-31451-3 (eBook) https://doi.org/10.1007/978-3-031-31451-3 © 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

This book is dedicated to my dear sister Rosana

Preface

This book results from a broad research on musical variation intended basically to systematize the analysis of organically constructed musical pieces. Since 2011, an analytical model, entitled Model of Derivative Analysis (identified by the acronym MDA), has been developed through the elaboration of original premises, concepts, symbology, and the incorporation of methodological tools and strategies. The practical application of MDA in a number of analytical studies since then has led to improvements and route corrections, aiming at the ideals of formalization, efficiency, and comprehensiveness.1 The research’s main theoretical framework is centred on the principles of Grundgestalt (normally translated as “basic shape”) and developing variation, both created by Austrian composer Arnold Schoenberg (1874–1951), associated with an organicist conception of musical creation based on a gradual derivation and concentrated economy of means. These concepts are among the most powerful and far-reaching of Schoenberg’s formulations on the fields of composition and analysis, forming a theoretical complex that has become an attractive academic subject in the last decades.2 Variation is certainly a central and pervasive aspect of human life. In a broad sense, all around us derives from something. The use of variation allows us to formulate original thoughts from others, to improve methods and game (or war) strategies, to draw conclusions and to make decisions based on previous experiences, to plan work and domestic routines, to project new versions of tools, to generalize situations from particular cases, to ameliorate medicine formulas,

1 For some theoretical formulations and analytical applications of the model, see Mayr (2018), Mayr and Almada (2016, 2017a, 2017b), Almada (2011, 2013a, 2013b, 2016a, 2021). For some studies directed to a compositional approach, see Almada (2015a, 2015b, 2016b, 2016c, 2017). 2 Since after Schoenberg’s death in 1951, a large number of theorists has been contributed to the expansion of Grundgestalt-Developing Variation theory, like Joseph Rufer (1952/1954), David Epstein (1980), Patricia Carpenter (1983), Walter Frisch (1984), Severine Neff (1984), Michael Schiano (1992), Jack Boss (1991, 1992, 2014), Ethan Haimo (1990, 1997), Stephen Collison (1994), Yuet Ng (2005), and Brent Auerbach (2005), among others.

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to enhance technologies, to add personal comments about some subject during a conversation, to obtain more efficient versions of procedures, to create new cooking receipts by changing the proportion of ingredients, and so on.3 A problematic issue normally faced by analysts concerns how to describe and label processes of developing variation and their outcomes (i.e., the variants created along) in a sufficiently precise, concise, and systematic manner. This difficulty is due mainly to the indefinite ways in which variation techniques can creatively be used by a composer, considering not only application of “canonic” operations (like inversion, augmentation, etc.) but also hybrid types or even purely idiosyncratic forms of transformation, fruit of contextual situations and of the composer’s own invention. On the other hand, there exists a considerable, inherent margin of subjectivity in the task of the interpretation of derivative relations, a quite complicating factor. The main motivation for the creation of MDA was precisely the search for minimizing the subjectivity of thematic-motivic analysis through the elaboration of a system sufficiently provided with a consistent conceptual corpus and methodological tools for this task, resulting in a process that has been gradually consolidated along the last years. The current version of the model proposes to correlate the notion of variation to some basic principles of the transformational theory. Under this perspective, variation can be considered as a special action which, when applied to a given object (as a musical motive), produces a transformed but somewhat related version of that object/motive. This lies in accordance with Steve Rings, for whom the emphasis in transformations “is on the relationships between musical entities, not on the entities as isolated monads. Transformational theory thematizes such relationships and seeks to sensitize the analyst to them” (Rings, 2011, p. 10, italics in the original). By exploring this new approach, I am mainly interested in a profound investigation of two aspects: (1) the nature of the relationships between a musical referential idea and its possible transformed versions (aiming specifically at similarity relations); and (2) the manners in which these transformations can be implemented, leading to a process of formalization inspired by the ideas of David Lewin (1987) and some of his followers.4 This new transformational-derivative model, whose theory is described in this book, is essentially dedicated to the analytical examination of tonal music, although 3 In this regard, in an essay entitled “Aesthetics and Cuisine: Mind over Matter” (inserted in the book Beauty and Brain) Elisabeth Rosin introduces the concept of variation as a paramount factor for the development of culinary inside a cultural context: “It seems to function primarily as an aesthetic principle, as a way of providing relief from monotony and the pleasure of novelty or the unexpected within a familiar framework. It is one of the clearest ways that culinary traditions expand, enlarge, and elaborate themselves” (Rosin, 1988, p. 318). These reflections seem to be a perfect example of the importance of variation in our lives. 4 As, for example, Edward Gollin (2000), Darin Hoskinson (2006), and Steven Rings (2011). To these works I would add the thesis written by Michael Schiano (1992), which though not explicitly associated to this theoretical framework, clearly examines the binomial Grundgestalt/developing variation under a transformational perspective, bringing many affinity relations with the present proposal.

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it could perfectly be applied to other harmonic languages, with minimal, punctual adjustments (a simple adaptation of the model to non-tonal contexts is briefly suggested in Appendix A). While formalization and systematization of the analytical procedures are among the most important goals of this book, I tried as much as possible, however, to prevent the text from becoming overly hermetic and complicated, by avoiding the overuse of mathematical jargon. After all, this is a book written by a musician and destined for musicians. In fact, although in some parts of the book math concepts (from trigonometry, algebra, geometry, groups, etc.) are eventually evoked and used, this is always done with very practical purposes, and in the most possible simple manner. With this, I am not meaning that musical variation cannot be studied with strict and rigorous mathematical formalization, not at all.5 Actually, apart from the fact that I do not consider myself capable of producing such formalization (due to the lack of the necessary theoretical knowledge), my intuition told me that the particular narrative that I planned for the book and the subjects it contains requires a simpler (and maybe excessively naive) approach. The structure of this book is segmented into three large parts. The first is dedicated to the presentation of the basic concepts and assumptions related to the notion of musical variation considered in the present conception, which corresponds, essentially, to what could be called “static”, or abstract variation. I basically mean with this to examine the subject “variation” by isolating it from any larger context or using a term coined by French-Greek composer Iannis Xenakis, treating it “out of the time”. Under this perspective, variation is seen as a unique process, unrelated to others, a strategy that aims to focus only on the mechanisms involved in the transformational phenomenon, disregarding any temporal connection and, therefore, decreasing analytical complexity. Chapter 1 presents the most elementary concepts that ground the theory, including a formal definition of variation and a spatial representation for the transformational process. Chapter 2 introduces the central notion of decomposable variation and of its main building blocks, the domains, and attributes. The description of a methodology for measurement of relations of similarity between musical ideas with the same cardinality is the subject of Chap. 3. Chapter 4 addresses the specific agents in the process of variation, the transformational operations. The operations are used in Chap. 5 as main support for an extension of the approach on similarity measurement, this time considering more complex situations, in which referential and derived ideas have different cardinalities. Part II addresses variation on a temporal perspective, focusing properly on Schoenbergian principles of Grundgestalt (Chap. 6) and developing variation (Chap. 7), including also a number of new concepts, complementary to those presented in Part I.

5 A remarkable example—among others—of how mathematics can be precisely applied to investigate and to formalize motivic theory can be found in the chapter “Motif Gestalt”, in the seminal book Topos in Music, written by Guerino Mazzola (2002, pp. 465–498).

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Part III is entirely dedicated to an analytical application of the theoretical fundamentals and methodological strategies described in the previous parts, with an in-depth investigation of derivative relations present in Brahms’s Intermezzo in A major for piano, Op. 118, no. 2. Chapter 8 examines the formal, harmonic, and metric structures of the piece, preparing the terrain for the derivative analysis of its motivic-thematic content, which is properly addressed in Chap. 9. In the realm of living beings, variation reaches maybe the highest level of importance. As conceived by Charles Darwin, it is the motor that propels evolution, as largely discussed in his The Origins of Species, published in 1859.6 Modern genetics and evolutionary biology7 have deepened even more Darwin’s considerations, through discoveries concerning intracellular mutations and the biochemical mechanisms that produce genetic variance, the low-level elements that aid to explain the emergence of new species from others, as well as their gradual divergence toward new genera, families, orders, classes, and so on.8 Variation is, evidently, also strongly present in Arts: painting, sculpture, architecture, literature, poetry, and, of course, music. At least considering what was documented in early counterpoint treatises and scores, the technology of variation (associated with one of its byproducts, the imitative techniques) is one of the most important factors for explaining the relatively fast expansion and sophistication of musical composition during the Ars Nova and Renaissance. Since then, variation has been turned a vital strategy for musical construction and structuring in both small and large scale, as it can be demonstrated in the forms of the ricercare, fugue, passacaglia and chacona, chorale prelude, theme and variations, opera, as well as in the developmental section of a sonata, jazz and choro improvisations, film and TV music, drums’ grooves, etc. Despite taken for granted by musicians, as a sort of “natural law”, variation is not so easily defined. Arnold Schoenberg—a name very present in this book—is perhaps the theorist who more profoundly studied the concept. As a matter of fact, he also employed largely variation in his compositional practice, throughout his tonal, atonal, and serial creative phases. Indeed, the common factor that connects all his highly diversified compositional work is certainly the priority given to motivicthematic treatment, in other words, to the use of variation as main propelling means for composition. As it will be constantly commented in the book, Schoenberg is perhaps who took the art of variation to the highest point, both compositionally and, especially, theoretically, with the elaboration of the aforementioned principles of Grundgestalt and developing variation.

6 As considered by Ernst Mayr, “The availability of variation is the indispensable prerequisite of evolution, and the study of the nature of variation is therefore a most important part of the study of evolution” (Mayr, 2002, p. 96). 7 Associated with geological discoveries about the real age of life on Earth. 8 The metaphorical connections between musical and biological variations are very powerful and will be explored in several points of the book, including in the adoption of terminology and conceptualization.

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Not surprisingly, one of the best definitions of variation, in my opinion, was coined by Schoenberg, who said that Variation is that kind of repetition which changes some of the features of a unit, motif, phrase, segment, section, or a larger part, but preserves others. To change everything would prevent there being any repetition at all, and thus might cause incoherence (Schoenberg, 1942, p. 15).

Here I propose an alternative (although by no means divergent) perspective for musical variation (and why not, a variation of the definition formulated by Schoenberg!), as a special type of action or “force” that, when applied to an object, is capable of transforming it into another object in such a manner that the latter maintains with the former some relation (in any degree) of similarity.

Acknowledgements The idea of this book, the complete planning of its structure, as well as most of the text result from a compact, three-month sabbatical leave, at the Music Department at Columbia University, in New York City. During this short period, I, fortunately, managed to concentrate a great portion of my days in an almost full-time routine of readings, researches, and intense writings, mostly inside the very well-equipped Music and Arts Library of Columbia. I am particularly very grateful to Dr. Walter Frisch, who so kindly sponsored my proposal to become a visiting scholar in his department. Meeting him, who I have known only from his wonderful, very important books, represented both an enormous honour and a great pleasure. Our talks about variation, Schoenberg, composition, popular songs, Brahms, opera, as well as not musical subjects (like museums and parks of New York, for example) were very nice and inspiring. I am also very grateful to my beloved Desirée Mayr, whose important research about developing variation in violin sonatas by Brahms and Brazilian Romantic composer Leopoldo Miguéz (1850–1902) has motivated me to initiate this project. I would like to thank to Springer Editor Robinson dos Santos, for all the support he gave me regarding the editorial process. All my gratitude also goes to Dr. Jack Boss for his careful and thorough reading of the manuscript and for his complimentary and encouraging words about the study. Rio de Janeiro, Brazil

Carlos de Lemos Almada

References Almada, C. (2011). A variação progressiva aplicada na geração de ideias temáticas. In: Proceedings of the 2nd International Symposium of Musicology (pp. 79–90). Rio de Janeiro: Federal University of Rio de Janeiro.

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Almada, C. (2013a). Considerações sobre a análise de Grundgestalt aplicada à música popular. Permusi, 29, 117–124. Almada, C. (2013b). Simbologia e hereditariedade na formação de uma Grundgestalt: a primeira das Quatro Canções op. 2 de Berg. Permusi, 27, 75–88. Almada, C. (2015a). Genetic algorithms based on the principles of Grundgestalt and developing variation. In: T. Collins, D. Meredith, & Volk, A. (Eds.), Mathematics and computation in music (pp. 42–51). New York: Springer Verlag. Almada, C. (2015b). Evolution in musical contexts: The software DARWIN. In: Proceedings of the 25th national conference of Brazilian Association of PostGraduation Research ANPPOM. Vitória: Federal University of Espírito Santo. Almada, C. (2016a). Derivative analysis and serial music: The theme of Schoenberg’s Orchestral Variations op. 31. Permusi, 33, 1–24. Almada, C. (2016b). Evolutionary variation applied to the composition of CTG, for woodwind trio. MusMat, 1(1), 1–14. Almada, C. (2016c). Artificial selection strategies implementation in a model for musical variation. Musica Theorica, 1(1), 1–15. Almada, C. (2017). Gödel-vector and Gödel-address as tools for genealogical determination of genetically-produced musical variants. In: G. Pareyon, S. Romero, & O. Agustin-Aquino (Eds.), The musical-mathematical mind: patterns and transformations (pp. 9–16). Cham: Springer. Almada, C. (2021). Correlations between musical and biological variation in derivative analysis. In: I.D. Khannanov, & R. Ruditsa (Eds.), Proceedings of the worldwide music conference 2021. WWMC 2021. Current research in systematic musicology, vol 247. Cham: Springer. https://doi.org/10.1007/978-3-030-858865_5. Auerbach, B. (2005). The analytical Grundgestalt: A new model and methodology based on the music of Johannes Brahms. Dissertation, University of Rochester. Boss, J. (1991). An analogue to developing variation in a late atonal song of Arnold Schoenberg. Dissertation, Yale University. Boss, J. (1992). Schoenberg’s op. 22 radio talk and developing variation in atonal music. Music Theory Spectrum, 14(2), 125–149. Boss, J. (2014). Schoenberg’s twelve-tone music: Symmetry and the musical idea. Boston: Cambridge University Press. Carpenter, P. (1983). Grundgestalt as tonal function. Music Theory Spectrum, 5, 15–38. Collison, S. (1994). Grundgestalt, developing variation, and motivic processes in the music of Arnold Schoenberg: An analytical study of the string quartets. Dissertation, King’s College. Epstein, D. (1980). Beyond Orpheus: Studies in music structure. Cambridge: The MIT Press. Frisch, W. (1984). Brahms and the principle of developing variation. Los Angeles: University of California Press. Gollin, E. (2000). Representations of space and conceptions of distance in transformational music theories. Dissertation, Harvard University.

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Haimo, E. (1990). Schoenberg’s serial odyssey: The evolution of his twelve-tone method 1914-1928. Oxford: Clarendon Press. Haimo, E. (1997). Developing variation and Schoenberg’s serial music. Music Analysis, 16(3), 349–365. Hoskinson, D. (2006). The Grundgestalt and network transformations in the late choral works of Anton Webern. Dissertation, University of Oregon. Lewin, D. (1987). Generalized musical intervals and transformations. New Haven: Yale University Press. Mayr, D. (2018). The Identification of developing variation in Johannes Brahms op. 78 and Leopoldo Miguéz op. 14 violin sonatas through derivative analysis. Dissertation, Federal University of Rio de Janeiro. Mayr, D., & Almada, C. (2016). Use of linkage technique in Johannes Brahms’ op.78 and Leopoldo Miguéz’s op.14 violin sonatas. Opus, 22(2), 429–449. Mayr, D., & Almada, C. (2017a). Geometric and vector representation of metric relations. In: Proceedings of the 2nd National Conference of Brazilian Association of Theory and Analysis TEMA (pp. 10–19). Florianópolis: State University of Santa Catarina. Mayr, D., & Almada, C. (2017b). Correlations between developing variation and genetic processes in the analysis of Brahms’ violin sonata op.78. In: Abstracts of the 9th European Congress of Musical Analysis EUROMAC. Strasbourg: Strasbourg University. Mayr, E. (2002). What evolution is? London: Phoenix. Mazzola, G. (2002). The topos of music: Geometric logic of concepts, theory, and performance. Basel: Birkhauser. Neff, S. (1984). Aspects of Grundgestalt in Schoenberg’s first string quartet, op.7. Journal of the Music Theory Society, 9(1–2), 7–56. Ng, Y. (2005). A Grundgestalt interpretation of metric dissonance in the music of Brahms. Dissertation, University of Rochester. Rings, S. (2011). Tonality and transformation. Oxford: Oxford University Press. Rosin, E. (1988). Aesthetics and cuisine: Mind over matter. In: I. Rentschler, B. Herzherger, & B. Epstein (Eds.), Beauty and the brain: Biological aspects of aesthetics (pp. 315–324). Berlin: Birkhäuser. Rufer, J. (1952). Die Komposition mit Zwölf Tönen, Max Hesses, Berlin. English edition: (1954) Composition with twelve notes related only to one another. Searle, H. (trad). Rocklife, London. Schiano, M. (1992). Arnold Schoenberg’s Grundgestalt and its influence. Dissertation, Brandeis University. Schoenberg, A. (1942). Models for beginners in composition. New York: Schirmer.

Contents

Part I Decontextualized Variation 1

Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Derivative Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Derivative Impact and Relations of Similarity . . . . . . . . . . . . . . . . . . . . . . . 1.3 Derivative Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Levels of Derivative Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4 7 8 10 11

2

Decomposable Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Phases of Decomposable Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Classifications of Decomposable Variation. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Harmony. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Matrix of Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 14 16 20 20 26 29 30 33

3

Measurement of Similarity (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Measurement of Similarity in the Pitch Domain . . . . . . . . . . . . . . . . . . . . . 3.3 Measurement of Similarity in the Temporal Domain . . . . . . . . . . . . . . . . 3.4 Measurement of Similarity in the Harmonic Domain. . . . . . . . . . . . . . . . 3.5 Global Similarity Between Musical Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Spatial Representation of Similarity Relations . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 36 38 45 50 52 61 63

4

Transformational Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 A Set-Functional Conception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Classifications of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Domains of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 66 67 67 68 xv

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4.2.3 Scope of the Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.4 Complementary Symbology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.5 Alteration of Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.6 Collateral Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3.2 Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.3 Change of Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.4 Change of Register . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.5 Chordal Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3.6 Chromatic Alteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3.7 Chromatic Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3.8 Chromatic Transposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3.9 Contextual Dyadic Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3.10 Deletion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.11 Diatonic Inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.12 Diatonic Transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3.13 Diminution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3.14 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3.15 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3.16 Merging of Durations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3.17 Metric Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3.18 Permutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3.19 Re-harmonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.20 Re-partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3.21 Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3.22 Rest Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.23 Retrogradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.24 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3.25 Split Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.26 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3.27 Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Composition of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.6 Transformational-Derivative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.6.1 Case 1: Simple/Pure Decomposable Variation . . . . . . . . . . . . . 97 4.6.2 Case 2: Compound/Pure Decomposable Variation . . . . . . . . . 98 4.6.3 Case 3: Simple/Hybrid Decomposable Variation . . . . . . . . . . . 99 4.6.4 Case 4: Compound/Hybrid Decomposable Variation . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5

Measurement of Similarity (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Similarity Between Non-compatible Cardinalities . . . . . . . . . . . . . . . . . . . 5.1.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 104 106

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5.1.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Some Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Part II Variation on Time Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6 Grundgestalt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Grundgestalt-Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Orders of Grundgestalten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Identification of a Grundgestalt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 118 119 120 121 123

7

125 126 128 130 132 133 136 137 138 143 143 146 148

Developing Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 A Transformational Typology of Developing Variation . . . . . . . . . . . . . 7.2 Relative and Absolute Developing Variation . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Genealogical Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Derivation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Crossover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Inter-Attribute Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Thematic Transformation, Linkage, and Metric Displacement. . . . . . 7.9 Speciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Teleology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part III Analysis: Brahms—Intermezzo in A Major Op. 118/2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8

Formal, Harmonic, and Metric Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 High-Level Harmony (Tonal Relations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Low-Level Harmony (Chord Progressions) . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Metric Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Metric Organization of Passages I–IX . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 156 158 165 166 167 176

9

Derivative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Grundgestalt, Variables, and Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Derivative Segment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Derivation of uA 02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Derivation of uA 03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Derivation of uA 05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Derivation of uA 06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 179 180 182 187 187 188 190

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9.5

9.6

9.7

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9.3.5 Derivation of uA 07 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.6 Derivation of uA 08 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.7 Derivation of uA 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.8 Derivation of uA 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.9 Derivation of uA 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.10 Derivation of uA 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.11 Derivation of uA 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.12 Overview of Segment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative Segment 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Derivation of uA 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Derivation of uA 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Derivation of uA 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Derivation of uA 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.5 Derivation of uA 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.6 Derivation of uA 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.7 Derivation of uA 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.8 Overview of Segment 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative Segment 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Derivation of uA 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Derivation of uA 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Derivation of uA 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 Derivation of uA 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.5 Derivation of uAs 31 and 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.6 Derivation of uAs 33 and 34. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.7 Derivation of uAs 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.8 Overview of Segment 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative Segment 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Derivation of uA 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Derivation of uA 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Derivation of uA 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative Segment 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Derivation of uA 45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2 Derivation of uAs 46 and 48. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.3 Derivation of uAs 47, 49, and 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.4 Derivation of uA 51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.5 Derivation of uAs 52 and 54. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.6 Derivation of uAs 53 and 55. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.7 Overview of Segment 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative Segment 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.1 Derivation of uA 56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.2 Derivation of uAs 57, 58, and 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.3 Derivation of uA 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.4 Derivation of uAs 61 and 62. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.5 Overview of Segment 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 194 196 197 198 200 202 202 211 213 214 215 217 217 217 219 219 223 227 227 228 228 229 230 231 232 234 235 235 236 237 237 239 242 245 247 248 250 250 252 253 254 254 255

Contents

9.9

Derivative Segment 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.1 Derivation of uA 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.2 Derivation of uA 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.3 Derivation of uA 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.4 Derivation of uA 67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.5 Derivation of uA 68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.6 Derivation of uA 69 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.7 Derivation of uA 71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.8 Overview of Segment 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Segment 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.1 A Qualitative Perspective of the Variants Production . . . . . . 9.11.2 High-Level Variants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.3 Developing Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.4 Low-Level Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.5 The Derivative Role of the Kopfnote . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

256 256 257 258 260 261 262 264 266 267 269 269 271 272 274 274 284

Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 A Variation in Non-tonal Contexts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 B MDA* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 C Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. Fig. Fig. Fig. Fig.

1.4 1.5 1.6 1.7 1.8

Fig. 1.9 Fig. 1.10 Fig. 2.1 Fig. 2.2

Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6

Fig. 2.7 Fig. 2.8

Example of a UDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of the variation space of UDS P (initial conception) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of the variation space of UDS P (second version) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of the action of function V in set S . . . . . . . . . . . . . . . Schematic representation of a derivative work . . . . . . . . . . . . . . . . . . . . Relation between P and C plotted as a system of vectors . . . . . . . . . Derivative space of referential idea P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative space of P, considering a point C positioned on edge QR at coordinates (x, x − 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of the bands of similarity on the derivative space of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of holistic variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of decomposable variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of sets S, Sp , St , and Sh , and the action of functions of abstraction abs-p, abs-t, and abs-h, producing abstracted parents Pp, Pt, and Ph . . . . . . . . . . . . . . . . . . . . . . . Representation of the actions of generic functions Vp , Vt , and Vh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of the action of function rnt . . . . . . . . . . . . . . . . . . . . . . . General representation of an abstract derivative work . . . . . . . . . . . . Schematic representations of possible cases of variation, presented as examples: pure/simple decomposable variation (a); pure/compound (b); hybrid/simple (c); hybrid/compound (d); and holistic variation (e) . . . . . . . . . . . . . . . . . . Three possible derivative trajectories for the abstract derivative work between P and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pitch attributes of a referential UDS P. Angled brackets indicate ordered content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5 5 6 6 7 8 9 10 11 14

15 16 16 17

18 19 20

xxi

xxii

List of Figures

Fig. 2.9 Fig. 2.10 Fig. 2.11 Fig. 2.12

Fig. 2.13 Fig. 2.14 Fig. 2.15 Fig. 2.16 Fig. 2.17

Fig. 2.18

Fig. Fig. Fig. Fig.

2.19 2.20 2.21 3.1

Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8

Representation of pitch space, considering midi-pitch numbering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of pitch-class space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of the melodic contour of Fig. 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Five archetypal pitch variants from P: by octave displacements (C1), chromatic transposition (C2), literal inversion (C3), free change of notes, but keeping the original melodic contour (C4), and free change of contour, but keeping the original ambit (C5). Mismatches are indicated by underlined numbers, and corresponding attributes are inserted in rectangles . . . . . . . . . . . . . . . . Temporal attributes of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of IOI equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two distinct local metric contexts for the rhythmic idea of Fig. 2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphic representation of the metric contour of Fig. 2.13 . . . . . . . . Four archetypal temporal variants from P: with rests inserted between some onset points (C1), metric displacement (C2), augmentation (C3), and diminution (C4). Mismatches are indicated by underlined numbers, and corresponding attributes are inserted in rectangles . . . . . . . . . . . Four archetypal harmonic variants from P: chromatic modulation (C1), diatonic chordal change (C2), mode interchange (C3), and inversion (C4). Mismatches are indicated by underlined numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the matrix of attributes of P . . . . . . . . . . . . . . . . . . . . . . . . . . Isolation of the three sections of matrix M . . . . . . . . . . . . . . . . . . . . . . . . Examples of identification of matrix subsets . . . . . . . . . . . . . . . . . . . . . . Representation of edit-distance method, after Toussaint (2013, p. 252) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General scheme of measurement of similarity between elements of P and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two contrasting pitch configurations, used for the calculation of a possible maximal pitch penalty (kpmax ) . . . . . . . . . Calculation of pitch penalty in the comparison between Pp and C1p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of pitch penalty in the comparison between Pp and C2p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of pitch penalty in the comparison between Pp and C3p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of pitch penalty in the comparison between Pp and C4p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of pitch penalty in the comparison between Pp and C5p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 24

25 26 26 27 28

29

31 32 32 33 36 37 40 41 43 44 45 46

List of Figures

Fig. 3.9

Fig. 3.10 Fig. 3.11 Fig. 3.12 Fig. 3.13 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 4.1 4.2 4.3 4.4

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12

Fig. 4.13 Fig. 4.14 Fig. 4.15 Fig. 4.16 Fig. Fig. Fig. Fig. Fig. Fig.

4.17 4.18 4.19 4.20 4.21 4.22

xxiii

Two contrasting temporal configurations, used for the calculation of a possible maximal temporal penalty (ktmax ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of pitch penalty in the comparison between Pt and C1t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of pitch penalty in the comparison between Pt and C2t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of pitch penalty in the comparison between Pt and C3t and Pt and C4t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of harmonic penalties in the comparison between Ph and C1h, C2h, C3h, and C4h . . . . . . . . . . . . . . . . . . . . . . . . . Determination of global penalty for variant C1 . . . . . . . . . . . . . . . . . . . Determination of global penalty for variant C2 . . . . . . . . . . . . . . . . . . . Determination of global penalty for variant C3 . . . . . . . . . . . . . . . . . . . Determination of global penalty for variant C4 . . . . . . . . . . . . . . . . . . . Determination of global penalty for variant C5 . . . . . . . . . . . . . . . . . . . Determination of global penalty for variant C6 . . . . . . . . . . . . . . . . . . . Model of derivative space (adapting Fig. 1.8) . . . . . . . . . . . . . . . . . . . . . Points C1–6 plotted in the derivative space of P . . . . . . . . . . . . . . . . . . Representation of the action of function υ on set Sx . . . . . . . . . . . . . . Set υ of operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Actions of functions υ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of the action of normal (a) and mutational operations (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of application of operator addition . . . . . . . . . . . . . . . . . . . . . Example of application of operator augmentation . . . . . . . . . . . . . . . . Example of application of operator change of mode. . . . . . . . . . . . . . . Example of application of operator change of register . . . . . . . . . . . Example of application of operator chordal inversion . . . . . . . . . . . . Example of application of operator chromatic alteration . . . . . . . . . Example of application of operator chromatic inversion . . . . . . . . . Example of application of operator chromatic transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of application of operator contextual dyadic transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of application of operator deletion . . . . . . . . . . . . . . . . . . . . . Example of application of operator diatonic inversion . . . . . . . . . . . Example of application of operator diatonic transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of application of operator diminution . . . . . . . . . . . . . . . . . . . Examples of application of operator extension . . . . . . . . . . . . . . . . . . . . Examples of application of operator interpolation . . . . . . . . . . . . . . . . Example of application of operator merging of durations . . . . . . . . Example of application of operator metric displacement . . . . . . . . . Examples of application of operator permutation . . . . . . . . . . . . . . . . .

48 49 50 51 53 55 56 57 58 59 60 61 62 66 66 67 69 71 72 73 73 74 75 75 76 77 78 78 79 80 81 82 82 83 84

xxiv

List of Figures

Fig. 4.23

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32

Fig. 4.33 Fig. 4.34 Fig. 4.35 Fig. 4.36 Fig. 4.37

Fig. 4.38 Fig. 4.39 Fig. 4.40 Fig. 4.41 Fig. 4.42 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. Fig. Fig. Fig.

5.7 5.8 5.9 5.10

Examples of application of operator re-harmonization, considering the three alternatives: insertion (a), deletion (b), and substitution (c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of application of operator re-partition . . . . . . . . . . . . . . . . . . Examples of application of operator replication . . . . . . . . . . . . . . . . . . Example of application of operator rest substitution . . . . . . . . . . . . . . Examples of application of operator retrogradation . . . . . . . . . . . . . . Examples of application of operator rotation . . . . . . . . . . . . . . . . . . . . . Example of application of operator split duration . . . . . . . . . . . . . . . . Examples of application of operator subtraction . . . . . . . . . . . . . . . . . . Examples of application of operator suppression . . . . . . . . . . . . . . . . . Examples of use of non-commutative composed operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operators distributed according to type (canonic or non-canonic), and domain of application . . . . . . . . . . . . . . . . . . . . . . . . . . Operations distributed according to scope. Operators forming the intersection of both sets are of dual scope . . . . . . . . . . . . Operations according to the capacity of altering the original cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operations that have mirrored versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chart of operations, considering type (canonic or non-canonic), domain (pitch, time, and/or harmony), and scope (normal, mutational, or dual) . . . . . . . . . . . . . . . . . . . . . . . . . . . Variant C (first case). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of TD analysis of case 1—pure/simple decomposable variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of TD analysis of case 2—compound/pure decomposable variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of TD analysis of case 3—simple/hybrid decomposable variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of TD analysis of case 4—compound/hybrid decomposable variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A variant of P with incompatible cardinality (example 1) . . . . . . . . TD analysis of variant of example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation of similarity between P and the variant of example 1 plotted in the DS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A variant of P with incompatible cardinality (example 2) . . . . . . . . TD analysis of variant of example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation of similarity between P and the variant of example 2 plotted in the DS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A variant of P with incompatible cardinality (example 3) . . . . . . . . TD analysis of variant of example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypothetical intermediary variant C’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of similarity between P and C’, according to the algorithms of Chap. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 86 86 87 88 89 89 90 91 93 94 95 95 96

96 98 99 100 101 102 104 105 106 106 107 108 108 109 110 111

List of Figures

Fig. 5.11 Fig. 5.12

Fig. 6.1

Fig. 6.2

Fig. 6.3

Fig. 6.4 Fig. 7.1

Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 7.6

Fig. 7.7 Fig. 7.8

Fig. 7.9

xxv

Variant of example 2 (a); Schenkerian analysis (b) . . . . . . . . . . . . . . . Alternative derivative analysis for example 2, considering holistic variation (harmonic contexts were omitted for simplicity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Idealized model of organic growth of musical piece through development of the elements presented in its Grundgestalt (the circle at the center). The squares (labeled with alphabetic letters) that emanate from it in concentric “waves” represent variants, appearing in successive generations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic representation of a Grundgestalt (Gr), formed by four GCs (A–D), in which are located six agents from of two distinct domains (represented by black and gray circles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic representation of a Grundgestalt (Gr), formed by four GCs (A–D), in which are located six agents from of two distinct domains (represented by black and gray circles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic example of multi-order-Grundgestalt structure in a hypothetical piece with four movements . . . . . . . . . . . Archetypal representation of a unit of developing-variation work considering steps 1 (first derivation), 2 (child-parent equivalence), and 3 (second derivation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General scheme of a nth-order developing-variation work (DVn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of a hypothetical “timeless” developing-variation process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of a hypothetical ordinary developing-variation process (first version) . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of a hypothetical ordinary developing-variation process (second version) . . . . . . . . . . . . . . . . . . . . Example of a derivative analysis of a hypothetical thematic fragment, considering variables and some possible variants (a); representation of the same theme using “genetic” description (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Four hypothetical fragments related by derivation from basic elements (variables Z, Y, X, and W) . . . . . . . . . . . . . . . . . . . . . . . . . Example of crossover of UDSs a and d of Fig. 7.7, by combination of permanent variables Z and X with temporary variable z, resulting into the hybrid UDS [a–d] . . . . . . . Example of inter-attribute equivalence, involving the replacement of p3 by p1 as referential attribute for derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

114

119

119

120 122

127 128 129 129 131

133 135

137

138

xxvi

List of Figures

Fig. 7.10

Fig. 7.11

Fig. 7.12

Fig. 7.13 Fig. 7.14 Fig. 7.15 Fig. 8.1 Fig. 8.2

Fig. 8.3 Fig. 8.4 Fig. 8.5 Fig. 8.6

Fig. 8.7 Fig. 8.8 Fig. 8.9 Fig. Fig. Fig. Fig. Fig. Fig.

8.10 8.11 8.12 8.13 8.14 8.15

Schematic representations of thematic transformation, with straight lines (a) and thematic development, with dashed lines (b). Adapted from Mayr (2018, pp. 53–54) . . . . . . . . . Schematic representations of linkage. UDS 2 is initiated with a transformation of the last element of UDS 1 (m → m’). Adapted from Mayr (2018, p. 65) . . . . . . . . . . . . . . . . . . . . . Schematic representation of a process of speciation. The derivation of a hypothetical lineage led by UDS a, whose main element is variable Z, is gradually undermined by the infiltration of variable w, until the complete substitution of Y, and the establishment of a new lineage, headed by UDS b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple example of teleology in a hypothetical process of artificial selection applied to rabbit creation . . . . . . . . . . . . . . . . . . . . Simple example of teleology in a hypothetical process of artificial selection applied to rabbit creation . . . . . . . . . . . . . . . . . . . . Representation of involution in a tree format . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—Basic formal structure, considering two levels of organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—Basic structure depicted in three levels of organization: sections (1), subsections (2), and phrases (3), associated with order numbers of the segments and respective measure numbers of entry. Gray cells indicate segments with non-regular extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—Network diagram, integrating form and tonal relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 - mm. 1–8 (reduction) . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 – mm. 17–24 (reduction) . . . . . . . Brahms’s Intermezzo Op. 118/2 – mm. 23–24 (a); hypothetical omission of an intermediary dominant stage (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 – mm. 25–28 (reduction) . . . . . . . Brahms’s Intermezzo Op. 118/2 – mm. 29–34 (reduction) . . . . . . . Brahms’s Intermezzo Op. 118/2 – mm. 35–38 (a); intervallic comparison of units of mm. 1–2 and 35–36 (b) . . . . . . . Brahms’s Intermezzo Op. 118/2 – mm. 39–48 (reduction) . . . . . . . Brahms’s Intermezzo Op. 118/2 – mm. 49–56 (reduction) . . . . . . . Brahms’s Intermezzo Op. 118/2 – mm. 57–64 (reduction) . . . . . . . Brahms’s Intermezzo Op. 118/2 – m. 60 (reduction) . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 – mm. 65–73 (reduction) . . . . . . . Brahms’s Intermezzo Op. 118/2 – mm. 73–76 (reduction) . . . . . . .

140

141

144 146 147 148 156

156 157 159 159

160 160 161 161 162 163 163 164 164 165

List of Figures

Fig. 8.16

Fig. 8.17

Fig. 8.18

Fig. 8.19 Fig. 8.20 Fig. 8.21

Fig. 8.22

Fig. 8.23 Fig. 8.24 Fig. 8.25 Fig. 8.26 Fig. 8.27 Fig. 8.28 Fig. 8.29 Fig. 8.30 Fig. 8.31 Fig. 8.32 Fig. 9.1

xxvii

Brahms’s Intermezzo Op. 118/2 (melodic line). Comparison between formal and metric segmentation, considering mm. 17–31. The “dissonance” between conflicted segments (5 and 6) and passages (II and III) is indicated by the shaded squares. Dashed lines in formal segmentation indicate upbeat beginnings. For the sake of simplicity, only the melodic line is shown . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—Passage I (mm. 1–8). Streams M (main melody) and H (harmony) configurations in relation to the notated bar lines . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 1–5)—melodic line re-written in order to accommodate an alternative metric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—passage II (mm. 17–23) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 19–24, reduction)—middleground hemiola involving M and H . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—passage III (mm. 24–30). The entrances of formal segments 6 and 7 are indicated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 28–31)—in a simplified and re-bared version, including the entrances of formal segments 7 and 8 and metric passage IV . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—passages IV (mm. 31–34) and V (mm. 35–38) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—passage VI (mm. 39–48) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 39–48)—rewritten in a simplified and re-bared version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic scheme of the foreground hemiola (right vs. left hand) in the entrance of section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—passage VII (mm. 49–56) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—passage VII (mm. 49–56) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—passage VIII (mm. 57–64) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—passage VIII (mm. 57–64), counterpoint between M and S . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—passage IX (mm. 65–76) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2—passage IX (mm. 65–76), counterpoint between M and S . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 0–1)—Grundgestalt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

168

169 169 170

170

171 171 172 172 173 173 174 175 175 175 176 180

xxviii

Fig. 9.2

Fig. 9.3 Fig. 9.4

Fig. 9.5 Fig. 9.6 Fig. 9.7 Fig. 9.8

Fig. 9.9 Fig. 9.10

Fig. 9.11 Fig. 9.12

Fig. 9.13

Fig. 9.14

Fig. 9.15 Fig. 9.16 Fig. 9.17

Fig. 9.18

List of Figures

(a) Matrix M of attributes related to Fig. 9.1; (b) pitch (left) and temporal (right) sections of matrix M, with variables assigned to attributes d-p3 (Z), 2 (Y), and t3 (X); (c) genetic representation of the Grundgestalt . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 0–8)—derivative segment 1 (first approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Units 01 and 11 in musical notation (and harmonic contexts); (b) their matrices of attributes; (c) pitch (kp), temporal (kt), harmonic (kh), and global penalties (k); (d) 11 plotted on the derivative space of 01 . . . . . . . . . . . . . . . . . . . . . . . Time-oriented transformational network related to segment 1, by including relations between uAs . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 02 from uA 01 (a); “Genetic” representation of uA 02 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 03 from uA 01 (a); “Genetic” representation of uA 03 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TD analysis of the derivation of uA 05 from uA 01, considering the intermediary hypothetical stage 01/05 (at right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 05 from uA 01 (a); “Genetic” representation of uA 05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation uA 06 from uA 02, considering variables involved (a); “Genetic” representation of uA 06 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Middle-level analysis as additional information for the derivation of uA 06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TD analysis of the derivation of uA 07 from uA 02, considering the intermediary hypothetical stage 02/07 (at right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 07 from uA 02, considering variables involved (a); “Genetic” representation of uA 07 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 08 from uA 06, considering variables involved (a); “Genetic” representation of uA 08 (b); (c) Middle-level analysis of uA 08, as related to uAs 06 and 01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 10 from uA 07, considering three basic editions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crossover derivation of uA 13, from combination of uAs 05 and 06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 13 from uA 06, considering variables involved (a); “Genetic” representation of uA 13 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possible derivation of uA 14 from uA 07 . . . . . . . . . . . . . . . . . . . . . . . . . .

181 183

184 185 187 188

189 190

191 192

193

194

195 196 197

198 199

List of Figures

Fig. 9.19

Fig. 9.20 Fig. 9.21 Fig. 9.22

Fig. 9.23

Fig. 9.24 Fig. 9.25 Fig. 9.26 Fig. 9.27 Fig. 9.28

Fig. 9.29 Fig. 9.30 Fig. Fig. Fig. Fig.

9.31 9.32 9.33 9.34

Fig. 9.35 Fig. 9.36

Fig. 9.37 Fig. 9.38

Fig. 9.39 Fig. 9.40

xxix

Low-level derivation of uA 14 from uAs 06 and 07, considering variables involved (a); “Genetic” representation of uA 14 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possible derivation of uA 15 from uA 01 . . . . . . . . . . . . . . . . . . . . . . . . . . Possible derivation of uA 15 from uA 06 . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 15 from uAs 01 and 06, considering variables involved (a); “Genetic” representation of uA 15 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 16 from uAs 01 and 08, considering variables involved (a); “Genetic” representation of uA 16 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updating of the uA network of segment 1 (based on Fig. 9.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative tree of segment 1 formatted according to genealogy established in Table 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative network of segment 1 depicting variants in musical notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative network of segment 1 depicting variants in “genetic” notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updating of the derivative network of segment 1 considering speciation (a); Distribution of the three middle-level variants of locution zx (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . DV paths in segment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lineage of a0 (DV path 4) considering absolute and relative developing variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Genealogical tree of variable Z in segment 1 . . . . . . . . . . . . . . . . . . . . . . Genealogical trees of variables Y and X in segment 1 . . . . . . . . . . . . Derivative networks related to variables Z, Y, and X . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 17–28)—derivative segment 2 (first approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-oriented transformational network related to segment 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 17 from b1.1 , considering variables involved (a); “Genetic” representation of uA 18 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TD analysis of the derivation of uA 18 from uA 17, considering the intermediary hypothetical stage 17/18 . . . . . . . . . . . Low-level derivation of uA 18 from uA 17, considering variables involved (a); “Genetic” representation of uA 18 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 19 from variant a1.2.1 . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 21 from uA 17, considering variables involved (a); “Genetic” representation of uA 21 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

200 201 201

203

204 204 205 206 206

208 209 210 211 212 212

213 213

214 215

216 216

217

xxx

Fig. 9.41 Fig. 9.42

Fig. 9.43

Fig. 9.44

Fig. 9.45 Fig. 9.46 Fig. 9.47 Fig. 9.48

Fig. 9.49 Fig. 9.50 Fig. 9.51 Fig. 9.52 Fig. 9.53 Fig. 9.54

Fig. 9.55

Fig. 9.56

Fig. 9.57

Fig. 9.58

List of Figures

TD analysis of the derivation of uA 22 from uA 21, considering the intermediary hypothetical stage 21/22 . . . . . . . . . . . Low-level derivation of uA 22 from uA 21, considering variables involved (a); “Genetic” representation of uA 22 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TD analysis of the derivation of uA 23 from crossover of uAs a0 and b0 , considering the intermediary hypothetical stage 23/[a0 +b0 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 23 from uAs a0 and b0 , considering variables involved (a); “Genetic” representation of uA 23 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 24 from uA 23 . . . . . . . . . . . . . . . . . . . . . . . . . . Updating of the uA network of segment 2 (based on Fig. 9.35) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated derivative tree (related to Fig. 9.25) considering segment 2. New variants are highlighted . . . . . . . . . . . . . . . . . . . . . . . . . . Absolute/relative DV graph considering lineage a0 → a1 → b0 → b1.1 → b13 → b13 .2 → b13 .2.1 , updating Fig. 9.30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative networks related to variables Z, Y, and X, updating Fig. 9.33. New variants are highlighted . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 29–38)—derivative segment 3 (first approach) . . . . . . . . . . . . . . Six manifestations of the relation between pitch classes B-A in segment 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative network related to the six manifestations of the block B-A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-oriented transformational network related to segment 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 25 from uA a0 , considering variables involved (a); “Genetic” representation of uA 25 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 26 from uA a2 , considering variables involved (a); “Genetic” representation of uA 25 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TD analysis of the derivation of uA 29 from crossover of uAs 27 and a0 , considering the intermediary hypothetical stage 29/[27+a0 +] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 29 from uA b0 , considering variables involved (a); “Genetic” representation of uA 29 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of collapsed uA 31.32 from collapsed uA 27.29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

218

219

220

220 221 221 222

223 224 224 225 226 226

227

228

229

229 230

List of Figures

Fig. 9.59

Fig. 9.60

Fig. 9.61

Fig. 9.62 Fig. 9.63 Fig. 9.64 Fig. 9.65 Fig. 9.66 Fig. Fig. Fig. Fig.

9.67 9.68 9.69 9.70

Fig. 9.71 Fig. 9.72 Fig. 9.73

Fig. 9.74 Fig. 9.75

Fig. 9.76 Fig. 9.77 Fig. 9.78 Fig. 9.79

Fig. 9.80

xxxi

Low-level derivation of uA 33 from uA a0 , considering variables involved (a); “Genetic” representation of uA 33 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 34 from uA a2 , considering variables involved (a); “Genetic” representation of uA 34 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 36 from uA 29, considering variables involved (a); “Genetic” representation of uA 36 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updating of the uA network of segment 3 (based on Fig. 9.53) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated derivative tree (related to Fig. 9.47) considering segment 3. New variants are highlighted . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative networks related to variables Z, Y, and X, updating Fig. 9.49. New variants are highlighted . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 39–48)—derivative segment 4 (first approach) . . . . . . . . . . . . . . Time-oriented transformational network related to segment 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 37 from b4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 41 from b13 .2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 42 from uA 41 . . . . . . . . . . . . . . . . . . . . . . . . . . Updating of the uA network of segment 4 (based on Fig. 9.66) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 49–56)—derivative segment 4 (first approach) . . . . . . . . . . . . . . Time-oriented transformational network related to segment 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 45 from variant b14 , considering variables involved (a); Middle-level interpretation of uA 45 (b); “Genetic” representation of uA 45 (c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TD analysis of the derivation of uA 46 from uA 45, considering the intermediary hypothetical stage 45/46 . . . . . . . . . . . Low-level derivation of uA 46 from uA 45, considering variables involved (a); “Genetic” representation of uA 46 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 48 from uA 46 . . . . . . . . . . . . . . . . . . . . . . . . . . Teleological derivation from uA 45 to uA 48 . . . . . . . . . . . . . . . . . . . . . TD analysis of the derivation of uA 47 from uA 45, considering the intermediary hypothetical stage 45/47 . . . . . . . . . . . Low-level derivation of uA 47 from uA 45, considering variables involved (a); “Genetic” representation of uA 47 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 49 from uA 47 . . . . . . . . . . . . . . . . . . . . . . . . . .

231

231

231 232 233 233 234 234 235 236 236 237 238 238

239 240

241 242 243 244

244 245

xxxii

List of Figures

Fig. 9.81 Fig. 9.82 Fig. 9.83

Fig. 9.84 Fig. 9.85

Fig. 9.86 Fig. 9.87

Fig. 9.88

Fig. 9.89 Fig. 9.90 Fig. 9.91 Fig. 9.92 Fig. 9.93 Fig. 9.94 Fig. Fig. Fig. Fig.

9.95 9.96 9.97 9.98

Fig. 9.99 Fig. 9.100 Fig. Fig. Fig. Fig.

9.101 9.102 9.103 9.104

Holistic derivation of uA 50 from uA 49 . . . . . . . . . . . . . . . . . . . . . . . . . . TD analysis of the derivation of uA 51 from uA 45, considering the intermediary hypothetical stage 45/51 . . . . . . . . . . . Low-level derivation of uA 51 from uA 45, considering variables involved (a); “Genetic” representation of uA 51 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TD analysis of the derivation of uA 52 from uA 51, considering the intermediary hypothetical stage 51/52 . . . . . . . . . . . Low-level derivation of uA 52 from uA 51, considering variables involved (a); “Genetic” representation of uA 52 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 54 from uA 52 . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 53 from uA 51, considering variables involved (a); “Genetic” representation of uA 53 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 55 from uA 53, considering variables involved (a); “Genetic” representation of uA 55 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updating of the uA network of segment 5 (based on Fig. 9.72) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated derivative tree of uAs (those incorporated in segment 5 are highlighted) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 57–64)—derivative segment 6 (first approach) . . . . . . . . . . . . . . Time-oriented transformational network related to segment 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 56 from variant c0 . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uAs 57 (from uA 56), 58 and 59 (from 57) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 60 from variant c3.2 . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 61 from uA 56 . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 62 from uAs 59 or 61 . . . . . . . . . . . . . . . . . . Updating of the uA network of segment 6 (based on Fig. 9.92) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brahms’s Intermezzo Op. 118/2 (mm. 65–73)—derivative segment 7 (first approach) . . . . . . . . . . . . . . Time-oriented transformational network related to segment 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 63 from variant c0 . . . . . . . . . . . . . . . . . . . . . . Holistic derivation of uA 64 from uA 63 . . . . . . . . . . . . . . . . . . . . . . . . . . Derivation of uA 65 from uA 63. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-level derivation of uA 65 from variant c2.1 , considering variables involved (a); “Genetic” representation of uA 65 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245 246

246 247

248 248

249

249 250 251 251 252 252 253 254 255 255 256 257 257 258 258 259

260

List of Figures

Fig. 9.105 TD analysis of the derivation of uA 67 from uA 65, considering the intermediary hypothetical stage 65/67 . . . . . . . . . . . Fig. 9.106 Low-level derivation of uA 67 from uA 65, considering variables involved (a); “Genetic” representation of uA 67 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.107 TD analysis of the derivation of uA 68 from uA 67, considering the intermediary hypothetical stage 67/68 . . . . . . . . . . . Fig. 9.108 Low-level derivation of uA 68 from uA 67, considering variables involved (a); “Genetic” representation of uA 68 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.109 Holistic derivation of uA 69 from uA 68 . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.110 Low-level derivation of uA 71 from uA 69, considering variables involved (a); “Genetic” representation of uA 71 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.111 Updating of the uA network of segment 7 (based on Fig. 9.100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.112 Updating of the uA network of segment 7 (based on Fig. 9.100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.113 Brahms’s Intermezzo Op. 118/2 (mm. 73–76)—derivative segment 8 (first approach) . . . . . . . . . . . . . . Fig. 9.114 Brahms’s Intermezzo Op. 118/2 (mm. 73–76)—derivative segment 8 (first approach) . . . . . . . . . . . . . . Fig. 9.115 Distribution of high- and low-level variants considering the eight derivative segments and the whole piece . . . . . . . . . . . . . . . . Fig. 9.116 Derivative curves corresponding to the production of variants in sections A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.117 Updated network of high-level variants . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.118 High-level network considering the derivative segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.119 Basic network related to the DV path a0 → c3.2.3.14 . . . . . . . . . . . . . . Fig. 9.120 Graphical representation of the relative/absolute DV path a0 → c3.2.3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.121 Low-level network considering the pool of variable Z . . . . . . . . . . . . Fig. 9.122 Genealogical tree of variable Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.123 Low-level network referred to variables Y (a) and X (b) . . . . . . . . . Fig. 9.124 Genealogical tree of variables Y (a) and X (b) . . . . . . . . . . . . . . . . . . . . Fig. 9.125 Schenkerian reduction of segment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.126 Schenkerian reduction of segment 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.127 Schenkerian reduction of segments 3 and 4 . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.128 Schenkerian reduction of segment 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.129 Schenkerian reduction of segment 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.130 Schenkerian reduction of segment 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.131 Schenkerian reduction of segment 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 9.132 Schenkerian reduction encompassing the whole piece . . . . . . . . . . .

xxxiii

261

262 263

263 264

265 266 267 268 268 269 270 271 272 273 275 276 277 278 279 279 280 281 282 282 283 283 283

xxxiv

List of Figures

Fig. 9.133 Basic pitch structure of the Op. 118/2 plotted as a network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. A.1 Basic structure of the non-tonal harmonic vector (a) and two examples (b, c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. A.2 Basic structure of the non-tonal harmonic vector (a) and two examples (b, c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. B.1 Pitch description of a UDS as the contour-word (descending leap, ascending skip, ascending step, descending step) in comparison with two other attributes (p3 and p4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. B.2 Two equivalent rhythms sharing the same IOI sequence . . . . . . . . . Fig. B.3 Example of rhythmic filtering, considering a context using four r-letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. B.4 The 12-divisor alphabet with two representations for the r-letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. B.5 Example of the use of micro- and macro-transformations in the production of variants of a referential UDS . . . . . . . . . . . . . . . .

284 292 292

296 297 297 298 301

List of Tables

Table 1.1 Angular bands for similarity relations in the derivative space of P. Second and third columns refer to values of .α . . . . . . . . . Table 2.1 Conversion of semitones into diatonic steps for the major scale . . . Table 2.2 Codes adopted for chordal qualities (attribute h3) . . . . . . . . . . . . . . . . . . Table 3.1 Penalties obtained for variants C1–6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 3.2 Coordinates of points C1–6, with respective angles α and β (rounded values) and degrees of similarity with P . . . . . . . . . . . . . . . Table 5.1 Cardinality-sensitive operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 7.1 Examples of genealogical notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 7.2 Q and Q* binary relations between P, C1, C2, and C3 . . . . . . . . . . . . . Table 8.1 Brahms’s Intermezzo Op. 118/2—passages of metric interest . . . . . Table 9.1 Brahms’s Intermezzo Op. 118/2—derivative segments. . . . . . . . . . . . . Table 9.2 Relations between uAs in the derivative segment 1. Bold-face relations refer to dual origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 9.3 Re-labels for uAs of derivative segment 1 using genealogical notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 9.4 Absolute quotient (Q*) between a0 and the variants of segment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 9.5 Values for σ (k) for absolute relations in DV path 4 . . . . . . . . . . . . . . . .

10 23 30 61 62 104 131 136 166 182 186 205 207 210

xxxv

Part I

Decontextualized Variation

Chapter 1

Basic Concepts

If you look at the history of the science [...] you will see that every idea is built upon a thousand of related ideas. Careful analysis leads one to see that what we choose to call a new theme is itself always some sort of variation, on a deep level, of previous themes. —Douglas Hofstadter, Variation on a Theme as the Crux of Creativity, 1985, p. 249

I start this journey by defining a unit of derivative signification (UDS, for short) as a brief melodic sequence,1 relatively complete in itself, in which pitch and temporal structures play a special role.2 A critical aspect associated with the identification of a UDS in analytical situations concerns segmentation, a problematic issue that affects musical analysis in general. Depending on the context considered, the segmentation of a musical tissue in clear autonomous UDSs can be satisfactorily done only by the use of a considerable margin of subjectivity. Fortunately, we will not have to deal with this problem here (it will be postponed to Part III), since—as aforementioned—the structures to be examined in this and the new chapters (i.e.,

1 Jack Boss, commenting some of Schoenberg’s notes on musical idea, developing variation, and motive, addresses the problem associated with the limitations of size for motive-forms, as argued by the composer himself: “Still, there is a variety of kinds and sizes of units that can fulfill these two criteria: units including a pitch succession and duration succession of two or three members, larger units formed by overlapping or juxtaposing such particles, and even units that comprise the structural pitches of a still larger unit such as a phrase or group of phrases” (Boss, 1991, p. 10). While I recognize the pertinence and accuracy of this statement in more general terms, for the sake of simplicity I choose to keep “brevity” as a basic feature for a UDS (sacrificing a little precision), in view of the fact that short musical ideas correspond to the most common case. 2 A UDS is roughly equivalent to one of Schoenberg’s definition for the musical idea: “(which), though consisting of melody, rhythm, and harmony, is neither one nor the other alone, but all three together” (Schoenberg, 1984, p. 288). For a complementary discussion about Schoenberg’s musical idea, see (Boss, 2014, pp. 5–7). I would also like to mention a definition proposed by Guerino Mazzola that seems quite similar to what I understand a UDS is: “A composition’s motif is a germ of a structural hierarchy, unfolding into its most diversified ramifications, variations, and fragmentations” (Mazzola, 2002, p. 465).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3_1

3

4 Fig. 1.1 Example of a UDS

1 Basic Concepts

 





    the UDSs) will conveniently taken as “out of time”, as resulting of a hypothetical, well-accomplished previous segmentation. The pitch and temporal structures of a UDS are classified as its primary domains and the harmonic context in which a UDS is (explicitly or not) inserted is considered as a secondary domain.3 Figure 1.1 presents a simple instance of a UDS (this example will be exhaustively used as a reference along with the book).

1.1 Derivative Work Let us now consider that all imaginable UDSs were brought together into a (conceptually infinite) set S. In the present context, variation is a generic function V: S .→ S, which when applied to a referential UDS P (stands for “parent”) maps it to another, unique member of S, the UDS C (“child”), such that both can be related by some degree of similarity. This basic condition implies that V can send P to virtually infinite possible transformational states: for example, an exact duplicate of itself (characterizing, informally, a “repetition”), an almost imperceptible modified copy (or a duplicate with some subtle “imperfection”), up to many and many slightly divergent versions, and so on, and even a very different, but still (in some way) related idea. This image is accomplished in Fig. 1.2. P is represented by a referential point in a plane, and all of its possible transformations (conceived as “C” points) are projected on a horizontal line (oriented by a decreasing of similarity, or, alternatively, an increasing of divergence) by the action of different “qualities” of function V. The space of C lies on a continuum delimited by a perfect copy of P (which is represented by a point on the orthogonal line which departs from point P) and a conceptual horizontal coordinate beyond which similarity between P and C would abruptly stop existing. 3 Of course, the model could also include other secondary domains, like timbre, dynamics, texture, etc., which would contribute to increase largely its complexity. Aiming at simplicity and concision, I opted therefore to select only the harmonic context for this category. The notion of domains can also be associated with the concept of (primary and secondary) parameters, proposed by Leonard Meyer (1989).

1.1 Derivative Work

5

Fig. 1.2 Representation of the variation space of UDS P (initial conception)

Fig. 1.3 Representation of the variation space of UDS P (second version)

As a matter of fact, to determine precisely a borderline separating similarity from absolute contrast between two musical units is extremely difficult, if not completely impossible. A more feasible representation of the variation space of P4 is depicted in Fig. 1.3: the borderline is here replaced by an indefinite area of very low similarity, inside which it is hardly distinguishable which musical ideas are related (and which are unrelated) to P. In sum, C is a transformation (or derivation) of P by the action of function V. C can also be named a variant of P or simply a derived idea. The adjective “generic” that qualifies the noun “function” in the definition of V reflects both assumptions that P is in some way transformed by the action of V (i.e., it does not matter yet

4 This

is a provisional spatial model, which will be elaborated ahead.

6

1 Basic Concepts

how)5 and that C, the output of this transformation, is related to the input P by some degree of similarity. In other words, V generically represents all possible manners in which P can be transformed in C.6 Figure 1.4 shows P and C as members of the set of musical ideas S, being connected by the action of function V. Algebraically, it is then possible to express C in function of P (Eq. 1.1). C = V (P )

(1.1)

The syntactic, ordered interaction of the three elements, P, V, and C will henceforth be referred to as a derivative work, a high-level conception of the process of variation. Figure 1.5 proposes a more concise representation of a derivative work, a scheme that will be adopted as preferential from now on. Notice how the similarity relation between P and C is evidenced by the dashed line and the symbol SP/C .

Fig. 1.4 Representation of the action of function V in set S Fig. 1.5 Schematic representation of a derivative work

SP/C

P

V(P)

C

5 The possible manners that transformations may occur is a part important of this proposal, to be examined in the Chap. 4. 6 Put another way, V could be seen as a set that congregates all possible types of actions (transformations) capable to send a given P to any possible variant C. From this, we infer that set V is also conceptually infinite. Thus, in a more formal manner, this could be expressed as: VP : = {f | f : P .→ C}, where VP is the set of all functions f capable of transforming an idea P into a related C.

1.2 Derivative Impact and Relations of Similarity

7

1.2 Derivative Impact and Relations of Similarity A derivative work is basically qualified according to the derivative impact that it causes on the referential idea P. Alternatively, it is possible to consider the derivative impact directly associated with the amount of similarity/divergence of C in relation to P, which is expressed by the relation of similarity between both musical ideas (SP/C ).7 Five basic types are considered: 1. Identity: when C is identical to P (or, in other words, the transformational action of V has no impact in the domains of P);8 2. High similarity (or, conversely, low divergence); 3. Medium similarity (or medium divergence); 4. Low similarity (or high divergence); 5. Null similarity (or total divergence). Given this, the action V can be considered as resulting from the opposition of two basic tendencies (or “forces”), intrinsic to the transformational process of P into C: similarity (or coherence, or even invariance) and divergence (or contrast, or variance). Their co-relations can be viewed as modeling a dynamic system, defined by the following vectors (Fig. 1.6). Fig. 1.6 Relation between P and C plotted as a system of vectors

P

(0,0)

x

divergence

V

Vy

C

y

similarity

7 The

Vx

(x,y)

quantitative aspect of this relation is examined in Chaps. 3 and 5. practical terms, this category can also encompass cases not as trivial as the complete identity, like, for example, changes of register, instrumentation, dynamics, etc. Because we disregard the domains related to these aspects, the outputs of such transformations will be also considered as— in practical terms—as identical to their respective inputs. 8 In

8

1 Basic Concepts

In this graphical representation of a derivative work, the horizontal, left-to-right axis was arbitrated as corresponding to the dimension of “divergence”, and the vertical, top-down axis to “similarity”.9 P and C are spatially represented by two points situated in the plan and V is a vector that connects them. Thus, V can be decomposed into two orthogonal vector components, Vx (representing “pur” divergence) and Vy (“pure” similarity).

1.3 Derivative Space I propose a refinement of this system by establishing a value of 1.00 for both the maximum similarity (or even, identity) and maximum divergence between P and C, resulting in an isosceles-right triangle plotted on a Cartesian plan and defined by the coordinates of its three vertices (see Fig. 1.7): P, representing the referential idea, at the origin (0,0); Q, positioned at the horizontal/divergence limit (1,0); and R, at the vertical/similarity limit (0,.−1). The area delimited by these points is called the derivative space (DS) of the referential idea P. As shown in Fig. 1.8, the vector V will reach C always on the edge QR in a point (x, x .− 1). Fig. 1.7 Derivative space of referential idea P

9 Since they result from arbitrary choices, both axes and orientations could be exchanged in alternative spatial configurations.

1.3 Derivative Space

9

f(x) = x-1 Fig. 1.8 Derivative space of P, considering a point C positioned on edge QR at coordinates (x, x .− 1)

Angles .α and .β are then associated with the derivative impact caused by V in P:10 the greater is .α, the lesser is the divergence between P and C (and, conversely, the greater will be the similarity between them). The opposite applies to angle .β. If .α = π/2, then |Vy| = 1 and |Vx| = 0 (total divergence, type 5), and if .α = 0, then |Vx| = 1 and |Vy| = 0 (identity, type 1). Excluding these extreme cases, any other possible derivative situation (in any combination of similarity/divergence relation), at least conceptually, will lie in a continuum of real values along the edge QR.11 Figure 1.9 proposes a segmentation of the derivative space into bands of similarity. Each band is obtained from an angular clockwise increment of .π/8 radians (or 22.5 degrees) over the previous one, beginning with the horizontal axis, which represents total divergence/null similarity, and closing with the identity relation (vertical axis). Table 1.1 presents the angular limits for the bands of similarity (for practical reasons, I adopt .α as main parameter, measured in radians). The notions of derivative space and bands of similarity will be better explored in Chap. 3.

10 An objective measurement of derivative impact is something not addressed in the current version of the model, due to the number of variables that could arise in this task. An adequate treatment of this aspect is planned for future investigation. 11 As one can perceive, QR corresponds to the horizontal axis of Fig. 1.3.

10 Fig. 1.9 Representation of the bands of similarity on the derivative space of P

Table 1.1 Angular bands for similarity relations in the derivative space of P. Second and third columns refer to values of .α

1 Basic Concepts

8

Similarity Null Low Medium-low Medium-high High identity

min 0 .> 0 .> π/8 .> π/4 .> 3π/4 .π/2

max 0 .π/8 .π/4 .3π/4 .< π/2 .π/2

1.4 Levels of Derivative Work A derivative work can operate in two basic levels: Holistic: when V acts on the indivisible structure of P (i.e., encompassing the three domains as a whole). Decomposable:12 when V acts on isolated domains of P, letting eventually (but not necessarily) the remaining unaltered.13 The notion of holistic variation is reasonably self-explanatory, corresponding to what common sense would consider “ordinary” variation (Figure 1.10 exemplifies some possible holistic transformations of a referential idea P). On the other hand, a decomposable variant is not so intuitive and, therefore, deserves a more profound examination, which will be properly accomplished in Chap. 2. 12 The

adjectives “holistic” and “decomposable” substitute for, respectively, “concrete” and “abstract”, used in previous versions of MDA. I am grateful to prof. Walter Frisch to draw my attention for this terminological question. 13 As it will be discussed in Chap. 4, it is also possible to consider a third level, an intermediary version of both categories.

References

11

Fig. 1.10 Examples of holistic variation

References Boss, J. (1991). An analogue to developing variation in a late atonal song of Arnold Schoenberg. Dissertation, Yale University. Boss, J. (2014). Schoenberg’s twelve-tone music: Symmetry and the musical idea. Boston: Cambridge University Press. Mazzola, G. (2002). The topos of music: Geometric logic of concepts, theory, and performance. Basel: Birkhauser. Meyer, L. (1989). Style and music. Chicago: The University of Chicago Press. Schoenberg, A. (1984). Style and idea: Selected writings of Arnold Schoenberg. London: Faber and Faber.

Chapter 2

Decomposable Variation

A central premise of MDA is that a large percentage of derivative work in a musical piece involves an isolated structural domain (think of a simple intervallic inversion of a melody maintaining the original rhythmic configuration, for example). In other words, the process of derivation in this case is mediated by a previous stage of abstraction. As shown in Fig. 2.1, pitch and rhythmic “essences” are abstracted from P, becoming in turn referential for independent types of derivation.1 For the sake of simplicity, let the secondary domain of harmony (omitted in the examples) be kept unaltered along the process. As a basic consideration, the primary domains of pitch and rhythm are taken as privileged in decomposable variation, letting the secondary harmony subordinated to them. Given this, it is my claim that, in general, pitch is the most permeable structural domain to the effects of variation. On the other hand, rhythmic figures have the tendency to be preserved (or only superficially transformed) in meaningfully related motivic/thematic units, since normally they are primarily responsible for the characterization of musical ideas themselves. More radical rhythmic variations can, of course, be applied throughout a derivative process, but almost inevitably they cause deeper differentiation between referential and derived forms (P and C, in the present terminology) in opposition to the more common cases of smooth or gradual transformations. From this, a logical conclusion can be drawn: temporal changes (i.e., rhythmic and metric), though statistically less common than those involving pitch, have potentially greater power for transforming a given referential idea. In other words, they are normally capable of causing more derivative impact and, maybe, for this reason, are less frequent.2 Consequently, it is necessary to provide the model of means for differing quantitatively the domains in regards to

1 Boss

(1991) refers to such situations as “specific variations”. is worth noting that harmony, as a secondary domain, is considerably less influential on the transformation of a musical idea than pitch and temporal domains. 2 It

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3_2

13

14

2 Decomposable Variation

Fig. 2.1 Examples of decomposable variation

the derivative impact that shall be expected to cause. This quantitative aspect will be addressed in Chap. 3, through the adoption of a system of relative weights.

2.1 Phases of Decomposable Variation The essential process of decomposable variation is performed by three sequential phases, described as follows: 1. Abstraction: in which the content of a domain is isolated from UDS P, becoming potential basis for further decomposable derivation; 2. Derivation: in which the process of variation properly occurs, being applied to the abstracted, referential unit(s) and resulting into other abstracted, derived unit(s); 3. Re-integration: in which the abstracted domains (either those transformed or not) are recombined to form a definitive derived (concrete) UDS C. In a more formal perspective that will be shown as useful later, it is also possible to consider these three phases represented by functions. Returning to set S (the set of all possible UDSs), let Sp , St , and Sh be the sets formed by abstractions of members of S (we can call them “abstracted” sets), which are related, respectively, to the pitch, temporal, and harmonic domains. Let abs-p be a function that abstracts from a given UDS P, member of S, its pitch structure, sending it to an “abstracted parent” Pp , member of set Sp (likewise, we can also consider similar functions for temporal and harmonic domains, as shown in Fig. 2.2). Derivation in this context is performed like a high-level derivative work, with the unique difference of being here based on abstracted units. Let Vp , Vt , and Vh be the generic functions that represent possible transformations associated with,

2.1 Phases of Decomposable Variation

15

V(C)

P

C

S

...

abs-p

abs-h

abs-t

Sp Pp

St

Sh Pt

Ph

Fig. 2.2 Representation of sets S, Sp , St , and Sh , and the action of functions of abstraction abs-p, abs-t, and abs-h, producing abstracted parents Pp, Pt, and Ph

respectively, sets Sp , St , and Sh .3 Their action on to corresponding abstracted parents will produce derived, “abstracted children” Cp , Ct , and Ch , as shown in Fig. 2.3. Concerning the basic condition of an ordinary derivative work, all derivations must keep some degree of similarity with the respective referential units (indicated by the dashed slurs). Finally, re-integration can be also viewed as a type of operation (labeled rnt) that picks the abstractions, recombines them, and sends the output back to S, transfigured as a concrete transformed musical idea C. As it is also suggested in Fig. 2.4, the action of V on P (with the consequent production of the variant C) can be then viewed as a high-level process of transformation, bypassing the three “intracellular” stages involved in abstract variation. After this brief digression, I will put temporarily aside the idea of sets and functions as a representative for decomposable variation (it will be resumed briefly in Chap. 4, slightly modified), and propose a simpler (and more practical) graphical scheme (Fig. 2.5). New conventions, symbols, and terminology are provided: concrete musical ideas (P and C) are represented by normal-line rectangles; abstracted ideas (p, t, and h, standing for the domains of “pitch”, “time”, and “harmony”) are represented by dashed-line rectangles; and derived abstract ideas are numbered inside gray, dashed-line rectangles. This representation will be elaborated in the next section.

3 Functions

Vx (“x” standing for any domain) act, therefore, as “decomposable” versions of V.

16

2 Decomposable Variation

V(C)

P

C

S

Sp

...

Cp Vp Pp

St

Vt

Ct

Sh

Pt

Vh

Ch

Ph

Fig. 2.3 Representation of the actions of generic functions Vp , Vt , and Vh

Fig. 2.4 Representation of the action of function rnt

2.2 Classifications of Decomposable Variation Decomposable variation can be classified in two distinct manners: 1. According to the number of stages of transformations required for its application. It can be:

2.2 Classifications of Decomposable Variation

17

Fig. 2.5 General representation of an abstract derivative work

• simple, when transformation occurs in only one stage, or • compound, when transformation is performed in two or more stages. 2. According to the number of domains involved in the process. It can be: • pure, when a unique domain (for example, pitch) is affected by the transformation, or • hybrid, if two (or, rarely, the three) domains are simultaneously involved in the derivative work. A possible question that may naturally arise at this point addresses the differentiation between holistic and hybrid-type variations, since, at first impression, they seem to mean the same thing. In fact, it is not always easy to distinguish both situations. In strict terms, the former case corresponds exactly to a transformation that involves simultaneously all domains (in a practice sense, both pitch and temporal), a definition that could also be applied to the latter type. The subtle difference between them lies in the manner in which the process occurs. The label “hybrid variation” is suitable to describe those specific situations in which a complete transformation between two musical ideas can be decomposed into stages of secondary transformations, while in concrete variation that is not possible or, at least, would demand non-logical, very complicated, or far-fetched descriptions.

18

2 Decomposable Variation

p

(a)

(b)

(c)

P

P

P

t

h

p

t

h

p

t

p

h

(d)

(e)

P

P

t

V(P)

h

V(P) C

1

1

V(P)

V(P)

1

1

V(P) 1

2

2

C 1

3 C

C

C

Fig. 2.6 Schematic representations of possible cases of variation, presented as examples: pure/simple decomposable variation (a); pure/compound (b); hybrid/simple (c); hybrid/compound (d); and holistic variation (e)

Using as models the schemes introduced in Fig. 2.5, Fig. 2.6 presents four examples of decomposable derivative work (a–d) combining the four possible combinations of classes (pure/simple, pure/compound, hybrid/simple, and hybrid/compound) in comparison with the holistic variation (Fig. 2.6d), which, obviously, does not present stages of abstraction and re-integration.4 While simple decomposable variation is relatively easy to be identified with precision in analytical situations, compound variation is normally more problematic. In most of the cases, the task of tracking back the intermediary abstract-derivative stages that map P onto C can only be achieved by the analyst as a sequence of plausible hypotheses. Figure 2.7 addresses this question. Let us assume it as a simplified version of the model of Fig. 2.6b (pure/compound abstract variation).5 Let us also suppose that an analyst considers variants 1, 2, 3, 4, 5, and 6 as possible intermediary, abstract “stations” between P and C.6 Considering that the edges in Fig. 2.7a represent valid connections between the stations, sixteenth distinct trajectories would be possible, three of them depicted in Fig. 2.7b, c, and d.7 Assuming that there is not a unique correct “answer” for the “problem”, how would be the best choice for the analyst? If disregarded the context (a very important factor in real situations), maybe the most logical, economic, and well-balanced approach should be the selection of the option with lesser number of stations (path b, in this case). Alternatively, the analyst may chose that option that provokes which more subtle and gradual changes

4 These

models will be resumed in Chap. 5. clarity, both stages of abstraction and re-integration were omitted in the scheme, letting only the stage of derivation. 6 These “stations” are formally named hypothetical intermediary stages in analytical situations. I will comment this concept with more details in due time. 7 Such trajectories, graphic representations of decomposable derivative work, will be treated as model for the derivative networks in Chap. 5. 5 For

2.2 Classifications of Decomposable Variation

19

Fig. 2.7 Three possible derivative trajectories for the abstract derivative work between P and C

(this decision would demand an evaluation of the specific musical conditions, not accessible in this simple example). An interesting and pertinent metaphor originated in the fields of paleontology and evolutionary biology can be useful for illustrating this aspect. The complete sequence of intermediary evolutionary stages that links two correlated forms of life (say, dinosaurs and birds) is virtually impossible to be concretely evidenced, due to the absence of the complete fossil record. Most of the myriad of variations, species, genera, etc. that would fill this gap simply lived and disappeared without traces. However, the meticulous study of the rare fossils preserved combined with the morphological and genetic-molecular mapping of possible descendants, powerful technology, cooperative researches, and solid experience, allow the scientists to reconstruct with a high degree of accuracy the links, despite the time span involved, measured in hundreds of millions of years. Although strongly supported by scientific methods, the reconstruction of the trajectory dinosaurs-birds is, strictly speaking, based on connective hypotheses.8 Aside, evidently, the huge differences separating biological and musical realms, undoubtedly there also exist some interesting points of contact between them, reinforcing the metaphorical links. In this particular aspect, considering the scope of MDA, the analytical task of identification of variants that act in a piece of music, as well as the explanation of their derivation from the respective referential ideas are both many times dependent on plausible conjectures.9 8 Ernst

Mayr (2002, pp. 70–72; see also pp. 84–85) discusses these questions associated with phylogeny with extraordinary clarity and accuracy. 9 Commenting a motivic analysis of Beethoven’s String Quartet Op. 135/4th movement, made by Schoenberg, Michael (Schiano, 1992, pp. 129–130) points out the fact that Schoenberg proposed some possible intermediary variants (that are not in the score) as plausible explanations of the connections between some of the ideas. In his words, “The events that Schoenberg wishes to

20

2 Decomposable Variation

Needless to say that the hypotheses eventually established by an analyst do not intend to reconstruct compositional lines of thought. They should rather be seen as part of a methodological strategy that ultimately aims to contribute for the systematization of the analysis. This discussion will be resumed in Chap. 4, as well as in Part II (as a central issue for developing variation) and in the analytical application of Part III, in this case under a “real-life” perspective.

2.3 Attributes The domains, primary and secondary, are composed of internal, mutually complementary structures, named attributes, whose primordial function is to describe specific aspects of a musical idea.10 Aiming at systematization, the attributes are formatted as numeric sequences.

2.3.1 Pitch The pitch domain has five attributes, as exemplified in Fig. 2.8. • p1: describes the “pitch sequence”, expressed in midi pitches. Midi-pitch convention uses integers for representing equally-tempered pitches, considering 0 as C.−1 (and 60 the middle C), and the unit as one semitone. Figure 2.9 depicts Fig. 2.8 Pitch attributes of a referential UDS P. Angled brackets indicate ordered content

connect exist on the musical surface. However, the intermediate steps are hypothetical (. . . ) a chain of logical steps between two events is made explicit.” As one can perceive, there are notable similarities between Schoenberg’s analytical procedures (as well as the methods developed by Schiano himself) and the present proposal. 10 In his doctoral dissertation, Jack Boss, based on some of Schoenberg’s writings about variation, defines a feature as “the most concrete quality of a motive affected by each variation kind” (Boss, 1991, p. 14), which can be compared with the concept of domain. Boss’s features are “duration succession” and “metric context” (inside the “rhythmic category”), “pitch succession”, “harmonic succession”, and “tonal context” (grouped in what the author calls the “intervallic category”). He also considers that the features can be subdivided into “aspects”, a conception very close to my idea of attributes.

2.3 Attributes Fig. 2.9 Representation of pitch space, considering midi-pitch numbering

21

22

2 Decomposable Variation

Fig. 2.10 Representation of pitch-class space

0

11 B

10

C

A B

1 C D

2 D D E

A

9 8

G A

3

E G

7

F G

4

F 5

6

a possible representation of the pitch space,11 as a conceptually infinite line, segmented into discrete units, the pitches, to which are assigned pitch-midi numbers. In the case of P of Fig. 2.8, p1 corresponds to the sequence , which when converted to midi pitches produces p1 = .12 • p2: describes the “pitch-class sequence”. It is equivalent to p1 under modulo 12.13 While pitch space is infinite, pitch-space is modular, and has only twelve elements, numbered from 0 (“C”, for convention) to 11 (“B”). Traditionally, it is represented as a “clock face”, as shown in Fig. 2.10. In this manner, p2 = mod12(p1) = = . • p3: describes the “intervallic sequence”. It conveys the arithmetic differences between contiguous elements of p1: p3 = = .14 Alternatively, p3 can be replaced by a diatonic version (properly labeled as d-p3). Especially in stable tonal contexts, this can be a more advantageous strategy, because it is a simpler, more direct, and, therefore, more efficient mode of describing intervallic relations in such situations. In this alternative attribute, intervals between the events are measured not in semitones but in scalar degrees

11 Originally

conceived by Robert Morris (1987), as well as the complementary notion of pitchclass space, to be introduced in the next topic. 12 Angled brackets “< >” indicate that the element s inside them are ordered in a specific manner. 13 Operation modulo 12 (mod12, for short) applied to an integer returns the reminder of the division of this integer by 12. For example, mod12(7) = 7; mod12(12) = mod12(24) .= 0; mod12(39) = 3; etc. 14 Negative numbers indicate descending intervals.

2.3 Attributes Table 2.1 Conversion of semitones into diatonic steps for the major scale

23 Semitones 0 1 2 3 4 5 6 7 8 9 10 11 12

Steps 0 1

Informal designations Unison Second

2

Third

3 3/4 4 5

Fourth Tritone Fifth Sixth

6

Seventh

7

Octave

or steps. Table 2.1 converts semitones into diatonic steps, considering as basis the major diatonic scale.15 So, in the case of the example of Fig. 2.8, the diatonic version d-p3 would be notated as (or, informally, “descending sixth, ascending third, ascending second, descending second”).16 Since p3 represents the general case, it will be adopted as default descriptor for intervallic sequence in theoretical formulations. The diatonic alternative d-p3 will be applied in Part III, in the analysis of Brahms’s Intermezzo Op. 118/2. • p4: describes the “melodic contour”, or the basic outline of the melody. It is the most abstract of the pitch attributes, being related to Theory of Contours, created by Robert Morris (1987). A melodic contour informs the relative movements of a melodic line, i.e., the eventual alternations of “ups” and “downs”, disregarding the specific intervallic magnitudes. Melodic contours can be expressed both graphically (see Fig. 2.11) or algebraically.17 The latter representation (which is adopted in the present proposal) considers the contour as a vector, in which “0” stands for the lowest point of a melody with n notes and n-1 (in case of no repetitions) for the highest, with the remaining notes assigned to intermediary numeration, according to the topography of the line. In the melody exemplified,

15 Other

scales (like minor harmonic, for example) will require adaptations. Rings (2011) deals ingeniously with the dual nature of a contextualized note, considered both the pitch-class and the diatonic spaces, by proposing an analytical identification of this note as the duple (sd, pc), where sd is the scale degree it occupies and pc is its pitch-class representation. ˆ 4). This same pitch class In this manner, a “E” in the key of G major is represented as the duple (6, ˆ 4), and so on. See also a correlated approach, in the context of F major would be notated as (7, proposed by John Clough (1979) and Clough and Myerson (1985), concerning their diatonic setclass theory. 17 For a simple algorithm for obtaining a melodic contour from a pitch sequence, see Appendix C. 16 Steve

24

2 Decomposable Variation

Fig. 2.11 Graphical representation of the melodic contour of Fig. 2.8

pitch E4 is the lowest point, receiving label “0” followed by G4 (1), A4 (2), and C5 (3), resulting in the algebraic contour p4 = .18 • p5: describes the “intervallic ambit”. It corresponds to the difference between the first and the last element of p1: p5 = 67 .− 72 = .−5.19 At first glance, one could consider unnecessary to maintain all five attributes as descriptors of the pitch information, since some of them seem to be mutually redundant (notably p1, p2, and p3). Why not, instead, just to select the most specific, and precise of them, p1, for this purpose? As a matter of fact, however, the five attributes do not describe exactly the same things. The strategy of considering all of them as integrated, correlated descriptors allows for the system to formally capture some subtle transformations between related musical ideas that otherwise would pass unnoticed.20 The following examples help to clarify this aspect, considering five cases of variants. Three of them (C1, C2, and C3) can be considered as “archetypal variations”: change of register, transpositions (chromatic and diatonic), and inversion (Fig. 2.12).21 None of the five variants shares with P the exact content of attribute p1 (mismatches are indicated as underlined numbers). Their relations of similarity with P are though evidenced by other attributes: pitch-class sequence in C1 (a simple case

18 Unlike

the remaining sequences, I decided to maintain for p4 the original notation proposed by Morris, namely depicting the numbers without commas for separation. 19 This attribute is intended to capture the abstract notion that musical ideas can be related by shared intervallic limits, even if their contents sharply differ. 20 In a very detailed study about musical expectations by a listener, David (Huron, 2006, p. 374) lists absolute pitch, pitch-class, intervals, contour, among other aspects, as complementary mental representations of pitch, which seems to be a solid support for my argument. The comparison between attributes is a central aspect in this book and will be examined in Chap. 3. 21 It is important to highlight that just the pitch domain is here in question. Rhythmic and metric dimensions are therefore considered fixed, as non-relevant parameters for this case.

2.3 Attributes

25

Fig. 2.12 Five archetypal pitch variants from P: by octave displacements (C1), chromatic transposition (C2), literal inversion (C3), free change of notes, but keeping the original melodic contour (C4), and free change of contour, but keeping the original ambit (C5). Mismatches are indicated by underlined numbers, and corresponding attributes are inserted in rectangles

of octave displacement of two events), intervallic configuration22 in C2 (chromatic transposition to a major third higher), metric contour in C4 (a more distant variation that shares with the referential form only the basic melodic outline), and intervallic ambit in C5 (a still more abstract relation with P, supported just by the maintenance of the distance between initial and final pitch events). On the other hand, C3 has none coincident attribute in relation to P. This suggests mistakenly that is a remote variant, contradicting the general intuition. In Chap. 3 we will return to these cases, examining them in more detail.

22 The

identities of melodic contour (p4) and ambit (p5) are a consequent result when two intervallic sequences correspond exactly.

26

2 Decomposable Variation

Fig. 2.13 Temporal attributes of P

Fig. 2.14 Example of IOI equivalence

2.3.2 Time The temporal domain is formed by four attributes (Fig. 2.13):23 • t1: describes the “sequence of durations”. It is expressed in multiples of the temporal unit (the sixteenth note is considered the default unit).24 • t2: describes the “sequence of inter-onset intervals”. An inter-onset interval (IOI), a concept proposed by David Temperley (2001), is the distance measured between the onsets of two rhythmic contiguous events, disregarding their durations. Figure 2.14 shows three rhythmic notations that are equivalent in IOI terms. • t3: describes the “metric contour”. The determination of the metric contour for a rhythmic fragment is somewhat similar to what is done in respect to the melodic contour for a pitch configuration. However, there is an important distinction between them: while a given pitch sequence will produce the same melodic contour not mattering which key supports it (or even if there is no key at all), the metric contour is strongly contextual-dependent. In fact there are always two metric contexts to be considered: global (represented by the time signature) and local (the metric positions that the rhythmic events occupy inside the measures). The global context determines a metric grid that allows for the assignment of distinct “weights” to the local events, depending on their actual positions. In other words, a given rhythmic configuration can present different metric contours, depending on its relation with the grid. Considering this complex nature, it is not so easy to elaborate an algorithm for obtaining automatically a metric contour

23 As

it can be observed, there are isomorphic relations between some of the pitch and temporal attributes: p1 and t1; p3 and t2; p4 and t3; p5 and t4. Attribute p2 (pitch-class sequence) is the only pitch attribute that has no counterpart in the temporal domain. 24 Evidently, other durations can be considered as unit, according to contextual conditions.

2.3 Attributes

27

Fig. 2.15 Two distinct local metric contexts for the rhythmic idea of Fig. 2.13

from a given rhythmic fragment.25 In practical terms, it is normally simpler and more direct to extract the contour by pure analytical observation of the context in question. Consider, for example, the case of our example, whose rhythmic-metric structure is detached in Fig. 2.15a. A metric grid was added below the figure, associated with the global context (determined by the 4/4 time signature). The cells correspond to the segmentation of the grid according to metric levels.26 The coincidences of the events with the segmentation and levels are scored with “1s”. Furthermore, since the measure presents a binary structure, it is expected that its first half is metricly more relevant than the second one. From this, the model assigns an additional score of 0.5 to the first-half events (1, 2, and 3). Likewise, looking at one higher level, we can conceive that the odd quarters (at beats 1 and 3) are stronger than the even ones (2 and 4), and

25 For

more details, see Mayr and Almada (2017). the shortest durational value present in the local context is the eight-note, this is the superficial level considered in this case, but other contexts can require other referential layers.

26 Since

28 Fig. 2.16 Graphic representation of the metric contour of Fig. 2.13

2 Decomposable Variation

4 3 2 1 0

therefore shall also be scored, this time by a quarter of one point (0.25). As shown in the computation table at the right of Fig. 2.15a, event 1 has the highest score (4.75), followed by events 3 (2.50), 5 (2.00), 2 (1.75), and 4 (1.25), determining then the contour (to better evidence the correlations with the melodic contour, t3 is also notated without separating commas). Figure 2.15b shows how a simple eight-note displacement of the rhythmic sequence implies in a dramatic modification of the metric contour. This illustrates a very effective type of transformation of the temporal domain, change of metric context. Graphic representations of metric contour are also possible, as shown in Fig. 2.16. • t4: describes the “durational span”. It is the sum of the durations, including eventual internal rests). Figure 2.17 presents some temporal archetypal variations, demonstrating the importance of taking the four attributes together also in this domain. In the first case, resulted from a simple shortening of some durations, the two attributes t1 mismatch, but the identity of both t2 and t3 evidence the strong similarity that exist between P and C1. The almost opposite happens with C2, in which the rhythmic configuration is metricly displaced by one quarter: while t1, t2, and t4 are perfectly preserved, the metric contour is completely modified.27 The last two variants (C3 and C4), despite not presenting any matching of the respective four attributes, are very close to P, since they result, respectively, from the canonic operations of augmentation and diminution. In fact, the proximity between variants and the referential unit in both cases is depicted only indirectly, through the relation between the temporal spans (t4): doubled (in C3) and halved (in C4), an aspect that must be taken into account in the calculation of similarity in the temporal domain.

27 Undoubtedly,

C2 is the variant that most diverges from P. As a matter of fact, metric displacement is a largely employed technique of variation (notably, by Brahms), which can be attributed to both an economy of means (with the preservation of the rhythmic configuration) and the contrast that results from the presentation of the idea in a different metric context.

2.3 Attributes

29

Fig. 2.17 Four archetypal temporal variants from P: with rests inserted between some onset points (C1), metric displacement (C2), augmentation (C3), and diminution (C4). Mismatches are indicated by underlined numbers, and corresponding attributes are inserted in rectangles

2.3.3 Harmony Harmonic information related to P and C (either explicitly or implicitly) is depicted as an algebraic vector, formed by five attributes: • h1: describes the “key” (expressed as a pitch class). • h2: describes the “mode” (by convention, 0 = major mode, and 1 = minor mode). • h3: describes the “chordal quality” (expressed as a code number between 0 and 9—see the adopted conventions in Table 2.2). • h4: describes the “chordal root” (expressed as a pitch class). • h5: describes the “chordal bass” (expressed as a pitch class). The harmonic accompaniment of P (see Fig. 1.1) is tonally centered in C major. Its first four events are harmonized by a C major triad in first inversion, and the last one by a second-inversion dominant seventh rooted in G. Therefore, there are for P two different harmonic vectors (that incidentally share same low-level harmonic information, related to key and mode), the first one associated with events 1 to 4 and the second with event 5. They are formatted as follows: h.1−4 = (to be read as “first-inversion of C major triad inside the key of C major”).

30 Table 2.2 Codes adopted for chordal qualities (attribute h3)

2 Decomposable Variation Codes 0 1 2 3 4 5 6 7 8 9

Qualities Major triad Minor triad Diminished triad Augmented triad Dominant seventh Major with major seventh Minor with minor seventh Half diminished Diminished seventh Other cases

h5 = (to be read as “second-inversion of G7 chord inside the key of C major”). As done previously for the primary domains of pitch and time, Fig. 2.18 shows four cases of archetypal harmonic variations (I am assuming here that the key signatures of the examples unambiguously inform their respective tonal contexts): chromatic modulation (C1); diatonic chordal change (C2), replacement of the C major triad (contextualized as the tonic I) by an E minor chord (iii); modal interchange (C3); and a simple inversion of the original chord (C4). Each situation presents a different configuration of the harmonic vector.

2.4 Matrix of Attributes The multidimensional nature of a UDS, expressed by the set of domains e attributes, can be better displayed in the format of a matrix (denoted by M). While the number of rows of M is fixed in twelve (corresponding to the complete set of attributes, encompassing primary and secondary domains), the number of columns varies according to the number of events present in the referential idea.28 The first four rows are related to the pitch domain: p1, p2, and p4 occupy respectively rows 1, 2, and 4. Because p3 is a sequence of differences between pitches, it always has one element less than p1, p2, and p4. So, by convention, it is positioned in the third row, with the ambit (p5), formed by a unique element, occupying the last cell.29 The following three rows refer to the temporal domain, t1, t2/t4, and t3. The third section of M (rows 8–12), related to the harmonic domain, is filled vertically, according to the correspondences between events and harmonic contexts. Figure 2.19 depicts the structure of the matrix of attributes of the referential idea P.

28 In

other words, there is a one-to-one correspondence between events and columns of M. Thus, the matrix of attributes related to Fig. 1.1 has dimensions 12 .× 5. 29 A similar strategy is used in the temporal domain, considering in this case attributes t2 and t4.

2.4 Matrix of Attributes

31

Fig. 2.18 Four archetypal harmonic variants from P: chromatic modulation (C1), diatonic chordal change (C2), mode interchange (C3), and inversion (C4). Mismatches are indicated by underlined numbers

A matrix of attributes can then be considered as an algebraic equivalent of the referential idea (as well as the derived idea, as will be shown) which it is related to. Eventually, in an analytical situation, matrix M can be “broken” in its three basic sections: Mp (pitch), Mt (time), and Mh (harmony), and depicted in a “chessboard” format, aiming at a clearer visualization, as shown in Fig. 2.20. It is also possible to consider subsets of the sections (a complete or partial row/column, or even isolated elements) for any analytical finality. In these cases, conventional matrix notation is called for, aiming at a precise identification of the subsets in question. The adopted conventions are the following: • The coordinates of a single element are indicated as an ordered pair (row, column), as depicted in Fig. 2.21a; • A colon is used for indicating a selected sequential segment of a row or column (Fig. 2.21b); • If a row/column subset is formed by a non-contiguous element, they are individually expressed inside embedded parentheses, separated by commas (Fig. 2.21c); • An alone colon in one of the matrix dimensions (column or row) indicates that it must be considered completely (Fig. 2.21d).

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Fig. 2.20 Isolation of the three sections of matrix M

event 5

ambit (p5)

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References

33

Fig. 2.21 Examples of identification of matrix subsets

References Boss, J. (1991). An analogue to developing variation in a late atonal song of Arnold Schoenberg. Dissertation, Yale University. Clough, J. (1979). Aspects of diatonic sets. Journal of Music Theory, 23(2), 45–61. Clough, J., & Myerson, G. (1985). Variety and multiplicity in diatonic systems. Journal of Music Theory, 29(2), 249–270. Huron, D. (2006). Sweet anticipation: Music and the psychology of expectation. Cambridge: The MIT Press. Mayr, D., & Almada, C. (2017). Geometric and vector representation of metric relations. In Proceedings of the 2nd National Conference of Brazilian Association of Theory and Analysis TEMA (pp. 10–19). State University of Santa Catarina, Florianópolis. Mayr, E. (2002). What evolution is? London: Phoenix. Morris, R. (1987). Composition with pitch classes: A theory of compositional design. New Haven: Yale University Press. Rings, S. (2011). Tonality and transformation. Oxford: Oxford University Press. Schiano, M. (1992). Arnold Schoenberg’s Grundgestalt and its influence. Dissertation, Brandeis University. Temperley, D. (2001). The cognition of basic musical structures. Cambridge: The MIT Press.

Chapter 3

Measurement of Similarity (I)

An important aspect of MDA is the possibility of measurement of similarity between referential and derived ideas. This is systematically accomplished through the interaction of concepts and premises that were so far introduced, in combination with the treatment for similarity evaluation that is proposed in the book The Geometry of Musical Rhythm, by Gottfried Toussaint (2013). In his chapter 33, Toussaint classifies the methods for measurement of similarity between objects (not necessarily musical ones) in two basic categories: feature-based methods and transformation-based methods. In feature-based methods, objects are compared in terms of the number of traits they have in common. In transformation-based approaches, similarity is measured by how little effort is required to transform one object to another. (Toussaint, 2013, p. 249, italics in the original).

Toussaint then describes five transformational approaches, commenting on their advantages from the standpoint of his objectives: Hamming distance, swap distance, directed-swap distance, many-to-many assignment distance, and edit distance. Because he is especially interested in comparing rhythms with either equal or different numbers of onsets and/or pulses (that is, with different cardinalities, according to the present terminology), his focus is concentrated on the most robust of them, the edit distance approach, also called Levenshtein distance.1 Basically, this method counts “the minimum number of edits (or mutations) necessary to convert one sequence to the other” (Toussaint, 2013, p. 252, italics in the original).2 Three types of operations are considered: insertion, deletion, and substitution. Toussaint exemplifies the application of the method in the transformation of the word “WAITER” into “WINE”, which have different “cardinalities” (respectively,

1 According

to Toussaint (2013, p. 253) this is due to “its inventor, Vladimir Levenshtein (1935– 2017), the father of Russian information theory”. 2 Although other possible methods for measurement of similarity could also be considered (Euclidean distance, for example), edit distance was the one with the best results and that was most suitable for the necessary adaptations in the creation of the algorithms. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3_3

35

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3 Measurement of Similarity (I)

Fig. 3.1 Representation of edit-distance method, after Toussaint (2013, p. 252)

6 and 4). The process involves two stages: (1) deletion of letters “A” and “R”: “WAITER” .→ “WITE”; and (2) substitution of “T” by “N”: “WITE” .→ “WINE”. Figure 3.1 proposes a graphic representation of this example, adapted to the MDA’s terminology. Due to a strong alliance of simplicity and efficiency, the edit distance is an ideal candidate for measurement of similarity between two related UDSs in MDA. I adopt here a somewhat modified version of the method, adapting it in order to deal with the structural particularities of the matrix of attributes. Considering a referential unit P and a possible variant of it, C, our adapted edition-distance approach basically counts the number of editions that are necessary to transform the matrix of attributes of P into the matrix of attributes of C. However, there is an important difference in relation to Toussaint’s description: since the matrices are multidimensional structures, involving musical attributes of distinct natures and meanings, the editions are weighted differently, depending on the rows considered (this will be detailed in due time). Let us consider first the simplest situation of derivative work, namely when P and C have the same cardinality. More complex cases will properly be addressed in Chap. 5.

3.1 General Information Assuming that P and C have compatible cardinalities, the transformations of types deletion and insertion are here out of question: only substitutions are applied in the

3.1 General Information

37

Fig. 3.2 General scheme of measurement of similarity between elements of P and C

mapping of P into C. In other words, the elements present in P’s and C’s matrices are always one-to-one related, and the “distance” between two corresponding elements will be measured as the “interval” (broadly, in David Lewin’s sense) between them.3 Figure 3.2 presents a general idea of the method. Since P and C have the same cardinality, their matrices of attributes are also equally formatted (say, with m rows and n columns). This means that a given element of the matrix of P (Pi,j ) will be related to one of the matrices of C positioned at the same coordinates (Ci,j ) and the “distance” between them (.σ ) will be proportional to the abstract difference of their contents. Thus, .σ will be null if both elements are equal (implying total similarity in the respective locus), and will have any positive value (depending on the magnitude of the interval) if they are different. As a convention aiming at a rapid visualization, let us indicate non-equal elements in C inside gray cells, as depicted in Fig. 3.2 (this convention will be used in further examples). An important aspect of the algorithm for measurement of similarity of P and C concerns the need to establish distinct weights for the differences between elements, according to the relevance of the musical descriptors associated with them. The assignment of weights is a central element of the algorithm, a process that involves multiple possibilities. The values that will be here adopted result from a relatively long empirical process, combining the try-and-error method with a number of intuitive assumptions. For clarity and simplicity, the matrices will be subdivided in sections (as shown in Fig. 2.20), corresponding to the three domains, pitch, time, and harmony.

3 The

specific conditions of this sort of measurement will be properly formalized ahead.

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3.2 Measurement of Similarity in the Pitch Domain Initially, it is necessary to delimit the section of the respective matrices of attributes that will be worked out. Thus, define Pp and Cp as sub-matrices corresponding to the first four rows of the matrices of attributes referential (P) and variant (C). Let us generically identify these structures as matrices of pitch attributes. The similarity between Pp and Cp is measured considering five vectors, corresponding to each one of the five attributes: • • • • •

v1: vector associated with the pitch sequence (p1); v2: vector associated with the pitch-class sequence (p2); v3: vector associated with the intervallic sequence (p3 or d-p3); v4: vector associated with the melodic contour (p4); v5: vector associated with the ambit (p5).

Vectors v1, v2, v3, and v4 have n elements, corresponding to the cardinality of Pp and Cp. Vector v5 is in fact a scalar since it is always formed by a single element. The process for obtaining the five vectors is described as follows: • Vector v1 results from the absolute difference between the first row of Cp and Pp. Formally: v1 = |Cp(1, :) – Pp(1, :)| • Vector v2 results from the absolute difference between the second row of Cp and Pp, considering the operation of modulo 12. Formally: v2 = |mod12(Cp(2, :) – Pp(2, :)| • Vector v3 results from the absolute difference between the third row of Cp and Pp, disregarding the respective last elements (since they are related to ambit, p5). Formally: v3 = |Cp(3, 1:end–1) – Pp(3, 1:end–1)| • Vector v4 results from the absolute difference between the fourth row of Cp and Pp. Formally: v4 = |Cp(4, :) – Pp(4, :)| • Vector v5 results from the absolute difference between the last element of the third row of Cp and Pp. Formally: v5 = |Cp(3, end) – Pp(3, end)| The five vectors give birth to a new structure called the pitch vector (vp), whose elements are the ordered sums of the elements of v1 to v4, followed by v5, such that vp =


The five elements of vp are then assigned to weights, which are organized also as a five-entry vector (wp), such that wp =< 15, 15, 40, 25, 5 >

.

This hierarchical disposition of values (obtained after empirical attempts, as aforementioned) reflects the intuition that the intervallic configuration of a UDS is the most important factor in the characterization of its pitch structure. For this reason, this attribute is assigned to the highest weight of wp, 40. Accordingly, the

3.2 Measurement of Similarity in the Pitch Domain

39

abstract melodic contour occupies the second place (25), followed by pitch and pitch-class sequences (both with 15), and the ambit, as the descriptor that exerts the least influence (5). A definitive index representing the edition-distance in the pitch domain is then calculated as the dot product of vectors vp and wp.4 Let us call this index a (provisional) pitch penalty (labeled as kp’), which is intended to express the degree of dissimilarity between P and C in the isolated domain of pitch.5 This process is formalized in Eq. 3.1: kp = vp.wp

.

(3.1)

Or kp = vp(1).wp(1) + vp(2).wp(2) + vp(3).wp(3) + vp(4).wp(4) + vp(5).wp(5)

.

As a final measure, kp’ shall be normalized inside values 0.00 (referring to maximal similarity) and 1.00 (representing maximal dissimilarity), in order to provide both coherence and standardization to the system, considering that analogous penalties will be also assigned to the temporal and harmonic domains. The normalized pitch penalty (kp) is obtained by the following formula (Eq. 3.2): kp =

.

kp − kpmin kpmax − kpmin

(3.2)

Where kp’ is the provisional pitch penalty, kpmin and kpmax are, respectively, the minimal and maximal pitch penalties. Assuming kpmin = 0.00, Eq. 3.3 is reduced to kp =

.

kp kpmax

(3.3)

While it is almost trivial to establish a relation of maximal similarity between two objects (formalized in the present proposal when the matrix of pitch attributes of C is a copy of that of P), the determination of a “maximally contrasting” structure to a referential one is an extremely problematic task. Obviously, contrast is not an absolute quality. Likewise, the idea of a clear-cut border separating similar objects from entirely dissimilar ones is almost nonsense, as already discussed in Chap. 1. As a matter of fact, we can conceive (conceptually) infinite possibilities for contrasting motives, in also infinite gradations of dissimilarity. From this, it

4 The dot product of two vectors of equal size is an operation that multiplies corresponding elements of the vectors (first .× first, second .× second, and so on) and sum the results, returning a scalar. 5 Jack Boss proposes in his doctoral dissertation a index of remoteness for measuring divergences of pitch variations of a atonal motive. This is index is conceptually similar to the notion of my penalties. For more details about the index of remoteness, see Boss (1991, pp. 40ff).

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Fig. 3.3 Two contrasting pitch configurations, used for the calculation of a possible maximal pitch penalty (kpmax )

is easy to conclude that many and many objects can be considered as “perfectly“ contrasting to a given reference. In other words, the pitch configuration that we search as “maximally” dissimilar for applying in the normalization process could be obtained in many ways, provided it contrasts sufficiently with the referential material.6 Given this, Fig. 3.3 proposes a possible “maximally-contrasting” pitch configuration related to the musical idea of Fig. 2.8, depicting also the stages for calculation of the pitch penalty, in this case corresponding to a maximum value (labeled, therefore, as kpmax ). Since only the domain of pitch is here in question,

6 Since we are dealing with a fuzzy, relative relation, is not necessary to create a specific contrasting limit for any referential idea one wants to analyze. This means that the maximal value here proposed for “maximal” dissimilarity will be adopted for all situations.

3.2 Measurement of Similarity in the Pitch Domain

41

Fig. 3.4 Calculation of pitch penalty in the comparison between Pp and C1p

both ideas are musically notated as open-headed notes, disregarding their respective rhythmic-metric contexts. The figure also presents the calculation of pitch penalty. The algorithm can be tested now with the five cases of archetypal variation of pitch domain presented in Fig. 2.12. Let us label the first variant of Pp as C1p, reproduced in Fig. 3.4 as a pitch abstraction. Before continuing, it is necessary to reflect a little on the particularity of this case. As previously stated, this variant was created by simple changing of the register of two pitches of Pp: the second one (E4 ), sent an octave higher to E5 , and the fourth (A4 ), transposed an octave lower to A3 . Intuitively, anyone understands these changes as very superficial transformations, a fact that is perfectly captured by the null values of v2 (which deals with pitch classes). On the other hand, these simple modifications disturb profoundly the intervallic vector (v3) which is incidentally the

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most weighted in the algorithm, resulting in a disproportionately high penalty, and distorting the perception of similarity between referential and variant UDSs. This leads us to consider the need of an adjustment of the algorithm, in order to enable it to treat similar situations in a more realistically manner. It is formulated as a rule (other similar rules will be eventually added in further stages): Adjustment rule # 1 (concerning the pitch domain): Any “12” inside v3 shall be reduced to one third (i.e., to “4”).7 The application of rule # 1 in the calculation of the pitch penalty for C1p results into a more appropriate, lower value: kp = 0.28. The second variant C2p is obtained from a chromatic transposition of Pp by four ascending semitones, as shown in Fig. 3.5: only the original pitch and pitch-class sequences are transformed. In spite of the lower value obtained for kp (corresponding to the intuition that both ideas are closely related), another adjustment in the algorithm becomes necessary, preventing eventual inconsistencies. Such an argument can be easily understood if we consider that chromatic transpositions by a different number of semitones would produce different values of penalties. This goes against the scientific knowledge8 that argues that distinct levels of transposition should affect equality (and at a very low rate) the evaluation of dissimilarity. As an illustration of this distortion, consider two different ascending-chromatic transpositions of Pp, by one and nine semitones. The application of the algorithm in both cases would result for kp, respectively, 0.04 and 0.33. Given this, a new adjustment rule must be proposed: Adjustment rule # 2 (pitch domain): If v1 and v2 are formed by the replication of a unique number, substitute its occurrences by “2” (in both vectors), not matter the magnitude of the original number. In the case of C2p, since v1 = , it is replaced by v1 = , just like v2 (this procedure will be adopted from now on for any possible transposition, that becomes, therefore, completely equivalent). Consequently, the pitch penalty of C2p is updated to 0.07. The literal pitch inversion C3p is shown in Fig. 3.6. Despite the number of gray cells in the variant’s matrix, and especially, the high value for the pitch penalty (0.57), this magnitude of divergence between parent and child is rather illusory. In fact, melodic inversion is in normal conditions easily perceived as a relatively close derivation, since the original intervallic pattern is strictly preserved, only with changed directions. As it can be observed in the exemplified case, a literal inversion is algebraically evidenced when v3 () matches exactly twice of the content of p3 (). So, another adjustment becomes necessary:

7 The choice of this rate of reduction results from empirical tests for adjusting the algorithm to many different cases. The same argumentation can be extended to the following adjustment rules. 8 See, for example, Huron (2006).

3.2 Measurement of Similarity in the Pitch Domain

43

Fig. 3.5 Calculation of pitch penalty in the comparison between Pp and C2p

Adjustment rule # 3 (pitch domain): In case of v3 equal to twice p3, set all of its internal values to “3”. A natural consequence of intervallic inversion is also the inversion of the melodic contour. Unlike the former, however, the latter is not so easily detected: inversion of the contour is only evidenced when the sum of the content of the p4 attributes of both referential and variant ideas results in a vector formed by the replication of a single integer, corresponding to the highest point of both contours. Since melodic and contour inversions can occur independently of each other, a complementary rule must be added:

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Fig. 3.6 Calculation of pitch penalty in the comparison between Pp and C3p

Adjustment rule # 4 (pitch domain): If the sum of the rows Pp(4, :) and Cp(4, :) results in a replicated integer (the highest melodic-contour point), set all values of v4 to “1”. Applying rules # 3 and 4 in the calculation, kp of C3p is reduced to 0.43, a more compatible penalty in this situation. The next variant (C4p) shares only the melodic contour with Pp, as shown in Fig. 3.7, which warrants a medium degree of similarity. Lastly, Fig. 3.8 depicts the calculation of pitch penalty for C5p, which keeps only the ambit as a common attribute with the referential material.

3.3 Measurement of Similarity in the Temporal Domain

45

Fig. 3.7 Calculation of pitch penalty in the comparison between Pp and C4p

3.3 Measurement of Similarity in the Temporal Domain Based on the edition-distance method for evaluating the pitch structure, an analogous algorithm was designed for the temporal domain. Firstly, we shall define the sections of P and C matrices that will be considered, namely, rows 5–7. Then we name these sections as, respectively, Pt (matrix of the temporal attributes of P) and Ct (matrix of the temporal attributes of C). Now, we can define four difference vectors: • v1: associated with the durational sequence (t1), such that v1 = |Ct(1, :) – Pt(1, :)|

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Fig. 3.8 Calculation of pitch penalty in the comparison between Pp and C5p

• v2: associated with the IOI sequence (t2), such that v2 = |Ct(2,1: end–1) – Pt(2,1: end–1)| • v3: associated with the metric contour (t3), such that v3 = |Ct(3, :) – Pt(3, :)| • v4: associated with the temporal span (t4), such that v4 = |Ct(2, end) – Pt(2, end)| As done before, the entries of the vectors v1 to v3 are summed and the results, together with the scalar v4, are ordered into a structure called temporal vector (vt), such that    .vt =< v1, v2, v3, v4 > .

3.3 Measurement of Similarity in the Temporal Domain

47

A four-entry vector of weights, calibrated for the temporal characteristics is then established as wt =< 15, 45, 30, 10 > .

.

This reflects the intuitive assumption that the intervals between onsets (IOIs) are the most relevant factor in the definition of a rhythm, followed by its metric configuration. According to this conception, the durations and the temporal span are only secondary descriptors. Like the case of pitch, a provisional penalty for temporal domain (denoted as kt’) is calculated as the dot product of vt and wt. Once again, for a matter of coherence, kt’ must be normalized, which requires a minimal and maximal value for temporal penalty. Since ktmin (corresponding to an exact copy of the rhythmic configuration of the referential UDS) can be set to zero, the normalized temporal penalty (kt) will result from the division of kt’ by a (possible) maximal value (ktmax ). As discussed in the last section, since the process of measurement of similarity involves always objects which are someway related, it would suffice to conceive a rhythmic configuration sharply contrasting in relation to Pt (among many possible alternatives), as shown in Fig. 3.9. As a test for the algorithm for measuring temporal similarity, I select the four variants introduced in Fig. 2.17 (let us label them as C1t, C2t, C3t, and C4t). Figure 3.10 reproduces C1t, comparing it with Pt. As it can be observed, this is a very close variant, obtained simply by the application of rests. Consequently, IOI sequence and metric contour are preserved, just some durations and the temporal span are affected, resulting, as expected, in a very low value for kt. Variant C2t is a perfect copy of Pt, only metricly dislocated by one eight-note (Fig. 3.11). However, this simple transformation is responsible for a medium value of dissimilarity in relation to Pt, evidencing the central importance of meter in rhythmic characterization. Variants C3t and C4t are compared to Pt in Fig. 3.12, since both involve similar types of derivation: while the former duplicates the original durations (an augmentation, in the compositional jargon), the latter halves them (characterizing a diminution). In spite of being substantially associated with the same procedure, the respective temporal penalties have disparate magnitudes (0.98 and 0.41). Moreover, such high values are very counter-intuitive, since in both cases durational and IOI proportions are kept unaltered. Indeed, the parenthood between a given rhythmic configuration and its augmented or diminished versions is easily perceived, everything else being constant (as, for example, in subject transformations of some of Bach’s fugues). A third point concerns the odd discrepancy between the metric contours of C3t and C4t. While the latter replicates the original attribute, the augmented rhythm is considerably distinct, also contrary to what would be expected (since durational proportions are strictly maintained). Let us examine firstly this last issue. As a matter of fact, the discrepancy of C3t’s metric contour is only apparent: because the durations were duplicated, it would be also necessary to consider the duplication of the metric contour, in other words,

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Fig. 3.9 Two contrasting temporal configurations, used for the calculation of a possible maximal temporal penalty (ktmax )

treat the variant in hypermetric terms.9 That is, the segment shall be considered as a unit, in this case, formed by two . 44 measures (or, alternatively, a . 84 measure). From this, the metric contour of C3t shall be rewritten as , thus mapping the original contour. This obviously impacts the calculation of the corresponding temporal penalty, which is then updated: kt .≈ 0.82. Allying this to the question of the high (and discrepant) values of dissimilarity of both variants, a new adjustment rule (with three stages) must be formulated, in order to improve the algorithm. Adjustment rule # 5 (temporal domain): If the division of the elements of row Pt(1, :) by the corresponding elements of row Ct(1, :) results into a replicated number x (typically, x = 2 or 1/2), then

9 Hypermeter

will be a topic to be addressed in detail in Part III.

3.3 Measurement of Similarity in the Temporal Domain

49

Fig. 3.10 Calculation of pitch penalty in the comparison between Pt and C1t

1. set all values of v1 and v2 to “1”. Formally, if = , then v1 = v2 = ; 2. replace v3 with the content of row Pt(3, :). Formally, v3 = Pt(3, :); 3. set v4 to “1/2”. Applying this rule to the algorithm, the penalties of both C3t and C4t are recalculated to a more realistic value (becoming obviously identical): kt .≈ 0.23.

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Fig. 3.11 Calculation of pitch penalty in the comparison between Pt and C2t

3.4 Measurement of Similarity in the Harmonic Domain Due to the particular structure of the harmonic section of the matrix of attributes, the algorithm for measurement of harmonic similarity between two UDSs presents some differences in relation to those designed for pitch and time. Firstly, there is just one vector (and not five or four, as in the pitch and temporal domains), called harmonic vector (vh) for measurement of the “distances” between the respective attributes of P and C. Moreover, contrarily to the previous algorithms, the harmonic’s does not express numeric differences, but relations of congruence and divergence. This is denoted by the use of binary notation (0 = congruence / 1 = divergence), and by mapping any entry of vh to one harmonic attribute (i.e., h1

3.4 Measurement of Similarity in the Harmonic Domain

Fig. 3.12 Calculation of pitch penalty in the comparison between Pt and C3t and Pt and C4t

51

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to h5). In this manner, a perfect copy of a harmonic context will be represented by vector vh = . Let the vector of harmonic weights be formatted as10 wh =< 45, 25, 15, 10, 5 > .

.

As done before, the provisional harmonic penalty (kh’) is calculated as the dot product of vh and wh. Likewise, the normalized harmonic penalty (kh) is equal to the division of kh’ by the maximal possible harmonic penalty (khmax ). However, contrarily to what was applied to the other algorithms and since vh is a binary vector, it is not necessary to elaborate a hypothetical contrasting example for the calculation of khmax , considering that a maximally-divergent harmony will be obligatorily expressed as vh = . Consequently, khmax =< 1, 1, 1, 1, 1 > . < 45, 25, 15, 10, 5 >= 45 + 25 + 15 + 5 = 100.

.

Let us now test the algorithm with the isolated harmonic variants presented in Fig. 2.17 and reproduced in Fig. 3.13 with the calculation of the respective harmonic penalties. Another difference in relation to the remaining algorithms concerns the fact that, in general, more than one event of a given UDS has the same harmonic context (see, for example, Fig. 1.1). This means that a replicated harmonic context in, say four events, will necessarily replicate the harmonic penalty four times. In fact, the harmonic penalty for a UDS as a whole will then be calculated as the arithmetic mean of the individual penalties. Due to their relative simplicity and similar structure, the three algorithms above presented can be computationally implemented without much effort. This initiative provides a lot of agility and efficiency in the calculation of the respective penalties. With this purpose, Appendix B suggests possible versions for each algorithm, formatted as pseudo-codes.

3.5 Global Similarity Between Musical Ideas After obtaining the penalties concerned to the three individual domains by the respective algorithms, a global measure of the dissimilarity k between two related UDSs can be calculated. This is made through a weighted combination of the three values (kp, kt, and kh), as depicted in Eq. 3.4. The distribution of weights attempts to capture two intuitions: firstly, the strong prominence of the primary domains

10 Like in the case of the adjustment rules, the weights of vector wh were obtained after a long phase of tests. Evidently, these are not at all fixed values. Eventual modifications either of the weight distribution or of the specific values can be made in the future, if necessary.

3.5 Global Similarity Between Musical Ideas

53

Fig. 3.13 Calculation of harmonic penalties in the comparison between Ph and C1h, C2h, C3h, and C4h

(pitch and time) in face of the secondary harmony (85% vs 15%), and secondly, the relative greater permeability of pitch structures to variation in comparison with rhythmic/metric configurations.11 k=

.

3.5kp + 5kt + 1.5kh 10

(3.4)

Some examples will now provide an adequate illustration for the process of determination of a global penalty in real musical situations. The next figures depict six possible variants of a referential idea P, considering transformations that

11 For

some support for these arguments under a cognitive perspective, see Dowling (1978) and McAdams and Matzkin (2001).

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3 Measurement of Similarity (I)

affect simultaneously (and in different degrees of intensity) elements of the pitch, temporal, and harmonic domains.12 Variant C1 (Fig. 3.14) presents very slight modifications in pitch and temporal structures in relation to P. A secondary dominant (V/V, chord label D7 ) is inserted as preparation for the last chord. In consequence of this, the individual penalties for dissimilarity are considerably low, resulting into a also low global value (k = 0.20). The melodic line of C2 (Fig. 3.15) is very close to the original configuration, with only a chromatic modification of the first pitch. However, this simple alteration obligates a drastic change of the harmonic context, suggesting a different home key, D minor, which results in an almost maximal harmonic penalty (kh = 0.97). Since there is none alteration in the temporal matrix of C2 (and therefore kt = 0.00), the computation of k yields also a relatively low value. As depicted in (Fig. 3.16), C3’s rhythm is formed by an almost exact augmentation of the original sequence of durations (except by the rest before the fourth event). As discussed before, this implies the application of adjustment rule # 5 (which deals with the cases of augmentation/diminution), resulting in a penalty of 0.23. In spite of the change to the minor mode, the pitch penalty is still lower (0.13), due mainly to the short intervals between events 2 and 4 (both of one semitone) and the register change of the first pitch C (which triggers adjustment rule # 1). More expressive modifications occur in the harmonic domain (kh = 0.44), though not impacting so much in the global dissimilarity (0.23). In the case of C4 (Fig. 3.17), is the pitch the domain more intensely modified (0.34); it is counterbalanced by a small rhythmic variation of the penultimate event. In the harmony, two passing chords are added, but this do not cause much impact on the calculation of kh. As shown in Fig. 3.18, the rhythmic configuration of variant C5 is entirely preserved (kt = 0.00), supporting an expansion of the melodic arc that keeps only the ambit and some of the original abstract contour. On the other hand, the harmony is radically modified, suggesting a transposition to the key of D minor, and reaching the maximal value for kh. At last, C6 (Fig. 3.19) brings a more substantial modification in the temporal structure, which affects decisively the global penalty (the highest value of the six cases, k = 0.40). In compensation, there is a single modification in the pitch domain and none in the harmonic context. Table 3.1 summarizes the k values assigned to the six variants.

12 Since we assume that the process was now computationally automatized, for a reason of simplicity, only the final results are provided, omitting all underlying calculations.

3.5 Global Similarity Between Musical Ideas

Fig. 3.14 Determination of global penalty for variant C1

55

56

Fig. 3.15 Determination of global penalty for variant C2

3 Measurement of Similarity (I)

3.5 Global Similarity Between Musical Ideas

Fig. 3.16 Determination of global penalty for variant C3

57

58

Fig. 3.17 Determination of global penalty for variant C4

3 Measurement of Similarity (I)

3.5 Global Similarity Between Musical Ideas

Fig. 3.18 Determination of global penalty for variant C5

59

60

Fig. 3.19 Determination of global penalty for variant C6

3 Measurement of Similarity (I)

3.6 Spatial Representation of Similarity Relations

61

Table 3.1 Penalties obtained for variants C1–6

Variant

Penalty (k)

C1 C2 C3 C4 C5 C6

0.20 0.22 0.23 0.19 0.33 0.40

Fig. 3.20 Model of derivative space (adapting Fig. 1.8)

3.6 Spatial Representation of Similarity Relations With these data and recalling to the derivative space’s model of Sect. 1.4, it is also possible to plot similarity relations between musical ideas, an interesting perspective. Figure 3.20 reproduces the graph of Fig. 1.8, replacing the variable x with the penalty for dissimilarity k. For this, the calculation of the penalty value is associated with the horizontal projection (now expressing coordinate k) of point C.13 Since the vertical projection is a function of k (i.e., f(k) = k – 1), the coordinates of the point C are then properly established. The angle .α is easily obtained through trigonometry (Eq. 3.5):14 α = arctan(

.

13 This

1−k ) k

(3.5)

association reflects the fact that the penalty represents ultimately the amount of divergence between referential and derived ideas. 14 As a convention, angle .α is measured clockwise. For this reason, Eq. 3.5 uses a negative value for f (k), that is, 1 – k, turning .α positive. The complementary angle .β equals 90.◦ minus .α.

62 Table 3.2 Coordinates of points C1–6, with respective angles .α and .β (rounded values) and degrees of similarity with P

3 Measurement of Similarity (I) Variant C1 C2 C3 C4 C5 C6

Coordinates

.α

.β

(0.20, .−0.80) (0.22, .−0.78) (0.23, .−0.77) (0.19, .−0.81) (0.33, .−0.67) (0.40, .−0.60)

76.◦

14.◦

74.◦ 73.◦ 77.◦ 64.◦ 56.◦

16.◦ 17.◦ 13.◦ 26.◦ 34.◦

Similarity High High High High High Medium-high

Fig. 3.21 Points C1–6 plotted in the derivative space of P

Table 3.2 updates Table 3.1 by including the coordinates of the six variants, the angular values (in degrees), and respective degrees of similarity. As one can observe on the derivative space of P (Fig. 3.21), the almost equal, low penalties for variants C1, C2, C3, and C4 imply that the corresponding points are plotted very closely each other on the edge QR. It is equivalent to say that they have high similarity with the referential idea, all inserted as a cluster in the “southernmost” band.15 On the other hand, variants C5 and C6 lay on the mediumhigh region. This chapter introduced a relatively simple and straightforward strategy for measuring similarity between two related musical ideas with the same number of elements (or equal cardinality), freely inspired on the method of Levenshtein distance, as proposed by Godfried Toussaint. The multidimensional nature of the musical descriptors that are in use in MDA, organized in the matrix of attributes, demanded the creation of weight vectors calibrated differently in each of the three domains. These weights attempt to capture intuitions about the greater importance of certain descriptors in front of others. Nonetheless, they should not be considered at all as fixed values. The same would be said about the five adjustment rules which were implemented in the algorithms. 15 It

is easy to conclude that the degree of similarity of a variant in relation to P can be calculated as the subtraction of its respective penalty from 1, that is s = 1 – k, or the coordinate “y” in the DS.

References

63

It is possible that new rules can be added in the future, aiming at the improvement of the system. In any case, the methodology presented here should be seen as an initial step in a process that can be made more effective and accurate with further analytical applications and eventual route corrections. The measurement of similarity of musical ideas in more complex situations, involving non-compatible cardinalities, as well as their corresponding spatial representation will be properly addressed in Chap. 5, after an indispensable examination of the concept of transformational operations, as it follows.

References Boss, J. (1991). An analogue to developing variation in a late atonal song of Arnold Schoenberg. Dissertation, Yale University. Dowling, J. (1978). Scales and contour: Two components of a theory of memory for melodies. Psychological Review, 85(4), 341–354. Huron, D. (2006). Sweet anticipation: Music and the psychology of expectation. Cambridge: The MIT Press. McAdams, S., & Matzkin, D. (2001). Similarity, invariance, and musical variation. In The biological foundations of music (Vol. 930, pp. 62–76). Annals of the New York Academy of Sciences. New York: New York Academy of Sciences. Toussaint, G. (2013). The geometry of musical rhythm: What makes a “good” rhythm good? Boca Raton: CRC Press.

Chapter 4

Transformational Operations

Up to this point, high-level variation was described as a generic function V, capable of transforming a given referential musical idea P. As it was also discussed, the outcome of such transformation (namely, the variant C) is related to P by some degree of similarity, which can, in the case of compatible cardinalities in the scope of our model MDA, be systematically measured. This measurement is accomplished through the isolation of P’s and C’s structural domains and respective attributes, organized in the format of matrices. Furthermore, the idea of derivative space provides a geometric representation of the similarity relations between P and C. In sum, in a derivative-work situation, we can affirm that C derives from P, maintaining with this some amount of similarity. However, up to now nothing has yet been saying about how P could be transformed into C, or by which manners the application of V could affect P. These questions will be properly addressed in the present chapter, with the aid of a new concept, transformational operations. A transformational operation, or simply, an operation, corresponds therefore to a specific manifestation of the generic function V formatted as an algorithm or set of rules to be applied to a referential element.1 Such referential element can address either the complete idea P (in the case of holistic variation) or, if we consider decomposable variation, a specific attribute, or even a subset of it.2 In more formal terms, one can express a given (decomposable) operation as a function .υ, whose argument—the parent .ρ—is an abstracted subset of the whole referential idea P, and whose output (the child) .γ , is a related transformation of 1 In

his Fundamentals of Musical Composition, Schoenberg (1967, p. 10) lists 19 types of variations which with a motive can be changed, including exemplification. Several of these types correspond to my operations. 2 Although it is also possible to consider operations in the realm of holistic variation, I will not explore this issue here, opting to center the focus on decomposable operations. This is mainly because holistic operations are normally strongly idiosyncratic and contextual-dependent, which prevents proper systematization. From now on, the use of the term “operation” in this section will refer exclusively to decomposable variation. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3_4

65

66

4 Transformational Operations

ρ and, therefore, also an abstracted element. Equation 4.1, derived from Eq. 1.1, exposes formally the basic syntactic structure of the action of an operation.

.

γ = υ(ρ)

.

(4.1)

4.1 A Set-Functional Conception At this point, it is possible to update the set representation of decomposable variation proposed in Sect. 2.2 by including the new elements .ρ, .γ , and .υ. Figure 4.1 replaces the three abstracted sets of Fig. 2.3 with the generic set Sx (x standing for p, t, or h). In this diagram, .υ represents, as before, a more specific transformational function. As a matter of fact, .υ (like V) can be seen as a conceptually infinite set (Fig. 4.2) that contains all possible transformations (operations, in the present terminology) capable to send a given .ρ to a related .γ in a domain-compatible set Sx . As a consequence, one can think of a number (conceptually also infinite) possible derivations (.γ ) from a given referential .ρ, through applications of operations which are members of set .υ. In more informal terms, it is possible to obtain an offspring of variants of any size from .ρ (a biological-like image that will be explored in Part II),

Fig. 4.1 Representation of the action of function .υ on set Sx Fig. 4.2 Set .υ of operations

4.2 Classifications of Operations

67

Fig. 4.3 Actions of functions .υ

as suggested in Fig. 4.3. In this scenery, the functions .υ1, .υ2, .υ3, etc. will represent real specific instances of operations, the very elements which will be examined and described in details in this chapter.

4.2 Classifications of Operations Operations can be classified according to three categories: (1) type; (2) domain of application; (3) scope of the argument.

4.2.1 Types Regarding the type, operations can be canonic and non-canonic. The former category corresponds to “classical” instances of transformation (some of their outputs were already commented in the previous chapter, then called “archetypal variations”). There are seven canonic operations considered in MDA: transposition (chromatic and diatonic), inversion (chromatic and diatonic), retrogradation, augmentation, and diminution. Non-canonic class contains any transformation out of canonic operations, forming a conceptually infinite set of possibilities (I select a small part of this set for use in MDA, which will be further on properly identified). A symbolic notation is used for designating both types in analysis, according to the following conventions: • Canonic operations are identified by a unique bold, uppercase initial of the operation’s name (lowercase letters will be used for denoting diatonic versions of chromatic operations). They are: augmentation (A), diminution (D), retrogra-

68

4 Transformational Operations

dation (R), chromatic transposition (T), diatonic transposition (t), chromatic inversion (I), and diatonic inversion (i); • Non-canonic operations are identified by an acronym formed by three bold, uppercase letters (preferentially, but not necessarily, the initial letters of the operation’s name), for example: addition (ADD), permutation (PER), metric displacement (MTD), change of register (OCT), etc.

4.2.2 Domains of Application Five possibilities are considered: (1) Operations of exclusive application to the pitch domain; (2) Operations of exclusive application to the temporal domain; (3) Operations of exclusive application to the harmonic domain; (4) Operations that can be applied to both the pitch or temporal domains; (5) Operations that are applied to both the pitch and harmonic domains. Categories 4 and 5 deserve some commentary. While the former comprises operations that share both name and algorithm, but behave distinctly according to the domain in question, the latter is associated with the notion of hybrid decomposable variation, introduced in Sect. 2.2. In operations of this kind, pitch and harmony are taken together, forming a unit of reference, and attributes of both domains are affected by a combination of their action.

4.2.3 Scope of the Argument As before stated, the argument of an operation .υ can be formed by the integral parent ρ or a subset of it (generically labeled as .μ). While the former case corresponds to the conventional situation in decomposable variation (and, accordingly, will be called “normal”), the latter will be classified as mutational operation, a very important element of the current version of MDA. Like mutations that occur inside the nucleus of living cells, mutational operations applied in a musical context promote micro-variations, normally a very gradual transformational process. The notation of the argument of a mutational operation differs from the normal case by the indication of the subset of application, after a comma (Eq. 4.2).

.

γ = υ(ρ, μ)

.

(4.2)

Where .μ is a subset of .ρ, the basic argument to be transformed by the operation.3 Figure 4.4 schematizes the actions of the two modes.

3 In

the case of normal operations .ρ is the only argument of .υ.

4.2 Classifications of Operations

69

Fig. 4.4 Representation of the action of normal (a) and mutational operations (b)

Some operations are dual, i.e., they can be applied in both versions, normal and mutational.

4.2.4 Complementary Symbology Some operations require special, complementary symbology for more precise description of their transformational actions. The complementary symbol is subscribed at the right of the identification letter (or acronym, in the case of non-canonic type) of the respective operation, and can be a number or a typographic sign. There are several specific situations (to be individually addressed, along with the descriptions of the operations) that the complementary symbols refer to: something that is also associated with the very nature of the operation, as, for example: T3 , OCT+ , ADD5 , etc.

4.2.5 Alteration of Cardinality Some pitch or rhythmic operations have the propriety of increasing or decreasing the cardinality of .ρ. They are potentially transformations of greater impact, since the compatibility of cardinalities is a basic requirement for measurement of similarity between two musical ideas.4 Notwithstanding this, some of such operations produce only superficial modifications, as inclusion of melodic embellishments in a refer-

4I

will return to this discussion in Chap. 5.

70

4 Transformational Operations

ential idea. These cases normally require integrated transformations affecting both pitch and temporal structures, not necessarily involving the same type of operation.

4.2.6 Collateral Effects Eventually, the application of an operation to a given attribute (say, p1) can indirectly modify related attributes of same domain (p3, and p4, for example).5 These secondary attributes are then said as collaterally affected by the operation, and therefore must also be rewritten in the matrix of attributes of child .γ .

4.3 Descriptions This section describes the twenty-seven operations that currently form the set .υ in MDA.6 They are listed in alphabetic order, being accompanied by information with respect of symbolic notation, type, scope, etc. Each operation is exemplified in one or two situations (when can be applied to more than one domain). When dual application is possible, just one mode (normal or mutational) is selected for illustration. In all cases, the attributes affected by the operations are highlighted in the respective matrix sections (in both parent and child), and the algebraic processes for transformation are also informed.

4.3.1 Addition • • • • • •

Symbolic notation: ADDx , where x is a positive integer. Type: non-canonic. Scope: dual. Target attributes: p3 or d-p3 (pitch) or t2 (time). Description: adds x units of melodic or durational intervals to the target attribute. Alteration of cardinality of γ : no.

While the application of ADD in the pitch domain implies an increase of an original ascending interval (for example, 3 + 1 = 4), descending intervals can be contracted (e.g. −3 + 1 = −2), have their directions reversed (e.g. −1 + 3 = 2), or even neutralized (e.g., −4 + 4 = 0), resulting into pitch repetition.7 5 In

rare cases, also of a different domain, as it will be presented. is opened to the eventual inclusion of new operations, insofar it is necessary. 7 In the mirrored version of this operation, subtraction, a supplementary complication can occur, as it will be seen. 6 MDA

4.3 Descriptions

71

Fig. 4.5 Examples of application of operator addition

Because temporal ADD (normal or mutational) is directly applied to the onset sequence (t2), it does not affect the original durational sequence (t1). The insertion of rests between elements (increasing, therefore, IOI sizes) becomes a natural consequence of its application (Fig. 4.5).

4.3.2 Augmentation • • • • • •

Symbolic notation: A. Type: canonic. Scope: dual. Target attribute: t1 (time). Description: duplicates the original durational values. Alteration of cardinality of γ : no.

Figure 4.6 exemplifies a mutational application of operator A, considering just the second duration, transformed from a eighth into a quarter note. How paradoxical this may sound, the derivative impact of this operation in mutational mode is normally higher than in normal case, in spite of affecting only a subset of the attribute. This is due to two factors: firstly, the augmented outcome is proportionally related to the referential configuration (an instance of archetypal variation, as seen before), and secondly, mutational augmentation inevitably results in metric disturbance (in unpredictable rates), like the sequence of syncopations in Fig. 4.6, what does not occur in normal-scope situations.

72

4 Transformational Operations

Fig. 4.6 Example of application of operator augmentation

4.3.3 Change of Mode • • • • •

Symbolic notation: MOD. Type: non-canonic. Scope: normal. Target attribute: p1 (pitch). Description: alters chromatically some pitches, in order to map them to the diatonic collection associated with the parallel key in relation to the original (ex: A Major → A minor). • Alteration of cardinality of γ : no. The operation MOD involves two integrated domains, pitch and harmony and, therefore, shall be accomplished in a pair of stages: (1) chromatic alteration of pitches that are exclusive of the original key (i.e., C major in Fig. 4.7) but not present in the diatonic collection of the parallel mode (C minor, in this case);8 (2) inversion of attribute h2 (dedicated to mode information in the harmonic vector), which is done through addition of one unit under modulo 12, maintaining fixed the remaining attributes, as shown at the bottom of the figure.9

4.3.4 Change of Register • • • •

Symbolic notation: OCT∗ .10 Type: non-canonic. Scope: mutational. Target attribute: p1 (pitch).

8 Because only a subset of the referential diatonic collection is affected by MOD (three elements, at most), this operation could be considered a hybrid category, laying between normal and mutational. 9 This stage must evidently be replicated for all events involved. 10 In analytical situations, the symbol “∗” is replaced by “+” or “−” for, respectively, ascending or descending octave displacement.

4.3 Descriptions

73

Fig. 4.7 Example of application of operator change of mode

Fig. 4.8 Example of application of operator change of register

• Description: changes the register of selected pitches, either ascending or descending. • Alteration of cardinality of γ : no. This operation is applied only in mutational mode (otherwise, it would produce an exact copy of ρ, but dislocated in register). In the example of Fig. 4.7, MOD affects only the first original note, transposing it an octave lower (Fig. 4.8).

4.3.5 Chordal Inversion • • • • •

Symbolic notation: CHI. Type: non-canonic. Scope: mutational. Target attribute: h5 (harmony). Description: changes uniquely the bass of a chord, keeping the remaining parameters unaltered.

74

4 Transformational Operations

Fig. 4.9 Example of application of operator chordal inversion

• Alteration of cardinality of γ : no. This operation is, in strict sense, of mutational scope, since just one element (namely, the bass note) is modified (Fig. 4.9).

4.3.6 Chromatic Alteration • • • • • •

Symbolic notation: ALT∗ Type: non-canonic. Scope: mutational. Target attribute: p1 (pitch). Description: alters chromatically selected pitches. Alteration of cardinality of γ : no.

Ascending or descending chromatic alteration can be considered similar to the mutational version of operation ADD1 (or SUB1 ). However, while the latter cases are applied to intervals (p3), the former is related specifically to pitches (p1). Nonetheless, a more subtle aspect makes chromatic alteration different from mutational chromatic transposition by one semitone (T1 or T−1 ), since both types are applied to pitches.11 In more precise terms, the output of ALT will be always a chromatic version of the original pitch (say, G-G or G-G), which is not mandatory in the case of the semitone transposition (or else, G-A or G-F would also be possible alternatives). A simple example of the application of ALT is provided in Fig. 4.10. An interesting collateral modification occurs in the melodic contour of γ , since the chromatic alteration eliminates the original repetition of pitch G. Observe, however, that despite the difference expressed by both the numeric sequences ( and ), the contour of ρ was substantially preserved in the variant.

11 Broadly

speaking, it is possible consider ALT as a special case of Tx , with x = 1 or −1.

4.3 Descriptions

75

Fig. 4.10 Example of application of operator chromatic alteration

Fig. 4.11 Example of application of operator chromatic inversion

4.3.7 Chromatic Inversion • • • • •

Symbolic notation: I Type: canonic. Scope: dual. Target attribute: p3 (pitch). Description: inverts chromatically (i.e., considering the chromatic scale) the direction of selected intervals. • Alteration of cardinality of γ : no. Chromatic inversion can affect the harmonic domain, especially in normal applications, as is exemplified in Fig. 4.11. As it can be noted, the tonal context was changed from C Major to F Minor, an inherent consequence of Hugo Riemann’s principle of harmonic duality. On the other hand, mutational inversion (chromatic or diatonic) of a given interval implies necessarily alteration in pitch (and, frequently, also contour and ambit).

4.3.8 Chromatic Transposition • Symbolic notation: Tx . • Type: canonic. • Scope: dual.

76

4 Transformational Operations

Fig. 4.12 Example of application of operator chromatic transposition

• Target attribute: p1 (pitch). • Description: adds a number x of units to selected pitches.12 • Alteration of cardinality of γ : no. Normal chromatic transposition (Fig. 4.12) preserves p3, p4, and p5, which turns out to be a very close relationship with the referential idea. On the other hand, mutational chromatic transposition, by altering just one (or a few) pitch(es), is capable of producing more substantial differences, since it disturbs also the melodic contour, besides the pitches themselves.

4.3.9 Contextual Dyadic Transformation • • • • • •

Symbolic notation: G or G’ Type: non-canonic. Scope: mutational. Target attribute: d-p3 (pitch). Description: inverts the selected diatonic interval. Alteration of cardinality of γ : no.

This is a special and unique operation, since it is applied only in mutational scope and has as exclusive target attribute d-p3, that is, it affects only diatonic intervals. Moreover, it is a contextual operation, since its output depends on which note of the interval is selected. The operation was created by American scholar Scott Murphy, who was based on correspondences detected in the openings of the piano sonatas in F minor composed by Brahms and Schumann, which can be associated with derivative processes explained by that operator. Basically, it inverts distonically a selected interval: a second becomes a seventh, a third becomes a sixth, and a fourth becomes a fifth (and vice-versa). It has two modes (Fig. 4.13): while G keeps the second note of the interval on the original position and moves the first one according

12 Transpositions

can be made by ascending (x > 0) or descending intervals (x < 0).

4.3 Descriptions

77 d-p3 = d-p3 = -5 2 1 -1

G(d-p3, 2:3)

59 11 -8 5 1 -1

d-p3 = G'(d-p3, 2:3)

72 64 72 74 72 -8 5 1 -1 1 0 1 2 1

Fig. 4.13 Example of application of operator contextual dyadic transformation

to the application of the inversion, the action of G’ affects the second note, keeping the first one on the original position.13

4.3.10 Deletion • • • • •

Symbolic notation: DEL. Type: non-canonic. Scope: mutational. Target attributes: p1 (pitch) or t1 (time). Description: deletes a selected internal member of a sequence of pitches or durations. • Alteration of cardinality of γ : yes. Deletion is the first operator of this list whose action affects the original cardinality, in this case, by reducing it in one element. Like addition, it can be applied to both pitch and temporal domains, as exemplified in Fig. 4.14. On the other hand, it operates only in mutational mode (otherwise, the complete target sequence would be deleted). Additionally, its action is restricted to the internal pitch/temporal events.14

4.3.11 Diatonic Inversion • Symbolic notation: i. • Type: canonic. 13 For more details, see Murphy (2022). I thank to Dr. Murphy for so kindly allowing the operation

to be incorporated into the MDA structure. 14 As it will be presented, a related operation (SUP) is employed exclusively for suppression of the

last event(s) of a given sequence.

78

4 Transformational Operations

Fig. 4.14 Examples of application of operator deletion

-5

2 1 -1 -3

= =

5 -2 -1 1 3

Fig. 4.15 Example of application of operator diatonic inversion

• • • •

Scope: dual. Target attribute: d-p3 (pitch). Description: inverts the directions of the original diatonic intervals. Alteration of cardinality of γ : no.

Because i is applied only to diatonic intervals, its output γ will always pertain to the same diatonic collection of the referential idea ρ (compare the diatonic inversion of Fig. 4.15 with the chromatic inversion of Fig. 4.11).

4.3.12 Diatonic Transposition • Symbolic notation: tx . • Type: canonic.

4.3 Descriptions

79

5

Fig. 4.16 Example of application of operator diatonic transposition

• • • •

Scope: dual. Target attribute: p1 (pitch). Description: adds a fixed amount of diatonic units15 to selected pitches. Alteration of cardinality of γ : no.

As aforementioned, a diatonic interval can have different numbers of semitones, depending on the case, which is due to the intrinsic asymmetry of diatonic scales. This sort of discrepancy is well illustrated in Fig. 4.16: in this case, a diatonic transposition by five ascending diatonic steps (t5 ) in mutational mode applied to the fourth and fifth events of ρ results in different numbers of semitones, respectively, eight and nine.16

4.3.13 Diminution • • • • • •

Symbolic notation: D. Type: canonic. Scope: dual. Target attribute: t1 (time). Description: divides selected durations by two. Alteration of cardinality of γ : no.

Figure 4.17 exemplifies a mutational application of D, whose output disturbs heavily the original rhythmic structure, causing strong a syncopation, similar to that resulted from mutational augmentation (see Fig. 4.6).17 Reinforcing what was commented before, mutational application of both operations causes normally a

15 Respecting

the function the pitches play on the diatonic context in question. an interesting approach concerning application of diatonic and chromatic operations in a distinct context, see Steven Rings (2011). 17 Not surprisingly, as one can easily perceive, diminution and augmentation are inversely correlated. They are said mirrored versions of each other (other pairs of mirrored operators will eventually be formed in this list). 16 For

80

4 Transformational Operations

Fig. 4.17 Example of application of operator diminution

much higher derivative impact than in normal mode, since the original proportions (and metric dispositions) are inevitably affected.18

4.3.14 Extension • • • • •

Symbolic notation: EXTx . Type: non-canonic. Scope: mutational. Target attributes: p1 (pitch) and t1 (time). Description: inserts x elements after the last event of a pitch or a durational sequence. • Alteration of cardinality of γ : yes. The subscript variable x indicates how many elements must be included, but not their specification, i.e., which pitches or durations are considered.19 The examples of Fig. 4.18 depict extensions of different sizes: one pitch (x = 1), and three durations (x = 3). Considering that the elements are inserted after the end of the original sequence, normally the derivative impact resulting from the application of the operation will be low since the first elements of a musical idea are almost invariably the main responsible for the establishment of its identity.

4.3.15 Interpolation • Symbolic notation: INTx . • Type: non-canonic.

18 An analogous situation, considering pitch domain, involves mutational and normal transpositions. 19 In this sense, EXT inaugurates the special subclass of non-qualitative operations, that is, those that do not specify the elements to be transformed. Other operators of this subclass will be presented in due time.

4.3 Descriptions

81

Fig. 4.18 Examples of application of operator extension

• Scope: mutational. • Target attributes: p1 (pitch) and t1 (time). • Description: inserts x elements in between two contiguous members of a sequence of pitches or durations. • Alteration of cardinality of γ : yes. Like EXT, INT increases the cardinality of γ in x elements. Once again, the notation does not reveal which element(s) must be inserted, just its/their position in γ .20 Nonetheless, it is possible to consider as a default norm that the interpolated elements tend to subdivide equally the space they are destined to. Something similar occurs in the examples of Fig. 4.19: in the pitch domain the interval of a minor third between the second and the third elements (E-G) is diatonically partitioned by the included F, while the four onsets inserted in between the fourth and fifth elements in the rhythmic case subdivide a quarter note into four equal durations.

4.3.16 Merging of Durations • • • •

Symbolic notation: MRG. Type: non-canonic. Scope: mutational. Target attributes: t1 (time).

20 INT belongs, therefore, to the non-qualitative-operation subclass. Non-qualitative operations are inherently more flexible than their counterparts (i.e., the qualitative ones), because involve an indefinite number of possibilities.

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4 Transformational Operations

Fig. 4.19 Examples of application of operator interpolation

Fig. 4.20 Example of application of operator merging of durations

• Description: merges two contiguous selected durations into a single one. • Alteration of cardinality of γ : yes. A necessary consequence of MRG is a reduction of cardinality of one unit in γ . The durational span (t4) is kept unaltered, as exemplified in Fig. 4.20.

4.3.17 Metric Displacement • • • • •

Symbolic notation: MTDx . Type: non-canonic. Scope: normal. Target attributes: t3 (time). Description: Dislocates the metric configuration of ρ by x durational units, x = 0. • Alteration of cardinality of γ : no.

4.3 Descriptions

83

strongest

Fig. 4.21 Example of application of operator metric displacement

Because t3 is rather a sequence of hierarchies than of concrete values (like pitches or onsets, for example), the argument of the operation must be viewed, differently from the remaining cases, as an abstract representation of what is concretely modified through a temporal dislocation of the whole rhythmic structure. In fact, the action of MTD causes “right” dislocation of the whole rhythmic sequence, taken as a block, when x > 0 (said as positive MTD), and “left” dislocation when x < 0 (negative MTD). In the former situation, a pause with a value equal to x shall be inserted before the beginning of the original sequence (as it occurs in Fig. 4.21). Incidentally, the output of the example is also equivalent to a twosteps rotation21 of the original metric contour, as suggested by the arrows below the rhythmic matrix of γ .

4.3.18 Permutation • • • • • •

Symbolic notation: PER. Type: non-canonic. Scope: mutational. Target attributes: p1 (pitch) or t1 (time). Description: re-arranges the original order of events.22 Alteration of cardinality of γ : no (Fig. 4.22).

21 Operator

to be described. permutation of an ordered sequence will be expressed using group-theory cycle notation. For example, let s be the original order of any sequential four events , and s1 the desired re-arranging of events. Then this special permutation will be noted as PER(s, (1)(234)), where the second argument informs that event 1 is sent to the first position (or else, it is not permuted with any other), event 2 is sent to the third position, event 3 to the fourth position, and finally, event 4 is sent to the second position. 22 A

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(p1, 135(24))

(t1, 123(45))

Fig. 4.22 Examples of application of operator permutation

4.3.19 Re-harmonization • • • • • •

Symbolic notation: RHA. Type: non-canonic. Scope: mutational. Target attributes: h3/h4.23 Description: reformulates a temporal span formed by two contiguous durations. Alteration of cardinality of γ : no.

This operation is unique among its pairs since it encompasses more than one possible action. Three situations are currently considered: insertion, deletion, or substitution of a chord.24 A special syntax was created for the naming this operation, namely, the specification of the case in question (“ins”, “del”, or “sub”) is informed inside parentheses, accompanied by the identification of the events which will be affected (see the examples of Fig. 4.23). It is important to mention that, in spite of what is suggested by its label, the deletion of a chord does not affect the cardinality of γ (the same is applied to insertion and substitution, of course). In fact, the re-harmonization types transform

23 Depending

on the specificity of the case considered. to chord deletion, chord substitution is an instance of non-qualitative operator, since no information is prior provided about the to-be-replaced chordal structure. 24 Contrarily

4.3 Descriptions

85

Fig. 4.23 Examples of application of operator re-harmonization, considering the three alternatives: insertion (a), deletion (b), and substitution (c)

only the original harmonic rhythm, as shown in Fig. 4.23.25 When such a transformation occurs (whether by insertion, deletion, or substitution) a different pattern of rhythmic articulation for the chordal accompaniment arises, although the number of melodic-rhythmic events is kept constant. For a better illustration of this particularity, Fig. 4.23 is formatted in an unusual manner. The referential ρ is subdivided into its two dimensional, abstract aspects, harmonic content, and rhythm, in a disposition that is replicated by the three variants (γ ) used as examples. Re-harmonization by chordal insertion is applied in (a), deletion in (b), and substitution in (c), resulting in different harmonic contexts, both in pitch and rhythmic aspects.

4.3.20 Re-partition • • • • • •

Symbolic notation: RPA. Type: non-canonic. Scope: mutational. Target attribute: t1 (time). Description: reformulates a temporal span formed by two contiguous durations. Alteration of cardinality of γ : no.

In a sense, RPA can also be considered as a non-qualitative operator, since its algorithm does not specify how the durational span occupied by both target 25 This is due to the fact that the harmonic rhythmic (associated with the columns of the harmonic section of the matrix of attributes) is normally more relaxed than the melodic rhythmic (established by the columns of both pitch and temporal sections of the matrix).

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Fig. 4.24 Example of application of operator re-partition

Fig. 4.25 Examples of application of operator replication

elements shall be re-partitioned; it just determines that a new temporal configuration (maintaining the original total duration) is to be created. In Fig. 4.24 the quarter-note span (= 4) related to the first two events of ρ, equally subdivided into two eights (2 + 2), is re-partitioned into unequal values (1 + 3), a possibility among others (3 + 1, 1.5 + 2.5, for example).

4.3.21 Replication • • • • •

Symbolic notation: RPLx . Type: non-canonic. Scope: mutational. Target attributes: p1, p3 or d-p3 (pitch), or t1 (time). Description: replicates one selected pitch or duration (or a group of contiguous pitch/durations) for x times. • Alteration of cardinality of γ : yes (Fig. 4.25).

4.3 Descriptions

87

strongest

Fig. 4.26 Example of application of operator rest substitution

4.3.22 Rest Substitution • • • • •

Symbolic notation: RST. Type: non-canonic. Scope: mutational. Target attribute: t1 (time). Description: substitutes a selected duration (or two contiguous durations) by its corresponding-value rest. • Alteration of cardinality of γ : yes. As shown in Fig. 4.26, the substitution of the first event by an eight-note rest causes not only a reduction of the cardinality in one unit, but also a displacement of the metric accent to the dotted quarter note, as suggested by the arrows below the matrix.

4.3.23 Retrogradation • • • • • •

Symbolic notation: R. Type: canonic. Scope: normal. Target attributes: p1 (pitch) or t1 (time). Description: takes backwards a sequence of pitches or durations. Alteration of cardinality of γ : no.

Normally, retrograded outputs diverge sharply from the referential idea, which constrains its use only to special situations (notably in non-tonal contexts). While this is true for the pitch domain of our example (top of Fig. 4.27), in which all attributes are affected, the transformation suffered by the temporal variant is, rather, somewhat slight. This can be explained by the quasi-palindromic configuration of ρ.

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Fig. 4.27 Examples of application of operator retrogradation

4.3.24 Rotation • • • • • •

Symbolic notation: ROTx . Type: non-canonic. Scope: normal. Target attributes: p1 (pitch) or t1 (time).26 Description: rotates x times a sequence of pitches or durations. Alteration of cardinality of γ : no.

When applied to pitch domain, ROT rotates not only the primary attribute (p1), but also of the secondaries p2 and p4. A complete rotation of p3 is not possible, since a new interval—measuring the distance of last and initial original pitches— must necessarily be included in the output matrix. For a similar reason, the original ambit is normally modified. With respect to the temporal domain, just t2 (besides t1) is automatically rotated. The durational span is obviously preserved, while any rotation of the metric contour will depend on the original configuration and/or the number of rotational steps (in the case exemplified in Fig. 4.28, t3 is not essentially altered).

26 This operation can also be suitable for application in attributes p4 and t3, melodic and metric contours.

4.3 Descriptions

89

Fig. 4.28 Examples of application of operator rotation

Fig. 4.29 Example of application of operator split duration

4.3.25 Split Duration • • • • • •

Symbolic notation: SPL. Type: non-canonic. Scope: normal. Target attribute: t1 (time). Description: splits a selected duration into two partitions. Alteration of cardinality of γ : yes.

Operator SPL is a mirrored version of MRG, and implies a cardinality increase in one unit. By default, it is expected the splitting of the original duration in two equal halves, although irregular subdivisions are also possible (Fig. 4.29).

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p3 =

Fig. 4.30 Examples of application of operator subtraction

4.3.26 Subtraction • • • • •

Symbolic notation: SUBx . Type: non-canonic. Scope: dual. Target attributes: p3 or d-p3 (pitch) or t2 (time). Description: subtracts a fixed number of (intervallic or durational) units from selected intervals or IOIs. • Alteration of cardinality of γ : no. Figure 4.30 presents two examples of the action of the operator subtraction, both in mutational (pitch) and normal modes (time). Although SUB seems to be an exact mirrored version of ADD, there is an important distinction that constrains its application in certain situations, considering exclusively temporal domain. In this case, the decrement x must be lower than the minimum numeric value of the argument, preventing the output from nonsense null or negative time spans.27

27 As an example, consider t2 = . Since the minimum value of t2 is 1, the application of SUB to t2 must use x > 1.

4.3 Descriptions

91

Fig. 4.31 Examples of application of operator suppression

4.3.27 Suppression • • • • • •

Symbolic notation: SUPx . Type: non-canonic. Scope: mutational. Target attributes: p1 (pitch)28 or t1 (time). Description: eliminates x events at the end of a pitch or a durational sequence. Alteration of cardinality of γ : yes.

SUP is the mirrored version of EXT, and implies a cardinality decrease in x units. Because this operation eliminates only final events of a sequence, letting the remaining unaltered, it normally affects collaterally only pitch ambit or temporal span. This is the case of the rhythmic transformation of Fig. 4.31 (in pitch transformation, coincidentally the suppression of the two last events resulted in the maintenance of attribute p5). As it can be observed in both examples, the essential information of melodic and metric contours was preserved, in spite of the differences in their numeric contents.

28 It

is possible also to apply it to attribute p3.

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4.4 Composition of Operators This brief section explores a possible, and very useful practical application of operations, namely the juxtaposition of two or more distinct operators for obtaining a compound transformation. This alternative is related to the mathematical property of associativity of functions. Basically, the associativity between two (or more) functions says that they can be combined (or composed). If .f (x) = y and .g(y) = z, then .g(f (x)) = g(y) = z. It is said then that f and g are composed. The same expression at the left of the equal sign can also be written as gof , where “o” represents an operator that performs the composition of f and g.29 This means that, in a composition of two functions, the second function uses the output of the first as input. Given these basic notions, and back to our musical context, the implications arisen from the usage of a composition of operators are quite clear. However, three guidelines must be enunciated for conditioning their correct employment in derivative situations: 1. The operators in composition must be consistent in regards to the domain of the attribute (that is, if one operator is applicable only to the temporal domain, so the other(s) also must be); 2. Moreover, the operators in composition must be consistent in relation to the attribute itself (i.e., if one attribute is applicable to pitch sequence, it is not possible to compose it with an operator applicable to intervallic sequence); 3. The function resulted from a composition of operators is normally noncommutative (but occasionally it may be). This means that if the application of the individual operators is re-ordered, the output will be different. The composition of operators is especially useful in analysis for abbreviating stages of transformation that are clearly implicit. Figure 4.32 illustrates this particular application with two examples. Three operators (retrogradation, chromatic transposition, and rotation) are composed in two particular orders, resulting in different variants when applied to a same parent.

29 Eventually,

implied.

 other symbols are employed, as for example .×, *, or . , or even suppressed, when

4.5 Summary

93

Fig. 4.32 Examples of use of non-commutative composed operators

4.5 Summary The next figures organize MDA operations according to distinct classifications.30 Firstly, Fig. 4.33 subdivides the twenty-seven operators in canonic (inside the gray circle) or non-canonic, as well as according to the domains they are associated with. There are seventeen operations related to pitch (six exclusively), seventeen to

30 It is important to add that the list of operations presented in this chapter by no means exhausts the possibilities, but shall be seen as a starting point. The set of operators can be enlarged by inclusion of new elements, emerging especially from analytical situations. This will be properly illustrated in Part III.

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G

Fig. 4.33 Operators distributed according to type (canonic or non-canonic), and domain of application

time (seven exclusively) and only four to harmony (three exclusively). Non-canonic operations largely outnumber the canonic (twenty vs. seven). Figure 4.34 informs that twenty operations can be applied in mutational mode (twelve of them, exclusively), against fifteen related to normal scope (seven exclusively). There exist eight dual operators. The operators are organized in two mutually exclusive sets in Fig. 4.35: those that do not affect the original cardinality (nineteen cases), and those that cause an increase or decrease of cardinality (eight). Figure 4.36 displays fourteen operations that have the special property of being mutually mirrored. Finally, the chart of Fig. 4.37 summarizes the main characteristics of the whole group of operations.

4.5 Summary

95

G Fig. 4.34 Operations distributed according to scope. Operators forming the intersection of both sets are of dual scope

G

Fig. 4.35 Operations according to the capacity of altering the original cardinality

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4 Transformational Operations

Fig. 4.36 Operations that have mirrored versions

contextual dyadic transf.

G

Fig. 4.37 Chart of operations, considering type (canonic or non-canonic), domain (pitch, time, and/or harmony), and scope (normal, mutational, or dual)

4.6 Transformational-Derivative Analysis Knowledge about operations is of paramount importance for determining derivative paths in analytical situations, in a process that from now on will be called

4.6 Transformational-Derivative Analysis

97

transformational-derivative analysis (TD analysis, for short). As it was briefly suggested in Chap. 2, it involves normally the elaboration of hypothetical intermediary stages of plausible derivations connecting a given UDS C and its presumed origin P. Methodologically, TD analysis is made through successive reductions (resembling those used in Schenkerian graphs) that gradually reveal the possible operations employed in the process. The next subsections examine some examples related to the basic cases of decomposable derivative work (modeled in Fig. 2.6). They will be the starting point for a discussion about the process of TD analysis.

4.6.1 Case 1: Simple/Pure Decomposable Variation In order to succinctly demonstrate the potentialities of the TD analysis, let us start with a very simple example. Consider C as a musical fragment identified as a possible variant of a referential idea P in a hypothetical analytical situation (Fig. 4.38). The methodology of transformational analysis makes to the analyst a basic question: “How could you systematically explain the chain of transformations that would map C onto P?” Figure 4.39 proposes a possible solution for the question. The analysis reveals a case as simple/pure decomposable variation, involving one-stage transformation applied to a unique domain, time in the case (it corresponds to the model of Fig. 2.6a). One can see this example as representing a hypothetical situation, in which the analyst intends to determine a possible derivative relation between a given foreground, concrete musical idea (C) and a referential one (P), based on a supposition that it is originated by some kind of transformation. For this reason, the arrows that correspond to both operations (the generic, high-level V, and the specific, low-level PER) point to the direction bottom-up, instead of top-down (as it happens in Fig. 2.6a).31 In sum, TD analysis can be seen as a sequence of proposed hypotheses whose goal is to reconstruct a possible derivative route, through logical reasoning and the use of simple means. In the exemplified case, one can easily perceive that the best (and simplest) manner of explaining the derivation of C from P is through a mutational permutation of the third and fourth durational values inside the temporal domain.

31 It would also be possible to think of inverse functions operating in top-down direction: P = V.−1 (C), and .ρ = .υ.−1 (.γ ). However, I will not intend to explore this possibility, because it would represent an unnecessary complication at this stage.

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Fig. 4.38 Variant C (first case)

4.6.2 Case 2: Compound/Pure Decomposable Variation Figure 4.40 presents a more complex example, this time involving sequential hypothetical transformations on the pitch domain. For a more concise expression, let us identify the hypothetical intermediary stages with the Greek letter .π, and enumerate them according to their appearance order. As shown in the bottom-up hypothetical analytical reconstitution, attribute p1 of P was affected by three sequential transformations: the first, a diatonic inversion (operation i) sends .ρ 1 to intermediary stage .π1, to which is then applied a diatonic transposition by a descending third (t.−3 ). The output, .π2, is base for another application of diatonic ascending transposition (t1 ), this time in mutational mode, encompassing only events 4 and 5. The new variant corresponds to the pitch configuration of C, being accordingly labeled as .γ .32

32 As

we will better discuss in Chap. 7, the model presented in this example can be considered as a sort of rudimentary process of developing variation, in this case, composed by hypothetical conjectures and inserted in a subjective timeline.

4.6 Transformational-Derivative Analysis

99

Fig. 4.39 Example of TD analysis of case 1—pure/simple decomposable variation

4.6.3 Case 3: Simple/Hybrid Decomposable Variation In the case shown in Fig. 4.41, two simultaneous one-stage transformations are performed in distinct domains, pitch and harmony. It is interesting to notice that the harmonic modification (caused by re-harmonization) is constrained by the pitch variation as an inevitable action to accommodate the replacement of note E by F. (resulting from mutational transposition), thus avoiding inconsistency between melody and accompaniment. As shown in the analysis, such simultaneity implies the existence of two referential and derived abstract units, respectively .ρ1 / .γ 1 (related to the pitch domain), and .ρ2 / .γ 2 (temporal domain).

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1

3

Fig. 4.40 Example of TD analysis of case 2—compound/pure decomposable variation

4.6 Transformational-Derivative Analysis

101

Fig. 4.41 Example of TD analysis of case 3—simple/hybrid decomposable variation

4.6.4 Case 4: Compound/Hybrid Decomposable Variation The fourth case (Fig. 4.42) addresses a still more complex situation of multiple-stage decomposable variation in two different domains (now, pitch and time). Notice that the intermediary pitch stage .π2 was arbitrarily positioned in the “timescale”, since it is completely independent of the sequence of derivative stages in the temporal domain.

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6

Fig. 4.42 Example of TD analysis of case 4—compound/hybrid decomposable variation

References Murphy, S. (2022). Contextual dyadic transformations, and the opening of Brahms’s op. 5 as a variation of the opening of Schumann’s op. 14. MusMat, 6(2), 1–16. Rings, S. (2011). Tonality and transformation. Oxford: Oxford University Press. Schoenberg, A. (1967). Fundamentals of musical composition. London: Faber and Faber.

Chapter 5

Measurement of Similarity (II)

I return to the comparison between a referential UDS P and one of its possible variants, C, considering the general case, namely, of cardinalities that are not necessarily compatible. Firstly, let qp represent the cardinality of P and qc the cardinality of C. There are three possible situations: 1. qp = qc 2. qp > qc 3. qp < qc If we put aside case (1), already covered in Chap. 3, we can conclude that it will be necessary, in the derivative process P.→C, the presence of at least one operation capable of altering P’s cardinality, either decreasing (in 2) or increasing it (3). As discussed in Chap. 4 (see especially Fig. 4.34), eight operations can affect the cardinality of a UDS. They are listed in Table 5.1, accompanied by some specific information. From now on, let us classify them as cardinality-sensitive operations or, more concisely, CS-operators.

5.1 Similarity Between Non-compatible Cardinalities In the cases where qp is different from qc , similarity shall be measured through TD analysis, being dependent on the number of reductions necessary to map C into P, as well as the number of events inserted and/or suppressed along the process. In other words, we are now able to implement the two basic procedures of Toussaint’s edit distance method which were left aside in Chap. 3, namely, insertion and deletion.1

1 Both shall be seen as high-level abstract operations, represented in our system for specific lowlevel CS-operators.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3_5

103

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5 Measurement of Similarity (II)

Table 5.1 Cardinality-sensitive operators Operation Deletion Extension Interpolation Merging of durations Rest-substitution Replication Split duration Suppression

Symbol DEL EXT INT MRG RST REP SPL SUP

qp < qc

qp > qc x

x x x x x x

Pitch x x x

x x

x

Time x x x x x x x x

Increase/decrease .−1

+ undefined + undefined .−1 .−1 + undefined +1 .− undefined

Fig. 5.1 A variant of P with incompatible cardinality (example 1)

C

The following didactic and progressive examples will help me to explain how derivative analysis is carried out when P and C have different cardinalities.

5.1.1 Example 1 Let us start with a very simple case, considering the variant shown in Fig. 5.1. In spite of being evidently a very close variation of P, the cardinality of C (6 events) prevents it from being compared to the referential idea (cardinality 5) by the use of the algorithms of Chap. 3. Therefore, the measurement of similarity will require a preliminary TD analysis, which is depicted in Fig. 5.2.2 This is a case of simple/hybrid decomposable variation, considering one-stage transformations in both pitch (through the insertion of the chromatic inflection A. between A and G) and temporal domains (by splitting the last eighth note into two sixteenths). These applications are said parallel (i.e., involve jointly two or more domains), which is denoted by a horizontal dashed line linking the labels of both operators. The calculation of similarity is accomplished in two steps: after the TD analysis is made, the CS-operations necessary to reach the cardinality of the referential idea are counted. In the present case, there is just one reduction, comprising two parallel

2 For visual clarity, the downward arrows connecting P and C to their abstracted forms were replaced by simple edges.

5.1 Similarity Between Non-compatible Cardinalities

105

Fig. 5.2 TD analysis of variant of example 1

operations. The second step is to determine a penalty due to the change of cardinality detected in the analysis. Let us label it as kq , whose calculation is determined by the following formula: kq =

.

δq 10 nr

(5.1)

where .δq is the difference between cardinalities of P and C (always a positive integer), n is the number of CS-operations employed, and r is the number of reductions. In the case of the example, .δq = 6 .− 5 = 1, n = 2, and r = 1. Plugging these values in Eq. 5.1 we obtain .kq = 0.05. A general formula for measurement of similarity is presented in Eq. 5.2, updating Eq. 3.5 by the inclusion of variable kq . k=

.

3.kp + 5kt + 1.5kh + kq 10

(5.2)

From now on, we will consider kq = 0 when qp = qc (as in all cases analyzed in Chap. 3). Conversely, since the two operations applied to change the cardinality (INT and SPL) are the only transformations suffered by P in the example, there are no pitch, temporal, and harmonic penalties to be assigned in this case and, therefore,

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Fig. 5.3 Relation of similarity between P and the variant of example 1 plotted in the DS

C

Fig. 5.4 A variant of P with incompatible cardinality (example 2)

the first member of Eq. 5.2 equals zero. Given this, the definitive penalty k equals kq (= 0.05), a very low value, confirming the initial impression that both UDSs are really very similar. Figure 5.3 provides a graphic representation of this relation, by plotting point C in the derivative space of P.

5.1.2 Example 2 A more intricate case is proposed in Fig. 5.4. As shown in TD analysis of Fig. 5.5, this time a compound/hybrid decomposable variation takes place, involving combined transformations of the three domains and an increase of eleven units of cardinality. In both pitch and temporal domains, five CS-operations are applied3 (therefore, n = 10), distributed into five levels of reductions (r = 5).4 The calculation of kq , by application of Eq. 5.2 produces the following result: 3 Pitch operators OCT and T do not affect cardinality and, for this reason, are not counted, likewise

the two harmonic operations of type RHA. transformations orient the establishment of these levels.

4 Rhythmic

5.1 Similarity Between Non-compatible Cardinalities

107

Fig. 5.5 TD analysis of variant of example 2

kq =

.

16 − 5 10 10 5

=

11 = 0.55 20

Like the previous case, the reductions to the original cardinality reach the referential idea P (implying kp = kt = kh = 0), then once again the definitive penalty is equal to kq = 0.55 (Fig. 5.6).

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Fig. 5.6 Relation of similarity between P and the variant of example 2 plotted in the DS

Fig. 5.7 A variant of P with incompatible cardinality (example 3)

C

5.1.3 Example 3 The third and last variant is depicted in Fig. 5.7. Unlike what happened in the previous examples, the original cardinality is reached in the transformational analysis some layers above5 the referential idea P, at stage C’, as indicated in Fig. 5.9. This hypothetical intermediary motive results from the combination of decomposable stages .π5, .π6, and .π7. For reasons of clarity, C’ is shown re-integrated in Fig. 5.8, inserted in a half-way between P and C (observe that, in this disposition, we can think of two new high-level operations—.VP→C’ and .VC’→C —connecting it to the extremities of the derivative process. An interesting metaphorical image is of an excavated fossil that allows explaining the links between two remotely related species. Figure 5.9 segments the TD graph into two distinct “derivative regions”: one above C’ (where lie the variants with higher cardinality than P), and the other below C’ (grouping the variants with the same cardinality of P). The measurement of similarity in each region is based on different types of penalties: kq (cardinalitychange penalty) for the “northern” region, and the domain penalties, kp , kt , and kh , for the “southern” region. Let us examine the first region. Just four levels imply alteration of cardinality (r = 4) and there are seven CS-operators (n = 7). Applying then Eq. 5.1, we have:

5 According

to the adopted analytical top-down route.

5.1 Similarity Between Non-compatible Cardinalities

Fig. 5.8 TD analysis of variant of example 3

kq =

.

10 − 5 10 74

=

5 ≈ 0.29 20

109

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5 Measurement of Similarity (II)

Fig. 5.9 Hypothetical intermediary variant C’

For the measurement of similarity related to the “southern” region it is only necessary to compare P and C’ with the use of the methodology introduced in Chap. 3, since both UDSs have compatible cardinalities (Fig. 5.10). As shown in Fig. 5.10, the change in the melodic contour is the main responsible for the pitch penalty (kp ) of 0.21. In spite of presenting an almost identical

5.1 Similarity Between Non-compatible Cardinalities

111

Fig. 5.10 Measurement of similarity between P and C’, according to the algorithms of Chap. 3

durational configuration comparing with P’s (returning a low penalty), the metric displacement of the rhythmic sequence of C’ is responsible for a divergence

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kt = 0.32. Finally, harmonic transformations by diatonic change of chords contribute with an individual penalty of kh = 0.13. Plugging these values in Eq. 5.2 and summing to them with the already calculated kq , we obtain the definitive penalty k = .0.25 + 0.29 = 0.54, replicating almost exactly the value of example 2 (for this reason, I decided to omit here the stage of plotting point C in the derivative space of P).

5.2 Some Concluding Remarks The three examples of this chapter, combined with those introduced in Sect. 4.5, were intended to provide an elementary overview of the methodology designed for the TD analysis. Nonetheless, how the reader must be thinking at this point, real musical situations will hardly be presented in such, so to speak, “laboratory” conditions. As elsewhere mentioned, context is paramount for determining which is the most adequate analytical approach to be adopted in any case. This has much to do with some aspects of Part II (in theoretical terms) and, especially of Part III (analytical application), but before going on, it is necessary to address a more problematic issue, which involves both conceptual and practical aspects of the transformational-derivative analysis. Not rarely an analyst faces situations in which, despite the clear existence of parental connections between a given musical fragment and a referential idea, the setting of hypothetical derivative stages (as accomplished in the standard cases above exposed) it is not at all an easy task. As a matter of fact, there is a fuzzy borderline separating, at one side, plausible and methodologically adequate analytical explanations (even on a hypothetical level, as we are working with) and, at the other, exaggeratedly complicated and far-fetched descriptions, which only distances the method from the musical phenomena it seeks to explain. This sort of questioning is especially compelling in analytical situations in which two or three domains suffer simultaneous transformation (as it occurred in some of the examples), bringing back the notion of holistic variation. Recall that this takes place when the musical substance of a motive is treated as an indivisible unit. Transformation in these cases affects the complete material, molding it in idiosyncratic ways, turning any attempt of decomposing the original idea into domains and attributes completely pointless. As an initial illustration of this argument, let us revisit the case of Fig. 5.4. For convenience, this variant is reproduced in Fig. 5.11, accompanied by a simple Schenkerian analysis that is intended to highlight the co-relations between its pitch structure and that of P. As one can easily perceive, the original pitch sequence (C-E-G-A-G) was perfectly preserved, just hidden by the inclusion of melodic ornaments (appoggiaturas, passing, and neighbor notes). An extension, replicating and transposing the last fragment (A-G-F.-G) is added as a prolongation of the dominant chord G7 . Evidently, it is a very superficial variation of P, which contrasts sharply with the complex analysis of Fig. 5.5. What is particularly striking in the

5.2 Some Concluding Remarks

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Fig. 5.11 Variant of example 2 (a); Schenkerian analysis (b)

latter is the high number of operations (fourteen) that were listed along the analytical process. It seems really extravagant that we need them to explain a derivation which could be informally described with few, simpler words, that is, “by inclusion of ornamentation and an extension”. Figure 5.12 proposes an alternative interpretation, considering now the presence of a holistic variation. By definition, holistic operations (denoted graphically by dashed arrows) are high-level functions, somewhat imprecise in terms of the range of action (they cannot be described as algorithms, for example, as some decomposable operators), which turns the analysis considerably laconic, but gives some vague idea of the transformations suffered. In the case of the example the derivation comprises two stages: a general addition of ornamental notes to the original pitch backbone (indicated by operation embellishment, EMB), applied to P, and a replication (RPL)6 of the concluding fragment of the varied form. Despite its inherent vagueness, holistic variation is not, of course, impermeable to analysis. Eventually, in real-music situations, it may be a better option to consider holistic rather than decomposable analysis as a more adequate approach. On the other hand, for the reasons elsewhere exposed, I do not intend to pursue here specific theory and methodological tools dedicated to it.7 The third part of this book will certainly provide a concrete dimension for what is being said here. Another important question to be mentioned concerns the fact that situations like that of example 3, namely, involving musical ideas separated by multiple-stage transformations, will hardly take place in real-life derivative analysis. Normally, the

6 In

this case, a holistic replication, that is, which does not give any detail about what is replicated, contrarily to decomposable RPL. 7 It will be left for a future study.

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Fig. 5.12 Alternative derivative analysis for example 2, considering holistic variation (harmonic contexts were omitted for simplicity)

intermediary stages (which were treated in this chapter as a series of hypotheses, uniquely due to a didactic strategy) will be in fact concrete ideas mediating remotely related forms. In other words, like in the biological realm, musical variation usually progresses gradually, in discrete steps, rather than jumping across wider derivative spans (as suggested in the analysis of Fig. 5.8). This naturally leads us to the notions of developing variation and Grundgestalt, whose theoretical formulations are exposed in detail in the chapters that form Part II.

Part II

Variation on Time

As pointed out by Damián Zanette (2008, p. 5) “in music, context is determined by a hierarchy of intermingled patterns occurring at different time scales.” By addressing variation in function of time, as a dynamic and potentially spreading process, the second part of the book is specially dedicated to a more contextual approach of the concept (approaching Zanette’s definition), and especially concerned with its possible structural implications, in contrast with Part I, in which variation was essentially isolated from the temporal dimension (at least, considering more larger periods of time). The conception of variation in function of time is closely related to Schoenbergian idea of developing variation, a process associated with gradual and progressive transformation of musical structures. Developing variation is, in turn, closely associated with Grundgestalt, another principle coined by Schoenberg, which represents essentially the source of referential material for further development. The interaction of both concepts in a theoretical binomial forms a powerful framework for supporting the large-scale temporal aspect of our model, whose structural elements, premises, conceptualization, and terminology are described in the next two chapters.

Reference Zanette, D. (2008). Zipf’s law and the creation of musical context. Musicae Scientiae, 10(1), 3–18.

Chapter 6

Grundgestalt

It is not easy to define Schoenberg’s concept of Grundgestalt. As it is well known, Schoenberg himself never was capable of producing a clear, precise, and unambiguous definition of the term along his career as a theorist (although his own music can be viewed as an extraordinary testing ground for the application for the concept).1 Most of his few comments about the principle, distributed along his writings (books as well as unpublished manuscripts) are frankly elliptic or contradictory, a common criticism among scholars interested in the subject.2 As argued by Michael Schiano (1992, p. 4), “given the term’s centricity to Schoenberg’s thinking, it is strikingly infrequent in his writings. It is the principle that survives and flourishes in the literature, not the term itself.” In a recently published study, Desirée Mayr (2018) examined profoundly the Grundgestalt’s literature, considering the vast amount of theoretical perspectives already produced along a period of almost 100 years by a group of thirteen authors (including, of course, Schoenberg himself).3 The comparison of distinct views allowed Mayr to create her definition, which I adopt as reference: The Grundgestalt consists of a group of a few elements presented at the beginning of a musical piece which has the potential capacity of generating a large amount of material by derivative processes. These elements can be considered as concrete units (conventional motives for example) and/or structures based on abstract attributes, involving intervallic or durational sequences, tonal relations, metric configurations, etc. The Grundgestalt acts, therefore, as a seed of an organically-constructed composition providing not only the needed

1 For

a more detailed discussion about this, see, for example Neff (1984), Frisch (1993), Haimo (1997), Simms (2000), and Almada (2011, 2016). 2 See, for example, the comments expressed by Neff (1984), Frisch (1984), and Haimo (1997) in this respect. 3 They are: Arnold Schoenberg (1967, 1984, 1994), Josef Rufer (1952/1954), David Epstein (1980), Patricia Carpenter (1983), Severine Neff (1984), Walter Frisch (1984, 1993), Stephen Collison (1994), Martin Leigh (1998), Richard Pye (2000), Devon Burts (2004), Norton Dudeque, Marie Colleen (2009), and Richard Taruskin (2010). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3_6

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material for themes and subsidiary elements but also the adequate means for its structuring (Mayr, 2018, p. 34).

From this, the concept of Grundgestalt can be associated with a source of primordial material, with which a musical piece can be (partially or, in rare cases, integrally) constructed through continuous derivative processes (related to developing-variation techniques, the main subject of the next chapter). As argued by some authors,4 the principle of Grundgestalt derives from an organicist conception of musical creation, a philosophical-scientific trend, inspired mainly by Goethe’s and Darwin’s theories, which influenced deeply Romantic music (especially written by Austro-German composers, like Mozart, Beethoven, Brahms, and, of course, Schoenberg).5 Organicism in music is essentially associated with a strong sense of economy of means and oriented by an equilibrium between coherence and variety, barely in equal proportions. As a very frequent metaphor, a musical work derived from a Grundgestalt could be compared to a plant that sprouts from a seed. Accordingly, the Grundgestalt, at least in an idealized conception, would contain implicitly the necessary instructions to the construction of the piece.6 An ideal model of the interaction of the idea of a Grundgestalt-seed and the processes of developing variation that acts in the generation of a musical composition is proposed in Fig. 6.1. The next sections of this chapter address the integration of the principle of Grundgestalt to MDA, under the transformational perspective that is here adopted.

6.1 Grundgestalt-Components A Grundgestalt (Gr, for short) is generally located at the very beginning of a musical piece. Regarding size, it can be presented as a brief UDS or, more commonly, be formed by the conjunction of two or more elements. These are named in MDA as Grundgestalt-components (GCs), identified with uppercase letters (A, B, C, . . . ). GCs behave like relatively autonomous “territories” inside the Grundgestalt, containing one or more agents (i.e., attributes or a subset of these),7 referential elements in the subsequent derivative process. It is noteworthy to add that GCs can refer to distinct domains/attributes. Figure 6.2 proposes a schematic representation of a hypothetical Grundgestalt segmented into four GCs (A–D), with a total of six agents.

4 Leonard

Meyer (1989), Sérgio Freitas (2012), and Nicole Grimes (2012). to Freitas and Grimes, the roots of organicist thinking can be traced back to the Greek Classical philosophy, with Plato and Aristotle. 6 For a very comprehensive and detailed discussion about the philosophical antecedents that probably influenced Schoenberg’s formulation of the principle of Grundgestalt, see Boss (2014, pp. 10–28). 7 This term is also employed by Jack Boss (1991) in a quite similar sense. 5 According

6.2 Variables

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Fig. 6.1 Idealized model of organic growth of musical piece through development of the elements presented in its Grundgestalt (the circle at the center). The squares (labeled with alphabetic letters) that emanate from it in concentric “waves” represent variants, appearing in successive generations

Fig. 6.2 Schematic representation of a Grundgestalt (Gr), formed by four GCs (A–D), in which are located six agents from of two distinct domains (represented by black and gray circles)

6.2 Variables Aiming at analytical applications, algebraic variables are assigned to the agents that form the GCs. In order to avoid confusions with the labeling of GCs, variables are

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Fig. 6.3 Schematic representation of a Grundgestalt (Gr), formed by four GCs (A–D), in which are located six agents from of two distinct domains (represented by black and gray circles)

identified with upper or lowercase letters8 in reverse alphabetic order (Z/z, Y/y, X/x, W/w, . . . ). Figure 6.3 ascribes variables to the agents of Fig. 6.2. The idea behind the use of variables is to allow straightforward and more precise identification in analytical situations, especially when it involves long-range derivation. Variants are said permanent (denoted by uppercase letters) when are pervasive regarding a whole piece, and temporary if their actuation is valid only in an isolated movement (in this case, identified by lowercase letters). An interesting biological metaphor involving variables is to consider them as genes that inhabit living bodies.9 According to Dawkins, living bodies may be seen as mere “mortal vehicles” which are “used” (and eventually substituted by others, when necessary) by the genes in order to conduct themselves as safely as possible along their virtually immortal journey for maximizing replication. Essentially, genes and bodies are always perfectly integrated into an entity (think of a rabbit, for example) in a given moment, but they live in very discrepant temporal scales. We can now project this image onto our musical/derivative universe and consider that, although abstract variables are part of musical (holistic) complexes, they can behave with a particular “logic”, so to speak. A given variable, for instance, can “inhabit” several motives (even those not clearly related by “phenetic” evidence) and be transmitted, through variation, to “containers” very different from those they originally integrated. Further analytical application, in Part III will be providential for illustrating this point of view.

6.3 Orders of Grundgestalten In a multi-movement musical work (as a piano sonata or a symphony), we can consider the presence of two or more Grundgestalten (in general, each one

8 Depending

on the scope which is covered by the variable, as explained bellow. Dawkins develops splendidly this view in his bestseller The Selfish Gene (Dawkins, 2006). I here follow his main line of thinking, adapted to musical contexts. 9 Richard

6.4 Identification of a Grundgestalt

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associated with a particular movement of the piece), hierarchically organized. I name a first-order Grundgestalt (1-Gr) the basic source of material for the whole work, which is introduced in the first bars of the initial movement. All remaining Grundgestalten, that normally occur at the beginning of the subsequent movements and are commonly less complex than the principal, are said of second-order (2-Gr). They can also be viewed, in a sense, as higher-order variants of the 1-Gr, since normally their constituents re-elaborate the original “genetic” material. In other words, part of a 2-Gr is formed by permanent variables (introduced in 1-Gr) and/or, more frequently, variants of these. It is also common that a 2-Gr presents some new elements, which will be developed along the corresponding movement, being confined to the movement’s boundaries. As said before, these elements are properly identified as temporary variables. However, eventually, temporary variables manage to be transmitted to a subsequent movement (not necessarily contiguous), becoming part of its particular Grundgestalt, which is then classified as of third-order (3-Gr). The transmitted temporary variables change their status, becoming semi-permanent variables, being identified with italicized-uppercase letters (Z, Y, X, . . . ). The scheme of Fig. 6.4 aims to clarify the functions of these new concepts. The case depicts the multi-order Grundgestalt structure of a hypothetical piece with four movements. Its first-order Grundgestalt (1-Gr) is composed by four GCs (A–D), “inhabited” by six permanent variables (Z–U). The second movement is associated with a 2-Gr formed by two components and four variables: two variants derived from permanent units (V3 and U1.2 ) and two new, therefore temporary variables, z and y. Another 2-Gr forms the basis of the third movement, containing other two variants of permanent variables introduced in 1-Gr (Z2.2.1 and W1 ) and three new temporary ones, z, y, and x (recall that being simple temporary variables, these new “z” and “y” have not to do with their second-movement counterparts). Finally, the fourth-movement presents an instance of third-order Grundgestalt, since one of its constituent (Z) is a semi-permanent variable, derived from the temporary z of the second-movement. Permanent variants and local temporary variables complete its structure.

6.4 Identification of a Grundgestalt The last section of this chapter addresses a central point for the methodology of derivative analysis, namely the proper identification of the Grundgestalt of a musical piece. Indeed, this task is expected to represent the initial stage of the analytical process, something that is not immune to problems. A first issue to be considered is the fact that, according to Desirée Mayr’s definition, the Grundgestalt is normally located at the beginning of the piece, but its multiple derivative elaborations are distributed along the composition in unpredictable and idiosyncratic manners. In practical terms, the analyst should first get acquainted with the meaningful derivations present in the piece before trying to obtain their source, in a retroactive prospection. It is not at all a simple task.

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Fig. 6.4 Schematic example of multi-order-Grundgestalt structure in a hypothetical piece with four movements

After this is accomplished, some additional questions can be posed, as for example: Which are the Grundgestalt’s “territorial” precise limits in the score (in terms of measures and beats)? In the case of these limits are determined, shall the analyst consider all musical events (notes, rhythmic figures, harmonies) inside such boundaries as potentially referential elements for further elaboration or some of them (which?) could be disregarded? How variables shall be assigned to the Grundgestalt’s elements? Is it possible that outside materials (i.e., considering the Grundgestalt’s limits) could also be treated eventually as referential? Because each piece is a particular case, with a unique derivative “history”, there are no easy answers for these questions. Nonetheless, in spite of the uncertainty that arises from this truism, a general principle can be adopted as north in the determination of a Grundgestalt, a principle that can be expressed in a single word: familiarization. Although not referred specifically to the search of relations between the Grundgestalt and its derived elements, the following argumentation of American theorist Candace Brower in respect of modes of apprehension of a musical structure

References

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after the cognition of the multiple relations that cohere superficial events seems perfectly applicable to the present discussion: When we perceive an aesthetic object, we do so on many different levels. If the object is a painting, we may view it first from a distance to take in its overall form, symmetry, and color composition. Moving closer, we may allow first one aspect and then another to come into focus, until we are able to observe the smallest detail. We may then retreat to a more distant perspective, allowing our apprehension of the whole to be enhanced by our greater familiarity with its parts. Likewise, when we hear a piece of music, we may first focus on local events, listening for subtle inflections of pitch, articulation, or rhythmic motive, while in subsequent hearings we may attend to the large-scale progression of themes and tonalities. Our conception of the piece is thus built up over repeated listenings, as the relationship between small- and large-scale structure is gradually clarified (Brower, 1993, p. 19).

In fact, only a careful, profound, and attentive knowledge of the musical object to be analyzed, gradually from local to global perspectives, may reveal the (normally very intricate) plot of motivic-thematic relations that animate an organically-constructed composition. This means that, in practical terms, the study of the piece through a repeated and careful exam of its score (and, possibly, of its recordings) is a paramount precondition for the search of ideological connections and transformations. This process of gradual familiarization with the work, in spite of the inherent subjectivity it conveys, could be seen as a necessary preliminary stage of the derivative analysis, something to be combined with a complementary structural investigation involving especially form and harmony in an integrated procedure. The application of this broad strategy will be the very starting point of the derivative analysis of Part III.

References Almada, C. (2011). A variação progressiva aplicada na geração de ideias temáticas. In Proceedings of the 2nd International Symposium of Musicology (pp. 79–90). Federal University of Rio de Janeiro, Rio de Janeiro. Almada, C. (2016). Derivative analysis and serial music: The theme of Schoenberg’s Orchestral Variations op. 31. Permusi, 33, 1–24. Boss, J. (1991). An analogue to developing variation in a late atonal song of Arnold Schoenberg. Dissertation, Yale University. Boss, J. (2014). Schoenberg’s twelve-tone music: Symmetry and the musical idea. Boston: Cambridge University Press. Brower, C. (1993). Memory and perception of rhythm. Music Theory Spectrum, 15(1), 19–35. Burts, D. (2004). An application of the Grundgestalt concept to the first and second sonatas for clarinet and piano, op. 120, no. 1, and no. 2, by Johannes Brahms. Dissertation, University of South Florida. Carpenter, P. (1983). Grundgestalt as tonal function. Music Theory Spectrum, 5, 15–38. Colleen, M. (2009). The lessons of Arnold Schoenberg in teaching: The musikalische Gedanke. Dissertation, University of North Texas. Collison, S. (1994). Grundgestalt, developing variation, and motivic processes in the music of Arnold Schoenberg: An analytical study of the string quartets. Dissertation, King’s College. Dawkins, R. (2006). The selfish gene (3rd ed.). Oxford: Oxford University Press.

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Dudeque, N. (2005). Music theory and analysis in the writings of Arnold Schoenberg (1874–1951). Aldershot: Ashgate. Epstein, D. (1980). Beyond Orpheus: Studies in music structure. Cambridge: The MIT Press. Freitas, S. (2012). Da mùsica como criatura viva: repercussões do organicismo na teoria contemporânea. Revista Científica FAP, 9, 64–82. Frisch, W. (1984). Brahms and the principle of developing variation. Los Angeles: University of California Press. Frisch, W. (1993). The early works of Arnold Schoenberg (1893–1908). Los Angeles: University of California Press. Grimes, N. (2012). The Schoenberg/Brahms critical tradition considered. Music Analysis, 31(ii), 127–132. Haimo, E. (1997). Developing variation and Schoenberg’s serial music. Music Analysis, 16(3), 349–365. Leigh, M. (1998). Grundgestalt, multipiece and intertextuality in Brahms’ op.117, 118, and 119. Dissertation, University of Nottingham. Mayr, D. (2018). The identification of developing variation in Johannes Brahms op.78 and Leopoldo Miguéz op.14 violin sonatas through derivative analysis. Dissertation, Federal University of Rio de Janeiro. Meyer, L. (1989). Style and music. Chicago: The University of Chicago Press. Neff, S. (1984). Aspects of Grundgestalt in Schoenberg’s first string quartet, op. 7. Journal of the Music Theory Society, 9(1–2), 7–56. Pye, R. (2000). ‘Asking about the inside’: Schoenberg’s ‘idea’ in the music of Roy Harris and William Schuman. Music Analysis, 19(1), 69–98. Rufer, J. (1952/1954). Die Komposition mit Zwölf Tönen. Berlin: Max Hesses. English edition: (1954) Composition with twelve notes related only to one another. Searle, H. (trad). London: Rocklife. Schiano, M. (1992). Arnold Schoenberg’s Grundgestalt and its influence. Dissertation, Brandeis University. Schoenberg, A. (1967). Fundamentals of musical composition. London: Faber and Faber. Schoenberg, A. (1984). Style and idea: Selected writings of Arnold Schoenberg. London: Faber and Faber. Schoenberg, A. (1994). Coherence, counterpoint, instrumentation in form. Neff, S. (Ed.) Neff, S., Cross, C. (trad.). Lincoln: University of Nebraska Press. Simms, B. (2000). The atonal music of Arnold Schoenberg (1908–1923). Oxford: Oxford University Press. Taruskin, R. (2010). Music in the early twentieth century. Oxford: Oxford University Press.

Chapter 7

Developing Variation

According to Bryan Simms, the first time Schoenberg mentioned the term “developing variation” was in 1917, in a manuscript intended to become a treatise on orchestration, Zusammenhang, Kontrapunkt, Instrumentation, and Formenlehre, translated and published in 1994 by Charlotte M. Cross and Severine Neff (Schoenberg, 1994). In this text, Schoenberg defines the new term “as a process of change applied to motives within a theme ‘allowing new ideas to arise”’ (Schoenberg, 1917, in: Simms 2000, p. 172). For being a more “technical” concept (and considerably less abstract) than Grundgestalt, developing variation does not lead to so much controversy in the specialized literature. Among the large number of definitions already produced for this term, two of them seem to me as adequately precise, comprehensive, and mutually complementary, and especially useful for the theoretical branch which is proposed in this book. Walter Frisch, certainly the main responsible for inspiring studies about the subject, after the publication of his seminal book Brahms and the Principle of Developing Variation, affirms that By ‘developing variation’, Schoenberg means the construction of a theme (usually of eight bars) by the continuous modification of the intervallic and/or rhythmic components of an initial idea.(. . . ) Schoenberg values developing variation as a compositional principle because it can prevent obvious, hence monotonous, repetition.(. . . ) And Brahms’ music stands as the most advanced manifestation of this principle in the common-practice era, for Brahms develops or varies his motives almost at once, dispensing with small-scale rhythmic or metric symmetry and thereby creating genuine musical prose (Frisch, 1984, pp. 8–9).

Another important theorist of the principle of developing variation is Ethan Haimo. Unlike Frisch, however, Haimo is mainly interested in the use of the technique by Schoenberg in his serial works, which results in a quite original and surprising approach, since it goes against the (equivocated) general intuition that

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3_7

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developing variation can be only applied to tonal contexts.1 The definition I select is extracted from one of these non-tonal studies (in the case, involving some thematic elaborations of the twelve-tone opera Moses und Aaron, composed in 1932). In Haimo’s words Developing variation is a special category of variation technique, one that implies a teleological process. As a result, later events – even markedly contrasting ones – can be understood as originate from, or grow out of, changes that were made in the repetitions of early musical units. Therefore, true developing variation can be distinguished from purely local varied repetitions that have no developmental consequences. Developing variation offers the possibility of forwards motion, permitting the creation of new or contrasting (but still related) ideas, while local variation affects only the passage in question (Haimo, 1997, p. 351, my italics).

A special concept in this definition, namely teleology, deserves some commentary. Being a combination of two originally Greek words, “telos” (objective, finality) and “logos” (reason), the notion of teleology in the present context expresses the possibility of developing variation being used to give some directionality to a composition. This brings a fascinating dimension to the idea of variation, whose potential application to analysis will be later explored in this chapter. The next sections examines the different aspects of the actuation of the principle of developing variation in MDA.

7.1 A Transformational Typology of Developing Variation Under the transformational bias of MDA, developing variation (DV, for short) can be concisely defined as a process that encompasses sequential, correlated variations occurring in function of time. In other terms, DV interconnects isolated transformations in a chain of related actions highly dependent on contextual conditions. Conversely, the “ordinary” derivative work (whose archetypal representation was introduced in Fig. 1.5) is inherently a non-contextualized kind of transformational action and, therefore, can be considered as a, so to speak, “out-of-time” process. Based on this vision, I propose here a new concept, the unit of developing-variation work, or UDV, whose archetypal representation is depicted in Fig. 7.1. A UDV presents the necessary and sufficient conditions for the establishment of a developing-variation process, and accordingly can also be classified as a first-order developing-variation work (DV1). A UDV is composed by three sequential steps: 1. First derivation (P0 .→ C0 ), as a conventional archetypal derivative work 2. Child-parent equivalence, where the child of the previous stage becomes the parent for the next (C0 = P1 ), and

1 In this regard, the doctoral dissertation of Jack Boss, entitled An Analogue to Developing Variation in a Late Atonal Song of Arnold Schoenberg (Boss, 1991) can be seen as a pioneering work.

7.1 A Transformational Typology of Developing Variation

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Fig. 7.1 Archetypal representation of a unit of developing-variation work considering steps 1 (first derivation), 2 (child-parent equivalence), and 3 (second derivation)

3. Second derivation (P1 .→ C1 ), another derivative work, and a consequence of the first. In sum, a UDV comprises the junction of two simple archetypal derivative works by an intermediary relation of equivalence, which functions analogously as a pivot chord in a tonal modulation. Four possible relations of similarity can be considered in a UDV: • S1: between P0 and C0 ; • S2: between C0 and P1 , necessarily equal to 1,2 considering their mandatory equivalence; • S3: between P1 and C1 ; • S4: between P0 and C1 . This last relation is the most meaningful in a developing-variation work. Although exceptions can occur, S4 is expected to be a lower value than S1 since, as a rule, DV tends to provoke continuous dilution of similarity between P0 (or the primordial parent) and its subsequent descendants.3 From this, it is possible to propose a general model of a nth-order developingvariation work (DVn)4 by “plugging” in series an indefinite number of UDVs, as shown in the model of Fig. 7.2. Notice that eventually “infertile” variants (indicated by empty rectangles) can be produced along with the “generations”. These are not

2 Recall

that “1” is the maximum value of divergence. the other hand, S1 and S3 are mutually independent and cannot be related based on a general assumption. 4 This first version of the model will be later improved. 3 On

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Fig. 7.2 General scheme of a nth-order developing-variation work (DVn)

relevant for DVn, which is represented only by the main branch of derivations, namely the UDV connections. The similarity relation between the primordial parent P0 and a nth descendant is denoted by the symbol .σ n . The measurement of this relation is associated with the notion of absolute developing variation, to be elaborated in the next section, and represents an important aspect of MDA, especially in analytical applications, as it will be worked in Part III.

7.2 Relative and Absolute Developing Variation At this point, we can review the cases of decomposable-compound variation (simple or hybrid, like those depicted in Figs. 5.6 and 5.7) as special instances of developing variation. Indeed, we observe in these cases the action of a progressive and recursive transformation of basic elements, an essential condition for characterizing a developing-variation work. Just one (though very important) factor distinguishes this type of situation from an ordinary DV process, namely the time scale consid-

7.2 Relative and Absolute Developing Variation

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Fig. 7.3 Graphical representation of a hypothetical “timeless” developing-variation process

Fig. 7.4 Graphical representation of a hypothetical ordinary developing-variation process (first version)

ered. While the latter case unfolds along the time, being evidenced by real events present in the score, the former, as mentioned in the previous chapters, corresponds to an analytical construct formed by a series of plausible and logical hypotheses proposed by the analyst, an inherently timeless process (maybe more precisely, they could be said as developing throughout a subjective time). Given this, consider the graph of Fig. 7.3 as a possible representation for such situation. In this onedimensional model (since time t is disregarded), variants follow each other on an axis oriented by the increase of divergence in relation to the referential idea P (positioned at the origin). Selecting any child Cn for examination, its similarity with the initial parent is expressed by .σ n . Observe that the limit 1.00 corresponds to the borderline that ideally separates P-related variants from contrasting ones. Now consider a typical case of developing variation in which transformations occur on the temporal dimension. This general model is presented in Fig. 7.4, which proposes an alternative conception (in a first version) for the scheme of Fig. 7.2. As one can observe, the successive variants are plotted in the individual derivative spaces of their direct predecessors. Let us call any of these isolated spacial dislocations as relative divergences. At a first impression, the general appearance of this new model may imply that divergence is cumulative (increasing according to the north-south direction), and

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that the absolute divergence between the initial parent and a given child (say, C3 ) could correspond to a sort of resulting of the previous transformations (and perhaps it could be measured through some geometric/trigonometric means). However, this hardly will be observed in the analysis of real music. As a matter of fact, divergence is not necessarily cumulative (although it may eventually be, especially during some amount of time): frequently, after some derivative steps, the expected dilution of similarity can recede, despite continuing to progress “southward” on the relativedivergence paths (in a sense, this phenomenon is somewhat similar to what occurs in biological contexts, when a given child is in some phenetic characteristic more similar to a remote ascendant than to his/her own parents). Given this, a new improvement of the model is required, as shown in Fig. 7.5. In this version P is positioned at the origin, separating the Cartesian plane in two parts, which are called the relative-divergence (at the south) and absolute-divergence (north) spaces. While the former is plotted as previously (with the unique difference that now the labels of the variants are positioned on the temporal axis, at the bottom of the graph), the latter brings a new type of information, namely the relations of divergence directly measured between any variant and the primary parent P. The example illustrates how a sudden recession (.σ 4 ) interrupts a series of increasing absolute divergence. The dotted line at the vertical coordinate 1.00 represents the ideal limit that separates similar from entirely contrasting ideas.

7.3 Genealogical Notation Before examining further concepts associated with developing variation in MDA, it is necessary to consider the problematic issue of labeling variants. In order to avoid long and confusing analytic designations, I propose a genealogical notation, which intends to reflect as clear as possible the different relationships between a referential unit and its descendants. The usage of this notation is ruled as it follows: the first step is to assign lowercase letters in alphabetic order (a, b, c, . . . )5 to name musical ideas (or segments of them) that are considered as references for derivation.6 These referential elements are generically named progenitors. Variants of first-generation are labeled with the same letter of the progenitor from which derive, being ordered according to their appearances with subscript numbers (ex: a1 , b3 , c1 , etc.). From the second generation on, dots are added as separation marks between further numbers. In sum, a given genealogical label is always initiated by a lowercase letter (indicating the progenitor of origin). If the letter is unaccompanied by other symbols, the label refers to the progenitor itself. If there is just one subscript number

5 Or

the respective letters related to the variables, if it is the case. that in the derivative analysis of a real musical piece it is very common the presence of more than one referential idea (P) that can produce, through variation, their respective lineages. 6 Notice

7.3 Genealogical Notation

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Fig. 7.5 Graphical representation of a hypothetical ordinary developing-variation process (second version) Table 7.1 Examples of genealogical notation

Label b a b3 a1.1 c2.1.1.4

Informal meaning Progenitor “b” Progenitor “a” Third child of “b”, first generation First child of the first child of “a”, second generation Fourth child of the first child of the first child of the second child of “c”, fourth generation

attached to the letter, the label refers to a first-generation variant. If there are two or more numbers (always separated by dots), the label refers to higher numbers of generations (which will correspond to the number of dots plus one). Table 7.1 provides some examples of genealogical notation. It is easy to perceive that, in spite of the multiple advantages of the genealogical notation (clarity, logic, simplicity, conciseness), certain special situations can result

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in very extended labels, which may turn the analysis somewhat clumsy and problematic. Consider, for example, a seventh-generation direct descendant of a given progenitor “a”, whose genealogical label must be expressed as: .a1.1.1.1.1.1.1 . In order to avoid lengthy strings like this one, an additional convention was created: if a genealogical label has a number with n repetitions, replace the sequence by indicating n in superscript to the number (considering the example above, the label would be simply re-written as .a17 ).7 Automatically, n informs also the number of the generation considered. This rule can be extended to labels with mixed numbers, as well. For example, compare the ninth-generation variant b1.1.1.1.2.1.1.1.3.3.3.3.3.1 with its concise reformulation: .b14 .2.13 .35 .1 .

7.4 Derivation of Variables As suggested in Chap. 6, variables are to be understood as abstract, low-level constructive units. Like conventional motives (high-level, holistic structures), they are subjected to derivation by application of transformational operations and, accordingly, can produce lineages of variants through processes of developing variation. The variants so produced are identified with the same letter of origin and genealogically numbered (e.g., Z2 , Y3.1.1 , etc.). Moreover, this labeling strategy provides an elegant and concise way for describing complex structures (like motives or themes) as strings of “genetic” information. Figure 7.6 depicts a hypothetical example of both types of representation. In (a) the motivic components of a given musical fragment of a hypothetical piece are identified as derivations of six basic units inserted into rectangles and labeled with variables Z to U. Note especially how the labels express the derivation from respective basic variables (obviously, the numbering adopted for them are also entirely hypothetical). Figure 7.6b proposes an alternative, chart-like notation for the fragment, in which only the labels are informed, respecting their order of occurrence. As a convention, variables associated with the pitch domain are displayed in the top row of the chart, while the temporal variables are listed in the bottom row. Reinforcing the metaphorical links depicted in the previous section, let both representations be called as “phenetic” (a) and “genetic” (b). In analytical situation (as some of those presented in Part III), derivative processes involving variables will be referred as low-level developing variation (lDV).8

7 Evidently, the number in superscript does not be seen as an exponent, but rather as an index that is intended to inform the number of repetitions of the “base”. In practical terms, this strategy is applied with three or more repetitions. 8 Analogously, the notion of high-level developing variation (h-DV) describes derivative processes involving UDSs.

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Fig. 7.6 Example of a derivative analysis of a hypothetical thematic fragment, considering variables and some possible variants (a); representation of the same theme using “genetic” description (b)

7.5 Quotient This is a concept that correlates with the notion of low-level DV, being a useful tool in derivative analysis for associating motivic units. Basically, the quotient between

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two UDSs C1 and C2 ,9 denoted as Q(C1 , C2 ), depicts the variables (permanent or temporary) shared by both units, disregarding eventual variants. We can also think of quotient as a kind of common denominator of the essential material of related musical ideas. Figure 7.7 illustrates the application of the concept in analysis. Consider four hypothetical musical fragments (a, b, c, and d), produced along a derivative process (neither the order of derivation nor the specific ways with which this process was accomplished are particularly relevant in this example). Consider also four referential variables, two associated with pitch domain (Z, Y) and two with time (X, W). The most evident derivations from them are indicated in the analysis (for simplicity, other possible more remote associations are not taken into account). Quotient relations between the fragments can be observed in different manners. Since the order is immaterial in this case, there are six possible binary relations: • • • • • •

Q(a, b) = {Z} Q(a, c) = {Z, Y, X, W} Q(a, d) = {W} Q(b, c) = {Z} Q(b, d) = {.∅} Q(c, d) = {W} Three ternary:

• Q(a, b, c) = {Z} • Q(a, c, d) = {W} • Q(b, c, d) = {.∅} And just one quarternary, namely Q(a, b, c, d) = {Z, Y, X, W}. Examining these relations, we can draw three meaningful conclusions: (i) motives a and c are the most closely related; (ii) variables Z and W are the most pervasive elements (present in three of the four fragments); (iii) motive d presents the most fragile parental relation with the remaining ones, and probably would not be considered part of their “genetic pool” in a real-music situation. It is also possible to consider in analytical situations a more rigid modality, named absolute quotient, denoted by Q*, which filtrates only the exact variable/variant versions which are shared. Aiming to illustrate the difference between Q and Q*, consider, for example, a referential idea P and three closely related variants C1, C2, and C3, whose contents are concisely described in “genetic” notation as it follows: • • • •

P = {Z, Y, X} C1 = {Z1 , Y} C2 = {Z1.1 , Y1 , X} C3 = {Z, X1 }

9 It

can also involve a larger number of units, although binary relations are more common.

Fig. 7.7 Four hypothetical fragments related by derivation from basic elements (variables Z, Y, X, and W)

7.5 Quotient 135

136 Table 7.2 Q and Q* binary relations between P, C1, C2, and C3

7 Developing Variation Pair P/C1 P/C2 P/C3 C1/C2 C1/C3 C2/C3

Q {Z, Y} {Z, Y, X} {Z, X} {Z, Y} {Z} {Z, X}

Q* {Y} {X} {Z} {.∅} {.∅} {.∅}

Considering, for simplicity, only binary relations, Table 7.2 compares the sets obtained from the application of Q and Q*. In sum, the absolute quotient provides the system of a formal apparatus for evaluation of the “longevity” of a given variable, a topic to be better explored in the next part of the book. On the other hand, it is also possible to think about musical elements that integrate a given musical unit but are not associated with any common variable. Such elements are, so to speak, filtered out when a quotient relation is established between two or more related UDSs and will be classified as alien material (AM, for short). It is important to state that AMs can be either purely residual (that is, of none consequence in musical development), or rather present real derivative significance. In the latter case, it is possible to envisage two basic functions (that may or not interact) for the insertion of alien materials: (a) dilution, which normally implies an increase of divergence related to some variable; (b) deviation, through the creation of a new derivative path (headed by the AM), propelling development in a different direction. This can be associated either with the entrance of contrasting formal moments or, in more radical cases, to lead to speciation (a topic to be adequately examined ahead).

7.6 Crossover This concept refers to the combination of abstract characteristics of two or more musical ideas as a basis for a single derivation. Crossover may involve the same or, more usually, distinct domains/attributes. A consequential crossover (that is more relevant in this study) takes place when the resulting form initiates a new derivative lineage, like a divergent branching in a tree. Figure 7.8 illustrates this process through a possible crossover of UDSs a and d of Fig. 7.7. Observe that the pitch structure of the new form (accordingly labeled as [a–d]) is based on two sequential manifestations of a local, temporary variable z (diatonic pitch sequence formed by an ascending third followed by a descending second),10 closing with a descending octave (a transformation of permanent variable Z, here hypothetically

10 Observe

also that, from this perspective, the second part of the pitch organization of d can be explained as the retrogradation of z, say z1 .

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Fig. 7.8 Example of crossover of UDSs a and d of Fig. 7.7, by combination of permanent variables Z and X with temporary variable z, resulting into the hybrid UDS [a–d]

labeled as Z1.2 ).11 Needless to say that variant [a–d] can become a basis for further derivation, eventually initiating an independent lineage (therefore, as the output of a consequential crossover).

7.7 Inter-Attribute Equivalence Another possibility during DV processes is the replacement of a given referential attribute by another one. This is called inter-attribute equivalence. A necessary condition for this to happen is that both attributes are of same domain and have some “synonymic” relation. The equivalence is more efficient between the pairs of attributes p1/p2, p1/p3 (or p3-d), t1/t2, and t1/t3. In practical terms, this results in a very subtle transference of derivative referential, which is substantially preserved, but allowing for expanding the range of available transformational operations.12 Figure 7.9 exemplifies the usage of the inter-attribute equivalence, in this case between p3 and p1, considering the process of developing variation of a hypothetical permanent variable Z. While three siblings of first-generation (Z1 , Z2 , and Z3 ) are

11 Also hypothetically, let us consider that a first elaboration of the unison Z was to expand it to an ascending octave, in a (Z1 , from which stems the descending octave Z1.2 ). 12 Equivalence is someway analogous to modulation between close tonal regions by the use if a pivot chord.

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Fig. 7.9 Example of inter-attribute equivalence, involving the replacement of p3 by p1 as referential attribute for derivation

obtained through distinct transformations applied to intervallic sequence, a secondgeneration variant arises through chromatic transposition applied to Z3 (an operation exclusively associated with attribute p1), after the establishment of inter-attribute equivalence (the change of referential is denoted in analysis by the symbol .≡).

7.8 Thematic Transformation, Linkage, and Metric Displacement This section addresses in a very concise manner a trio of concepts considered by Walter Frisch in his celebrated book Brahms and the Principle of Developing Variation (1984) as formative of Brahms’s style.13 At a first glance, these concepts may seem not directly associated with developing-variation techniques, however the knowledge about their scopes contributes decisively for the understanding of the large-scale processes employed by Brahms in his music, and therefore deserve to be introduced in the present chapter as adequate preparation for the analysis that will be presented in Part III. Due to their importance, these concepts will be as better as possible incorporated to the present theory. Although it could be considered, in a sense, as a kind of an antipode of “intracellular”, developmental procedures, the creative usage of thematic transformation (TT) by Brahms along his career became, according to Frisch, one of the factors that characterize his rich and peculiar developmental modus operandi. Besides Brahms, 13 For

a more detailed discussion about these subjects, see, besides Frisch (1984), the compilation proposed by Desirée Mayr (2018), on which most of this section was based.

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139

the use of TT in the creation of related versions of themes can be observed in the music of Mozart, Beethoven, Schubert, Schumann, etc.14 However, is certainly Liszt the composer who more intensively has employed this practice.15 As defined by Frisch, thematic transformation involves a radical change in mood and character, while the theme’s basic shape, especially its pitch contour or configuration, is retained. Tempo, rhythm, meter, harmonization, dynamics,articulation all these may be altered, but the theme still retains a recognizable Gestalt. To use a more concrete analogy: the various garments with which the theme is clothed do not affect its basic anatomical structure, its flesh and bones. (Frisch, 1984, p. 42)

TT techniques are in general considered superficial or ornamental, preserving most of the “essence” of P (or we can say that they yield highly-similar Cs). Some of the most common thematic transformations are: • • • • •

change of mode; change of mood and or tempo; insertion of melodic embellishing; change of meter; melodic and rhythmic “distortions”.

Therefore, according to MDA’s terminology, it is not extravagant to associate these procedures with the idea of holistic derivation, which affects a theme as a whole (or, in the biological metaphor, its “phenetic” structure). In opposition to TT, Frisch presents the concept of thematic development (TD), a transformation that is applied to basic units (a “genetic” procedure), whose combination ultimately forms larger blocks (mostly themes). Both models are confronted in Fig. 7.10. In (a) a theme, taken as a unit, is transformed in a number of alternative ways, in such a manner that its overall aspect is maintained in each variant, albeit superficially diversified, in some aspect. Thematic development’s representation, in (b), involves the elaboration of themes through assembling of variants of basic blocks (or, in the present terms, by derivation of variables). The innovative thesis of Frisch is based on the assumption that in his mature phases Brahms practiced a very sophisticated version of thematic transformation, allied to thematic development, a distinctive trait of the composer’s developingvariation style. This line of reasoning led Mayr to propose a new category for defining this combination of apparently (but not in reality) incoherent compositional processes, namely thematic transformation-development (TTD).16 As well summarized by her,

14 As

it is also proposed by Rudolph Réti (1978). especially the remarkable study made by Lyle John Anderson (1977) about this subject. 16 In MDA, TTD situations will correspond to hybrid cases of decomposable/holistic variation, as before suggested. 15 See

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Fig. 7.10 Schematic representations of thematic transformation, with straight lines (a) and thematic development, with dashed lines (b). Adapted from Mayr (2018, pp. 53–54)

although at first impression thematic transformation and development may seem antagonistic procedures, this bibliographical revision has demonstrated how Brahms assimilated the best of both principles to benefit his constructive intentions. Moreover, his compositional trajectory is marked by an oscillating balance between TT and DT: favoring the former in the earlier period, giving greater importance to the latter in the intermediary phase, culminating with a perfect synthesis of both in the last works. Based on this (. . . ) it is also possible to conclude that Brahms managed to create a sort of hybrid process, combining TT and DT. This can be characterized when a basic intention for the transformation of a theme produces derived ideas that are able to progress as autonomous forms, eventually becoming referential forms for new derivations. This process (. . . ) will be analytically identified in this study with the label TDT (thematic-development transformation) (Mayr, 2018, pp. 58–59).

The concept of linkage (originally in German, Knüpftechnik) was firstly formulated by Heinrich Schenker, consisting of a little-known element of his theory. It is defined by Orlando Fraga as a “technique of motivic association in which a new phrase repeats the motivic idea of the precedent phrase, aiming both to provide continuity and to initiate something anew” (Fraga, 2011, p. 76). In its adaptation to developing-variation contexts, linkage is seen by Peter Smith, a theorist that studied deeply the subject, as a “transformation of a gesture of conclusion into one of initiation” (Smith, 2001, p. 109). Basically, linkage provides a subtle,

7.8 Thematic Transformation, Linkage, and Metric Displacement

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Fig. 7.11 Schematic representations of linkage. UDS 2 is initiated with a transformation of the last element of UDS 1 (m .→ m’). Adapted from Mayr (2018, p. 65)

fluid connection between ideas, which becomes a special tool for keeping the developmental momentum.17 As before mentioned, linkage is considered by Frisch as a legitimate landmark of Brahms’s thematic treatment, and “one of the most efficient manners to provide a continuous and organic growth in a musical piece”, in the understanding of Desirée Mayr (2018, p. 65).18 Figure 7.11 depicts a simple schematic model for linkage. As stated by Mayr (2018, pp. 70–71), the concept of linkage in its several meanings, applications, and possibilities can be summarized as follows: (a) linkage is an efficient technique intended to promote developing variation; (b) its employment by Brahms is a characteristic trait of his thematic construction modus operandi, present in most of his compositional career; (c) it can operate on different structural levels and consider three domains: melody, harmony, and rhythm/metric; (d) melodic linkage, which is the most salient type and of easiest understanding, occurs when a variation of the conclusive motivic fragment of a musical idea is employed for initiating the subsequent thematic idea; (e) harmonic and rhythmic linkage involve more vague and abstract associative relations (and, consequently, relatively harder to detect in analysis), resulting in structural reinterpretation, due to the ‘blurring’ of formal boundaries. This can be caused by frustrating harmonic expectancy (omission of chords, use of inversions, etc.) or displacement and/or manipulation of metric or hypermetric structures; (f) linkage may also be employed for highlighting expressive connotations with narrative plots, as in songs; (g) the main specific purposes of linkage use for organic construction can be summarized as: to achieve cohesiveness by continuity ([Oswald] Jonas [one of the first followers of Schenker]), to generate new thematic ideas (Frisch), to ‘provide continuity [and contrast] in an economic way’ (Smith), to be employed as structuring means in a larger-scale (Frisch and Smith), ‘to produce confusion and disorientation and the expressive implications therein’ (Rahn).

17 Smith contributes to an expansion of the subject by proposing that there exist other types of linkage besides melodic (the most evident), namely, harmonic and metric. According to the author, many times two or more types can interact in certain situations, as he demonstrates in several examples extracted from the literature. 18 Linkage in the music of Brahms is also studied by Steven Rahn (2015), with an interesting and original approach associated with literature, focusing on texts by French writer Alain RobbeGrillet.

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As mentioned in previous chapters, a strong preference for metric ambiguity is one of the most distinctive traits of the compositional personality of Brahms, as recognized by Schoenberg in his famous broadcast lecture “Brahms the Progressive”, transmitted in 1933 by Frankfurt Radio, celebrating Brahms’s centennial. In this lecture (later published as an essay in the book Style and Idea) Schoenberg exalts the extraordinary capacity of Brahms for avoiding quadrature in musical phrases (even in songs), and his inclination toward metric ambiguity, accomplished through the use of strategies like hemiolas (in several structural levels), superimposition of metricly mismatched strata, and by the employment of “musical prose” (a term coined by Schoenberg, in opposition to “musical poetry”, associated with regularly subdivided phrase lengths). These elements provide the necessary material and conditions to Frisch, who argues that the ingenious, creative manipulation of metric resources by Brahms corresponds to an important facet of his developing-variation style. In his book, Frisch examines a number of examples of what he classifies as metric “development”, “conflict”, and “fluctuation”, bringing the necessary evidence for supporting his arguments. In a PhD dissertation published in 2005, Yuet Ng proposes am interesting expansion of the theory of Grundgestalt, by including meter as a possible parameter to be considered (besides intervallic configuration, rhythmic ideas, and even harmonic relations). Ng bases his proposal on the concept of metric dissonance, coined by Harald Krebs (1987), which allows for the establishment of a detailed typology and analytical methods. The author supports his formulations with the help of a number of examples of Brahms’s excerpts, which results in a very consistent study.19 Essentially, the most attractive aspect of the manipulation of meter (by the superimposition of incongruous rhythmic lines, use of hemiolas, by the contradiction of the notated barring, etc.) in the context of developing variation lies in the possibility of obtaining distinct musical constructions (and effects) with a minimal of transformation, something that Brahms explores intensively. This is directly associated with the principle of economy, a fundamental element of developing variation. As it will be demonstrated in Part III, meter plays a special role in the identification of the developing-variation processes in Brahms’s Intermezzo Op.118/no.2. Some ideas introduced in this section will be properly resumed and elaborated along with the analysis, addressing specific situations.

19 Mayr

(2018) applies Ng’s theory in her own investigation of Brahms’s violin sonata Op.78, pointing out that the striking metric instability of the initial bars of the piece carry the “seed” for further elaborations.

7.10 Teleology

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7.9 Speciation The notion that new living species are created from others due to a combination of geographical isolation with cumulative variation during large amounts of time was proposed by Darwin as one of the most far-reaching theses of his revolutionary The Origins of the Species, published in 1859. Based on this line of thought, Darwin provocatively suggested to his contemporaries that the overwhelming amount of species that already lived on Earth could be ultimately connected to each other by derivative links.20 In 1906, American biologist Orator Cook proposed the term speciation for denoting this process of the birth of a new species. Like its biological counterpart, musical speciation21 takes place when a given variant radically diverges from its immediate parent, founding its own lineage. In general, rhythmic divergence is the main responsible (but not uniquely) for the occurrence of speciation, especially when accompanied by changes in cardinality. In practical terms, the arousal of speciation is evidenced in analysis through the measurement of the absolute quotient Q* between a referential and derived UDSs. The gradual decrease in importance of a given shared variable (which “conducts” derivation up to this point) is a strong signal that a speciation process is about to happen. A common complementary procedure is to infiltrate a related, but subsidiary element (generally a temporary variable), which becomes also gradually more and more relevant, replacing ultimately the role of the previous material. The combination of both strategies leads almost naturally to speciation, implying in the creation of a, so to speak, detour of the derivative path, which not rarely is associated with the settling of a new formal boundary. An abstract scheme of speciation is provided by Fig. 7.12.

7.10 Teleology A teleological perspective in the realm of musical variation is based on the belief in the existence of an objective or goal orienting the derivative process. In Darwinian terms, “subjective” teleological trajectories22 can be viewed as resulting from the

20 A modern and comprehensive study about Darwin’s extraordinary insight is provided by Richard

Dawkins in his acclaimed book Tale of the Ancestor (Dawkins, 2004). 21 For an interesting study concerning metaphoric relations between music and biological evolution in some writings of Béla Bartók, including the idea of speciation in both peasant Hungarian and art music, see Bennett (2015). 22 Strictly speaking, the idea of a “real”, “intended” teleology in natural selection is nonsense. As pointed out by Ernst Mayr, “in earlier periods many authors thought that a perfection-giving process was involved in evolution. Before the discovery of the principle of natural selection, one could not imagine any other principle than teleology that would lead to such seemingly perfect organs as the eye, annual migrations, certain kinds of disease resistance, and other properties of organisms. However, orthogenesis and other teleological explanations of evolution have now been

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Fig. 7.12 Schematic representation of a process of speciation. The derivation of a hypothetical lineage led by UDS a, whose main element is variable Z, is gradually undermined by the infiltration of variable w, until the complete substitution of Y, and the establishment of a new lineage, headed by UDS b

action of evolutionary pressures that affect the living species along a given span of time. In natural selection, these evolutionary “paths” depend on random mutations and environmental conditions and are normally measured in geological time scales.

thoroughly refuted, and it has been shown that indeed natural selection is capable of producing all the adaptations that were formerly attributed to orthogenesis” (Mayr, 2002, p. 319).

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145

On the other hand, in artificial selection (i.e., made under human supervision, aiming at animal or agricultural improvements) the evolutionary teleological process is more evident (and considerably faster), which makes it more adequate for our comparison with music. Figure 7.13 proposes a very simple illustration of the process of artificial selection combined with a teleological planning. Suppose that a farmer, moved by market conditions, aims to produce rabbits one-third bigger than those he/she currently posses. A good strategy is to mate ordinary-sized individuals with those of opposed sex with a genetic predisposition for becoming slightly bigger. By selecting among the resulting brooding the biggest individuals and, at the appropriate moment, mating the pairs, and repeating successively the process, after a few generations the intended goal is reached. As a matter of fact, like the case of the farmer, a composer can project specific developmental processes for musical ideas, and follow them by recurrently applying variation techniques and selecting those forms more promising for further transformation, according to his/her creative intentions.23 In spite of being a difficult (in some cases even impossible) task, the search for these intentions in an analysis corresponds precisely to the determination of the goal—or telos—of a given derivative process. Normally, only retroactively—through careful observation—an analyst can reconstruct a transformational path linking an elaborated variant to its predecessors. As in other DV aspects, context is a paramount element for detecting in analysis that a teleological planning is taking place. Therefore, isolated musical examples would be very limited, and incomplete descriptors. Nonetheless, I propose a basic illustration which maps very roughly the above presented metaphor of the rabbits. Consider UDS a (Fig. 7.14) as a referential motive, and that the composer plans to gradually expand its original and characteristic initial intervallic configuration (assigned with variable Z) in subsequent variants and to reach the definitive version after four successive transformations, accordingly labeled as .a14 (let us assume that each of these variants is employed in some specific thematic situation, and that the transformations are spaced throughout with several measures of the piece). For simplicity, other possible musical attributes of a (like durational configuration), subject to simultaneous (or not) transformations were not considered. The proposal of teleological hypotheses can become an efficient tool for the analyst who tries to understand the nexus of a series of choices made by a composer in a given organically-constructed piece.24

23 For the description of an experiment addressing recurrent “matting-selecting” musical motives, through the use of computational program named DARWIN, see Almada (2015). 24 An interesting example of the presence of telos in real music is presented by James Bennett in his aforementioned dissertation about evolutionary treatment in Bartók’s compositions. Although not using specifically the term (Bennett refers to the situation as “evolving motive”), there is a clear teleological planning in the fifth of the Improvisations on Hungarian Peasant Songs Op. 20, such that parallel perfects fifths in the accompaniment from m. 21 on are gradually contracted into augmented fourths and thirds (for a detailed analysis of this passage, see Bennett 2015, pp. 60–64).

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Fig. 7.13 Simple example of teleology in a hypothetical process of artificial selection applied to rabbit creation

7.11 Involution Different from what is normally thought, evolutionary processes do not always unfold in linear routes, progressively, along the arrow of time. Although rare, eventually “regressive” movements take place reproducing “ancient” versions of a given form in development, as returning to a previous point of its genealogical tree.

7.11 Involution

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Fig. 7.14 Simple example of teleology in a hypothetical process of artificial selection applied to rabbit creation

The musical equivalent of this biological phenomenon, in MDA’s terms, will be called involution. Figure 7.15 proposes a schematic representation of a case of involution in a hypothetical developing-variation process, here depicted in a tree format for a more intuitive explanation. In the example, a fifth-generation variant (a2.1.2.1.1 ), instead of keeping the forward movement away from the root, produces unexpectedly the “revival” of an “ancestor”, a1 , which then follows branching a new second-generation variant, properly labeled as a1.4 (highlighted in gray). This topic closes the exposition of MDA’s theoretical framework. Allied to the basic concepts introduced in Part I (decomposable/holistic variation, derivative space, attributes, measurement of similarity, transformational operations, TD analysis, among others), the elements of Part II, associated with the principles of Grundgestalt and developing variation (mainly, Grungestalt-components, variables, genealogical notation, quotient, crossover, inter-attribute equivalence, linkage, speciation, telos, and involution) form the necessary support for practical application in analysis, to be accomplished in Part III.

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Fig. 7.15 Representation of involution in a tree format

a

3

... a

a

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a 2.1.3 a 2.1.2.1

a a a

1.4

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a

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involution

1.2

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a0 References Almada, C. (2015). Evolution in musical contexts: The software DARWIN. In Proceedings of the 25th National Conference of Brazilian Association of Post-Graduation Research ANPPOM, Federal University of Espírito Santo, Vitória. Anderson, L. (1977). Motivic and thematic transformation in selected works of Liszt. Dissertation, Ohio State University. Bennett, J. (2015). Explosions of diversity: Béla Bartók’s evolutionary model of folk music. Dissertation, University of Wisconsin. Boss, J. (1991). An analogue to developing variation in a late atonal song of Arnold Schoenberg. Dissertation, Yale University. Darwin, C. (2009/1859). The origins of species. London: Penguin Classics. Dawkins, R. (2004). The ancestor’s tale: A pilgrimage to the dawn of evolution. Boston: Houghton Mifflin Harcourt. Fraga, O. (2011). Progressão Linear. Londrina: Eduel. Frisch, W. (1984). Brahms and the principle of developing variation. Los Angeles: University of California Press. Haimo, E. (1997). Developing variation and Schoenberg’s serial music. Music Analysis, 16(3), 349–365. Krebs, H. (1987). Extensions of the concepts of metric consonance and dissonance. Journal of Music Theory, 31(1), 99–120.

References

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Mayr, D. (2018). The Identification of developing variation in Johannes Brahms op.78 and Leopoldo Miguéz op.14 violin sonatas through derivative analysis. Dissertation, Federal University of Rio de Janeiro. Mayr, E. (2002). What evolution is? London: Phoenix. Ng, Y. (2005). A Grundgestalt interpretation of metric dissonance in the music of Brahms. Dissertation, University of Rochester. Rahn, S. (2015). Knüpftechnik: Coding narrative in the music of Brahms and the experimental fiction of Robbe-Grillet. Dissertation, University of North Carolina. Réti, R. (1978). The thematic process in music. Westport: Greenwood. Schoenberg, A. (1994). Coherence, counterpoint, instrumentation in form. Neff, S. (Ed.) Neff, S., Cross, C. (trad.). Lincoln: University of Nebraska Press. Simms, B. (2000). The atonal music of Arnold Schoenberg (1908–1923). Oxford: Oxford University Press. Smith, P. (2001). New perspectives on Brahms’s linkage technique. Intégral, 21, 109–154.

Part III

Analysis: Brahms—Intermezzo in A Major Op. 118/2

An adequate understanding of the derivative relations in a musical composition demands necessarily formal-harmonic contextualization. Moreover, as discussed in the last chapter, I consider that metric organization (both from global and local perspectives) normally plays also a role in the establishment of a contextual basis for derivative analysis. This becomes particularly important in Brahms’s music, as it is the present case. As a matter of fact, the Intermezzo Op. 118, no. 2 in A Major (henceforward identified as Op. 118/2, for simplicity) is notably provided by a highly sophisticated, and complex metric structure, whose impressive effects contribute largely to the impressions of fluidity and ambiguity that characterize this piece.1 With these considerations in mind, the third part of the book encompasses two complementary and correlated chapters, a structural analysis combining formal, harmonic, and metric organizations (Chap. 8), concluding with a detailed examination of the web of derivative relations that form the piece (Chap. 9). The work selected for analysis is the second of the six piano pieces reunited in the collection Sechs Klavierstück Op. 118, composed in 1893, being identified by the title Piano Intermezzo Op. 118/no. 2. The Op. 118/2 has an especially beautiful and enchanting—yet melancholic— character, perfectly adjusted to a sophisticated harmonic accompaniment, which is enriched by a combination of characteristically Brahmsian textural contrasts and metric ambiguities. Maybe it is the conjunction of such attributes that makes the piece so attractive to analysis, motivating studies concerning different aspects of musical structure. I selected for a short examination four of these studies. They will provide, in a lesser or a greater extent, some basic references for the present analysis. The first study is found in the fourth part of the book Beyond Orpheus, originally published in 1980 by David Epstein. As an important connection with this proposal, Epstein’s analysis is inserted in a framework concerned with music structure, in

1 Both characteristics are, in lesser or greater extent, reinforced by formal and harmonic elements, as it will be shown.

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III

Analysis: Brahms—Intermezzo in A Major Op. 118/2

which the principle of Grundgestalt plays a crucial role. Also central in Epstein’s text is the notion of ambiguity and its importance for Brahms. In spite of being very short (it is depicted entirely in the second half of page 175 of the book), the analysis of Op. 118/2 addresses an interesting aspect of ambiguity, related to the expression of tonal organization. As stated by the author, “the tonic (. . . ) is never heard as a point of rest until the closing measures of [the] outer sections. (. . . ) The presence of the tonic is generally inverted and unstable” (Epstein, 1980, p. 175). Another argument added to this question (as it will be demonstrated in the next chapter) is that the initial point of true repose, or structural downbeat, in this music is not found until the close of the first section, when the tonic explicitly arrives. (. . . ) By contrast, the middle section begins downbeat oriented, but in a weaker tonal region – stating clearly the local key center of F. minor (p. 175).

The article “Foreground Motivic Ambiguity: Its Clarification at Middleground Levels in Selected Late Piano Pieces of Johannes Brahms”, by Allan Cadwallader (1988) examines ambiguous construction in piano compositions by Brahms. Among other pieces, Cadwallader selects the Op. 118/2 for his investigation. As suggested by this author, the initial motive of the piece “can be ambiguously foreshadowed because of the pitches of the tonal system [that Brahms chose for the motive] are themselves inherently ambiguous”. He also adds that “contrapuntal shifts and other, rhythmic devices often obscure surface and foreground levels” (Cadwallader, 1988, p. 59). In another passage he concludes that “Brahms sets up ambiguous qualities at the surface which are realized and clarified at deeper levels and different contexts as the piece unfolds” (p. 60). In 1999 James Bass dedicates his master’s thesis entirely to the study of the techniques used by Brahms to construct cadences in the Op. 118/2, where pervades “an ambivalence of commitment of a tonal center for long periods of time” (Bass, 1999, p. v), essentially corroborating Epstein’s main argumentation. After displaying his analytical methodology associated with melodic and rhythmic aspects, Bass addresses metric placement as an independent, rather decisive factor for measuring “cadential strength”, in his terms. To this he adds three subsidiary elements, namely, timbre, dynamics, and expressive gesture. The combination of these attributes provides the necessary conditions to Bass for the elaboration of a typology for cadences, used then as basis for his analysis (displayed in the format of graphs), which is presented in the last chapter of the study. The discussion of the results allows for him to conclude that, interestingly, all (but one) cadences analyzed share one distinct characteristic. A general slowing occurs in some or all of the factors accompanying the cadential moment. Notes occur in duration relatively longer than preceding ones, harmonic rhythm slows, dynamic levels diminish, textures thin, and tempos broaden (p. 34).

More recently, Steven Rings analyzed the Op. 118/2 with considerable detail in the second part of his book Tonality and Transformation, (Rings, 2011). Focusing on transformational aspects associated with motivic construction, melodic functions, rhythmic-metric organization, as well as harmonic and tonal plans, Rings’s analysis

References

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reveals how these dimensions interact in function of an artistic expression notably marked by the sign of ambiguity. A brief illustration of his approach is how he proposes to examine some harmonic progressions under a binary perspective, considering the dialogue between a pair of strata, formed by thoroughbass (TB) and fundamental bass pitches (FB). According to this view, eventual discordances between both streams are associated with harmonic ambiguity, and almost always correlated to other domains in decisive passages of the piece. Basically, this strategy aims to evidence the duel between the putative tonic A and its subdominant region D, which seems to be for Rings at the center of the tonal uncertainty that animates the Op. 118/2, forming a duality referred in some moments as “D-instead-of-A” (p. 196) or “D-apotheosis” (p. 197). Besides these, a number of other considerations proposed by Rings will be especially useful as a reference and/or confirmation of some specific aspects addressed in the present analysis, and will be properly commented in due time.

References Bass, J. (1999). An examination of cadential activity and its relationship to structure in Brahms’s intermezzo opus 118, no. 2, in A Major. Dissertation, Southeastern Louisiana University. Cadwallader, A. (1988). Foreground motivic ambiguity: Its clarification at middleground levels in selected late piano pieces of Johannes Brahms. Music Analysis, 7(1), 59–91. Epstein, D. (1980). Beyond Orpheus: Studies in music structure. Cambridge: The MIT Press. Rings, S. (2011). Tonality and transformation. Oxford: Oxford University Press.

Chapter 8

Formal, Harmonic, and Metric Structure

The formal, harmonic, and metric structures of the intermezzo are separately examined in the following sections.

8.1 Form The piece presents a two-level, fractal-like ternary basic formal organization, schematized in Fig. 8.1.1 A more detailed view of this architecture is provided in the chart of Fig. 8.2. A third level (phrasal) is superimposed to the other two (sectional and subsectional). For practical purposes of analysis, the twenty-eight phrasal segments are sequentially numbered.2 A remarkable aspect of the formal third level is its almost completely regular segmentation into even-extension phrases (mostly with four, and eventually two bars), with the exception of just three cases (shaded in Fig. 8.2).3 This quasiperfect quadrature contrasts sharply with a highly complex and turbulent metric organization, as it will be later discussed.

1 Notice

that in the sub-sectional level the letters “a” and “b” are employed as generic categories, meaning, respectively, “statement” and “contrast” (the same applies to the recapitulatory subsection a’). In other words, subsection a of section A, for example, is entirely distinct from its counterpart, subsection a of section B. From now on, the subsections will be individually referred through the relations which maintain with the corresponding sections, in this manner: [A.a], [A.b], [A.a’], [B.a], [B.b], and [B.b’]. 2 Since all formal segments are initiated in anacrusis (a special characteristic of this intermezzo), for simplicity and consistency with analytical conventions, I indicate their beginnings at the first downbeat of the respective bars (i.e., a beat after the actual upbeat beginnings). As it will be seen in the section dedicated to meter, a different criterion for locating metric segments will be established. 3 They will be examined in due course. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3_8

155

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Fig. 8.1 Brahms’s Intermezzo Op. 118/2—Basic formal structure, considering two levels of organization

Fig. 8.2 Brahms’s Intermezzo Op. 118/2—Basic structure depicted in three levels of organization: sections (1), subsections (2), and phrases (3), associated with order numbers of the segments and respective measure numbers of entry. Gray cells indicate segments with non-regular extensions

8.2 High-Level Harmony (Tonal Relations) Figure 8.3 includes tonal relations to the tripartite formal structure, resulting in a network diagram.4 Nodes represent keys (the inserted uppercase letters refer to

4 The

present scheme adapts the models proposed by David Lewin (1987), David Kopp (2002), and Steven Rings (2011).

8.2 High-Level Harmony (Tonal Relations)

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Fig. 8.3 Brahms’s Intermezzo Op. 118/2—Network diagram, integrating form and tonal relations

the keys in major mode and lowercase to those in minor) and arrows depict neoRiemannian operations.5 Structural cadences (PAC and HC)6 are represented by equilateral triangles, whose spatial orientation define their types: a triangle based on an edge (representing, therefore, a stable state) denotes a PAC; a triangle based on a vertex (unstable state) corresponds to a HC. The gray rotated triangles at the end of segments 5 and 23 refer to ambiguous cases (I will return to this later). The determination of the relative importance of a PAC in the structure combines the size of its corresponding triangle with its “geographical” position: the bigger the triangle, the more stable is the PAC. Thus, the cadence that concludes segment 28 is unsurprisingly the most important of the piece, followed by that of segment 12 (which closes section A, the first structural repose of the piece, as stated by Epstein), and the remaining ones. The information conveyed by the twenty-eight segments can be further compacted, if we consider that some of the phrases are repetitions (or slight variations) of others (see correspondences in the left box of Fig. 8.3). Thus, for simplicity, I will disregard segments 3 and 20–28 in the following analyzes.

5I

adopt conceptualization and symbology established by Richard Cohn (1998, 2012) in his expanded “family” PLR. 6 Acronyms popularized by James Hepokoski and Warren Darcy (2006), referring, respectively, to the Perfect Authentic Cadence and to the Half Cadence.

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8.3 Low-Level Harmony (Chord Progressions) Figure 8.4 shows an analytical reduction of the main thematic material of the Op. 118/2 (mm. 1–8), organized as a period, according to Schoenberg’s terminology,7 comprising phrases 1 and 2. Interestingly, as one can observe, Brahms defers a more convincing conclusion for the passage by using a HC in the place of the expected PAC, which can be attributed to the general idea of avoiding closure which pervades the whole piece. Harmonic ambiguity plays a role also in a more local level: the accompaniment of the two initial measures seems to indicate D as a more stable point than the tonic itself.8 This uncertainty between A and D is intensified not only by metric position but also by the absence of the differential pitch G., introduced only in m. 3. As a matter of fact, the whole subsection [A.a], encompassing segments [.1 + 2 + 3], can be considered as a double period, in which the antecedent [.1 + 2] is a period itself, and the consequent (3) is an exact repetition of the former (symptomatically, also lacking a PAC). The beginning of subsection [A.b] is shown in Fig. 8.5. The conjunction of fluid modulations, pedal-points, and the metric mismatch of melodic and harmonic rhythms9 contributes to the contrasting character of the subsection. The two-phrase passage of Fig. 8.5 is structured as a period with weakened cadential articulations, in both cases by conflicting tonic and dominant information. While the antecedent ends with a HC supported by the tonic pedal (m. 20), the closure of the consequent (m. 24) puts together the leading tone of the key and an arpeggio of the tonic chord,10 resulting in the improbable chordal combination i.7 . Figure 8.6a reproduces the two-bar connection as it occurs in the score, proposing an interpretation for the incongruity between melody and harmony of m. 24: this would be explained by the omission of the expected V between both chords (Fig. 8.6b). In other words, while the melody “aims” at the dominant repose of a HC, the accompaniment is directed to a PAC.11 By omitting the intermediary chord,

7 In

its standardized configuration, a period is an eight-bar-long musical structure, subdivided into correlated halves, named antecedent and consequent. The former is responsible to present the main motivic material (mm. 1–2), which is followed by a contrasting motive (mm. 3–4), closing with a half cadence. The consequent resumes the beginning of the antecedent (mm. 5–6) and then provides a proper closure for the whole structure, normally a PAC (mm. 7–8). For more information about the period structure and its counterpart, the sentence, see Schoenberg (1967). For a more recent approach on these Schoenbergian concepts, see Caplin (1998). 8 An interpretation supported the analyzes by Epstein, Cadwallader, and Rings. 9 This aspect will be properly examined in the metric analysis. 10 Rings (2011, p. 195) also highlights this moment in his analysis. As already mentioned, this ambiguity seems to be at service of an intentional escape from closures, being clearly associated with the metric fluctuation that characterizes the piece, as it will better discussed bellow. 11 This ambiguity, depicted in the analysis of Fig. 8.5 by a rotated triangle (referring either to a “PAC-and-HC” or “non-PAC-neither-HC”), could be compared to the visual/cognitive effect

8.3 Low-Level Harmony (Chord Progressions)

159

4/3

Fig. 8.4 Brahms’s Intermezzo Op. 118/2 - mm. 1–8 (reduction)

Fig. 8.5 Brahms’s Intermezzo Op. 118/2 – mm. 17–24 (reduction)

Brahms seems to force the combination of the two contradictory closures, creating a boundary-blur effect. This strange cadence is immediately followed by a brief re-transitional passage (Fig. 8.7), marked by a dominant pedal (contextualized inside the modal mixture of A minor/major) and a chromatic ascending line that leads to C.5 , preparing the entry of subsection [A.a’] (notice especially how V and I are chromatically connected in mm. 27–28). The recapitulatory subsection [A.a’] has an unusual organization, being subdivided into three blocks. The first is represented by semi-phrase 7 (Fig. 8.8), corresponding to a slightly-varied recapitulation of mm. 1–2. This segment is then

known as “figure-background”. For a detailed study on some interesting relations between musical and visual ambiguities and paradoxes, see Brower (2008).

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Fig. 8.6 Brahms’s Intermezzo Op. 118/2 – mm. 23–24 (a); hypothetical omission of an intermediary dominant stage (b) Fig. 8.7 Brahms’s Intermezzo Op. 118/2 – mm. 25–28 (reduction)

followed by a brief elaborative passage (segment 8), based on a two-bar dialogue between motives of subsections [A.a] (in the lower register) and [A.b] (the quarter notes in the right hand). The unit of mm. 31–32 is restated in minor mode in mm. 33–34, causing an effect classified by Hepokoski and Darcy (2006) as “lights-out”. The third block (Fig. 8.9) recapitulates the consequent of [A.a] (mm. 5–8), this time involving a quasi-literal inversion of the two initial bars (both versions

8.3 Low-Level Harmony (Chord Progressions)

161

Fig. 8.8 Brahms’s Intermezzo Op. 118/2 – mm. 29–34 (reduction)

Fig. 8.9 Brahms’s Intermezzo Op. 118/2 – mm. 35–38 (a); intervallic comparison of units of mm. 1–2 and 35–36 (b)

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Fig. 8.10 Brahms’s Intermezzo Op. 118/2 – mm. 39–48 (reduction)

are compared in Fig. 8.9b).12 Harmonically speaking, there are no substantial differences between passages 2 and 9, except by the fact that now the previously deferred PAC finally is required for closing the theme. Section A is finalized by a codetta (Fig. 8.10). It begins with a four-bar phrase (mm. 39–42), clearly based on motivic material of subsection [A.b], closing with a relatively weak PAC (due to a pedal-point through the segment) with prolongs the cadence of m. 38. The phrase is then reiterated (mm. 43–46) with the counterpoint to the main line being transposed an octave lower and with the suppression of the pedal from this repetition. The expected reiteration of the cadential resolution (m. 46) is frustrated by a second-inversion tonic degree, which overlaps with the beginning of segment 12, bringing the proper closure to section A, the strongest PAC so far. As it is expected for the central section of a ternary form, B contrasts sharply with A in tonal ambiance (the relative region, F. minor), rhythmic/metric configurations (this will be examined in details in the metric analysis), and textural diversity. Its first subsection [B.a], shown in Fig. 8.11, is structured as a sixteen-bar period. The antecedent phrase (repeated exactly in the consequent) is formed by two semiphrases (13+14), closing with a HC. An incisive change to a chorale texture in combination with a modulation to the parallel major key of F. opens subsection [B.b] (Fig. 8.12). However, this key is not clearly established in the antecedent (segment 15), since unusual and insistent interchanges between the minor degrees III and VI suggest an alternative (and more plausible) interpretation for the passage, as part of a remote

12 Rings (2011, p. 198) points out this interesting transformation of the basic motive of the intermezzo. I will return to this with more detail in the derivative analysis.

8.3 Low-Level Harmony (Chord Progressions)

163

Fig. 8.11 Brahms’s Intermezzo Op. 118/2 – mm. 49–56 (reduction)

4/3

Fig. 8.12 Brahms’s Intermezzo Op. 118/2 – mm. 57–64 (reduction)

key, A. minor.13 In m. 59.2, a half-diminished chord rooted in C, followed by a F major triad seems to imply definitively this key (in spite of being enigmatically

13 This duality illustrates perfectly the Schoenbergian concept of floating tonality, i.e., the coexistence in equal terms of two keys (Schoenberg, 1969, pp. 111–113). Interestingly, in his Structural Functions of Harmony Schoenberg translates the original German term schwebend Tonalität as “suspended tonality”. However, in the English version of Harmonielehre (Theory of Harmony), published in 1978, the adjective schwebend is translated by Roy Carter as “floating”, and “suspended tonality” is used for denoting another concept (subtly distinct from the other), namely, aufgehobene Tonalität. Considering both the precedence of the latter book and the fact that the two concepts are there described in more details (see Schoenberg, 1978, pp. 128–131 and pp. 370–380), contrary to what happens in the former book, I adopt here the designation “floating” as a more adequate translation for “schwebend”.

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Fig. 8.13 Brahms’s Intermezzo Op. 118/2 – m. 60 (reduction)

Fig. 8.14 Brahms’s Intermezzo Op. 118/2 – mm. 65–73 (reduction)

oriented by the enharmonic key of B. minor). However, the apparent IAC14 in A. minor is suddenly transformed in a HC in F. major, closing the antecedent of the period. Figure 8.13 details the voice leading of this passage. Interestingly, resolved the dispute between both tonal poles, the consequent (segment 16) follows unambiguously in F. major, leading to a closure with another HC. Although formally identified as a recapitulation of [B.a], resuming its textural configuration and key, subsection [B.a’] seems more a brief elaboration of the previous ideas, being structured in two four-bar segments (.17+18), and beginning in canonic imitation, as shown in Fig. 8.14. Phrase 18 is emphasized by the presence of both the highest (climax) and lowest (nadir) melodic points of the piece, comprising an interval of two octaves and a perfect fifth (thirty-one semitones).15 Another remarkable aspect involving the last phrase is its odd extension (five bars), sharply contrasting with the previous seventeen four- and two-bar segments. A PAC closes properly section B.

14 Acronym for Imperfect Authentic Cadence, in which the melodic line does not repose in the tonic. Brahms takes here advantage of the fact that the C. is a common tone between both keys, another aspect of the pervading ambiguity in the piece. 15 The manner by which this wide melodic arc is filled in such a narrow span of time, ending with an incisive PAC under the nadir note, represents a quite dramatic and energetic gesture, the very expressive climax of the whole intermezzo.

8.4 Metric Structure

165

Fig. 8.15 Brahms’s Intermezzo Op. 118/2 – mm. 73–76 (reduction)

A short passage functions as a retransition (Fig. 8.15), both in tonal and motivic terms, preparing the recapitulation of the main section.16 The closure of the passage brings another striking ambiguity. As it can be observed in Fig. 8.15, the strong expectation created by the cadential I6/4 (m. 75) is frustrated by a conjunction of three factors: (a) the dominant-of-dominant harmony on the strong beat of m. 76 (in the place of V, the very target-chord of the expected HC); (b) the delayed chordal resolution (intensified by the fermata) in the third beat of the bar, a “territory” that belongs, as upbeat, to the next recapitulatory events;17 and (c) the “wrong” bass notes in m. 76, that seem to express contradictory cadential information (“V-I”). Once again, Brahms’s intention of keeping the intermezzo’s melodic flux somewhat independent from the formal/harmonic plan becomes evident, which is accomplished by the loosening of lower-level constraints. This view will be still more patent in the examination of meter, in the next section of this chapter.

8.4 Metric Structure As previously stated, Op. 118/2’s metric structure is extraordinarily complex, deserving, therefore, a detailed analysis in a separate section. Furthermore, such

16 Interestingly, like the previous phrase, segment 19 has also an odd extension (three-bar long), as if intended to restores the omnipresent quadrature (.5 + 3 = 8) of the intermezzo. 17 It is noteworthy to add how cleverly the melodic resolution of the passage anticipates the motivic anacrusis C.-B.

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complexity seems to play a functional purpose in the structure of the piece, in intimate alliance with the several formal/harmonic ambiguities implemented by Brahms, as discussed in the previous sections.18 Aiming to properly deal with the multiplicity of aspects and implications involved, the present analysis addresses three complementary approaches: • High-level hypermetric organization, considering three superimposed textural streams: main melody, harmonic accompaniment, and, when is the case, a secondary, internal melodic line; • Low-level aspects, directly influenced by rhythmic events, also considering combined streams; • Internal metric/hypermetric organization of the individual streams. These three approaches will provide the necessary information for the elaboration of new concepts, analytical methods, and a typology of the distinct metric situations present in the piece.

8.4.1 Segmentation At first, I propose to subdivide the piece into nine passages of special metric interest (ordered with Roman numerals for differing them from formal segments), whose localization in the score is shown in Table 8.1.19 Contrarily to what was used for form, the points of entrance of the metric segments are exactly positioned by the use of a more precise notation that informs measure and beat numbers (separated by a dot). Table 8.1 Brahms’s Intermezzo Op. 118/2—passages of metric interest

18 For

Passage I II III IV V VI VII VIII IX

mm. 0.3–8.2 16.3–23.3 24.1–30.2 30.3–34.2 34.3–38.2 38.3–48.2 48.3–56.2 56.3–64.2 64.3–76.2

Correspondence with formal segments [1+2] [4+5] .− m. 24 m.24 + [6] + m. 30 [8] [9] [10+11+12] [13+14] [15+16] [17+18+19]

some additional literature concerning metric ambiguities in Brahms’s music, see Cohn (2001) and Murphy (2001, 2009). 19 Like their counterparts (the formal-harmonic segments), the nine passages cover the essential intermezzo’s “territory”, omitting only the repetitions (in this case, mm. 9–16, and mm. 77–116).

8.4 Metric Structure

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Fig. 8.16 Brahms’s Intermezzo Op. 118/2 (melodic line). Comparison between formal and metric segmentation, considering mm. 17–31. The “dissonance” between conflicted segments (5 and 6) and passages (II and III) is indicated by the shaded squares. Dashed lines in formal segmentation indicate upbeat beginnings. For the sake of simplicity, only the melodic line is shown

As one can observe, the main divergence between formal and metric segmentation lies in the discrepancy of the borderlines established for segments 5/6 and passages II/III. The beginning of passage III (highlighted in bold in Table 8.1) gains special importance for being the only coincidence of a metric boundary with a downbeat.20 Such a, so to speak, “segmentation dissonance” (analogous to a minorsecond clash) is depicted in musical notation in Fig. 8.16. The nine passages are analyzed in the next subsection.

8.4.2 Metric Organization of Passages I–IX A first aspect to be observed in the metric analysis of the Op. 118/2 concerns the relation between the individual streams (melody, harmony, and, eventually, a secondary melodic line) and the grid established by the barlines in the score, which supposedly would orient the perception of strong/weak beats. However, as it will be presented, an expressive part of the musical events of the piece are almost continuously in a relation of discordance with this “default” grid.21 The following graphical schemes are intended to evidence the different situations in

20 The

reason for this interpretation will be clarified ahead. approximately to the idea behind the concepts of metric dissonance, coined by Harald Krebs (1999), and the relations of noncongruent metric, proposed by Wallace Berry (1987).

21 This corresponds

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Fig. 8.17 Brahms’s Intermezzo Op. 118/2—Passage I (mm. 1–8). Streams M (main melody) and H (harmony) configurations in relation to the notated bar lines

which divergences (or convergences) between streams and the grid occur along the piece. Figure 8.17 introduces the basic model for such schemes: vertical lines represent the notated barlines, while the sequences of rectangles indicate how the musical material is metricly organized in the measures (of course, according to present analytical interpretation) in the stream in question (M stands for main melody and H for harmony). Integers inside the rectangles/measures inform the number of beats that they contain. Shaded rectangles denote the occurrence of non-ternary dispositions. This situation is unique in the piece (for analytical purposes it will be classified as “situation #1”).22 While the harmonic stream matches the barline grid, the melody is entirely out-of-phase (one beat dislocated).23 Contrary to what is suggested by their notated anacrusis, the semi-phrases of the right-hand line seem rather initiate in the strong beats (especially in the bars which contain the main motive), as proposed in Fig. 8.18, which re-writes the melody in these new terms. In spite of respecting clearly the ternary configuration expressed in the time signature, M and H progress in a non-integrated manner, as independent, displaced “tracks”. This kind of “tectonic” relation constitutes a very peculiar, and striking effect, and of special derivative significance, as will be discussed in Chap. 9. The tectonic mismatch between M and H leaves the scene with the beginning of passage II (Fig. 8.19), but not the impression of metric dislocation in relation to the grid, which is now provoked by both integrated streams (let this new event be labeled as “situation #2”).

22 This

initial metric discrepancy between melody and accompaniment is evidenced by the change of chordal functions and, especially, by the half cadences that firmly articulate antecedent and consequent phrases (see Fig. 8.4). 23 The mismatch between a given stream and the notated meter is classified by Krebs as subliminal dissonance.

8.4 Metric Structure

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Fig. 8.18 Brahms’s Intermezzo Op. 118/2 (mm. 1–5)—melodic line re-written in order to accommodate an alternative metric interpretation Fig. 8.19 Brahms’s Intermezzo Op. 118/2—passage II (mm. 17–23)

Locally, in mm. 19–20, a hemiolic disturbance between M and H arises from an asymmetric binary division of the harmonic space (.6 = 4 + 2 beats),24 as shown in Fig. 8.20. To differentiate it from other possible cases, I will call this type a middleground hemiola. In m. 23, both streams converge to another 4/4 bar (once again, the harmonic consistency orientates the interpretation), preparing the entrance of the important passage III. As before mentioned, passage III (shown in Fig. 8.21) corresponds to the first metric concordance between the main melody (and harmony, as well) and the barline grid (situation #3). Indeed, the seven bars that form the passage are the only in the entire intermezzo (disregarding its posterior recapitulation, mm. 92–98) in which the melody matches the notated grid. This, obviously, is anything but casual. The passage concludes with two binary bars that have different functional meanings. While m. 30 (in 2/4) clearly results from a one-beat shortening of the respective phrase,25 the compound binary 6/8 of m. 29 creates a subtle hemiolic 24 One could argue that the 4/4 bar might be subdivided in two 2/4 bars, constituting a “classical” instance of hemiola, but the fact that the four beats are occupied by the same dominant chord seems to corroborate the first interpretation. 25 The following beat belongs to the next phrase (segment 8), as its anacrusis.

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Fig. 8.20 Brahms’s Intermezzo Op. 118/2 (mm. 19–24, reduction)—middleground hemiola involving M and H Fig. 8.21 Brahms’s Intermezzo Op. 118/2—passage III (mm. 24–30). The entrances of formal segments 6 and 7 are indicated

effect in relation to the previous ternary flux.26 By proposing a re-baring version of mm. 29–30, Fig. 8.22 aims at clarifying this point. Passages IV and V (Fig. 8.23) have an identical extension and metric organization, corresponding to a return to situation #2.

26 Notice

that this case is different from that of passage II, since now the streams do not conflict, with binary and ternary superposition. I will classify this specific situation as a contextual hemiola, because the effect here is caused by the frustration of the expectancy of maintenance of the triple meter.

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Fig. 8.22 Brahms’s Intermezzo Op. 118/2 (mm. 28–31)—in a simplified and re-bared version, including the entrances of formal segments 7 and 8 and metric passage IV Fig. 8.23 Brahms’s Intermezzo Op. 118/2—passages IV (mm. 31–34) and V (mm. 35–38)

Figure 8.24 shows the codetta (passage VI). Like the previous two passages, it has a basic organization corresponding to situation #2. However, its internal distribution of beats is quite idiosyncratic, with three four-plus-two hypermetric blocks. Figure 8.25 details the three blocks, re-baring the original score. For simplicity, left-hand rhythmic figuration is reduced to long notes, evidencing as harmonic organization corroborates the metric changes. The first two blocks are mutually related, since both are intended to close the phrases of the period, respectively, antecedent and consequent.27 The third block, corresponding to the cadential closure of section A (segment 12), can be explained rather differently. The arrival of the melodic tonic A coincides not only with the strongest PAC of the whole section (c.f. Fig. 8.10), but also with the notated barline (see the arrow in Fig. 8.24), a rare event in the intermezzo, as it has been demonstrated in this analysis. Here, this structural fact is highly meaningful, because it evidences a kind of resolution of the ambiguities accumulated so far (especially, but not exclusively, in the metric realm).

27 They

could also be interpreted as instances of contextual hemiolas, in this case considering a hypermetric perspective.

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Fig. 8.24 Brahms’s Intermezzo Op. 118/2—passage VI (mm. 39–48)

Fig. 8.25 Brahms’s Intermezzo Op. 118/2 (mm. 39–48)—rewritten in a simplified and re-bared version

Two elements play a special role in the determination of the metric distribution of the third block: the arrival of the tonic at m. 48 and the re-contextualized resumption of Op. 118/2’s main motive.28 Since both events are expected to be associated with strong beats, the configuration suggested for the two final bars becomes the most logical alternative.

28 For

the first time re-harmonized by a dominant, cadential chord.

8.4 Metric Structure

173

Fig. 8.26 Basic scheme of the foreground hemiola (right vs. left hand) in the entrance of section B

Fig. 8.27 Brahms’s Intermezzo Op. 118/2—passage VII (mm. 49–56)

The entrance of section B corresponds to an increase of metric complexity and instability.29 This is due basically to two factors: (a) the inclusion of a new stream (S), acting primordially in imitative dialogues with the main melody; and (b) the introduction of a new level of rhythmic conflict, involving melody (in eight notes) and harmonic accompaniment (in triplets). This results in what will be here called foreground hemiola (Fig. 8.26). Passage VII, corresponding exactly to subsection [B.a], is shown in Fig. 8.27, and can be referred as the inauguration of a new metric situation (#5), to be developed during section B. Figure 8.28 provides a more detailed view of the metric complexities of the passage, focusing on the imitative dialogue between the top streams M and S (here extracted from the accompaniment and re-written as an independent voice). For clarity, the accompaniment line (H), performed in the score in eight-note triplets, is omitted from the figure, being represented only by the extensions associated with the harmonic territories.30 Primordially, a clear metric sense for the melodic line is established only retroactively, by the strong-weak pattern associated with the important descendingsecond motive E-D (which induces the re-barring of m. 49 proposed in the analysis).

29 Contributing to enhance the contrasting effect of B in relation to section A, together with the changes in tonal ambiance and texture, as already discussed. 30 Accent marks are added to both lines in order to indicate group beginnings.

174

8 Formal, Harmonic, and Metric Structure

Fig. 8.28 Brahms’s Intermezzo Op. 118/2—passage VII (mm. 49–56)

This configuration is transmitted to S, through an almost exact canonic imitation. After this initial out-of-phase dialogue, both streams meet at m. 51, but another misalignment takes place, again in the hemiola (mm. 52–53). Meanwhile, the harmony, which has been obeyed a different metric organization so far, meets M in m. 53, anticipating the three-stream metric match of m. 54, which is prolonged until the end of the passage. In spite of the choral-like, simpler texture of section [B.b] (mm. 57–64), the metric complexity (corresponding to the situation #6) is increased, as depicted in Fig. 8.29. Its twenty-four-beat length is subdivided into two phrases (antecedent and consequent), whose beginnings take place at different points in each of the three strata (indicated by the dashed lines). Only at the end of the phrases, for a brief moment, the three streams match metricly (see the arrows). As shown in the graph, the harmonic accompaniment is now perfectly adjusted to the notated bars. This contrasts sharply with the dialogue between M and S, which are not only temporally dislocated, but also present quite distinct metric organizations. Such configurations are once again dictated by a canonic-like imitation between M and S and by a clear perception of the pitch C. as a strong metric event.31 Figure 8.30 provides a thorough view of the counterpoint which involves both lines. Observe how compression (from four to three beats) of the third resumption of the obsessive quarter-note motive C.-F.-E.-D. (see the brackets) in M, at the beginning of the consequent, provokes a special perturbation in the dux-comes flux. Passage IX (Fig. 8.31) closes section B, being associated with the situation #7. After reaching its highest latitude, metric ambiguity is finally resolved. Symptomatically, this takes place at the very climactic moment of the piece, and just after the hemiola that involves the three strata.

31 This

will be evidenced and discussed in great detail at the last section of Chap. 9.

8.4 Metric Structure

175

Fig. 8.29 Brahms’s Intermezzo Op. 118/2—passage VIII (mm. 57–64)

Fig. 8.30 Brahms’s Intermezzo Op. 118/2—passage VIII (mm. 57–64), counterpoint between M and S

Fig. 8.31 Brahms’s Intermezzo Op. 118/2—passage IX (mm. 65–76)

As shown in Fig. 8.32, the two-voice dialogue is also present in the subsection, but this time the main melodic line (initiating again with the leap C.-F.) is headed by S, being imitated by M (this order is inverted in the consequent phrase, as shown by the arrows). A calm closing of the whole section takes place on a

176

8 Formal, Harmonic, and Metric Structure

Fig. 8.32 Brahms’s Intermezzo Op. 118/2—passage IX (mm. 65–76), counterpoint between M and S

completely stabilized ternary meter (mm. 73–76), involving a liquidation process based on sequences of the descending-second motive: E-D, D-C.-C. (prolonged by a fermata)-B. The natural continuation in direction to the expected tonic (B-A) overlaps the anacrustic motive that brings recapitulation of section A. Closing this chapter, it is possible to consider that harmony and meter seem to interact along with the piece as sources of structural ambiguity. Basically, both dimensions cooperate in roughly equal terms in section A in order to provide a combined effect of uncertainty (especially headed by the duality A/D and the “tectonic” superimposition). In section B, the departure from the tonic to the mediant parallel regions is counterbalanced by an unambiguous harmonic context. Meter is then left as the unique responsible for sustaining the uncertainty of the musical discourse. As compensating this, foreground hemiola and the entrance of a new line, always acting as a canonic response to the main melody, provoke an increase of metric complexity, which is led to the highest point of the dissonance, just before the melodic climax, which initiates the following process of resolution. The structures of form, harmony, and meter of Op. 118/2 that were analyzed in details in this chapter integrate the necessary context for the application of the derivative analysis in Chap. 9.

References Berry, W. (1987). Structural functions in music. New York: Dover. Brower, C. (2008). Paradoxes of pitch space. Music Analysis, 27(1), 51–106. Caplin, W. (1998). Classical form: A theory of formal functions for the instrumental music of Haydn, Mozart, and Beethoven. Oxford: Oxford University Press. Cohn, R. (1998). Introduction to neo-Riemannian theory: A survey and a historical perspective. Journal of Music Theory, 42(2), 167–180. Cohn, R. (2001). Complex hemiolas, ski-hill graphs and metric spaces. Music Analysis, 20(3), 295–326. Cohn, R. (2012). Audacious euphony: Chromaticism and the triad’s second nature. Oxford: Oxford University Press. Hepokoski, J., & Darcy, W. (2006). Elements of sonata theory: Norms, types, and deformations in the late-eighteenth-century sonata. Oxford: Oxford University Press. Kopp, D. (2002). Chromatic transformations in nineteenth-century music. New York: Cambridge University Press.

References

177

Krebs, H. (1999). Fantasy pieces: Metrical dissonances in the music of Robert Schumann. New York: Oxford University Press. Lewin, D. (1987). Generalized musical intervals and transformations. New Haven: Yale University Press. Murphy, S. (2001). Metric cubes in some music of Brahms. Journal of Music Theory, 53(1), 1–53. Murphy, S. (2009). On metre in the rondo of Brahms’s op. 25. Music Analysis, 20(3), 323–353. Rings, S. (2011). Tonality and transformation. Oxford: Oxford University Press. Schoenberg, A. (1967). Fundamentals of musical composition. London: Faber and Faber. Schoenberg, A. (1969). Structural functions of harmony. New York: W.W. Norton. Schoenberg, A. (1978/1911). Theory of harmony. Carter, R. (trad.). London: Faber and Faber.

Chapter 9

Derivative Analysis

I begin with the description of the methodological protocol that will be adopted in the derivative analysis of the Intermezzo Op. 118/2.

9.1 Methodology The protocol comprises eight sequential stages: 1. Determination of the limits of the Grundgestalt; 2. Segmentation of the Grundgestalt into Grundgestalt-components (Gcs), if it is the case; 3. Assignment of permanent variables to meaningfully derivative elements inside the Grundgestalt or the Gcs; 4. Segmentation of the piece (or its movements) into relatively short “territories”, which will be called derivative segments; 5. Identification of units of analysis (uAs, for short) inside the derivative segments. An uA (numbered sequentially) corresponds to a minimal, relatively autonomous element that conveys derivative information (likely) subject to transformation. It can be a single UDS or more complex structures formed by the concatenation of two or more UDSs; 6. Determination of derivative relations between uAs inside the derivative segment in question; 7. Determination of eventual lineages involving uAs (in high level) and variables (low level);

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3_9

179

180

9 Derivative Analysis

8. In multi-movement works, the protocol shall also be extended to the movements that follow the first one. After this, it is also necessary to determine high-level relations between them.1 Besides these basic procedures, a number of complementary strategies (related to concepts introduced in Parts I and II of this book) will contribute for the systematization of the derivative analysis of the intermezzo. Such specific tools will be presented in due course in the chapter. The next sections address the methodological stages described above.

9.2 Grundgestalt, Variables, and Segmentation The Op. 118/2s Grundgestalt corresponds to the UDS shown in Fig. 9.1, the very initial gesture of the intermezzo. As it will be demonstrated along the derivative analysis, this brief fragment (a, so to speak, monolithic Grundgestalt, whose subdivision into components would be meaningless) contains in latent state most part of the implications of the to-be-realized piece, which can be seen as a remarkable example of the economy of means that characterize Brahms’ style. Figure 9.2.a depicts the matrix of attributes related to the Grundgestalt and identify precisely their most essential abstracted elements. Observe that, due to the particular nature of the piece (concerning specifically melodic derivative relations), the attribute associated with the intervallic sequence is better represented by its diatonic version (d-p3). The importance of this attribute for further elaboration is evidenced by the assignment of the permanent variable Z, whose content can be described as a descending diatonic step (-1) followed by an ascending leap Fig. 9.1 Brahms’s Intermezzo Op. 118/2 (mm. 0–1)—Grundgestalt

1 Since

the Op. 118/2 is a single-movement piece, stage 8 will not be used in the present analysis.

9.2 Grundgestalt, Variables, and Segmentation

181

Fig. 9.2 (a) Matrix M of attributes related to Fig. 9.1; (b) pitch (left) and temporal (right) sections of matrix M, with variables assigned to attributes d-p3 (Z), 2 (Y), and t3 (X); (c) genetic representation of the Grundgestalt

encompassing two steps (2). Two other variables refer to temporal attributes: the IOI sequence t2 (Y), and the metric contour t3 (X).2 In (c) a “genetic” scheme for the Grundgestalt is proposed. This kind of representation is intended to depict the essential material (i.e., the variables which are component) of a musical unit in a derivative context. The piece is subdivided into eight segments of derivative relevance, as shown in Table 9.1. The boundaries of thesee segments correspond approximately to the formal sectioning. The eight segments are analyzed in the next sections.

2 Alternatively, i is also possible to describe the metric contour of the Grundgestalt in more abstract terms, as a sequence of three metric states: “neutral (or middle)–weak–strong”, as it will be done in some specific situations.

182 Table 9.1 Brahms’s Intermezzo Op. 118/2—derivative segments

9 Derivative Analysis Segment 1 2 3 4 5 6 7 8

mm. 0.3–8.2 16.3–28.2 28.3–38.2 38–48.2 48.3–56.2 56.3–64.2 64.3–73.2 73.1–76.2

Correspondence with formal structure [A.a] [A.b] [A.a’] [A.codetta] [B.a] [B.b] [B.a’] [RT]

9.3 Derivative Segment 1 As a starting point for the analysis, Fig. 9.3 proposes a preliminary examination of the first derivative segment (for concision, this step will be bypassed in the analysis of the next segments). The passage in question (mm. 0–8) reproduces the events in the score in a slightly reduced format (shaded symbols indicate non-relevant elements for derivative analysis). Brackets and sequential numbering identify sixteen candidates for being considered as units of analysis (uAs). Harmonic analysis is added beneath the score. Before continuing, it is necessary to consider two basic questions. First, it is reasonable to conclude that not all selected fragments can be considered as sufficiently relevant for analysis. The examination of the derivative relation between fragments 01 (that represents the Grundgestalt itself) and 11 (a non-exact repetition of 01, at mm. 4) illustrates this point. Observe that 01 differs from 11 uniquely by their harmonic contexts. Figure 9.4 introduces a model used for comparing same-cardinality units. They are depicted both in musical notation (including their respective harmonic contexts) and as matrices of attributes (recall that gray cells indicate discrepant elements). As seen in Chap. 3, the calculation of the dissimilarity between both units is made by entering the matrices’ data into the corresponding algorithms.3 The resulting pitch, temporal, and harmonic penalties are then plugged in Eq. 3.4, which returns the global penalty. In the present case, k .≈ 0.02, a very low value (the calculation is omitted). Finally, the relation of similarity is graphically represented in the derivative space of 01 (Fig. 9.4d), evidencing how close both fragments are (more precisely, the method tells us that 01 and 11 share circa of 99.8.% of their musical substance). Aiming at a more concise and efficient approach, highly similar units will be from now on considered as equivalent, and therefore disregarded as potential uAs (in practical terms, they are seen as kinds of “imperfect copies”). Two other pairs of

3 Appendix C presents some suggestions, written as pseudocodes, for the elaboration of algorithms intended to computational implementation.

9.3 Derivative Segment 1

183

Fig. 9.3 Brahms’s Intermezzo Op. 118/2 (mm. 0–8)—derivative segment 1 (first approach)

units—02/04 and 03/12—can be also considered as equivalent. Given this, units 11, 04, and 11 will be removed from the group of potential uAs. Transposition (chromatic or diatonic) is another type of equivalence considered in MDA, since similarity in such cases is also computed as very high.4 The application of this new norm in segment 1 lead us to remove unit 09, since it corresponds to a diatonic transposition of 08. Figure 9.5 updates the configuration of Fig. 9.3, proposing a more abstract scheme. For practical purposes, let us call it a time-oriented transformational network. The remaining non-equivalent twelve uAs keep the original numbers5 and are depicted in a manner that attempts to optimize spatially the mutual relations. The architecture of the network raises another issue. Since music is a temporal phenomenon, one could assume a given derivative relation between two uAs should involve some amount of time separating them. Put another way, it is logical to expect a cause-effect relation as a precondition for derivation. Given this, the putative relation 01.→02 should not be considered for analysis, since the two uAs occur simultaneously in the score. Despite the clarity of this argumentation, however, the decisive importance of uA 02 in the subsequent process of developing variation led me to consider this case of (purely theoretical) “simultaneous variation” as a remarkable exception and, accordingly, I decided to incorporate it to the analysis. This view apparently contradicts the concrete notion of derivation as a sequential process (which is especially linked to the theory introduced in Chaps. 6 and 7), as observed either in the score or performances of a piece. As a matter of fact, we can think rather of an extrapolation of this perspective, in order to encompass also the composer’s mind. In such “territory” time is not necessarily an inflexible constraint for the flux of musical events: cause-and-effect relations can be simply subverted, subordinated uniquely to imagination and idiosyncratic structuring processes. By following this line of reasoning, the temporal positions of the multitude of elements in a finished

4 Provided

that temporal structure is preserved, of course.

5 Actually, the numbering here used is provisional. The labels of the uAs will be later reformulated,

evidencing properly the mutual derivative relations that link them to the Grundgestalt.

184

9 Derivative Analysis

Fig. 9.4 (a) Units 01 and 11 in musical notation (and harmonic contexts); (b) their matrices of attributes; (c) pitch (kp), temporal (kt), harmonic (kh), and global penalties (k); (d) 11 plotted on the derivative space of 01

9.3 Derivative Segment 1

185

Fig. 9.5 Time-oriented transformational network related to segment 1, by including relations between uAs

work represent uniquely the manner with which they were arranged in the final product (fruit of the inscrutable compositional processes), not necessarily the order of their creation neither, especially, the ways that the composer intends they shall be understood. Table 9.2 lists the fifteen definitive derivative relations of segment 1, ordered according to their emergence in the score. Not surprisingly, uA 01 (the Grundgestalt) is the most recurrent of the parents, heading five cases. On the other hand, it is worth noting how four children are associated with two possible origins (it is the case of uAs 13, 14, 15, and 16). Let us, therefore, assign to them a dual origin. Two possible causes for these situations can be conjectured: alternative interpretations or crossover. While the former case demands only the choice of the best option for the definitive representation, the latter corresponds to a true dual derivation, and must be accordingly analyzed (I will return to this in due course). The methodology associated with the analysis of the derivative relations that integrate this and the next derivative segments follows a new protocol, which is based on seven steps: 1. Analysis of the high-level6 relations of unique origin. In this step uAs are the actors. If the two related uAs have the same cardinality, the analysis is simple,

6 Or,

metaphorically, “phenetic”.

186 Table 9.2 Relations between uAs in the derivative segment 1. Bold-face relations refer to dual origins

2. 3.

4.

5.

6.

9 Derivative Analysis Parent 01 01 01 02 02 06 07 05 06 06 07 01 06 01 08

Child 02 03 05 06 07 08 10 13 13 14 14 15 15 16 16

accomplished according to the strategies described in Chap. 3.7 Otherwise, a TD analysis is required (see Chap. 5). In both cases, the similarity between parent and child is calculated and stored for further usage; Low-level analysis (or “genetic”), considering the relations examined in step 1. In this case the focus is on the derivation of variables; Analysis of dual-origin relations. Firstly, alternative interpretations are compared aiming at the choice of the best one, based on criteria of simplicity and logic (I will return to this aspect later). Chosen an origin, the relation is treated as in steps 1 and 2, accordingly to the case in question (i.e., considering same or distinct cardinalities); Analysis of relations resulting from crossover processes. A specific method is used for this (as it will be presented). As in the previous relations, both high- and low-level analyzes shall be provided; Re-evaluation of the segment’s relations. With all relations properly mapped, a higher-level analysis considering the whole segment shall be prepared. The network of derivative relations is then updated, with eventual suppression or reformatting of relations. This provides the necessary conditions for relabeling the uAs, by the use of a proper hierarchical, “genealogical” notation; Contextualization of the segment. The updated configuration of the segment is integrated into those related to previous ones (when it is the case), aiming at a still higher level of understanding of the derivative structure as a whole;

7 For avoiding long calculations, only the final penalty k will be informed, assuming that all the necessary computations will automatically be done by the comparative algorithms, feed with the data from the matrices of attributes of the corresponding uAs.

9.3 Derivative Segment 1

187

Fig. 9.6 Low-level derivation of uA 02 from uA 01 (a); “Genetic” representation of uA 02 (b)

7. Plotting of the analytical data as phylogenetic-like schemes of several types (genealogical trees of uAs and variables, absolute-relative DV graphs, etc.), aiming to provide complementary overviews of the derivative relations present in the piece.

9.3.1 Derivation of uA 02 UAs 01 and 02 have the same cardinality and are, as one can easily perceive, closely related. This is formally confirmed by the application of the algorithms for measurement of similarity, which returns a penalty .k = 0.18. Figure 9.6 proposes a model for low-level analysis (step 2 of the analytical protocol) considering relation 01.→02. As shown in the figure, variable Z is transformed by a mutational operation of subtraction (of three diatonic steps) applied to the second element of attribute d-p3.8 On the other hand, the temporal variables Y (IOI sequence) and X (metric configuration) are kept unaltered, thus there is not necessary to include them in the analysis. A “genetic” representation of uA 02 is depicted in (b).

9.3.2 Derivation of uA 03 The first actual sequential relation in the segment (.01 → 03) can also be seen structured as a statement-response block [.1 + 3], functioning as a sort of high-level

8 I recall that variable Z is only associated with the intervallic-sequence (attribute d-p3), thus the absolute pitches that form the two fragments are, so to speak, only byproducts of the operation and, therefore, they are not relevant in this low-level analysis.

188

9 Derivative Analysis

Fig. 9.7 Low-level derivation of uA 03 from uA 01 (a); “Genetic” representation of uA 03 (b)

ADD4

2 d-p3

03 2

unit of “semantic” meaning inside the motivic-thematic plot of the piece.9 Like in the previous case, both units have the same cardinality, then their similarity is calculated by the algorithms (.k = 0.19, calculations omitted). Likewise, it involves an economical low-level transformation of variable Z (again, keeping constant X and Y), a mutational operation applied to the second element, this time an intervallic addition of four steps (Fig. 9.7).

9.3.3 Derivation of uA 05 This derivation involves units with incompatible cardinalities and, therefore, shall be analyzed by the use of the TD methodology, as mentioned in Chap. 5. Figure 9.8 presents the TD analysis of relation 01.→05, which models the case exemplified in Fig. 5.10. As shown in the analysis, both pitch (represented by variable Z) and rhythm (Y) suffer gradual transformations. Initially, a mutational inversion of the first interval of Z (.−1 → +1) gives birth to variant Z3 . By the use of attribute equivalence (d-p3 10 In the temporal domain, .≡ p1), Z3 is then diatonically transposed down two steps. variant Y1 results from unequal re-partition of Y’s pair of eight notes (.2 + 2 → 3+1).11 A re-harmonization by substitution of the two original chords completes the formation of the intermediary (and hypothetical) fragment shown at the rightmost of Fig. 9.8, whose hybrid nature is captured by the label 01/05. This is ultimately 9I

will return to this interpretation at the end of the chapter.

10 Recall that normal transpositions do not affect the structure of the variant Z

3 and, therefore, their outcomes are not treated as another variant. On the other hand, a mutational transposition change the intervallic configuration of a unit and, then, produce a true variation. This is the case of the transformation that yields Z3.1 from Z3 . 11 Observe that since t2 (the attribute associated with variable Y) refers to onset intervals, the last actual duration of a rhythm is not relevant because it can not actually be determined by the respective vector. For this reason, as a convention, in the representation of t2 using musical notation, the last event will be always written as a shaded rhythmic figure.

9.3 Derivative Segment 1

p1

3

189

3.1

t1

3.1.1

t1

3

RPL (t1, 1:2) p1

p1

1.1

3.1

equivalent forms (no true variation)

Fig. 9.8 TD analysis of the derivation of uA 05 from uA 01, considering the intermediary hypothetical stage 01/05 (at right)

the last transformational stage before the change of cardinality that proceeds by the mutational operations split and replication in both pitch and temporal domains. The inclusion of a third chord to the harmonic context concludes the process. As discussed in Chap. 5, the measurement of similarity between uAs 01 and 05 shall be accomplished in three steps: 1. Determination of the penalty due to change of cardinality (kq ), through the application of Eq. 5.1, here reproduced, for convenience: kq =

.

δq 10 nr

Let us recall that kq is the penalty associated with the difference of cardinalities of the two units (.δq = 5−3 = 2), n is the number of CS-operations employed (.n = 3), and r is the number of reductions necessary to reach the intermediary form (.r = 2). Plugging these values in the equation, we have:

190

9 Derivative Analysis * attribute equiv.

t (p1, 3) *

RPL1(p1, 2)

3.1

d-p3

3.1.1

RPL1(t1, 1:2)

SPL1(t1, 3) 1.1

t2

t2

RPL1(p1, 4)

d-p3

3

3

d-p3

3

t2

3

3 3

Fig. 9.9 Low-level derivation of uA 05 from uA 01 (a); “Genetic” representation of uA 05

kq =

.

2 10 32

≈ 0.13

2. Determination of the penalties due to the dissimilarity between uAs 01 and 01/05, using the usual method for comparison of ideas with compatible cardinalities. The algorithms return .k ≈ 0.13. 3. The definitive penalty of relation .01 → 05 is then equal to the sum of penalties of stages (1) and (2): .k = 0.13 + 0.13 = 0.26 The low-level derivative analysis of the relation .01 → 05 is depicted in Fig. 9.9.

9.3.4 Derivation of uA 06 Given the long-range implications that arise from the introduction of a new rhythmic configuration (the quarter-note motive), the relation .02 → 06 can be considered as the most important in the context of segment 1. As both units have the same cardinality, their similarity can be calculated by using the edit-distance algorithms. In this manner, we obtain k = 0.28, a higher value than the previous penalties, due especially to the rhythmic change. In respect to the low level, as depicted in Fig. 9.10, uA 06’s rhythmic profile results from a durational addition, that doubles the original values of variable Y, producing variant Y2 .12 Variable X (associated with metric configuration) is also affected, through a unary rotation of the original metric contour, from which is 12 Alternatively, one could also think of a mutational augmentation applied to the two first elements

of attribute t1. However, in this case, it would also be necessary to apply a mutational diminution to the third element, in order to halve its duration to a quarter and to map the durational sequence of t1. As always, the simplest alternative is the best choice.

9.3 Derivative Segment 1

191

Fig. 9.10 Low-level derivation uA 06 from uA 02, considering variables involved (a); “Genetic” representation of uA 06 (b)

1

i (Z1)

d-p3

1.1

d-p3

ADD D2(Y) 2

t2

X t3

ROT1(X)

X1 t3

06 1.1 2

1

obtained variant X1 . Considering the pitch domain, a second-generation variant Z1.1 results from inversion of the intervallic content of Z1 . Relation 01.→06 is also marked by another very relevant kinship linkage. Although implicit and almost imperceptible, such a relation has important derivative implications that will be evidenced in future events. Actually, a new distinctive element, associating here the abstract attributes of pitch intervals and meter, gives birth to another level of affinity between both ideas. In a sense, this combination integrates variables Z and X, forming an additional level of derivative significance, a sort of middle layer in between low (variables) and high (units of analysis) levels.13 From this new perspective, the middle-level block is formed by the interaction of a descending diatonic second and the strong-weak metric pattern,14 both inherited from the initial pair of elements of the Grundgestalt. I define this special dual

13 In a sense, this can be compared with the notion of conceptual motive, coined by Jeffrey Swinkin,

a sort of abstract musical idea that can be materialized in different ways in a piece of music, but that has not a concrete, definitive format. In his analysis of Brahms’s Variations on a Schumann’s Theme Op. 9, Swinkin uses this principle to assign the broad idea of “bass-as-theme” as a conceptual motive present in three different variations (in very distinct manifestations) throughout the piece. For more details, see Swinkin (2012, p. 70). 14 Observe that I am not talking exactly about variables Z and X (that refer to the complete idea), but of a sort of partial (in case of the intervallic sequence) or “miniaturized” (in the case of the metric attribute) copies of them.

192

9 Derivative Analysis

01

06

d*i1(

)

"weak"

2

"strong"

2

"weak"

zx1

"strong"

zx 0

Fig. 9.11 Middle-level analysis as additional information for the derivation of uA 06

combination of variables as a locution, using the symbol zx0 for identifying the attributes they are related to (the idea behind the adoption of lowercase versions of the variables is to capture the “middleness” condition, previously evoked). Figure 9.11 proposes a close examination of this special relationship. In this kind of hybrid, “genetic/phenetic” level, transformation can be applied to both elements or can affect just one of them, as it is the present case. As suggested in the analysis, there is a composition of two middle-level operations acting in the transformation: a dislocation (symbol d) of the entire block from the beginning to the end of the variant, followed by an intervallic inversion (i) which affects only z, the first element of the locution (denoted by the subscribed “1”), denoting a kind of mutational application. Adopting the same notational system used for the low and high levels, the variant is properly labeled as zx1 . In short, consider locution zx as a kind of property that when assigned to a given idea suggests the existence of an implicit “blood tie” with the Grundgestalt, even if there is no clear evidence of this at a first glance. In an attempt to avoid unnecessary complications for the analysis, I decided not to create a special analytical apparatus destined to examine middle-level situations. Instead, I will simply point out their occurrence and make eventual commentary, when necessary. More important, I think, is to emphasize the emergence of this new type of cooperative relation, and how it acts in the enhancement of developingvariation process.

9.3 Derivative Segment 1

193

9.3.5 Derivation of uA 07 Because relation 02.→07 involves increasing of one unit of cardinality, a TD analysis is required (Fig. 9.12). Three new variable variants (Z1.2 , Z1.2.1 , and Y3 ) arise from high-level transformation. Observe especially how the use of interattribute equivalence was necessary, due to the employment of some operations related to attributes p1 (t,15 RPL and OCT) and t1(SPL). The low-level derivation of these variables is summarized in Fig. 9.13.16 Like done with uA 05, the calculation of k for relation .02 → 07 is a two-step process. Addressing firstly the penalty due to the change of cardinality (kq ), we have .δq = 2, .r = 2 and .n = 4 (obtained in the analysis of Fig. 9.12), thus

-10 p1

Fig. 9.12 TD analysis of the derivation of uA 07 from uA 02, considering the intermediary hypothetical stage 02/07 (at right)

15 Recall

that t is a diatonic transposition, so the subscript “-12” added to the symbol refers to twelve descending scalar steps in the key of A major. 16 Notice that, although in strict terms the metric contour was altered by the inclusion of another duration, this transformation is substantially irrelevant in derivative terms (the metric pattern of uA 06 can still be informally described by the prefix “medium-weak-strong”, followed by another medium or weak event).

194

9 Derivative Analysis

p1, 1

Fig. 9.13 Low-level derivation of uA 07 from uA 02, considering variables involved (a); “Genetic” representation of uA 07 (b)

kq =

.

2 10 42

= 0.10

To this value is added the penalty of dissimilarity referred to the relation between uA 02 and the intermediary form 02/07 (equal to 0.13), calculated by the use of the algorithms. Finally, we have .k = 0.10 + 0.13 = 0.23.

9.3.6 Derivation of uA 08 Relation 06.→08 is considerably evident and straightforward. Both units have the same cardinality, which makes the similarity between them to be calculated by the algorithms (.k = 0.12). Figure 9.14 depicts the low-level derivative analysis. A new variant .Z14 ,17 whose generation is preceded by the intermediary form .Z13 , is incorporated to the “genetic pool” of the pitch domain. Observe also a middle-level link between uAs 06 and 08 (Fig. 9.14c). The locution zx1 is transformed into the variant zx1.1 , through an inversion applied to the ascending-step gesture. From another perspective, this inversion “undoes” that sent 17 As

discussed in Chap. 7, this alternative notation is intended to turn more compact long genealogical labels (in this case, Z1.1.1.1 ).

9.3 Derivative Segment 1

195

zx 0

08

06

2

2

d*i1(

)

"strong"

i1(

d(

)

"weak"

zx1.1

"weak"

zx1

"strong"

(c)

"weak"

"strong"

2

)

zx 0 2

"weak"

"strong"

01

Fig. 9.14 Low-level derivation of uA 08 from uA 06, considering variables involved (a); “Genetic” representation of uA 08 (b); (c) Middle-level analysis of uA 08, as related to uAs 06 and 01

zx0 to zx1 (in this manner, zx0 could be directly related to zx1.1 by pure dislocation, as suggested at the bottom of the figure).

196

9 Derivative Analysis

9.3.7 Derivation of uA 10 The derivation of uA 10 from 07 is also quite evident. In fact, it is so clear that, supported by the argumentation presented in Sect. 5.2, I would dare to say that, rather than resulting from a series of decomposable transformations, it could be better (and quite more simply) seen as a case of holistic variation. In order to demonstrate this interpretation, Fig. 9.15 proposes a more flexible model of analysis for situations like this. In this model, the “editions” (in the terms used in Chap. 3) are performed as somewhat “informal” operations applied directly to the concrete events (and not in their abstracted images), similarly to the way an artisan shapes a vase by manipulating the clay that will form it. In the present case, five distinct events of uA 07 (here treated in a more abstract conception) are taken as referential: (1) the overall rhythmic Gestalt; (2) the first pitch (D1 ); (3) the second pitch (D3 ); (4) the gesture associated with the pitch repetition D3 –D3 ; (5) the melodic descending trajectory that is suggested after the repetition. Given this, the transformation 07.→10 can be described as the preservation of events (1) and (3), while a sequence of three editions is applied to the remaining events, namely transposition of (2), dislocation of (4), and inversion and expansion of (5). Although it is a considerably simpler manner to treat this particular case (it would be enough to compare it with an alternative TD analysis to perceive this), holistic analysis (or, maybe, it could be better named a semi-holistic analysis) is not always the best interpretation of similar situations. Context still is the strongest support for the judgment about the suitability of analytical tools in any case.18 For the measurement of similarity of uAs 07 and 10, it is necessary to establish a new specific procedure, since the conventional methods are not applicable for holistic derivation. As a first attempt in this direction, I propose an equally

ed.#1: transpose

07

fixed

10 ed.#2: dislocate

ed.#3: invert and expand

Fig. 9.15 Holistic derivation of uA 10 from uA 07, considering three basic editions

18 Needless

to say that, by definition, there is no low-level derivation in holistic variation, since abstractions are here out of question.

9.3 Derivative Segment 1

197

simple method, based on the intuition that the penalty of dissimilarity shall be directly proportional to the number of editions (n) necessary to accomplish the transformation of the referential unit into the variant. So, k can be obtained from the product of the total of editions by a constant c. This constant could be calibrated through empiric attempts, but I think that 0.07 can be a good first estimation. Thus, in the present case, .k = n × c = 3 × 0.07 = 0.21. The next derivations to be examined involve dual origins.

9.3.8 Derivation of uA 13 The derivation of uA 13 can be explained as a particularly simple example of a crossover process, through combination of uAs 05 and 06, each one contributing with a primary domain, time (05) and pitch (06). Due to the fact that both parents occur in the piece exactly at the same moment, they naturally share the harmonic context. Let us formally notate this type of relation as [05+06].→13, whose basic model is shown in Fig. 9.16. Observe that its rhythmic/metric configuration is inherited from uA 05 without modifications. Apart from a re-harmonization by substitution of chords, the unique transformation occurs in the pitch domain, firstly by replication of pitches G.4 and A4 , and then diatonic transposition. Two inter-attribute equivalences are required in

p1

13/[05+06] p1

Fig. 9.16 Crossover derivation of uA 13, from combination of uAs 05 and 06

198

9 Derivative Analysis

Fig. 9.17 Low-level derivation of uA 13 from uA 06, considering variables involved (a); “Genetic” representation of uA 13 (b)

the extremities of the process. The derivation involves a change of cardinality, so the elaboration of an intermediary stage (13/[05+06]) is required.19 The measurement of similarity in crossover is a little more complicated than in the other cases, since now three penalties for dissimilarity shall be calculated (let us label them as k1, k2, and k3): • k1: Referred to relation 05.→13 (they have the same cardinality); • k2: Referred to the difference of cardinalities between uAs 06 and 13; • k3: Referred to relation 13.→13/06. Penalties k1 and k3 are calculated using the habitual algorithms. Both are very low, confirming the intuition that the related units are very similar: .k1 = 0.05 and .k3 = 0.07. Entering the values of .δq (2), n (1), and r (1) in Eq. 5.2, we obtain .k2 = 2/10 = 0.20. The definitive penalty k will then result from the sum of k3 with the arithmetic mean between k1 and k2: k = 0.07 +

.

(0.05 + 0.20) = 0.17 2

The low-level derivative analysis of relation [05+06].→13 is shown in Fig. 9.17.

9.3.9 Derivation of uA 14 Contrary to what occurred in the previous case, here we have an alternative interpretation for the derivation of uA 14 rather than properly a crossover origin.

19 Notice

that, due to the decreasing of cardinality, the arrow related to the analytical reductions (q) is inverted. For the same reason, the similarity with the intermediary 13/[05+06] is calculated with respect to variant (13), instead of the parent 06.

9.3 Derivative Segment 1

199

14

t1

t12(p1) 1.2.1.1.1

p1 p

14/07 i(Z Z1.2.1.1 ) 1.2.1.1

SUP (Z

,1)

MRG (t1,1:2) t1

Fig. 9.18 Possible derivation of uA 14 from uA 07

This means that both uAs 06 and 07 could be plausibly assigned as a parent for 14. Let us denote this type of relation as [06.|07].→14. Accordingly, two individual, possible derivative analyzes can be proposed, namely 06.→14 and 07.→14. The former is a simple case of same-cardinality relation, which allows us to use the comparison algorithms and get k = 0.06. On the other hand, the relation 07.→14 involves a decreasing of cardinalities, demanding a TD analysis (Fig. 9.18). In the face of this new type of situation, it is necessary to establish a clear criterion for the choice of which analysis would be the best representative of the derivation. As a logical option, I adopt the simplest alternative, that is, 06.→14.20 Concerning low-level derivation (Fig. 9.19), however, it seems adequate to consider the two possible generations of the pitch structure of uA 14, as shown in Fig. 9.19. According to this new analytical approach, both derivative paths from Z1.1 (via Z1.1.2 ) and Z1.2.1 are plausible descriptors for the pitch transformation. Since it is necessary to opt for one of them in order to properly label the variant, a new rule shall be formulated: from now on, in cases with two possible derivations, that which results from the simplest process will be chosen. Following this new rule, the label of the new variant should be Z1.1.2.1 . However, this specific intervallic configuration 20 This

choice makes unnecessary the calculation of the penalty of relation 07.→14.

200

9 Derivative Analysis

(see Figure 9.13)

Fig. 9.19 Low-level derivation of uA 14 from uAs 06 and 07, considering variables involved (a); “Genetic” representation of uA 14 (b)

(.< 0, −1 >) had already been produced by a different derivative process as an intermediary stage during the formation of uA 07 (see the low-level analysis of Fig. 9.13), where it was labeled as Z1.2 .21 Thus, for simplicity and coherence, this label will be chosen to nominate the present intervallic variant.

9.3.10 Derivation of uA 15 The derivation of uA 15 can also be interpreted according to two possible origins: uAs 01 and 06 (thus being formally notated as [01.|06].→15). Unlike the previous case, however, both alternatives involve change of cardinality, which makes the choice of the simplest (and therefore most adequate) option a bit more difficult. The two TD analyzes are depicted in Figs. 9.20 and 9.21. Let us assume then that the smaller penalty will determine the choice of the definitive derivation of uA 15. Bypassing, for simplicity, the various stages involved in the process of calculation of the respective values, we have:

21 Evidencing,

therefore, a case of involution.

9.3 Derivative Segment 1

Fig. 9.20 Possible derivation of uA 15 from uA 01

Fig. 9.21 Possible derivation of uA 15 from uA 06

201

202

9 Derivative Analysis

• Relation .01 → 15: .kq = 0.05; .k15→01/15 = 0.27; thus, .k01→15 = 0.05 + 0.27 = 0.32; • Relation .06 → 15: .kq = 0.05; .k15→06/15 = 0.09; thus, .k06→15 = 0.05 + 0.09 = 0.14. Since the latter is the smallest penalty, relation .06 → 15 is chosen for representing the derivation of uA 15. Regarding now the level of variable transformation, we observe again a sort of “parental dispute” in respect to the origin of the variants, this time involving both pitch and rhythm (i.e., Z and Y lineages). According to the convention which is here adopted, labels Z4 and Y2.1 are chosen for the respective identification of the new variants (Fig. 9.22).

9.3.11 Derivation of uA 16 The origin of uA 16, the last unit to be analyzed in segment 1, can also assigned to two sources, uAs 01 and 08, which implies a dual relation denoted as .[01|08] → 16. Since the three units have the same cardinality, the penalties of the two derivative relations are calculated by the algorithms: 0.29 (for .01 → 16) and 0.07 (for .08 → 16). As the latter relation presents the lowest value for k, it will properly represent the derivation of uA 16. Low-level derivation considers only pitch transformation (Fig. 9.23). Once again, there are two possible manners for explaining uA 16’s intervallic configuration, which is labeled, according to the criterion of simplicity, as the first-generation variant Z5 .

9.3.12 Overview of Segment 1 After the analyzes of the uAs that form segment 1, it is necessary to compile the individual data and to provide a global overview of the derivative process. This includes a more adequate identification for the uAs, which involves the establishment of their specific genealogies (as it was done in the low-level analysis, by labelling the variants of the variables). Finally, I will propose here some summarizing schemes and phylogenetic-like representations that will become a model for the analyzes of the next segments. Figure 9.24 updates the network introduced in Fig. 9.5. The updated network differs from the original in three important aspects: (a) since now a more abstract perspective is intended, all information related to barlines and harmonic context was omitted; (b) dual derivation is indicated by the use of two different styles for the arrows that to lead to a two-parent child: a straight line depicts the main relation, while a dotted line informs a possible but not preferred alternative; (c) crossover

9.3 Derivative Segment 1

203

RPL (p1, 1)

Fig. 9.22 Low-level derivation of uA 15 from uAs 01 and 06, considering variables involved (a); “Genetic” representation of uA 15 (b)

derivation is denoted by a double-line arrow departing from the two combined parents (there is a single case in segment 1, namely, involving uAs 05 and 06). This initial revision of the derivative relations acting in segment 1 allows the elaboration of a genealogy for the uAs, which, consequently, must be re-labeled (Table 9.3), according to the rules for genealogical notation (presented in Sect. 7.3). Regarding the genealogy of the high-level variants, there is a single issue to be examined in more detail: Which should be the most adequate identification uA 13 (see interrogation marks in Table 9.3)? As a matter of fact, due to its crossover origin, both parents (a3 and a1.1 ) could be taken as a basis for naming the variant. To

204

9 Derivative Analysis

Fig. 9.23 Low-level derivation of uA 16 from uAs 01 and 08, considering variables involved (a); “Genetic” representation of uA 16 (b)

AD

D

Fig. 9.24 Updating of the uA network of segment 1 (based on Fig. 9.5)

properly addressing this situation, let us establish a new criterion: the chosen parent will be that is nearest to the root, in this case, the first-generation a3 (former uA 05). Therefore, uA 13 shall be genealogically notated as a3.1 .

9.3 Derivative Segment 1

205

Table 9.3 Re-labels for uAs of derivative segment 1 using genealogical notation uA 01 02 03 05 06 07 08 10 13 14 15 16

Parent – 01 01 01 02 02 06 07 05.|06 06 06 08

Child 02, 03, 05 06, 07 – 13 08, 13, 14, 15 10 16 – – – – –

Generation 0 1 1 1 2 2 3 3 2 or 3 (?) 3 3 4

Genealogical label a0 a1 a2 a3 a1.1 a1.2 .a13 a1.2.1 a3.1 or a1.1.2 (?) a1.1.2 a1.1.3 .a14

Fig. 9.25 Derivative tree of segment 1 formatted according to genealogy established in Table 9.3

Figure 9.25 reformulates the network of Fig. 9.24 including the new genealogical labels.22 Observe that now, in this new abstract format temporal organization becomes pointless. Because only relations are in question, the disposition of nodes is conditioned only by a search for the clearest and simplest visualization. This basic structure can be used for representing the relations between the variants of two equivalent manners. Let us call the first a phenetic-like representation (Fig. 9.26). It simply depicts the units/nodes in their respective musical format,

22 For

a similar visual representation of derivative relations in a biological-like approach, see Hanninen (2009, pp. 18–20), named as an “association graph”.

206

9 Derivative Analysis a 1.2.1

a 1.1.3

a 1.2

a 1.1

a1

a2

a1

3

a1

4

a 1.1.2

a0

a 3.1

a3

Fig. 9.26 Derivative network of segment 1 depicting variants in musical notation

3

3 3

Fig. 9.27 Derivative network of segment 1 depicting variants in “genetic” notation

keeping their positions in the graph and edge structure.23 The second alternative is a genetic-like representation (Fig. 9.27), in which the variants are described as combinations of their low-level constituents (Z, Y, X, and variants).24

23 The

original uA labels were kept to facilitate the identification. that in this case variant a1.2.1 cannot be decomposed into “genetic” elements because it was obtained from holistic variation (see Fig. 9.15).

24 Notice

9.3 Derivative Segment 1 Table 9.4 Absolute quotient (Q*) between a0 and the variants of segment 1

207 Pair a0 / a1 a0 / a2 a0 / a3 a0 / a1.1 a0 / a1.2 a0 / a3.1 a0 / .a13 a0 / a1.1.2 a0 / a1.1.3 a0 / .a14

Generation 1 1 1 2 2 2 3 3 3 4

Q* {Y, X} {Y, X} {X} {.∅} {X} {X} {.∅} {.∅} {.∅} {.∅}

This latter representation allows the mapping of the relations of permanence of the three variables considering the absolute quotient (see Sect. 7.5).25 Table 9.4 measures the absolute quotient of the ten variants considering a0 as reference. The absence of shared basic characteristics between a0 (the Grundgestalt) and a1.1 , as well as the derivations of the latter, suggest that speciation (recall Sect. 7.9) could be a possible analytical interpretation. Under this perspective, a1.1 may be considered a new derived “species”, inaugurating a new lineage. Figure 9.28 updates the network of Fig. 9.25 from this new bias, including also the values of k for each individual relation, as calculated along with the analysis. Besides simplifying the system, this change is consistent with the importance of the head of the new lineage “b” (former uA 06), which is evidenced by the number of direct descendants it has, four (in contrast, a0 has just three). This fact, in itself, denotes the derivative influence of the form, which is reinforced by the propagation of b-descendants in further segments. In Fig. 9.28b the structure of the network is filtered by the middle-level prism, evidencing three regions occupied by the three variants of locution zx. Let us now consider the question of developing variation. Firstly, recall that, by definition a UDV,26 a basic condition for developing variation, demands the recursion of two contiguous derivations, at least. Given this, it is possible to observe in the graph of Fig. 9.28a that four developing-variation lines meet this basic condition in segment 1 (refer to them as DV paths): 1. 2. 3. 4.

a0.→a3.→a3.1 a0.→a1.→a1.2.→a1.2.1 b0.→b1.→b1.1 a0.→a1.→b0.→b1.→b1.1 The four DV paths are highlighted in Fig. 9.29.

25 Since

all variants, but the holistic a1.1.2 , contain versions of variables Z, Y, and X, to examine them using relative quotient would be redundant. 26 Unit of developing variation, see Sect. 7.1.

208

9 Derivative Analysis

(a)

d

(b)

zx0

zx0

d* i 1

zx0

zx1

i1

zx0

zx1

zx1.1

zx1

zx0

zx0

zx1.1

zx0

Fig. 9.28 Updating of the derivative network of segment 1 considering speciation (a); Distribution of the three middle-level variants of locution zx (b)

Let us now examine in details DV path 4, the longest and, probably, the lineage with the most far-reaching consequences in the piece. The model introduced in Fig. 7.5, regarding relative/absolute developing variation, is the most adequate analytical tool for the task. Before plotting a relative/absolute DV graph (according to the model introduced in Fig. 7.5), it is necessary to calculate the values related to the accumulated dissimilarity between the head of the lineage (a0 ) and its descendants, or, in other terms, the absolute divergence (denoted by the symbol .σ ), considering that contiguous (or relative) values were already determined (and depicted above the respective arrows in Fig. 9.29, path 4). Given this, it is necessary to measure similarity (put another way, to calculate k) for three absolute relations:

9.3 Derivative Segment 1

209

Fig. 9.29 DV paths in segment 1

• • • •

σ 1, referred to relation a0.→a1 σ 2, referred to relation a0.→b0 .σ 3, referred to relation a0.→b1 27 .σ 4, referred to relation a0.→b1.1 . .

Incidentally, all of them are of same-cardinality relations. In this manner, the respective penalties for dissimilarity are calculated in the ordinary manner, by the algorithms. Table 9.5 summarizes the relations considered in DV path 4. The lineage of a0 in segment 1 can now be plotted, as shown in Fig. 9.30. As the graph informs, initially, absolute divergence increases when the new “species” (b0 ) arises, and then is kept almost stable at 0.28, in spite of the local (relative) fluctuations. We can also examine low-level derivation in segment 1 from a global perspective. The genealogical trees of variables Z, Y, and X are depicted in Figs. 9.31 and 9.32 (dotted-line arrows indicate alternative derivations). The comparison of the graphs

27 In

fact, the penalty of this relation was already computed in the dual-origin derivation of uA 16.

210

9 Derivative Analysis

Table 9.5 Values for .σ (k) for absolute relations in DV path 4

Type Relative/absolute Relative Absolute Relative Absolute Relative Absolute

Relation a0.→a1 a1.→b0 a0.→b0 b0.→b1 a0.→b1 b1.→b1.1 a0.→b1.1

Fig. 9.30 Lineage of a0 (DV path 4) considering absolute and relative developing variation

0.188

k 0.18 0.28 0.28 0.12 0.28 0.07 0.29

0 8 0.29 0.28

0.18

0.28 28

00.12

0 0.07

tells us that the diversity of the pitch domain (expressed by Z’s variants) is considerably greater than the others (Y and X).28 Finally, like was done with the “phenetic” relations, low-level networks are provided in Fig. 9.33.

28 Notice

that in their musical representation, Z variants are “normalized”, that is, by convention, they always initiate with pitch C5 .

9.4 Derivative Segment 2

211

3.1

d-p3

3.1.1

d-p3

3

3

d-p3

Fig. 9.31 Genealogical tree of variable Z in segment 1

9.4 Derivative Segment 2 Segment 2 corresponds to the middle contrasting subsection of section A (i.e., [A.b], according to the formal terminology). The departure from the central key implies naturally some tonal instability, which coincides with a brief motivic elaboration based on variants of segment 1. Figure 9.34 presents an initial view of the passage, highlighting candidates for uAs and including provisional numeric labels. A first draft for a derivative network (as done in segment 1) is proposed in Fig. 9.35. Observe that, aiming at a more concise perspective, two of the initially considered uAs were now discarded by similar reasons: a copy of uA 17 and a transposed version of it (uA 20). This preliminary analysis also suggests that some uAs can be interpreted as derived from previous units, which is indicated by dashed-line boxes arrows. Let us name this type of relation as associative. On the other hand, the link between

212

9 Derivative Analysis

1.1

t2

3

t2

Fig. 9.32 Genealogical trees of variables Y and X in segment 1 13

3.1.1

3

3

Fig. 9.33 Derivative networks related to variables Z, Y, and X

9.4 Derivative Segment 2

17

213 20

18

17

22

21

24

23

19

pedal on

pedal on

^

^

5 pedal on ^

Fig. 9.34 Brahms’s Intermezzo Op. 118/2 (mm. 17–28)—derivative segment 2 (first approach) 22 18

b 24 17

21

19

23

b0

Fig. 9.35 Time-oriented transformational network related to segment 2

b1.1 and uA 17 is represented by a filled-line arrow, intending to denote a direct, elaborative relationship between both ideas. This lead us to consider that time could be a decisive factor for the differentiation of these two situations: relations between similar uAs separated by short gaps of time tend to be derivative, rather than associative; conversely, a pair of similar uAs relatively distant on time will probably relate by association.

9.4.1 Derivation of uA 17 Given this, the first element of the passage (uA 17) clearly integrates an elaborative process, initiated by the last variant of segment 1, b1.1 (former uA 16). There is evident a kinship linking between both units, expressed by a relatively high degree of similarity (82%).29

29 Since k = 0.16, as algorithmically calculated. Actually, the similarity could be still higher if the two harmonic contexts were not so contrasting.

214

9 Derivative Analysis

Z14

2

4

Z15 d-p3

d-p3

AD D 2( 3

Z1

) ,1

Z13

originally obtained in Figure 9-14

17

Z13 2

1

Fig. 9.36 Low-level derivation of uA 17 from b1.1 , considering variables involved (a); “Genetic” representation of uA 18 (b)

Figure 9.36 presents the low-level derivative analysis and the “genetic” representation of uA 17. Note how the apparent derivation of variant .Z15 from .Z14 is, in fact, a case of involution,30 that is, the mutational application of an intervallic subtraction on the first element corresponds to the exact inversion of the operation ADD2 applied to the first element of .Z13 .31 In situations like this, the original variant form is considered as the official representative of the configuration, which includes the maintenance of the original label.

9.4.2 Derivation of uA 18 The derivative relation 17.→18 is considerably more complex, which is due not only to the differences of cardinality. Pitch is the domain most affected, by the application of three sequential transformations, as shown in Fig. 9.37. This, of course, impacts the calculation of the divergence between both units. As algorithmically calculated,

30 See

Sect. 7.11.

31 Interestingly, .Z

13 arose as an intermediary stage (i.e., hypothetically) in the low-level derivation of uA 08 (see Fig. 9.14.) Thus, is only in the “genetic” structure of uA 17 that the variant is—so to speak—brought to “life”.

9.4 Derivative Segment 2

215

18

p1

Y2.1.1

3 Z1.2.1.1

q=6

INT2 (p1, 2:3) p1

3 Z1.2.1

RPL 1(Y2, 2:3)

Y2.1

RHA (ins, 1) q=4

RPL 1(p1, 2) 3 Z1.2

t 3( Z 13 ,2)

Z13

RPL 1(Y2, 2)

17/18

p1

Y2

kq=

q

10 n r

=

3 = 0.15 10 4 2

17 Fig. 9.37 TD analysis of the derivation of uA 18 from uA 17, considering the intermediary hypothetical stage 17/18

the penalty for dissimilarity between uA 17 and the intermediary form 17/18 is very low (0.07). It is added to the penalty due to cardinality divergence, kq = 0.15 (depicted in Fig. 9.37),32 and we obtain .k = k17→17/18 + kq = 0.07 + 0.15 = 0.22. In the level of variables three new variants are incorporated into the Z lineage (and one to Y’s), as shown in Fig. 9.38.

9.4.3 Derivation of uA 19 The uA 19 is clearly derived from variant a1.2.1 (former uA 10). Since the latter was obtained from holistic derivation, it is quite logical to consider that variants from it will also have a holistic nature. Figure 9.39 proposes an analysis of this derivative process by modeling the method originally used in Fig. 9.15. Three editions of the same type (which could be described as “upwards diatonic transposition by interval

32 From

now on, the calculation of kq will be added to the corresponding TD analyzes.

216

9 Derivative Analysis

Z13

ADD3 ( Z 13 , 1)

Z1.2 3

3 , 1) RPL1( Z 1.2

d-p3

d-p3

d-p3

INT2(Z 1.2.1 , 1) 3

Z1.2.1 3

RPL1(Y2.1 , 2:3)

Y2.1 t2

Z1.2.1.1 3 d-p3

Y2.1.1 t2

18 Z1.2.1.1 3 Y2.1.1

1

Fig. 9.38 Low-level derivation of uA 18 from uA 17, considering variables involved (a); “Genetic” representation of uA 18 (b)

higher d

ed.#2: transpose 2

ve

ed.#1: transpose 8 higher

19

ed.#3: transpose 3 d higher

a 1.2.1 ed.#4: delete

Fig. 9.39 Holistic derivation of uA 19 from variant a1.2.1

i”) plus a simple “note deletion” describe the transformation that maps a1.2.1 onto uA 19. As done before, divergence k is calculated as the product of the number of editions (4) by constant c (= 0.07): k = 4 .× 0.07 = 0.28.

9.4 Derivative Segment 2 Fig. 9.40 Low-level derivation of uA 21 from uA 17, considering variables involved (a); “Genetic” representation of uA 21 (b)

217

T (p1, 2) dp-3

9.4.4 Derivation of uA 21 The same-cardinality relation 17.→21 is considerably straightforward (k = 0.16, by the algorithms). Low-level analysis brings a new variant (.Z13 .3 ) to the Z pool (Fig. 9.40).

9.4.5 Derivation of uA 22 Figure 9.41 analyzes the derivative relation 21.→22. The most relevant novelty in this process is the emergence of a new rhythmic figure (dotted quarter-eighth note), resulting from the re-partition (operation RPA) of the durational span occupied by the two last quarter notes in 21’s rhythmic sequence. As usual in such cases, k results from the sum of the cardinality penalty kq (0.13, see Fig. 9.41) with that one due to the divergence between uA 22 and the intermediary form 21/22 (0.12, by the algorithms). Thus, .k = 0.13 + 0.12 = 0.25. Four new low-level variants are created (Fig. 9.42).

9.4.6 Derivation of uA 23 Like in the case of uA 13 in segment 1, the derivation of uA 23 involves a crossover of two basic units. Meaningfully, these are not ordinary parents, but the heads of the two lineages considered so far, namely a0 and b0 . The combination of both heads (as in a marriage between heirs of two feudal lords) seems to give to uA 23 (or, at least, to the very moment in which it arises) special importance. In fact, the return of the “a-family”, so to speak, after a relatively long elaborative gap based

218

9 Derivative Analysis

RPA

Fig. 9.41 TD analysis of the derivation of uA 22 from uA 21, considering the intermediary hypothetical stage 21/22

on “b” ideas, coincides with the end of the contrasting formal subsection [A.b] and the near recapitulation [A.a’]. Particularly, the subtle and unexpected manner by which the basic eighth-note idea reentries the rhetorical discourse, emerging from the development of its close “rival”, reveals much of Brahms’s sophisticated motivic treatment. Unlike uA 13’s case, in which each parent contributed to the derivation with a single primary domain, here the basic units are responsible almost in equal parts for the formation of the child’s pitch structure.33 In this regard, the TD analysis of Fig. 9.43 introduces some new procedures. Firstly, observe how Z (from a0 ) is primarily “edited”, by losing the last element of the corresponding pitch sequence p1, before being combined with Z1.1 (from b0 ). This is accomplished by a new-created operation CON (standing for concatenation). By adding the cardinality-change penalty kq (0.13, calculated in Fig. 9.43) to the value 0.15 (calculated by the algorithms) we obtain the total dissimilarity .k = 0.28.

33 On

the other hand, the temporal configuration is entirely originated from transformations of a0 ’s rhythmic organization.

9.4 Derivative Segment 2

Z1.3 3

INT1 (p1, 1:2)

d-p3

Z1.3 3.1

EXT 1(p1)

d-p3

RDV(t1, 2:3)

Y2.1

219

t2

Y2.2.1 t2

INT1 (t1, 1:2)

Z1.3.1.1 3 d-p3

Y2.2.1.1 t2

22 Z1.3.1.1 3 Y2.2.1.1

1

Fig. 9.42 Low-level derivation of uA 22 from uA 21, considering variables involved (a); “Genetic” representation of uA 22 (b)

Figure 9.44 depicts the new derived variables produced in the relation. Observe especially a new situation involving variant .Z6.13 . It results from a transformation applied to Z6.1.1 after a d-p3/p3 equivalence, i.e., considering two distinct versions of the same attribute, in order to capture the chromatic alteration of the last element

9.4.7 Derivation of uA 24 The last unit of segment 2 is visibly associated with uA 23, as a slightly transformed sequence. As shown in Fig. 9.45, holistic derivation provides a simple and straightforward analytical explanation for it. In fact, a unique edition was necessary, sending the chromatic block a minor third (three semitones) above the original configuration. For this reason, penalty for dissimilarity is very low, .k = 0.07.

9.4.8 Overview of Segment 2 As done previously, this subsection summarizes and organizes the analytical findings obtained in segment 2, depicting them in the format of networks and genealogical trees.

220

9 Derivative Analysis

Fig. 9.43 TD analysis of the derivation of uA 23 from crossover of uAs a0 and b0 , considering the intermediary hypothetical stage 23/[a0 +b0 ] Z

DEL1(p1, 3)

Z6

CON (Z6, Z1.1)

d-p3

d-p3

Z6.1

t (Z6.1, 3:5)

d-p3

Z6.1.1 d-p3

T-1(Z6.1.1, 5) p3

Z6.13 p3

Z1.1 d-p3

Y t2

RPL(t1)

Y4 t2

MRG(t1, 4:5)

Y4.1 t2

23 Z6.1 3 Y4.1

Fig. 9.44 Low-level derivation of uA 23 from uAs a0 and b0 , considering variables involved (a); “Genetic” representation of uA 23 (b)

Figure 9.46 updates the basic configuration of the units of analysis of the segment, shown in Fig. 9.35. Aiming at a more concise representation, the new network replaces the original numeric identification (kept in shaded fonts for

9.4 Derivative Segment 2

221

fixed

24

ed.#1: transpose 3 semitones higher

23 Fig. 9.45 Holistic derivation of uA 24 from uA 23 22

b 1.3 2.1 18

b 14 24 17

ab1

21

b 1.3 2

b 13 19

23

a 1.2.1.1

b0 ab

Derivative segment 2 Derivative segment 1 b 1.1

a 1.2.1

a0

b0

Fig. 9.46 Updating of the uA network of segment 2 (based on Fig. 9.35)

reference) with genealogical labels, according to the respective derivations. Since the time scale has no relevance in this scheme (in contrast with the original graph), the segment-1 units from which some variants branch are freely disposed, in favor of a clearer visualization. Note especially the emergence of a new sort of variant, a hybrid form, resulting from the crossover of two basic units, a0 and b0 . This new element, formerly identified as uA 23, is in the graph properly labeled as ab0 , from which stems variant ab1 . This scheme is inserted in the more abstract network of Fig. 9.47, updating the graph presented in Fig. 9.25. The new variants are highlighted in the tree in order to evidence more clearly the expansion of the network occurred in segment 2. Concerning developing variation, among several possible alternatives, let us consider the path examined in segment 1 (see Fig. 9.30): a0.→a1 .→b0.→b1.1 .

222

9 Derivative Analysis

Fig. 9.47 Updated derivative tree (related to Fig. 9.25) considering segment 2. New variants are highlighted

Figure 9.48 plots a continuation for this lineage, by connecting the last element (b1.1 ) to three descendants.34 Lastly, considering the level of variable derivation, Fig. 9.49 updates the Z-YX networks (see Fig. 9.33), highlighting the new variants incorporated. As it can be observed, no metric transformation (related to lineage X) was applied in the segment.35

34 The similarity values between the same-cardinality variants are automatically calculated by the algorithms. Only one relation involves different cardinalities: that between the extreme poles, namely a0 and .b13 .2.1 . In strict terms, the calculation of similarity between these uAs should involve a TD analysis. However, aiming to avoid this additional complication, I try here a new, simpler protocol: (1) determine the penalty for dissimilarity between a0 and intermediary uA 21/22 (both are compatible in cardinality), by the use of the ordinary algorithms, with result .≈ 0.40; (2) adopt as a penalty for cardinality change (3 to 5) the same value obtained in the case of relation 21.→22, that is, kq = 0.13; (3) Sum both penalties, thus k = 0.40 + 0.13 = 0.53. 35 Aiming at a more agile and concise dynamic, other diagrams presented in the discussion of the derivative processes performed in segment 1 were here omitted. I will return to them at the end of the analysis, in order to provide a global view of the whole intermezzo.

9.5 Derivative Segment 3

2

3 4

0.53

0.38

1

0.33

0.5

0.28 0.28 0.29

1

0.18

Fig. 9.48 Absolute/relative DV graph considering lineage .a0 → a1 → b0 → b1.1 → b13 → b13 .2 → b13 .2.1 , updating Fig. 9.30

223

5

6

7

0.18

0.28

0.12

0.07

0.18

0.16

0.25

3

1.2

3

3

1.2.1

9.5 Derivative Segment 3 The third derivative segment coincides with the recapitulatory subsection [A.a’] (comprising formal segments 7, 8, and 9). This is evidenced by the strong return of the basic elements of the Grundgestalt, represented especially by the eight-note rhythmic/metric configuration. Despite this, it is clearly visible in Fig. 9.50 that what is taking place is not a simple restatement of the initial material, but a kind of reformulation of the uAs of segment 1, merged with the elaboration that occurred

224

9 Derivative Analysis

1.2.1.1 3

1

1.2.1

1.2

4

4.1

2.2

2.2.1

1.3

1.3.1 2.1.1 1.3.1.1

6 6.1

6.1.1

3.1.1

3

6.1

3

3

Fig. 9.49 Derivative networks related to variables Z, Y, and X, updating Fig. 9.33. New variants are highlighted 25

26

27

IV I6

IV

I6

ii 6/5

a

33

32

28

28

28 V4/3 /IV IV6 V

31

29

34

35

36

30

I6

I6 iv

i6

IV I V6/5 /V I6/4 iii 6 ii V7

I

ii°/6/5

Fig. 9.50 Brahms’s Intermezzo Op. 118/2 (mm. 29–38)—derivative segment 3 (first approach)

in the contrasting central subsection. Put this in derivative terms, segment 3 is characterized by an interaction of a- and b-derived ideas. This fact maybe explains the profusion of new uAs which were identified, twelve in total, disregarding the repetitions. However, this number can be relativized, if we evoke the principle of derivative equivalence discussed in the beginning of the analysis (see Sect. 9.3). According to this, uA 28—a re-harmonized resumption of a0 —can be simply discarded.36 Moreover, it is possible to consider that the contiguous units 27 and 29 form

36 The

same is applicable in relation to uA 35, which is technically a copy of variant a3.1 .

9.5 Derivative Segment 3

1

225

5

2 3

4

6

Fig. 9.51 Six manifestations of the relation between pitch classes B-A in segment 3

a high-level block that, in turn, becomes referential for the conjunct (holistic) transformation [31+32]. By seeing the events from such a global perspective we can simplify the analysis, evidencing distinct derivative processes. Quite interestingly, derivation in segment 3 can also be viewed from another prism, considering special relations between two specific pitch classes, B and A, in this case disregarding any explicit association with the “genetic” elements detected so far in the analysis. So, I propose at this point to open a pair of parentheses and to explore such relations in a separate prospection (though I intuit that, in a sense, both analyzes could be closely connected).37 Figure 9.51 identify six distinct manifestations of the block B-A, depicting them as Schenkerian-like reductions of corresponding melodic events. Given this, it is possible to see the six units as distinct transformations of the abstract basic idea that expresses the relation between pitch classes B and A. The correlations between these units can be formally represented with the help of operations and visualized in a network, as depicted in Fig. 9.52. According to this conception, unit 2 can be seen as derived from 1 by a mutational change of register affecting its second element (pitch A4 ).38 A more complex metamorphosis explains unit 3. In fact, it comprises a composition of three operators applied to 2: it is firstly sent an octave lower (OCT- ), and then suffers retrogradation. The third operation, labeled as FIL (a new one), acts in a melodic interval (in this case, the descending minor seventh A4 -B3 ) filling the gap with diatonic steps.39 Unit 3, in turn, gives rise to 4 by a simple change of mode of its constituents, from A major to A minor. Unit 5 corresponds to the retrogradation of 2, while 6 can be considered as resulting from a freer version of operation FIL (with the modification denoted by the use of italics). It is also possible to conceive other secondary relations between the units, as that suggested by the dashed-line arrow between units 3 and 6.

37 An

attractive possibility that I decided not to pursue in the present study, aiming at simplicity. that now I am dealing with concrete pitches rather than pitch classes, in order to apply the necessary transformations. 39 Steven Rings (2011, p. 197) points out the relation between the ascending leap B-A (unit 2, in this analysis) and the two descending scalar segments (3 and 4), calling them Septzüge (adopting Schenker’s terminology). However, he does not perceive the remaining derivative connections (1, 5, and 6). 38 Note

) CT

+

,2 (1

MOD(3)

9 Derivative Analysis

(2) R*FIL * T C O

226

3

4

O

2 )

2 R(

1

1)

F

( IL

5 ?

6 Fig. 9.52 Derivative network related to the six manifestations of the block B-A 32

31

29

b0

27

a2

a0

36

26

34

25

33

30

28

V4/3 /IV IV6 V IV I 6 IV I 6 ii6/5

a

I6 IV I V6/5/V I6/4 iii6 ii V7 I i 6 iv

/ 6/5

i6 ii°

Fig. 9.53 Time-oriented transformational network related to segment 3

Back to the uAs of segment 3, Fig. 9.53 presents the time-oriented version of the derivative network for segment 3. This case differs from the previous ones in two aspects: firstly, the number of associative relations (denoted by dashed-line arrows) involving units from segment 1 is greater than that of actual elaborative relations (filled-line arrows). The second aspect concerns the presence of a new

9.5 Derivative Segment 3

227

Fig. 9.54 Low-level derivation of uA 25 from uA a0 , considering variables involved (a); “Genetic” representation of uA 25 (b)

Z d-p3

ADD1 (Z, 2)

Z7 d-p3

25 Z7 Y convention, double-line braces connecting pairs of units. It introduces the idea that some uAs can be seen as forming higher-level structures, in a similar way that atoms are combined into molecules. This new technology can be used not only for evidencing how a linked material behaves (like a0 -3 ) when its components are individually transformed (in blocks 25-26 and 33-34), but also for proposing highlevel derivation, when the pair as a whole is treated as a parent. This is the case of the block 31-32, derived from 27-29 (observe the curved arrow connecting the two braces). Doing in this way, we bypass three individual, lowerlevel relations: 27.→31, 29.→32, and 31.→32 (omitted in the network), providing concision and a more global view for the analysis.

9.5.1 Derivation of uA 25 As it is visually evident, there is a very close association between uA 25 and the basic unit a0 . As demonstrated by the algorithimic calculation, they diverge by only 8%, which is mainly due to a subtle mutational transformation applied to the intervallic sequence, resulting in variant Z7 , as shown in Fig. 9.54.

9.5.2 Derivation of uA 26 Likewise, uA 26 is very similar to its associated parent, a2 , from which differs only slightly in the rhythmic configuration, bringing a new variant to the Y pool (Fig. 9.55).40 40 Since rhythm is more sensitive to variation than pitch and harmony, according to the quantitative

conventions adopted in the measurement of similarity, divergence here is a little higher than the previous case: k = 0.11.

228

9 Derivative Analysis

Fig. 9.55 Low-level derivation of uA 26 from uA a2 , considering variables involved (a); “Genetic” representation of uA 25 (b)

Y t2

RPL(t1)

5

t2

26 Z Y5

9.5.3 Derivation of uA 28 Association with the Grundgestalt (a0 ) is still more intense in the case of uA 28. In fact, it can be seen as an almost exact copy (disregarding only pitch register and slight differences in harmonic context), which makes its derivation not worth for analysis (recall the criteria previously discussed concerning equivalence between units). However, its presence in the segment is relevant for explaining uA 30, a sort of minor-mode reflex of the basic idea. Most of the low divergence between both units (.k = 0.08) arises from harmony (more specifically, caused by the opposition of modes). Regarding the variable level, none variant is produced, because the transformation does not affect Z, associated with the diatonic intervallic sequence (in fact, the third rows of both matrices of attributes are identical each other, in spite of the differences in respect of pitches and pitch classes).

9.5.4 Derivation of uA 29 Unit 29 stems directly from 27, but also can be associated to a0 , which denotes a case of dual derivation through a crossover. As the transformation results in an increase of cardinality, a TD analysis is required (Fig. 9.56). As habitual, the similarity between the child and the combined parents is calculated by the sum of penalties .k29→29/[a0 +27] (0.04, from algorithmic calculation) and kq (0.10, from Fig. 9.56). Thus, .k = 0.10 + 0.04 = 0.14.

9.5 Derivative Segment 3

229 29

p1

4 (involution) 3

t1

t-3 (p1) 1.4

p1

RHA (sub, 2) SPL (t1, 3) t1

29/[a0+ 27]

EXT1 (p1) 1

p1

27 kq=

q

10 n r

=

2 = 0.10 10 2 1

Fig. 9.56 TD analysis of the derivation of uA 29 from crossover of uAs 27 and a0 , considering the intermediary hypothetical stage 29/[27+a0 +] Fig. 9.57 Low-level derivation of uA 29 from uA b0 , considering variables involved (a); “Genetic” representation of uA 29 (b)

Z1 d-p3

EXT1(p1)

Z1.4 d-p3

29 Z1.4 Y3

1

Variant .Z14 branches from Z by the inclusion of one element at the end of the original intervallic sequence (operation EXT), as shown in the analysis of Fig. 9.57.41

9.5.5 Derivation of uAs 31 and 32 As suggested before, the best (since its the simplest) way to address the derivation of these two units is to consider them as forming a unique block. Put another way, they

41 Observe

in Fig. 9.56 that, in the rhythmic domain, the transformation of Y yields Y3 , a variant that is already part of the pool, therefore, another instance of involution.

230 Fig. 9.58 Holistic derivation of collapsed uA 31.32 from collapsed uA 27.29

9 Derivative Analysis

31.32

ed.#1: modulate from A major to A minor

27.29 k = 1*0.07 = 0.07

will collapse into a higher-level unit (labeled as uA 31.32),42 which largely simplify the derivative analysis, considering the also collapsed unit 27.29 as its actual parent. Likewise, it is logical to see this derivative relation through the prism of holistic variation (Fig. 9.58).

9.5.6 Derivation of uAs 33 and 34 The block formed by uAs 33 and 34, as previously commented in the formalharmonic analysis (see Fig. 8.9), corresponds to an inverted recap of the initial gesture of the intermezzo. As pointed out then, they cannot be considered as strict (chromatic) inversions, since the actual magnitude (measured in semitones) of the intervals diverge. However, as in the present derivative analysis the intervallicsequence attribute is calibrated to the diatonic scale of A major, both transformations can be, in this context, indeed strict (diatonic) inversions. Concerning derivation, unlike what was done in the previous case, it seems more logical to consider the individual relations in question, namely .a0 → 33 and .a2 → 34. Since there is no change of cardinality in both cases, the algorithms are used for calculating their penalties .ka0 →33 = 0.09 and .ka2 →34 = 0.12. Figures 9.59 and 9.60 show the low-level derivative analysis of the two cases, which bring two new Z variants.

42 The intermediary dot denotes that the two unites were “welded” and are will not any more treated

as individual entities.

9.5 Derivative Segment 3

231

Fig. 9.59 Low-level derivation of uA 33 from uA a0 , considering variables involved (a); “Genetic” representation of uA 33 (b)

i(Z)

Z

Z8 d-p3

d-p3

33 Z8 Y Fig. 9.60 Low-level derivation of uA 34 from uA a2 , considering variables involved (a); “Genetic” representation of uA 34 (b)

i(Z)

Z2

Z 2.1 d-p3

d-p3

34 Z2.1 Y 1

≡

Z1.1.4.1

i (Z1.1.4.1 , 3)

d-p3

Z1.1.4.1.1

t3(p1, 1)

d-p3

≡

Z1.1.4.13 d-p3

36

Z1.1.4.13 3

Fig. 9.61 Low-level derivation of uA 36 from uA 29, considering variables involved (a); “Genetic” representation of uA 36 (b)

9.5.7 Derivation of uAs 36 The last unit of the segment (36) is derived from uA 29 (.k = 0.16, from the algorithmic calculation). Figure 9.61 proposes a possible chain of low-level derivations for explaining the pitch configuration of the variant.

232

9 Derivative Analysis 31

32

b 4.1

b4

25

a4

26

34

33

a5

a 2.1

a 2.2

27

b4

29

b 4.1

36

b 4.1.1

Derivative segment 3 Derivative segment 2 Derivative segment 1 a0

a2

b0

Fig. 9.62 Updating of the uA network of segment 3 (based on Fig. 9.53)

9.5.8 Overview of Segment 3 As proceeded in the previous segments, Fig. 9.62 updates the derivative relations of the basic uA’s network (c.f. Fig. 9.53), including also proper labeling. Some commentary must be added with respect to particular aspects of this network’s architecture: • The recapitulative nature of this passage is evidenced by the associative relations that connect the uAs to segment-1 basic ideas (a0 , a2 , and b0 ). This situation contrasts with what occurs in segment 2, where elaborative relations (denoting a notion of continuation) predominate; • Shadings are used for indicating elements discarded from the derivative process. This involves uAs 28 and 30 (as almost exact copies of a0 ) and the block 31.32, a mode-changed resumption of the conjunction formed by uAs 27.29 (genealogically relabeled as b4 and b4.1 ); • Curved-dotted arrows recall associative, more remote relations between a0 and the two b-related elaborated units in the segment (b4.1 and b4.1.1 ). This structure is captured in the general high-level network (Fig. 9.63): the new variants are concentrated around the main referential poles (a0 and b0 ). Lastly, Fig. 9.64 updates the low-level networks by including the variants produced in segment 3.

9.5 Derivative Segment 3

233

1.2.1.1

0.28

0.16 4.1.1

4.1

4

0.1

0. 08

4

0.18

0.22 3

4

0.08

4 4.1

4

08 0.

0.16

0.11

0.25

2 2.1

3 1.2

9 0.0

2 0.1 5

2 2.2

3 1.2.1

0.28

ab0 0.07

ab1

Fig. 9.63 Updated derivative tree (related to Fig. 9.47) considering segment 3. New variants are highlighted

11.2.1.1 3

1

11.2.1

1 1.2

4.1

4 2.2

2.2.1

1.3 1

1.1.4

5

11.3.1

2.1 11.3.1.1

1.1.4.1

2.1.1

8 7

1.1.4.1.1

3

6

1.1.4.1

6.1

6.1.1

3.1.1

6.13 3

3

Fig. 9.64 Derivative networks related to variables Z, Y, and X, updating Fig. 9.49. New variants are highlighted

234

9 Derivative Analysis

9.6 Derivative Segment 4 The fourth derivative segment corresponds to the codetta that closes section A (mm. 39–48). As done previously, a first view of the acting uAs in the segment is provided (Fig. 9.65). The apparent proliferation of new elements is but illusory since there is considerably derivative redundancy in the passage. As depicted in the time-oriented network of Fig. 9.66, most of the units (shaded in the figure) are pure copies and will be disregarded. Moreover, uAs 38, 39, and 40 result from simple diatonic transpositions and, therefore, are considered as equivalent to their respective parents and will be also excluded from the derivative analysis.43 As a matter of fact, the formal role of the passage (conclusion of the intermezzo’s first part) coheres perfectly with the

38

41

38

37 37

39

42

38

38

39

43

44

40

40

pedal on ^1

pedal on ^5

vi

IV

ii

V7

I

7

vi

IV

ii 7

I 6/4 V IV V 7

I

Fig. 9.65 Brahms’s Intermezzo Op. 118/2 (mm. 39–48)—derivative segment 4 (first approach)

Fig. 9.66 Time-oriented transformational network related to segment 4

43 Excluded the equivalences (repetitions plus transpositions), only three uAs are left to be examined in this segment: 37, 41, and 42.

9.6 Derivative Segment 4

235

37

ed.#1: halve duration k = 2*0.07 = 0.14

b4.1 ed.#2: delete

Fig. 9.67 Holistic derivation of uA 37 from b4.1

expected, retrospective resumption of the main motives so far worked, hence the redundancy of material.

9.6.1 Derivation of uA 37 Despite the fact that an indirect connection with the basic unit a0 can be evoked (as suggested in Fig. 9.66), b4.1 is clearly the most immediate reference for uA 37. In this case, holistic variation provides a simple and straightforward derivative explanation (Fig. 9.67).44

9.6.2 Derivation of uA 41 Like in the previous case, the associative relation between uA 41 and variant .b13 .2.1 is also very clear. The transformation of the latter into the former seems also to result from holistic variation, since the basic original format is preserved. Of the four editions implemented, only the last one is a new type of action, deserving special comment: consider it as an informal instruction that creates a middle-level locution (zx1.1 , in this case) in substitution of an inputted note (Fig. 9.68).

44 Observe

that transposition was not included among the editions considered.

236

9 Derivative Analysis

ed.#3: transpose 1 diatonic step

ed.#2: chromatize

41 ed.#1: transpose 2 diatonic steps

Fig. 9.68 Holistic derivation of uA 41 from .b13 .2.1

ed.#4: create a "zx1"

b1.3 2.1 k = 4*0.07 = 0.28

ed.#2: transpose 1 diatonic step

ed.#1: transpose 2 diatonic steps

42

41 k = 2*0.07 = 0.14

Fig. 9.69 Holistic derivation of uA 42 from uA 41

9.6.3 Derivation of uA 42 Once again, holistic derivation is called for explaining this very simple elaboration of uA 41 (Fig. 9.69). Contrarily to what occurred previously, the scarcity of derivative relations in segment 4 turns unnecessary a more detailed overview. In place of this, Fig. 9.70 solely updates the derivative network and renames the units with proper genealogical notation.

9.7 Derivative Segment 5

237 42 2 3 3 b 1.2.1

a0

43

a2

44

38 8

3 b 1.2.1.1

41

b1

b 4.1.2

37

Derivative segment 4 b 4.1

Derivative segment 3 Derivative segment 2 b1

3 b 1.2.1

Derivative segment 1 a0

a2

Fig. 9.70 Updating of the uA network of segment 4 (based on Fig. 9.66)

9.7 Derivative Segment 5 Segment 5 corresponds exactly to subsection [B.a] of the intermezzo, which introduces, as previously pointed out, a considerable amount of contrast in the piece, considering tonal space, texture, and meter. As also discussed, especially with respect to the metric analysis, a secondary voice is added in imitative dialogue with the main line, and it is kept along the whole B section. Figure 9.71 parses both lines into relevant units of analysis, which provides the basis for the elaboration of the related time-oriented network (Fig. 9.72). As one can observe, the architecture of the network is quite distinct from those of the previous derivative segments. There is clearly a flux of progressive transformations taking place, subdivided into two streams (main and subsidiary lines), in total accordance with the imitative logic.

9.7.1 Derivation of uA 45 The derivative process in segment 5 is initiated with uA 45, which maintains an associative relation with variant .b14 , originally arisen in segment 2. This derivation reinforces the linkage that exists between both elaborative territories, subsection [A.b] (segment 2) and section [B.a] (segment 5). Although, at the first glance, the two units appear to be considerably close each other in rhythm and contour, the algorithmic measurement of similarity returns a medium degree divergence, k = 0.50, due to the “distances” between the corresponding elements of their respective matrices of attributes. This is the highest value so far obtained in the analysis considering direct derivation, a fact that meaningfully reinforces the idea of a departure, which seems to be consistent with a compositional intention of contrast, adequate for initiating the B section. On the other hand, the low-level derivation is considerably simple (Fig. 9.73): while in the pitch domain it is necessary to apply two recursive diatonic transposi-

238

9 Derivative Analysis 49

47

45

51

50

48

46

55

53

54

5 52

I

f:

(= III)

i

iv

i

VI ii

V

i

VI

V7

v4 V /V

6

Fig. 9.71 Brahms’s Intermezzo Op. 118/2 (mm. 49–56)—derivative segment 4 (first approach)

49

50

45

55

51

47 53

46 52 48 54 I

f:

(= III)

i

iv

i

VI ii

V

i

VI

6

v4 V /V

V7

Fig. 9.72 Time-oriented transformational network related to segment 5

tions (and one intermediary stage) to transform the parent into the child (.Z13 .2.14 ), the rhythmic derivation is straightforward, by re-partition of the first pair of onsets. Another difference between uA 45 and its parent .b14 is observed in the middlelevel organization: it reintroduces variant zx1.1 as a concluding gesture (Fig. 9.73b); furthermore and interestingly, it brings back the same original pitches C.5 and B5 .45 Unit 45 can be seen as the derivative head of the segment, disseminating its “genetic” material in two derivative directions: “horizontally”, through a sequentiallike process toward uA 47 (and from this to the next variants of the same stream), and “vertically”, with the quasi-canonic imitation of the the secondary voice, initiating with uA 46 and following with subsequent variants.

45 This variant can be seen as an expanded version of the original format, in which the precedent event is replaced by a melodic line whose approximate contour is represented by the line in the figure. The same gesture is subsequently transmitted to the remaining uAs in the segment.

9.7 Derivative Segment 5

239

t-3((p1, 5)

3

1.2.1.1

3

t-1(p1, 2:6)

3

1.2.1

d-p3

d-p3

RPA P (t1, 1:2)

3

4

1.2.1

d-p3

3

2.1

t2

t2

x

t3

("medium-weak-stronger")

t3

zx1.1

(c)

"weak"

"strong"

2

45 3

4

1.2.1 3

2.1

Fig. 9.73 Low-level derivation of uA 45 from variant .b14 , considering variables involved (a); Middle-level interpretation of uA 45 (b); “Genetic” representation of uA 45 (c)

9.7.2 Derivation of uAs 46 and 48 In a sense, uA 46 can be seen as a compacted copy of uA 45, constructed in such a way that both ideas conclude at the same time, even so the latter starts after the former. Because of their different cardinalities, a TD analysis is required (Fig. 9.74). Essentially, uA 46 branches from 45 by a simple deletion of its two last pitch and rhythmic events (and the first chord of the harmonic context). Interestingly,

240

9 Derivative Analysis

t4

2.1

3 5 1.2.1

2.14

4

3 1.2.1

2.13

Fig. 9.74 TD analysis of the derivation of uA 46 from uA 45, considering the intermediary hypothetical stage 45/46

a further rhythmic re-partition of the time span occupied by the two initial onsets (.8 = 6 + 2 = 4 + 4) results in a copy of a variant previously produced (Y2.1 ), characterizing another case of involution. After this stage, a metric displacement applied to attribute t4 (accessed by the principle of attribute equivalence) leads to the definitive rhythmic configuration of uA 46. As usual in such cases, the definitive similarity between both units is measured as the sum of penalties due to the differences of cardinality kq (calculated in Fig. 9.74) and divergence with the intermediary “lost link”, in the case uA 45/46 (by algorithmic calculation):46 k = kq + k46→45/46 = 0.07 + 0.16 = 0.23.

.

Instead of following the chronological flux of events (which would lead us to the analysis of uA 47), it seems more logical to continue to examine the next unit of the imitative stream, uA 48. One can easily perceive that the relation 46.→48 is proportionally equivalent to the previous one, 45.→46. Indeed, they could be

46 The

related low-level analysis is shown in Fig. 9.75.

9.7 Derivative Segment 5

241

DEL (p1, 5:6)

3

4

3

5

1.2.1

1.2.1

d-p3

d-p3

DEL (t1, 5:6)

3

2.1

4

2.1

t2

t2

46 3

5

1.2.1 4

2.1

Fig. 9.75 Low-level derivation of uA 46 from uA 45, considering variables involved (a); “Genetic” representation of uA 46 (b)

grouped and described as a progressive reduction of the referential idea (and may be expressed as 45.→(46).→48). Due to the simplicity with which this process is done, I think that holistic (rather than TD) analysis is, this case, a better and more direct means for addressing uA 48’s derivation, as shown in Fig. 9.76a. This process of progressive motivic compression matches almost exactly a specific compositional technique, properly called liquidation. As defined by Arnold Schoenberg, who idealized the concept, “liquidation consists of gradually eliminating characteristic features until uncharacteristic ones remain, which no longer demand a continuation” (Schoenberg, 1967, p. 58). However, unlike Schoenberg’s definition, the remaining feature (i.e., uA 48 itself) is all but “uncharacteristic”, since it “personifies” one of the most essential elements of the piece, the middle-level zx locution. As suggested in Fig. 9.76b, the present manifestation of the locution can be seen as another transformation of the original material, through a new-created operation iso, whose action isolates the argument (i.e., zx0 ), thus returning a “purer” version of itself. In this line of reasoning, we can consider the double-stage process that maps the referential unity 45 into uA 48 (mediated by uA 46) as a clear— though very brief—example of teleological derivation (as presented in Sect. 7.10), schematically modeled in Fig. 9.77.

242

9 Derivative Analysis

48 (a) k = 1*0.07 = 0.07 ed.#1: delete

46

(see Figure 9-14)

zx1

zx1.1 2

d*i1(

i1(

)

"strong"

"weak"

"strong"

2

d(

)

"weak"

(b)

also possible:

)

iso (

zx 0

)

2

"weak"

"strong"

01

48

zx2 2

"weak"

) "strong"

iso (

Fig. 9.76 Holistic derivation of uA 48 from uA 46

9.7.3 Derivation of uAs 47, 49, and 50 The elaborative stream that departs from uA 45 reaches initially uA 47, propagating then to uAs 49 and 50. As shown in the TD analysis in Fig. 9.78, the derivation of uA 47 (like what occurred with uA 46) begins with the deletion of pitch and rhythmic elements (just one of each domain, in this case). And once again, a metric displacement concludes the series of temporal transformations.

9.7 Derivative Segment 5

243

"liq uid ati on

"

Fig. 9.77 Teleological derivation from uA 45 to uA 48

Concerning similarity, in contrast with a very low penalty for change of cardinality (0.03), relation 47.→45/47 scores a relatively high value (0.47, calculated by the comparison algorithms), resulting in a total penalty .k = 0.03 + 0.47 = 0.50. Figure 9.79 presents the low-level analysis related to the unit. This time, the metric transformation leads to a new X variant (X2 ). As done before, it is better described in more abstract terms, considering the initial metric gesture. As suggested by the parentheses, the new version changes the original binary anacrustic configuration (“medium-weak”) into a ternary (“weak-medium-weak”), aiming, in both cases. at the strongest element: the next event. Relation 47.→49 is very close to a sequence. Actually, just the minimal rhythmic transformation of the penultimate event justifies not to consider both units as equivalent under diatonic transposition. Thus, the derivation is explained by a very simple holistic analysis, as shown in Fig. 9.80. Like in the imitative stream, the first elaborative phrase concludes with the liquidation of uA 49, giving rise to uA 50, also a transformation that is better described as holistic (Fig. 9.81). In this manner, uA 50 can be seen as a “deformed” version of uA 49, in which, after the deletion of the first event (responsible by the “liquidation effect”), the distance between the two eight-note blocks is shortened by the action of two opposed diatonic-second transpositions, while the last event is kept as a fixed point.

244

9 Derivative Analysis

p1

t4

MTD D-2 (t4)

t1(p1) 2.1.3 2.1.1 2.1

t4

3 1.2.1.2.1 1 .2 14 1

p1

SUB2(

RHA(sub, 1:2)

, 3)

2.1.3 2.1 2.1

SUB 3

, 1)

t1

4

5

p1, 4)

DEL 2.13

Fig. 9.78 TD analysis of the derivation of uA 47 from uA 45, considering the intermediary hypothetical stage 45/47 DEL (p1, 3)

3

4

1.2.1

3

4

SUB2 (

, 3) 3

1.2.1.2

d-p3

DEL (t1, 5)

3

2.1

3

2.1.2

t2

4

1.2.1.2.1

d-p3

t2

d-p3

SUB4 (

2.1.2

, 1) 3

2.1.2.1

t2

SUB2 (

2.1.2.1

, 3) 3

2.1.2.1.1

t2

MTD-2(t4)

X t4 "medium-weak-strong"

X2 t4 "(weak-medium-weak)-strong"

47 3

4

1.2.1.2.1 3

2.1.2.1.11

2

Fig. 9.79 Low-level derivation of uA 47 from uA 45, considering variables involved (a); “Genetic” representation of uA 47 (b)

9.7 Derivative Segment 5 Fig. 9.80 Holistic derivation of uA 49 from uA 47

245

49

ed.#1: halve duration

47 k = 1*0.07 = 0.07 Fig. 9.81 Holistic derivation of uA 50 from uA 49

50 ed.#2: transpose diatonically 2

49

ed.#3: transpose diatonically 2

fixed

ed.#1: delete

k = 3*0.07 = 0.21

9.7.4 Derivation of uA 51 The second half of segment 5, corresponding, formally, to the consequent of the period, is launched by uA 51, which is clearly another branch from the basic uA 45. Its derivation, depicted in the TD analysis of Fig. 9.82, is quite similar to those of uAs 46 and 47, which creates an additional link between these units. The three derivations initiate equally with pitch and rhythmic deletions and conclude with a (“negative”) metric displacement. This sort of “blood link” is also evidenced by lowlevel involutions (i.e., the arrival to already-produced variants by a different path) occurring in both pitch and temporal domains.

246

9 Derivative Analysis

51

p1

t4

MTD-4 (t4) ADD2 (

, 1)

(involution)

t4

3 2.1.2.1

RHA(sub, 2) SUB4 ( (involution) p1

, 1)

(involution)

5

3 11.2.1

45/51

3

2.1.2 1

5

DEL(p1, 6)

DEL(p1, 6) 4

3 11.2.1

2.13

0.03

Fig. 9.82 TD analysis of the derivation of uA 51 from uA 45, considering the intermediary hypothetical stage 45/51 Fig. 9.83 Low-level derivation of uA 51 from uA 45, considering variables involved (a); “Genetic” representation of uA 51 (b)

ADD2(

3

5

3

5

1.2.1

, 1) 3

6

1.2.1

1.2.1

d-p3

d-p3

51 3

6

1.2.1 3

2.1.2.1

The very low penalty due to cardinality change (0.03) is summed to that referred to the divergence between uA 51 and the intermediary form 45/51 (0.29, algorithmically calculated). We, therefore, obtain .k = 0.03 + 0.29 = 0.32. Due to the presence of the involutions, just a new variable-variant is produced, as shown in Fig. 9.83. Observe that the metric re-arrangement in uA 51 can be also considered as “involutionary” (in more abstract terms, of course), since it provokes a “return” to the basic configuration X, “middle-weak-strong”.

9.7 Derivative Segment 5

247

52

p1

t4

OCT- (p1)

MTD-4 (t4)

(involution) p1

7

3 1.2.1 1

t4

51/52

2.1

q=4

p1, 5) p1

MRG((

, 1:2)

3 2.1.2.1

q=5

51

2

0.05

Fig. 9.84 TD analysis of the derivation of uA 52 from uA 51, considering the intermediary hypothetical stage 51/52

9.7.5 Derivation of uAs 52 and 54 As it happens in the antecedent, both imitative and elaborative streams emanate from a “pivot” unit, in this case, uA 51. The derivations involved in the former stream (considering uAs 52 and 54) are addressed in the present subsection, while the elaborative relations will be examined in the next. Figure 9.84 depicts the TD analysis of relation 51.→52. Observe how it is similar (considering both procedures and output) to relation 45.→46 (Fig. 9.74), that initiates the imitative dialogue in the antecedent. This reinforces the idea that a consistent compositional strategy is in action.47 Once again, rhythmic derivation reaches an involution. Therefore, only the pitch domain contributes to expanding the low-level pool of derivations, as shown in the analysis of Fig. 9.85. Like in the antecedent, a second imitation yields a shortened version (54), whose derivation is also described as a holistic transformation, once again based on a pure

47 The penalty for dissimilarity related to the derivation of uA 52 is .k

calculation) = 0.18.

= 0.05+0.13 (by algorithmic

248

9 Derivative Analysis

Fig. 9.85 Low-level derivation of uA 52 from uA 51, considering variables involved (a); “Genetic” representation of uA 52 (b)

Fig. 9.86 Holistic derivation of uA 54 from uA 52

54

48

k = 1*0.07 = 0.07 ed.#1: delete

52 deletion of the first two events of the referential form (uA 52). As suggested in Fig. 9.86, both outputs are considered equivalent, according to the adopted analytical criteria.

9.7.6 Derivation of uAs 53 and 55 This subsection examines the derivation of the last two units of segment 5, 53 and 55, that form the elaborative stream of the consequent, branching from uA 51. Since the three units share the same cardinality (five events), the measurements of similarity concerning the two relations (51.→53 and 53.→55) are properly calculated through the use of the comparison algorithms, that return, respectively, the values 0.35 and 0.17. Figure 9.87 depicts three new low-level variants that arise from the derivation of uA 53, involving variables Z, Y, and X. The latter case is the first second-generation metric variant to be detected in the analysis, an variable-attribute that has proved to be quite resistant to variation in the intermezzo, if compared to the intervallic (Z) and IOI (Y) sequences. X2.1 derives from a 4-step rotation applied to variant X2 (ultimately associated with uA 47’s rhythmic configuration). Just one pitch variant is produced in the derivation of uA 55 (Fig. 9.88).

9.7 Derivative Segment 5

249

DIM(t1,3:4)

Fig. 9.87 Low-level derivation of uA 53 from uA 51, considering variables involved (a); “Genetic” representation of uA 53 (b)

(

1

, 1:3) d-p3

55

Fig. 9.88 Low-level derivation of uA 55 from uA 53, considering variables involved (a); “Genetic” representation of uA 55 (b)

250

9 Derivative Analysis 50 0 c1.1.1

c1.1

c1

55 c3.2.1

54

48 c2.1

49 9

47 7

c2

c3.1

52 2

46 6

c3.2

53

51 1 c3

45

b 15

c0

Derivative segment 5

Derivative segment 4

Derivative segment 3

b 14 Derivative segment 2 Derivative segment 1

Fig. 9.89 Updating of the uA network of segment 5 (based on Fig. 9.72)

9.7.7 Overview of Segment 5 The derivative network related to segment 5 is depicted in Fig. 9.89. The graph makes clear the fact that uA 45, associated with segment-2’s variant .b14 , acts as a local “patriarch” from which new sub-branches arise. As suggested in the analysis, this seems a necessary and sufficient condition for the establishment of a new speciation. Therefore, .b15 is relabeled as c0 , initiating a new lineage. A global view of the derivative paths in high-level mode is provided by Fig. 9.90. Notice how the network’s area is expanded due to the plugging of the sub-tree related to the new lineage c.

9.8 Derivative Segment 6 Figure 9.91 depicts a basic overview of segment 6, identifying the units of analysis that act in its territory. Clearly, the head of the segment (uA 56), is associated with segment 5’s head, variant c0 . Another associative relation links uA 60 to variant c3.2 . Despite the sharp contrasts in texture (chorale) and key (with the change of mode), segment 6 is considerably more conservative in derivative terms than the previous subsection. Besides the fact that only nine units act in the segment (a relatively modest number), as shown in Fig. 9.91, two of them (uAs 57 and 58) are repeated at the end. Both repetitions are not considered in the time-oriented

9.8 Derivative Segment 6

251 c 1.1.1

1.2.1.1

0.21

c 2.1

c 1.1

0.28

c 3.1

0.07 0.07

4

0.1 4.1

4

0 .1

0.1

6

4.1.2

0.18

c2 c1 0.50

4.1.1

c3

0.23

c 3.2

0.17

0.35

c0

c 3.2.1

0.32

08

4

0.

0.50 0.18

0.22 3

4

0.08

4 4.1

4

0.

0.16

08

0.11

0.25

2 2.1

3 1.2

9 0.0

2

0.1

5

2 2.2

3 1.2.1

0.28

3 1.2.1.1

0.14

3 13 1.2.1

0.28

ab0 0.07

ab1

Fig. 9.90 Updated derivative tree of uAs (those incorporated in segment 5 are highlighted) 58

56

57

5 59

61

60

57

58

62

Fig. 9.91 Brahms’s Intermezzo Op. 118/2 (mm. 57–64)—derivative segment 6 (first approach)

transformational network of Fig. 9.92, depicting a derivative scheme similar to that of segment 5, i.e., it can be basically described as formed by two streams, elaborative and imitative. However, unlike what occurs in segment 5 in which there is a considerable development at the level of variables (especially in pitch domain), the construction of the both streams in segment 6 proceeds almost entirely according to two processes, namely, replication and metric displacement. Such fact leads me to conclude that holistic analysis may be the most adequate approach for the examination of the derivative relations in the segment.

252

9 Derivative Analysis

c

60

58

61

56

62

57

59

pitches rre-spelled s enharrmonically a

Fig. 9.92 Time-oriented transformational network related to segment 6

ed. #2: reduce duration

ed. #3: delete

k = 3*0.07 = 0.21

Fig. 9.93 Holistic derivation of uA 56 from variant c0

9.8.1 Derivation of uA 56 As shown in Fig. 9.93, the derivation of uA 56 from c0 is quite straightforward. Actually, this can be seen as a perfect example of thematic transformation, as discussed in Chap. 7. The most salient aspect of this variation is due to the change of mode, a very common technique applied by classical and romantic composers in order to present themes in new, but very similar clothes.

9.8 Derivative Segment 6

253

58

k = 2*0.07 = 0.14 ed.#2: dislocate metrically (one beat)

ed.#1: double duration

56 ed.#2: dislocate metrically (two beats)

57

k = 2*0.07 = 0.14

ed.#1: dislocate metrically (one beat) ed.#2: transpose 1 semitone lower

k = 2*0.07 = 0.14

59 Fig. 9.94 Holistic derivation of uAs 57 (from uA 56), 58 and 59 (from 57)

Observe that the third edition applied to the parent (the deletion of the last event) causes in the child the suppression of the middle-level variable zx1.1 , the appoggiatura-like closing gesture. As it will be presented in subsequent analyzes, other units in the segment also omit this middle-level characteristic from their structures.

9.8.2 Derivation of uAs 57, 58, and 59 As above commented, the two-stream (imitation/elaboration) structure of segment 5 is replicated (in a simpler version) in segment 6. This becomes clear in Fig. 9.94, which depicts a kind of complex of holistic derivations branching from the antecedent’s head, uA 56. As proposed in the analysis, the same strategy is applied to both imitated (57) and “elaborated”48 (58) units: the doubled duration of the second event (F.) and the metric displacement of the whole motive by, respectively, one and two beats. UA 59 branches from uA 57 (also by a metricly dislocated imitation), gaining a chromatic closure in order to fit harmonically to the dominant of A. minor.

48 The quotes denote the fact that, actually, it is not a true elaboration (in the normal sense for the term).

254

9 Derivative Analysis

60

zx1.1 ed.#1: transpose 1 semitone lower

ed.#2: delete

k = 2*0.07 = 0.14 Fig. 9.95 Holistic derivation of uA 60 from variant c3.2

9.8.3 Derivation of uA 60 Acting as an isolated unit in segment 6, uA 86 can be associated with variant c3.2 . Actually, this unit is inserted in the context as a short melodic bridge connected to the half cadence that closes the antecedent of the period. Observe that, contrary to what occurred in the relation c0.→56, in this case the deletion of the last element of the parent causes the emergence of the middle-level variant zx1.1 at the closure of the variant (Fig. 9.95).

9.8.4 Derivation of uAs 61 and 62 As occurred in segment 5, the head of the consequent (uA 61) derives from the head of the antecedent (56). The transformation is, in fact, almost imperceptible: a simple contraction of the first event, which preserves the original metric configuration (Fig. 9.96). As suggested in the network of Fig. 9.92, it is possible to consider a dual-holistic derivation for uA 62. Figure 9.97 analyzes both alternatives.

9.8 Derivative Segment 6

255

61 k = 1*0.07 = 0.07

ed.#1: halve duration

56 Fig. 9.96 Holistic derivation of uA 61 from uA 56 ed.#1: duplicate durations

61 k = 2*0.07 = 0.14 ed.#2: insert a closing note

62 k = 2*0.07 = 0.14

zx1.1

ed.#1: fill the gap

59 ed.#2: replace by the diatonic version

Fig. 9.97 Holistic derivation of uA 62 from uAs 59 or 61

9.8.5 Overview of Segment 6 Not surprisingly, the updated derivative network of segment 6 (Fig. 9.98) is considerably simple if compared to the previous ones and—as will be seen—to the next, segment 7. If considered the development of the musical ideas throughout with the piece as a derivative trajectory, segment 6 could be seen as a sort of “rest stop”, in which previously developed material is assimilated, instead of “pushed” forward, in a developmental sense. Such a one-step-back strategy is well articulated with the more accentuated changes of mode and texture, as a kind of compensation, revealing another facet of the extraordinary sense of form of Brahms.

256

9 Derivative Analysis

c4.1.1

c4.1

c3.2.2

c4.3.1

c4.2

c4.3

60

Derivative segment 6

Fig. 9.98 Updating of the uA network of segment 6 (based on Fig. 9.92)

9.9 Derivative Segment 7 As proposed in the formal analysis of the op. 118/2 (Chap. 8), the passage of mm. 65–73 (corresponding to the boundaries of segment 7) can be considered as an elaborated recapitulation of the first subsection of B. Figure 9.99 identifies thirteen units of analysis. Disregarding the repetitions of uAs 65, 66, and 70, the total drops to nine distinct units. Actually, the time-oriented network of Fig. 9.100 evidences two additional exclusions (uAs 66 and 70), since they are resumptions (by the principle of equivalence) of two variants of the pool, respectively, c2.1 and c3.2 . The segment presents roughly the internal architecture of a period, shared also by the other two subsections of B (corresponding to segments 5 and 6). In the present case, however, the consequent seems to address more a (very brief) elaboration than properly a reaffirmation of the basic idea (as expected in conventional periods).

9.9.1 Derivation of uA 63 The head of segment 7, uA 63, is an almost exact octave-equivalent copy of c0 , the “patriarch” variant of section B, from which is distinguishable only by a simple

9.9 Derivative Segment 7

64

65

i

iv

68

66

66

63

f

257

65

i6

69

67

ii 6

V7

i

70

70

71

66

VI

iv

i 6/4

V7

i

Fig. 9.99 Brahms’s Intermezzo Op. 118/2 (mm. 65–73)—derivative segment 7 (first approach)

2.1

0

3.2

Fig. 9.100 Time-oriented transformational network related to segment 7

contraction of the initial duration. Given this, its derivation is better described as holistic, as shown in Fig. 9.101.49

9.9.2 Derivation of uA 64 Holistic transformation is also applied to uA 63 to produce uA 64, a nearly-similar reduced version of it, The two units dialogue in the imitative stream, replicating the constructive procedure also applied in segments 5 and 6 (Fig. 9.102).

49

Unlike the previous heads (c0 and c4 ), uA 63 is presented in the middle piano register, which seems associated with an expressive intention of producing a fast traveling of the wide melodic arc of two octaves between C.4 and C.6 (m. 69), the climax of the whole intermezzo, as previously mentioned.

258

9 Derivative Analysis

63 ed.#1: reduce duration in 1/3

0 k = 1*0.07 = 0.07 Fig. 9.101 Holistic derivation of uA 63 from variant c0

64 ed.#1: duplicate duration

63 ed.#2: metric displacement (1.5 beat)

ed.#3: delete

k = 3*0.07 = 0.21 Fig. 9.102 Holistic derivation of uA 64 from uA 63

9.9.3 Derivation of uA 65 Contrary to what occurs in the previous units, the derivation of uA 65 is considerably more complex. In fact, we can see it as a kind of key-unit in the segment, which triggers a series of transformations that lead to the climax of m. 69. As suggested in Fig. 9.100, uA 65 is assigned to a dual origin: variant c2.1 (an associative relation) and uA 63 (through elaboration). Unlike normal cases of crossover and dual derivation, however, the two putative parents are not responsible for donating, each one in isolation, pitch and rhythmic

9.9 Derivative Segment 7

259

structure of the child. Actually, the two units contribute so distinctly for the formation of uA 65 that I prefer to examine both relations separately. Consider firstly relation 63.→65. As it is clearly depicted in Fig. 9.99, both units overlap, sharing, as we will see, an especially meaningful melodic fragment, formed by the pitches C.4 and B4 . Actually, this appoggiatura-like gesture intersects the two units, functioning both as a closure for uA 63 (therefore, associated with middlelevel variant zx1.1 ) and the beginning of uA 65 (which depicts the basic variable zx). On the other hand, 63.→65 can be seen as a perfect example of transformation through linkage, one of the most characteristic of Brahms’s developing-variation techniques, according to Walter Frisch, as already commented in Sect. 7.8. Recall that linkage occurs when the end of a given musical idea is used for initiating the development of another idea, providing fluid and gradual derivation. In the present case, the ideological importance of the connective element gives still more relevance for this special moment. In addition, it is worth to note that the three initial pitches of uA 65 replicates the pitch structure of the Grundgestalt (or unit a0 ), a strong associative relation, even disguised by the new rhythmic-metric context.50 Figure 9.103 summarizes the elements involved in the derivative relation 63.→65. Let us now consider variant c2.1 as an alternative referential source for uA 65. Because they are cardinality-compatible, their similarity degree is algorithmically calculated and expressed by the general penalty .k = 0.40. The specific pitch and temporal transformations involved in this derivation are depicted in Fig. 9.104. Fig. 9.103 Derivation of uA 65 from uA 63

50 Probably,

this association is not intentional.

260

9 Derivative Analysis

(p1, 2:5)

5

1 1.2.1.2.2

d-p3

2.1.2.1.2 1

(p1, 4:5) 1 1.2.1.2.2.1

d-p3

2.1.2.1.2.1 1

t2

ROT T4 (t1)

t2

2.1.1

"strong-(medium-weak)-(weak-weaker)"

65 1 1.2.1.2.2.1 2.1.2.1.2.1 .1.2.1.2.1 1 1

2.1.1

Fig. 9.104 Low-level derivation of uA 65 from variant c2.1 , considering variables involved (a); “Genetic” representation of uA 65 (b)

9.9.4 Derivation of uA 67 Instead of proceeding as in the previous B-segments (5 and 6), the so-defined “imitative” stream in segment 7 presents, actually, an progressive elaboration of the related ideas. This can be observed in the derivation of uA 67, depicted in the TD analysis of Fig. 9.105. While rhythmic derivation is very simple (by splitting the first quarter into two eighth notes), the more complex pitch transformation is explained by the use of four recursive operations. Three of these affect the intervallic structure (variable Z), as shown in low-level analysis (Fig. 9.106). As the two units have distinct cardinalities, the measurement of their dissimilarity corresponds to the sum of kq (0.05, Fig. 9.105) and the penalty referred to the relation 67.→65/67 (0.15, calculated by the comparison algorithms): .k = 0.15 + 0.05 0.20.

9.9 Derivative Segment 7

261

p2

q=6

RPL (

, 4)

d-p3

65/67

p1

(p1, 3:4)

(p1, 12(534)) p1

Fig. 9.105 TD analysis of the derivation of uA 67 from uA 65, considering the intermediary hypothetical stage 65/67

9.9.5 Derivation of uA 68 Elaboration goes further with the next unit (68), which clearly derives from uA 67, despite its dual origin, as preliminarily suggested in the network of Fig. 9.100. The hypothesized connection 63.→69 was then proposed, according to the antecedentconsequent model established in segments 5 and 6 (which supposedly segment 7 should replicate). However, this is not what occurs in the present passage: unlike the two previous cases, the head figure does not properly return (albeit transformed) at the halfway, but it is carried forward, gradually becoming more distant from the origin. As a matter of fact, according to the present interpretation, uA 68, which contains the melodic apex of the Op. 118/2, represents also the furthest latitude in derivative terms, considering not only segment 7, but also the entire piece. Once again, a TD analysis is required for explaining the relation, due to the difference of cardinality between parent and child (Fig. 9.107). The divergence

262

9 Derivative Analysis

RPL1(

(p1, (1)(2)(534)) 1 1.2.1.2.2.1

1 1.2.1.2.2.1

d-p3

d-p3

, 4) 4 1 1.2.1.2.2.1

d-p3

2.1.2.1.2.1.1 1

2.1.2.1.2.1 1

t2

t2

67 1.2.1.2.2.14 1 2.1.2.1.2.1.1 1.2.1.2.1.1 1

2 2.1.1

Fig. 9.106 Low-level derivation of uA 67 from uA 65, considering variables involved (a); “Genetic” representation of uA 67 (b)

between the two units, .k = 0.20 (= 0.05 + 0.15), repeats coincidentally the values obtained in relation .65 → 67. Once again, the two units in question have almost identical rhythms, while are considerably diverse in the intervallic configuration. The sequence of pitch transformations that links them expands considerably the pool of forms related to variable Z, as it can be observed in the low-level analysis of Fig. 9.108. Concerning metric, although in a strict sense one could consider that the original pattern was modified with the addition of a strong-beat quarter note at the end of the variant. In a practical sense, this would be seen as negligible, a transitory step in direction of a more meaningful state to be established with the next unit (69). For this reason, supported by the principles of simplicity and concision, I prefer to consider that variant X2.1.1 is maintained in the “genetic” representation of uA 68.

9.9.6 Derivation of uA 69 After reaching the climax, the derivative momentum suffers a natural slowdown. This is evidenced by uA 69, which can be defined as a canonic-like restatement of uA 68, as shown in the holistic analysis of Fig. 9.109. This brings back the imitative stream, as a signal of the returning of the formal “normality”. Despite its simplicity, the three-edition transformational process (deletion + displacement + duplication)

9.9 Derivative Segment 7

263

68

8

p1

t1

q=7 (p1)*INT1 (p1, 3:4) p1

EXT1 (t1)

67/68

7

q=6 (p1, 5:6) p1

6

RHA (sub) (p1, 2:5) p1

5

ROT1(p1) 4

p1

t1

Fig. 9.107 TD analysis of the derivation of uA 68 from uA 67, considering the intermediary hypothetical stage 67/68

5 1 1.2.1.2.2.1

d-p3

6 1 1.2.1.2.2.1

d-p3

INT1(p1, 3:4) 7 1 1.2.1.2.2.1

d-p3

8 1 1.2.1.2.2.1

d-p3

1 2.1.2.1.2.1 1

2.1.2.1.2.1.1 1

t2

t 1 (p1, 5:6)

t 2 (p1, 2:5)

ROT1(p1) 4 1 1.2.1.2.2.1

d-p3

EXT1 (t1)

t2

68 8 1 1.2.1.2.2.1

2.1.2.1.2.1 1

2 2.1.1

Fig. 9.108 Low-level derivation of uA 68 from uA 67, considering variables involved (a); “Genetic” representation of uA 68 (b)

264

9 Derivative Analysis

zx1.1

69

k = 3*0.07 = 0.21 ed.#3: duplicate durations

68 ed.#2: delete ed.#1: metrical displacement (one beat)

Fig. 9.109 Holistic derivation of uA 69 from uA 68

brings up the middle-level variant zx1.1 , personified by the appoggiatura G.-F., that will be retained in the next units as a kind of common denominator.

9.9.7 Derivation of uA 71 Segment 7 closes with uA 71, a direct variant from uA 69. Despite the fact that the two units present a relative high similarity,51 their pitch configuration differs considerably, which requires not less than four intermediary stages of transformation, as depicted in the low-level analysis of Fig. 9.110.52 Observe also that, because the parent is a holistic unit, the low-level derivation of uA 71 is based on “ancient” versions of the variables Z and Y: .Z13 .2.16 .2.2.18 (from uA 68) and .Y2.13 .2.1.2 (from c3.2 ). For the same reason, X2.1 is adopted as a metric reference in the “genetic” representation of the unit.

51 Penalty k = 0.22 (algorithmic calculation) which is especially due to the maintenance of rhythmic configuration. 52 This tortuous derivation is especially intensified by two applications of the operator PER, which normally implies radical divergences between parent and child, since the original order of intervals is in someway scrambled. At this point, it is worthy to recall that the derivative analysis (in any level) in no way intends to reconstruct the composer’s mental processes. The idea behind the method is rather to propose a possible (among many others) explanation—that ultimately reflects the personal, idiosyncratic interpretation of the analyst –, in this case considering a systematic transformation of the pitch material. Evidently, it would be nonsense to argue that Brahms could be planned to apply five successive modifications to the motive here identified as uA 69 in order to shape it into its definitive format, uA 71. This book does not propose such a thing at all, it is always important to emphasize. I will return to this paramount question in the conclusive chapter.

8 1 1.2.1.2.2.1

3

2 2.1

PER (p1, (12)(3)(4)(5)(6)) 10 1 1.2.1.2.2.1

d-p3

t 2 (p1, 2) 11 1 1.2.1.2.2.1

d-p3

t 3 (p1, 4) 12 1 1.2.1.2.2.1

d-p3

PER (p1, (1324)(5)(6))

Fig. 9.110 Low-level derivation of uA 71 from uA 69, considering variables involved (a); “Genetic” representation of uA 71 (b)

2.1 1. 2.1.2.2

13 1 1.2.1.2.2.1

71

t2

t2

3

9 1 1.2.1.2.2.1

d-p3

2.1. 2.1.2.2

RPL1 (t1, 5)

DEL (p1, 7)

2.1. 2.1.2

3

d-p3

13 1 1.2.1.2.2.1

d-p3

9.9 Derivative Segment 7 265

266

9 Derivative Analysis

9.9.8 Overview of Segment 7 Figure 9.111 updates the temporary network of Fig. 9.100, proposing genealogical labels for the seven units of segment 7. An interesting, additional aspect revealed by the analysis concerns the presence of lines of transformations that seem to be planned in order to provide a gradual ascent to the apex. Albeit quite brief (from m. 65 to m. 69), these lines can be associated with the notion of teleological derivation, that, as discussed in Sect. 7.10, is a process through which a given goal is achieved by progressive variations. In the present case, we can observe the mutual cooperation of two distinct, though complementary processes of intensification (involving pitch and temporal domains) whose goal is clearly the arrival of the climax of m. 69 (Fig. 9.112). While the melodic leaps present in the series of motives become gradually wider

4

3.2.3.1

3

3.2.3.1

3.2.3.1.1

"zx1.1relation"

5.1

5

Derivative segment 7

Derivative segment 6

2.1

Fig. 9.111 Updating of the uA network of segment 7 (based on Fig. 9.100)

3.2.3.1

3.2.3

9.10 Segment 8 5

267

3.2.3

7

5

3.2.3.1.1

3.2.3.1

12

10

telos 5

1

3.2.3

4

3.2.3.1

6

3.2.3.1.1

6

Fig. 9.112 Updating of the uA network of segment 7 (based on Fig. 9.100)

(5.→7.→10.→12 semitones), rhythmic intensification is achieved by a progressive increase of the number of eighth notes (from one to seven) at each station.53

9.10 Segment 8 Segment 8 corresponds formally to the short re-transition (mm. 74–76) that prepares the recapitulation of the main section [A.a]. Although very simple, there are some interesting aspects that deserve to be commented. As shown in Fig. 9.113, the essential material of the passage refers to a unique source, namely the concluding fragment of segment 7’s last unit (genealogically notated as .c3.2.3.14 ). Despite this apparent simplicity, the sequential depiction of this single element conveys meaningful derivative/associative implications. The network of Fig. 9.114 provides a clear view of the relations involved. The arrows at the top of the figure describe the descending scalar trajectory of the fragment.54 Observe that this interpretation requires that we consider the elision of a supposed intermediary stage on F.. The very goal of the sequence is reached at m. 76, but the resolution of this last appoggiatura C.-B is shortly postponed by a dramatic fermata. The immediate repetition of the fragment (depicted by the dashed-line arrow) almost magically teleport the listener from the end of a typical liquidation (in Schoenbergian terms)

53 Although the maximum of six is reached in the penultimate variation, we can consider that the last one is a bit more intense rhythmically, due to the additional quarter note which closes the idea. 54 For convenience, registral differences were normalized in order to turn this point clearer and more direct.

268

9 Derivative Analysis

[A.a]

[segment 8] (closure of c

)

71

f

i

A:

vi

73

72

73

^ pedal on 1

^ pedal on 5

IV V6

a0

V7/V

I 6/4

V7

IV V6/4

Fig. 9.113 Brahms’s Intermezzo Op. 118/2 (mm. 73–76)—derivative segment 8 (first approach)

2 2 zx1.1

2 "elided"

2 repeat

.

c 2.1 zx2

zx0

Fig. 9.114 Brahms’s Intermezzo Op. 118/2 (mm. 73–76)—derivative segment 8 (first approach)

to the resumption of the Grundgestalt of the piece, which is accomplished in a very elegant manner by Brahms.55 The arrows below the scores in Fig. 9.114 proposes another sort of relationship, considering low-level organization, that is, with a focus on meter: the sequence is thus initiated with the middle-level variant zx1.1 (that could be informally described as “something preceding the appoggiatura-like gesture”) which is transformed into variant zx2 (the isolated appoggiatura), shared by the sequenced fragments, reaching finally the primordial element zx0 , with the recapitulation.

55 In a sense, it is also possible to see this short passage with teleological lens, even more explicitly than in the previous case.

9.11 Discussion

269

9.11 Discussion The above discussion about of the events that form segment 8 closes properly the derivative analysis of the intermezzo Op. 118/2.56 The main purpose of this last section of Part III is to summarize and discuss the most relevant findings and results of the analysis, as well as to present a detailed overview of the complete derivative scenario of the piece.

9.11.1 A Qualitative Perspective of the Variants Production Figure 9.115 provides a global view of the derivative territory covered by the analysis addressing the distribution of both high-level (uAs) and low-level (variables) variants throughout the segments. This scheme reinforces the idea that some 88 4

low level high level (uAs)

24

X Y Z scale 1:3

53

26 1 7 19 scale 1:5

18

Z

10 7

13

7

6

8 1 7

A.a

A

segment 2

segment 3

A.b

A.a'

7

7

13 9

3 0

segment 1

60

4

12 18

18 1

3

6

segment 4

A.cdt

0 segment 5

segment 6

B.a

B.b

0 0 segment 7

B.a'

segment 8

Op.118/2

RT

B

Fig. 9.115 Distribution of high- and low-level variants considering the eight derivative segments and the whole piece

56 Strictly

speaking, there are in the recapitulated main melody of subsection [A.a] a couple of slight modifications in relation to their original format. Since these are only simple, superficial (and holistic) metamorphoses, with no true derivative/associative implications in the context of the thematic ideas or in the variable level, I consider them, in practice, as negligible. This implies that the entire recapitulation that takes place after segment 8 will be considered as equivalent to the exposition and, therefore, not suitable for the derivative analysis.

270

9 Derivative Analysis

variants (low + high)

30

B 20

A 10

0 1

2

3

4

5

6

7 segments

Fig. 9.116 Derivative curves corresponding to the production of variants in sections A and B

segments are openly elaborative (especially, segments 1, 2, 5, and 7) in functional terms, while others, with lower production of variants (like 3, 4, 6, and 8), present a clear associative profile. Another interesting piece of information conveyed by the figure concerns how variation evolves in function of the form. In this aspect, the piece presents two basic models of a curve of derivation (see Fig. 9.116). After the initial “explosion” of variants of segment 1 (a somewhat expected phenomenon in an organicallyconstructed piece), the rate of variation decreases slowly until the codetta of section A (segment 4). The beginning of the contrasting section B is marked by the emergence of a new derivative peak, which is followed by a sudden drop of variants in segment 6. Instead of continuing to descend, however, the curve changes the direction, returning to the previous level. These two profiles (let call them “heavy tail” and “V”) seem to be associated with distinct derivative-compositional strategies, a potentially promising topic to be explored in future analyzes.

9.11 Discussion

271 c 3.2.3.1

4

0.22

c 3.2.3.1

3

0.21

c 3.2.3.1.1 0.20

c 3.2.3.1 c 3.2.2

1.2.1.1

c 1.1.1

c 1.1 4.1

4 0.1

0.1 6

0.14

0.07

0.21

c2 0.23

0.07

c1

4.1.1

0.50

0.32

c0

c4

08 0.

0.08

4

0.14

c 4.1.1

c 4.2

0.07

c5

0.22 3

c 3.2.1 0.14

c 4.1

0.14 0.21 0.07

0.50 0.18

0.17

0.35

c3

4.1.2

0.40

c 3.2

0.18

4

4.1

c 3.2.3

c 3.1

0.28

4 0.1

0.20

c 2.1

c 4.3

0.14

c 4.3.1

0.21

4

0.

0.16

08

c 5.1

0.11 2.1

3 1.2

9

2

0.0

0.1 2.2

5

0.28

0.25

ab0 3 1.2.1

0.07

0.28

ab1 3 1.2.1.1

0.14 3 3 1.2.1

Fig. 9.117 Updated network of high-level variants

9.11.2 High-Level Variants The updated network of high-level variants is shown in Fig. 9.117. The relations of similarities between any contiguous units are informed next to the respective arrows. Figure 9.118 rewrites the network with the respective segmentation. This view suggests an interesting “geographic/migratory” perspective to our analysis, involving the distinct populations of high-level variants. In this sense, a metaphoric description of the derivative process in the Op. 118/2 could be the following: after an initial westward impulse (in segment 2), departed from the main nucleus (segment 1), a movement in the opposed direction takes place (segment 3), colonizing a new territory, under the command of the b lineage. In section B (segments 5, 6, and 7) this eastward expansion is intensified, and a new, wide area is occupied by the c-descendants.57

57 It is amazing how this simple description reproduces on a very small scale a real biologicalevolutionary scenario. Basically, populations geographically isolated tend to form ecological niches. After very long spans of time, accumulated variation inside the niches leads to speciation, that is to the emergence of new species. In the present analyzes, the three related “species” a, b, and c “live” and develop themselves inside very precise formal boundaries, which suggests that derivation can be used as a powerful segmentation force, likewise texture, harmony, agogic, and rhythm.

272

9 Derivative Analysis

c 3.2.3.11

4

c 3.2.3.1

3

c 3.2.3.1.11

6 1.2.1.1

c 3.2.3.1 c 3.2.2

2

c 1.1.1

28 0.28

c 2.1 c 3.1

5 1

4

c 1.1

c 3.2

c 3.2.1

c2 c3

4.1.2

c 4.1

c 4.1.1

4.1

4

3

4.1.1

c1

c4

c0

3

6

c 4.2 c 4.3

c5 4 4.1

7

c 3.2.3

c 4.3.1

7

4

4

c 5.1 2 2.1

3

3 1.2

2 2.2

5

ab0

2

3 1.2.1

ab1 3 1.2.1.1

4 3 3 1.2.1

Fig. 9.118 High-level network considering the derivative segmentation

9.11.3 Developing Variation The complex network of high-level variants has potentially a very huge amount of possible developing-variation paths. I propose here to explore one of these—may be the longest—connecting the patriarch a0 to the last variant of segment 7, .c3.2.3.14 . Figure 9.119 depicts an initial representation of the relations involved in this DV path, considering absolute and relative types (see Sect. 7.2). While all contiguous, absolute similarities were already calculated along the analysis (and are available in Fig. 9.117), the determination of the absolute values demands new, specific derivative analyzes. Such a task is in itself not especially problematic, since the process can be automatized with the help of the algorithms for measurement of similarity. However, due to the fact that almost all comparisons involve different cardinalities, additional TD graphs would also be required, increasing considerably the complication of the process and the amount of work to do. For this reason, I propose here to introduce an alternative, simpler model for comparing musical units with distinct cardinalities. This model is based on the claim that most of the “personality” of a given musical configuration is established by its initial elements. Because of this, the similarity between two musical ideas A and B will depend mainly on how close or distant are their beginnings. The algorithm is properly accomplished as it follows: (1) prepare the matrices of attributes of A and B: (2) determine the difference of cardinalities q between A and B (consider for

9.11 Discussion

273

c 3.2.3.1

4

0.5

c 3.2.3.1

3

7

4 0.5

c 3.2.3.1.1

0.69

0.20

c 3.2.3.1 0.68

0.20

0.51

c 3.2.3 0.40

0.44

c 3.2 0.35

0.46 29

0.

c3 0.3

0.5

0.6

9

3

3

0.32 3

0.18

4

0.22

0.50

c0

Fig. 9.119 Basic network related to the DV path .a0 → c3.2.3.14

example the cardinalities of A and B as respectively 3 and 6, thus .q = 6−3 = 3); (3) cut off from the larger matrix the columns whose numbers are greater than q (in our example, columns 4, 5, and 6 of B’s matrix will be disregarded); (4) the two matrices are now compatible in cardinality and, therefore, comparable algorithmically; (5) to compensate for the loss of cardinality, add 0.07 for each column deleted from the largest matrix to the global penalty for dissimilarity calculated by the use of the algorithms (in the exemplified case, the additional value corresponds to .0.21 = 3 × 0.07).

274

9 Derivative Analysis

Applying this strategy to all pairs in question we obtain the relative values shown in Fig. 9.119 (for the sake of concision and simplicity the calculations were omitted from the text). With these data we can plot a relative/absolute graph related to the DV path (Fig. 9.120). It is particularly interesting to observe how relative divergence goes down after climbing a primary peak of 0.63 (with .b14 ), reaching further the highest point (0.69) with c3.2.3.1.1 , precisely the variant which contains the melodic climax of the piece (m. 69).

9.11.4 Low-Level Variants Almost 70% of all low-level derivation in the piece (58 variants out 88) refers to variable Z, something that reflects how flexible in derivative terms are the pitch relations in the piece.58 In comparison, rhythm (represented by variable Y) is considerably more modest (27 variants), while the meter (variable X) is almost negligible. Figure 9.121 makes clear the impressive diversity of Z’s pool, representing it in a network format. The same information is presented in Fig. 9.122 as a genealogical tree, depicting the intervallic variants in musical notation (recall that the variants were normalized such that all initiate with pitch C5 ). Notice that branches/lineages differ quite a lot from each other with respect to size (i.e., number of descendants), a phenomenon that intensifies in the latter sections of the piece. Likewise, rhythmic low-level derivation (involving Y and X) is addressed in Figs. 9.123 and 9.124.

9.11.5 The Derivative Role of the Kopfnote In this last, albeit very important topic I proposes a global overview of the piece considering the structural role played by its initial melodic event, the pitch class C.. The particular relevance of this examination concerns an associativity quite uncommon in organically-built works. In the present case, as it will be demonstrated, this individual element governs the derivative organization of the intermezzo in a very special manner: C. is simply the motivic head of all main ideas that permeate the piece and the point from which the respective elaborations emerge (Fig. 9.125).

58 As previously discussed, we can consider pitch and, especially, intervallic configurations as the most privileged variation vehicles used by composers, does not mattering style, epoch, or nationality.

9.11 Discussion

0.54

0.57

0.69

0.51

0.68

0.44

0.46

0.53

0.18

0.5

0.63

1 0.39 0.29 0.28 0.28

Fig. 9.120 Graphical representation of the relative/absolute DV path .a0 → c3.2.3.14

275

0

0.18

0.28

0.12

0.07

0.18

0.22

0.50

0.32

0.35

0.40

0.20

0.20

0.21

0.22

a

b4

b b b

b

3

c0

c3

c 3.2 c 3.2.3

c 3.2.3.1.1 c3.2.3.1

4

c 3.2.3.1 c 3.2.3.1

3

276

9 Derivative Analysis

1.2.17

6 1.2.1.2.1 6 1.2.1.2

1.2.16

6 1.2.1.2.2

6 1.2.1.2.2.1

6 1.2.1.2.2.1.1

6 1.2.1.2.2.1

4 6 1.2.1.2.2.1

5 6 1.2.1.2.2.1

1.2.15 6 6 1.2.1.2.2.1

4 1.2.1.2

1.2.14

4 1.2.1.2.1 7 6 1.2.1.2.2.1

1.2.1 8 6 1.2.1.2.2.1

1.2.1.1 9 6 1.2.1.2.2.1

1.2.1 10 6 1.2.1.2.2.1

1.2 11 6 1.2.1.2.2.1

12 6 1.2.1.2.2.1

1.4 1.3

1.1.4

13 6 1.2.1.2.2.1

1.3.1

2.1 1.3.1.1

1.1.4.1 8 7

1.1.4.1.1

3

1.1.4.1

6 6.1

6.1.1

3.1.1

3

6.1

3

3

Fig. 9.121 Low-level network considering the pool of variable Z

A secondary, also unusual association to be examined in this subsection concerns the presence of literal allusions to the Grundgestalt’s pitch events (i.e., involving the exact sequence of pitch-classes C.-B-D) throughout the piece.59 The better method to address properly both aspects is certainly Schenkerian analysis. For reasons of concision and objectivity, and directed by the specific goals of the current approach, the following graphs propose only initial middleground analyzes of each derivative segment in question, avoiding deeper reductions. These graphs will provide the basis for the final network (Fig. 9.132), which summarizes

59 Evidently,

I mean here those mentions that are not expected thematic-formal recapitulations.

6

d-p3

6.1

d-p3

3.1

d-p3

2.1

3.1.1

1.4.1

d-p3

p3

6.1.1

d-p3

Fig. 9.122 Genealogical tree of variable Z

d-p3

8

d-p3

7

d-p3

1.4

d-p3

3

3

d-p3

1.4.1.1

d-p3

1.3

p3

6.1

3

d-p3

3

3

1.2

d-p3

d-p3

1.4.1

3

d-p3

1.3.1

3

3

1.2.1

d-p3

d-p3

3

1.3.1.1

3

1.2.1.1

d-p3

3

3

1.2.1

4

6

6

d-p3

1.2.1.2.2.16

3

d-p3

1.2.1.2.2.15

3

d-p3

3

1.2.1

d-p3

5

3

4

3

6

6

6

13

1.2.1.2.2.1

6

d-p3

3

d-p3

1.2.1.2.2.112

3

d-p3

1.2.1.2.2.17

3

d-p3

1.2.1.2.2.14

d-p3

1.2.1.2

d-p3

3

1.2.1

6

3

4

3

6

3

6

6

d-p3

3

1.2.1.2.2.111

d-p3

1.2.1.2.2.18

3

d-p3

1.2.1.2.2.1

d-p3

1.2.1.2.1

d-p3

3

1.2.1

3

7

1.2.1

6

3

6

6

6

d-p3

3

1.2.1.2.2.110

d-p3

3

1.2.1.2.2.19

d-p3

1.2.1.2.2.1.1

d-p3

3

1.2.1.2

d-p3

3

6

1.2.1.2.1

3

6

3

6

d-p3

1.2.1.2.2.1

d-p3

1.2.1.2.2

d-p3

9.11 Discussion 277

2

2.1

2.1.1

t2

t2

t2

5

4.1

4

t2

t2

t2

Fig. 9.123 Low-level network referred to variables Y (a) and X (b)

(b)

(a)

2.2.1

t2

3

2.2

t2

1.1

t2

3

2.1

t2

t2

t2

2.1. 2.1

3

3

2.1. 2

2.1

4

t2

t2

2.1. 2.1.2

3

3

2.1. 2.1.1

3

3

t2

2.1. 2.1.2.2

t2

3

2.1. 2.1.2.1

t2

3

2.1. 2.1.2.1.1

t2

3

2.1. 2.1.2.1

278 9 Derivative Analysis

9.11 Discussion

279

(a)

3

4

4.1

2.2

2.2.1

5 2.13

2.1.1

2.14

3 2.1.2

3 2.1.2.2

3 2.1.2.1

3 2.1.2.1.2

3 2.1.2.1.1

3 2.1.2.1.2.2

3 2.1.2.1.2.1

3 2.1.2.1.2.1.1

3

3 2.1.2.1.2.1

(b)

2.1

2.1.1

Fig. 9.124 Genealogical tree of variables Y (a) and X (b) Grundgestalt

A.a

"expanded" Grundgestalt

(segment 1) 3^

A:

I

m.5

IV4

6 3

V2

ii

V

v IV I

V65/V

vii 7

vi 6

V7/V V

Fig. 9.125 Schenkerian reduction of segment 1

the most structural aspects of the isolated reductions, depicting a global view of the pitch-derivative architecture.

280

A.b

9 Derivative Analysis

(segment 2) m. 21

N

e:

m. 24

N

(

)

A: VI

V/iv

a:

Np6 VI

6

Ger.6 V 5 /V

V

Fig. 9.126 Schenkerian reduction of segment 2

The first graph, corresponding to the subsection [A.a] (and segment 1),60 is shown in Fig. 9.122, which introduces two new visual conventions: (a) a triangle, intended to register the presence of the ordered pitch-class triple C.-B-D that forms the Grundgestalt (needless to say that eventual octave displacements of the sequence will be treated as equivalent) wherever it may occur; (b) a larger rectangle (in which the triangle is always inserted), used for representing what we can call the “extended” Grundgestalt, which encompasses also the next three events, closing with the high A. In Schenkerian terms, the initial note of the intermezzo is also the first Kopfnote, that is, the primary or head tone which initiates the primordial melodic line (Urline), being properly identified by the label .3ˆ (the half-note attached to this pitch denotes its hierarchical importance among the subsequent events). As suggested in the analysis, both antecedent and consequent that form the main theme can be described as structural descending lines from C. to G., whose ends are supported by half cadences (with the second being the “heaviest”, due to a V/V preceding the cadential closure on V). Segment 2 (Fig. 9.126) initiates with E, unfolded from the latter G.. The neighbor-motive that characterizes the b-lineage (indicated by the horizontal bracket) is transposed to A (m. 21), bridging the return of G. at m. 24, and signaling the beginning of the retransition, which coincides with the incisive resolution of the metric dissonance (see Chap. 8). Then, a chromatic ascent leads to segment 3 and to the resumption of the Kopfnote, with the varied re-exposition [A.a’] and codetta, territory covered by segment 3 (Fig. 9.127).

60 As done in the derivative analysis, the repetition of the period (mm.9–16) is not taken into account.

9.11 Discussion

281 (segment 4)

3) A.a' (segment 3^

3^

m. 43

m. 39

m. 35

2^ 1^

( )

A:

4

V3//IV IV6

6 3

6

6

ii 5

a:

I IV V 5/V V IV

iv

V

I

IV6

ii

V

8

7

4

3

I

ii 65

Fig. 9.127 Schenkerian reduction of segments 3 and 4

Four associations with the Grundgestalt occurs in this passage: a literal recapitulation, two mentions in the bass (the second one in the minor mode), and another inside the “extended” Grundgestalt, closing the A section.61 As already commented in the harmonic analysis (Chap. 8), this point of closure is the first moment in the piece where a firm PAC is established. The Schenkerian conventions, associated with the data obtained in the derivative analysis, reinforce this view, forecasting the resolution of the Ursatz, to be definitely accomplished at the end of the piece (see Fig. 9.131). Pitch C. also launches the three segments of section B (now labeled as .5ˆ in Fsharp minor), heading the c-based lineage. Figure 9.128 presents the Schenkerian graph related to segment 5. Quite interestingly, in spite of the evident differences in the musical surface, segment 5 shares the same structural backbone of segment 1, namely C.-A-G., closing once again with a HC. Segment 6, corresponding to the contrasting [B.b] section, also initiates with ˆ but in F-sharp major) and closes with a HC (Fig. 9.129). the Kopfnote (also as .5, However, unlike segment 5, the head C. is now prolonged along the passage. Observe how this central pitch class is reinforced by the successive (almost obsessive) canonic-like imitations (indicated by diagonal arrows). The analysis also highlights the ambiguity of the tonal context, arisen from the “dispute” between Fsharp major and A-sharp minor (interpreted as subordinated to the former) and from the enharmonic re-spelling of mm. 59–60. As expected, the secondary Urline of section B, initiated in segment 5, is resolved at the end of segment 7 (Fig. 9.130), forming the descent line .5ˆ − 4ˆ − 3ˆ − 2ˆ − 1ˆ in F-sharp minor. Considering the aspect of associativity, the pitch content of the Grundgestalt is subtly infiltrated in the c-territory, as a sort of glimpse of the near recapitulation, through the important variant c3.2.3 (highlighted by the gray ellipse).

61 Variant

a5 (m. 36), the inverted version of a0 , maintains with this an elaborative relation, rather than associative and, therefore, it is not here considered.

282

9 Derivative Analysis

B.a

(segment 5) 5^ N

f:

i

iv

ii V i

VI

v

V/V V

Fig. 9.128 Schenkerian reduction of segment 5

B.b

(segment 6) 5^

5^

m. 59

N

(a :

F:I

i

6

6

ii5 V i)

vi6

enharm.

V7 I

V 64

5 3

Fig. 9.129 Schenkerian reduction of segment 6

Finally, Fig. 9.131 depicts the retransition (segment 8). The graph reinforces what was discussed when this passage was analyzed, now under Schenkerian lens: the appoggiatura G.–F. that closes segment 7—the middle-level variant zx1.1 , in derivative terms—is used as a pivotal element for the descent line that aims at the primary Kopfnote C., launching the recapitulation. The graph also includes an interpretation for the conclusion of the intermezzo (mm. 114–116), in which the definitive Ursatz of the piece is accomplished. Figure 9.132 provides an overview of the pitch structure of the intermezzo (the recapitulation is omitted).

9.11 Discussion

283

B.a' (segment 7) 5^

10

4^ 3^

m. 69

2^ 1^

10 10

N

f:

i ii7 V7 i

v iv

i

5 3

iv V 64

VI

i

A:

vi

Fig. 9.130 Schenkerian reduction of segment 7 (segment 8) 1^ = 6^

R RT

3^

5^ 4^

A'.a

^^ 21

^ 3

()

(...) A:

V6 IV

vi

I4

V/V V

IV4

V

8

7

4

3

I

Fig. 9.131 Schenkerian reduction of segment 8

B

A a

b

a'

a

b

RT

a' 6^

3^ 2^

1^

^ = 5

4^

=

3^ 2^ ^1

2^

| Rec. 2^ ^1

3^

...

N

F /f :

A:

I

V

I V I

i

V

I

i

vi

Fig. 9.132 Schenkerian reduction encompassing the whole piece

iv V

i

V I

V I

284 Fig. 9.133 Basic pitch structure of the Op. 118/2 plotted as a network

9 Derivative Analysis

A

unison

^ 3

second

C A

third

(a) G

fourth

E

(b)

B G

B

A

B

C

(a')

B

A

B

(a)

^ 5

C G

A

C

(a') ^ 1

RT

^ 6

(b)

C

F E D

C ^ 3

As an alternative, a more schematic representation of the piece is plotted as a network in Fig. 9.133 in order to emphasize the central role played by pitch class C. in the derivative structure. The arrows in the graph differentiate the intervals connecting the pitch nodes, suggesting some interesting relations. Brahms ingeniously chose for the head of the three basic sources of material (here labeled as a-, b-, and c-lineages) the only common element of the tonic triads of the three main keys in his tonal plan.

References Hanninen, D. (2009). Species concepts in biology and perspectives on association in music analysis. Perspectives of New Music, 47(1), 5–68. Rings, S. (2011). Tonality and transformation. Oxford: Oxford University Press. Schoenberg, A. (1967). Fundamentals of musical composition. London: Faber and Faber. Swinkin, J. (2012). Variation as thematic actualisation: The case of Brahms’s op. 9. Music Analysis 31(1), 37–89.

Afterword

At the end of this long journey through the universe of musical variation, I would like to summarize the main aspects visited in the chapters and add some general comments, as well as propose possible developments and avenues to be explored in future works. As the primordial premise of this book, I suggested that variation can be seen as a sort of algebraic function capable of mapping some (relatively short and autonomous) musical object into another one, such that both can be mutually related by some degree of similarity. A theoretical-methodological apparatus, centered on a group of concepts and constructs (derivative space, domains, attributes, the matrix of attributes, distance algorithms, weights, etc.), was then elaborated in order to quantify these relations. A spatial representation of variation was also provided. Another central aspect of the proposal concerned the possibility of decomposing musical ideas into pitch, rhythmic, and harmonic components, which can eventually become referential structures, subject to variation. This led to the idea of transformational-derivative analysis, as a sequence of hypothetical, but plausible stages of gradual transformations of these components. On the other hand, “holistic” variation can explain, in analytical terms, situations where indivisible units are clearly remolded as whole objects, without the need to separate their individual components. The manners with which such decomposable variation can take place were discussed in Chap. 4, dedicated to the notion of transformational operations. Correlated concepts were added to the theory, among them the idea of mutational application and composition of operations. The second part of the book addressed variation on time, providing a necessary bridge between the previous abstract approach and the concrete field of analytical applications. This intermediary platform was especially grounded on Schoenbergian theory of Grundgestalt and developing variation, merged with original formulations, especially based on biological-evolutionary concepts (e.g., phylogeny, speciation, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3

285

286

Afterword

involution, teleology, and crossover). Such conceptual combination gave rise to a number of new constructs, like the idea of variables, high- and low-level derivation, genealogical notation, quotient, and absolute/relative developing variation. Such a theoretical framework supported the detailed derivative analysis of Brahms’s Intermezzo Op. 118/2 in Part III. The choice of this piece as a case study was due to some special conditions: its relatively short duration, the explicit organicity of its motivic-thematic structure, and, why not say, its extraordinary beauty, lyricism, and personality. A concise exam of form, harmony, and meter prepared the derivative analysis of the piece. After some initial considerations about the methodological process to be applied, the Grundgestalt and its basic components were properly identified, with the assignment of variables to the source attributes, the very agents of the derivative work to be mapped. Eight segments (broadly corresponding to formal subsections) were then demarcated. In each of these, relations (either associative or elaborative) involving their actors—units of analysis in the high level, variables in the low—were mapped, qualified, and quantified through the methods and algorithms previously introduced. Finally, phylogenetic-like graphs and other kinds of networks were elaborated intended to clarify long-range relations and to provide a global overview of the derivative processes. I am aware that the theory and analytical methodologies presented in this book are not exempt from questioning. There are probably a number of aspects that can be seen as problematic by someone, and others that deserve improvements or to be addressed in alternative manners. Anyway, I would like to anticipate some possible questions and answer them according to my points of view. “It is hard to believe that a composer may envisage sequences of variations as those presented in many of the derivative analyzes.” Yes, it would be at least reckless to suggest that the analyzes of this book are intended to somehow reconstruct the creative processes of a composer. However, this is by no means the case. As stated in Chap. 2, TD methodology and, especially, the concept of hypothetical intermediary stages are just resources that aim to contribute for enhancing the analytical systematization. In sum, they have to be seen as useful constructs for establishing plausible nexuses between related musical objects, and not faithful descriptors of the processes employed by composers to produce them. “The algorithms destined to the measurement of similarity seem to be grounded on arbitrary premises and choices.” Indeed, many theoretical and/or methodological decisions seem to be imposed throughout with the book: the choice of domains and attributes, the weights established for the pitch, temporal, and harmonic vectors, the structure of the formula for calculation of the global penalty, and so on. I agree that, taken together, these decisions are potential contributors for turning the theory more fragile. After all, different choices perhaps would produce different results, one could argue (and I agree, too). I would like, however, to add some counter-arguments: Firstly, the choices made are not arbitrary. All of them were based either on intuitive knowledge

Afterword

287

(many times supported by cognitive studies, as mentioned elsewhere) or on a number of empirical tests. Moreover, I see them as chosen paths in crossroads. In case of disagreement with the results obtained from some methodological procedure, analysts are stimulated to return and to try alternative possibilities. As I mention in some parts of the book, the theory it conveys is still on construction and the method of trial and error is the best way for consolidating it (particularly, I intend to pursue this way). I would add, as a third argument, that maybe the technology for the measurement of similarity could be seen in the context of MDA just as a complementary artifact. In this sense, it could be unplugged from the main analytical body without causing significant losses to the analysis. Actually, if one is interested mainly in mapping the derivative relations between variants, the distances between them (in the similarity/divergence real scale) matter very little or even nothing. So, we can maybe consider these two modes for applying MDA in analysis: a complete version (including similarity measurement) and, optionally, a much simpler (and less problematic) alternative, in which just relations are taken into account. “The method of derivative analysis is too much complicated, involving a large number of conventions and graphs.” It is true, but I would rather use the adjective “complex” instead of “complicated”, since all this analytical apparatus is intended to accomplish a task inherently very intricate, namely, a systematical mapping of derivative relations in an organiclike musical context. Due to the myriad of transformational paths and processes that can take place in a composition, including the possibility of “hidden” variation (i.e., affecting abstract musical structures, like ours attributes or segments of these), complexity is an unavoidable price to be paid if one desires to keep a rigorous control on the analytical processes. However, as I commented above, here it is also possible to consider simpler alternatives. For example, I wonder if the method used for addressing holistic variation (considering transformations as a sum of combined editions) could be extended to other cases or, in a more radical scenario, to replace all derivative-analysis procedures. Both alternatives can be tested in future analyzes. Another way for simplifying MDA was suggested at the conclusive section of Part III, with the determination of the developing-variation lineage of a0 . In this context, a new, simpler method for comparing units with non-compatible cardinalities was introduced as a shortcut alternative for the use of TD analysis, which demands complex graphs and conventions. Anyway, like the previous case, this provides the model with a sort of “two-way switch” to be turned on in each alternative depending on contextual conditions or even the analyst’s intentions. In this sense, I also recommend the reading of the Appendix B, where a considerably simpler and more concise version of MDA (labeled as MDA*) is introduced. Lastly, I would like to add some words about possible future work associated with this variation theory. Certainly, there are some avenues that seem for me open to being explored. One of these concerns the expansion of the model in order to include other musical domains and attributes. Dynamics, timbre, and texture, for example, are some obvious candidates.

288

Afterword

Another possibility is to expand MDA’s scope (originally centered on commonpractice period) to encompass other repertoires, like Post-Tonal pieces (as is briefly suggested in Appendix A), electroacoustic or popular music, for example. Besides analysis, composition is also a very promising source of applications for the model. As it was done in MDA’s previous versions, the theoretical framework can be used as support for the construction of the necessary apparatus (mainly computational), in a reverse-engineering strategy. As I said in the introductory section, since the book is intended essentially for musicians (more precisely theoreticians, composers, and students of composition), I tried to avoid, as far as possible, excessive mathematical work in the text. Despite this, I consider that the idea of transformation as a possible manner to treat musical variation—the very core of this study—can certainly be developed in the near future in more formal fields, as linear algebra and group theory. In this regard, a special and quite attractive possible way for further improvement could be the use of topology for providing the model with a more formal structure for measurement of similarity, becoming an (possibly better) alternative for the methods that are currently applied in MDA. On the other hand, I also think that a more close approximation with evolutionary biology can become a very rich territory to be expanded, especially considering those aspects which the similarities between musical variation and the development of organisms (in the several possible levels, as species, populations, individuals, genes) are more intense. In this sense, I foresee the a more deep and formal association between graph theory and phylogeny as a fruitful path.

Further Reading Arndt, M. (2013). Schoenberg – Schenker – Bach: A harmonic, contrapuntal, formal braid. Zeitschrift der Gesellschaft für Musiktheorie, 16(1), 67–97. Arndt, M. (2017). Toward a renovation of motivic analysis: Corrupt organicism in Berg’s piano sonata, op. 1. Theory and Practice, 42, 101–140. Arutyunova, V., & Averkinb, A. (2017). Genetic algorithms for music variation on Genom platform. In: Proceedings of the 9th International Conference on Theory and Application of Soft Computing, Computing with Words and Perception (pp. 317–324). Cahn, S. (1996). Variations in manifold time: Historical consciousness in the music and writings of Arnold Schoenberg. Dissertation, State University of New York. Chaitin, G. (2012). Proving Darwin: Making biology mathematical. New York: Pantheon Books. Chung, A. (2012). Lewinian transformations, transformations of transformations, musical hermeneutics. Dissertation, Wesleyan University. Dahlhaus, C. (1990). Schoenberg and the new music. Cambridge: Cambridge University Press. Deutsch, D., & Feroe, J. (1981). The Internal representation of pitch sequences in tonal music. Psychological Review, 88(6), 503–522. Embry, J. (2007). The role of organicism in the original and revised versions of Brahms’s piano trio in B major, op. 8, mvt. I: A comparison by means of Grundgestalt analysis. Dissertation, University of Massachusetts. Graham, P. (1983). A response to Schenker’s analysis of Chopin’s etude, opus 10, no. 12, using Schoenberg’s Grundgestalt concept. The Musical Quarterly, 69(4), 543–569.

Further Reading

289

Green, E. (2008). ‘It don’t mean a thing if it ain’t got that Grundgestalt!’— Ellington from a motivic perspective. Jazz Perspectives, 2(2), 215–249. Guerra, S. (2019). Toward a theory of structuring rhythm in improvisation in timeline-based musics. MusMat, 3(2), 44–60. Knoll, C. (2006). Prolongation, expanding variation, and pitch hierarchy: A study of Fred Lerdahl’s Waves and Coffin Hollow. Dissertation, Bowling Green State University. Lerdahl, F., & Jackendoff, R. (1983). A generative theory of tonal music. Cambridge: The MIT Press. McConnell, S. (2018). Developing variation and melodic contour analysis: A new look at the music of Max Reger. Dissertation, University of North Texas. Nakatami, M. (2017). The effect of the developing variation technique on Brahms’ early piano solo works in the form of theme and variations. Dissertation, University of Miami. Sampaio, M. (2012). A teoria de relações de contornos musicais: Inconsistências, soluções e ferramentas. Dissertation, Federal University of Bahia. Sloboda, J. (1985). The musical mind: the cognitive psychology of music. Oxford: Oxford University Press. Trudeau, R. (1993). Introduction to graph theory. New York: Dover. Tymoczko, D. (2011). A geometry of music: Harmony and counterpoint in the extended common practice. Oxford: Oxford University Press. Zovko, M. (2007). Twelve-tone technique and its forms: Variation techniques of Arnold Schoenberg’s “Variations for Orchestra op. 31”. International Review of the Aesthetics and Sociology of Music, 38(1), 39–53.

Appendix A

Variation in Non-tonal Contexts

The conceptual and methodological corpus necessary for the analysis of nontonal contexts in MDA requires a unique, and simple adjustment concerning the secondary domain of harmony. The five-entry vector used so far for describing harmonic information related to the events of a tonally-based musical idea will be here substituted by a new vector, formed by four elements, as shown in Fig. A.1. Like its tonal counterpart, the new vector intends to depict the specific harmonic context of each event. However, instead of expressing hierarchically-organized harmonic information (key, mode, chordal quality, root, and bass), the non-tonal vector refers uniquely to pitch-class sets. The information is encoded as follows: the first two entries (a–b) are related to the prime form of the set, adopting Allen Forte’s well-known notation.1 The third entry (c) is a pitch class associated with the specific set class in question (i.e., the set that occurs in the musical surface). The specific content of the set is finally determined through the combination of pitch class related to entry (c) and one of the two conventional operations of post-tonal theory, transposition (in case of .d = 0) or inversion (.d = 1). As a direct consequence of this new structure, non-tonal matrices of attributes have one row less than their tonal counterparts. Figure A.2 shows an example. After a simple harmonic reduction, one obtains two trichords, the first one encompasses events 1 to 4, the second events 5 to 7. The first trichord, {0, 6, 9}, corresponds exactly to the prime form of set 3–10/(069), therefore, both entries (c) and (d) are equal to 0, and the respective harmonic vector (labeled as M/H1-4) is notated as . The second trichord, {2, 6, 7}, is obtained, in turn, by a seven-semitone inversion/transposition of the prime form of set 3–4/(015). Consequently, entries c and correspond, respectively, to 7 and 1, and the second harmonic vector (M/H5-7) is notated as .

1 See

Forte (1973). This notation is formatted as two-digit code (e.g., 3–7, 5–8, etc.) in which the first element is associated with the cardinality of the set (“3” for trichords, “4” for tetrachords, and so on) and the second an order number, according to a list elaborated by Forte.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3

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292

A Variation in Non-tonal Contexts

Fig. A.1 Basic structure of the non-tonal harmonic vector (a) and two examples (b, c)

Fig. A.2 Basic structure of the non-tonal harmonic vector (a) and two examples (b, c)

Evidently, the previous transformational operations related to the harmonic domain are no longer applicable to non-tonal contexts. It is possible to propose new alternatives, both related to concrete, foreground affinities (based on the principle of invariance/variance between specific contents of sets), or more abstract relations

A

Variation in Non-tonal Contexts

293

(connecting prime forms of sets that are related by inclusion, for example). Although a very attractive perspective, I do not intend to pursue this issue here (since the book is primarily dedicated to the variation on tonal music), letting such an exploration into future studies.

Reference Forte, A. (1973). The structure of atonal music. Yale University Press, New Haven.

Appendix B

MDA*

This appendix introduces a recent elaboration in the scope of the research, called MDA*, a considerably simpler version of the model of derivative analysis. MDA* was designed for application in analytical situations in which the complete MDA apparatus might be considered, so to speak, excessive. Another possibility is the usage of the elements of MDA* as a complementary tool in normal derivative analysis, applying them in specific, local situations (as, for example, in isolated sections of a large piece). Basically, MDA* considers only the primary domains (pitch and time), which are abstracted (as in MDA) and then manipulated by a process called UDS filtering. The filtering of a UDS involves different strategies, depending on the domain in question. These are briefly described in the following sections.

Pitch Filtering The idea behind this process is to produce a new abstract attribute of a pitch structure, called sequence of contour gestures (or contour letters, c-letters, for short). A sequence of c-letters represents a generic descriptor of a melody that, in a gradation of abstraction, lays roughly in between the sequence of intervals (p3 or dp3) and the melodic contour. In this sense, a c-letter can be seen as an abstraction on an interval. Conversely, a melodic contour can be described as an abstracted version of a sequence of c-letters. The c-letters, like melodic intervals, encode binary relations between contiguous pitches. They are adapted from the four basic melodic categories proposed by W. Jay Dowling (1978) in a systematical study of folk melodies, namely: 1. unison (labeled as u)—for repetition of a pitch (or zero semitones); 2. step (t)—comprises minor and major seconds (1 or 2 semitones);

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3

295

296

B MDA*

p3

c-word p4

Fig. B.1 Pitch description of a UDS as the contour-word (descending leap, ascending skip, ascending step, descending step) in comparison with two other attributes (p3 and p4)

3. skip (s)—corresponds typically to movements inside a chord arppeggio (3 to 5 semitones); 4. leap (l)—all intervals greater than a perfect fourth. Instead of use Dowling’s signals “.+” and “.−” for denoting ascending and descending movements, I prefer write the c-letters in upper and lowercase, respectively. From this, a contour alphabet Ap can be defined as the set of the seven instances of c-letters: Ap = {u, T , t, S, s, L, l}

.

The pitch structure of a UDS in MDA* is described as a vector formed by the sequence its c-letters, which is called a contour-word (or c-word), as shown in Fig. B.1.

Rhythmic Filtering The rhythm of a UDS is filtered by the use of another strategy. It is based on two conditions: • The actual durations of a rhythmic configuration are not considered, but just the intervals between the onsets (that is, we are talking about IOIs). This creates a level of abstraction that makes a given rhythmic figure a member of a class of equivalences (Fig. B.2); • The onsets of the rhythms are framed in one-beat windows. In other terms, the filtering of a rhythm returns a sequence of one-beat units of IOI patterns. In MDA* these IOI units are named as rhythmic letters (r-letters, for short). For a proper identification and classification of the r-letters that act in a given context, the beats shall be internally segmented by a divisor whose magnitude will ultimately

B MDA*

297

Fig. B.2 Two equivalent rhythms sharing the same IOI sequence

b

a

d

c

etc. one-beat window r-letters: b

d

d

c

a

b

b

c

b

a

a

d

Fig. B.3 Example of rhythmic filtering, considering a context using four r-letters

depend on the rhythmic diversity of the music to be analyzed.1 Let me clarify this point. Consider, for example, the application of MDA* to the analysis of a piece in which the eighth note is the lowest durational value present in the rhythmic configuration (as it occurs in most of rock music). In this case, the beat divisor will be 2 and, consequently, only four r-letters (labelled as a–d) will be enough to describe any possible rhythmic situation in this specific context (Fig. B.3): none onset (a), one onset at the onbeat (b), one onset at the upbeat (c), and two onsets (d). Analogously what was done with the pitch domain, let Ar be the rhythmic alphabet considered in this example, so Ar = {a, b, c, d}

.

More diversified rhythmic situations will require, evidently, larger alphabets. Although this system, in thesis at least, could be applied to any possible rhythmic situation, even the very complex ones (combining, for example, 32th notes with septuplets), the size of the resulting alphabet Ar (which ultimately conditions

1 In arithmetic terms, the divisor will be equal to the least common multiple of the rhythmic values present in the context in question.

298

a b c d e f g h i j k l m

B MDA*

n o p q r s t u v w x y z

Fig. B.4 The 12-divisor alphabet with two representations for the r-letters

the feasibility of the filtering process itself) is a paramount limiting factor. In practical terms, I observed that 12 can be considered an optimal value for the beat divisor. Since 12 is the least common multiple of 4 and 3, it provides many (but not an excessively large number) of distinct r-letters, a set that covers rhythmic configurations present in numerous repertoires (especially those that MDA* is destined to). Figure B.4 shows the content of this alphabet, depicting the r-letters in two representations: as “checked” blocks in a 12-position grid, and as a possible musical realization.2 It has 26 r-letters, which conveniently allows us to map them to the letters of the Latin alphabet.3 For practical purposes, let r-letters c, e, g, h, j, l, m, n, p, q, r, t, u, and v integrate the subset Ar4 , formed by the patterns that result from the division of the grid by 4. Accordingly, the subset Ar3 = d, i, k, o, and s contains the patterns resulted from

2 Another alternative representation of the r-letters is as binary vectors (for example, j = ), whose main advantage is facilitate computational implementation. 3 Rigorously, the number of possible combinations is greater than 26, but I decided to disregard some possible r-letters (as those that mix triplets and 16th notes or most of the possible—but idiosyncratic—figures with the 16th triplets). Moreover, it is possible to observe that this alphabet does not explore all the rhythmic potentialities provided by the 12-divisor, putting aside r-letters based on 32th triplets (the unit of the grid), due to their relative rarity.

B MDA*

299

the division of the grid by 3. Two r-letters are spacial cases: a is considered the neutral element considering the grid (corresponding to a quarter-note rest), and b, that pertain to both subsets. Lastly, I define a r-word as a structure used for describing the rhythm of a UDS as a sequence of r-letters (considering, of course, the scope of the adopted Ar ). Thus, the UDS Fig. B.1 can be encoded as the r-word .

Variation in MDA* The system encompasses two basic classes of operations:4 1. Micro-transformations (mT)5 —they include the operations that affect the level of (c-/r-)letters. There are specific operators for each domain considered: (a) Expansion (EXP)—applied to c-letters, sending them to a more “ascending” category, according to the following sequence: l > s > t > u > T > S > L. Example: EXP(S) = L; (b) Contraction (CON)—applied to c-letters, sending them to a more “descending” category, according to the following sequence: L > S > T > u > t > s > l. Example: CON(T) = u; (c) Inversion (INV)—applied to c-letters, changes the direction of the movement, according to the following scheme: L > l, l > L, S > s, s > S, T > t, t > T, and u > u. Example: INV(l) = L; (d) Transposition (Tn )—applied to r-letters, moving them n positions “rightwards” on the metric grid; n = 3 (for r-letter b and members of subset Ar4 ) or n = 4 (for r-letter b and members of subset Ar3 ).6 Example: T3 (q) = u;7

4 MDA* also considers two additional, higher classes (I do not intend to detail them here, since only the most basic elements of the system are in question): (3) Meta-transformations (MeT), that act at the level of the concatenation of words (considering both separate domains). Given this, it is possible to relate a word to others, contiguous or not. Put more precisely, this level reveals the derivative relations that occur in a given context; (4) Contextual transformations (cT), that are related to a possible contextual referential, previously established. These references can be associated to any domain besides those considered in the filtering process, for example, harmony or metric position. Due to their somewhat vague nature, cTs can be created and defined according to specific situations. 5 For the sake of illustration, only the main operations are here listed. 6 This operation is not applicable to r-letter a and to the last four r-letters of A . r 7 If expressed as binary vectors, T (100100100000) = (0000100100100). 3

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(e) Inclusion (INCn )—applied to a r-letter, includes a new onset in its metric grid, resulting into another valid r-letter.8 Example: INC10 (b) = l;9 (f) Exclusion (EXCn )—applied to a r-letter, excludes a new onset in its metric grid, resulting into another valid r-letter. Example: EXC1 (s) = o;10 2. Macro-transformations (MT)—they act at the level of (c-/r-) words and are indistinctly used by both domains: (a) Addition (ADD.γ )—inserts a letter .γ at the end of of a word. Example: ADDb () = ; (b) Deletion (DELn )—excludes the n-th letter of a word. Example: DEL4 () = ; (c) Retrogradation (RET)—rewrites a word backwards. Example: RET() = ; (d) Rotation (ROTn )—rotates the internal order of a word by n times. Example: ROT2 () = ; γ (e) Insertion (INS.n )—inserts a letter .γ in between the (n-th .− 1) and the n-th j position of a word. Example: INS.3 () = ; (f) Replication (RPLm:n )—replicates a letter or a sequence of letters of a word, defined by the interval m:n, where m and n are position numbers.11 The replication must be inserted just after the informed interval. Example: RPL2:4 () = ; Figure B.5 presents a very simple example of UDs related by micro- and macro transformations. Notice especially how the system allows many possible concrete manifestations for a UDS, from the re-integration of a pair of compatible12 words (contour + rhythm), which is due to the flexibility of these descriptors.13

8 This means that “ill-formed” patterns—like hybrids created by combination of members of subsets Ar3 and Ar4 —are not allowed in the system. 9 INC (100000000000) = (1000000000100). 10 10 EXC (100010001000) = (000010001000). 1 11 In the case of replication of a single letter, m = n. 12 That is, with same cardinality. 13 Measurement of similarity is an aspect in MDA* that will be here omitted, since its implementation is still being studied. A simpler possible model considers the establishment of a constant value (as a penalty) for each edition applied. However, this not take into account the comparison of related units, which seems for me an essential element that needs be incorporated to the system in the near future.

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filtering c-word

r-word

l S T t

j b e b

EXP

mT

T3

l S S t

n b e b

INS u2 MT

RPL 2:3

l u S S t

n b e b e b DEL6

ROT3

S t l u S

n b e b e b

re-integration Possible realizations:

... Fig. B.5 Example of the use of micro- and macro-transformations in the production of variants of a referential UDS

Reference Dowling, J. (1978). Scales and contour: Two components of a theory of memory for melodies. Psychological Review, 85(4), 341–354.

Appendix C

Algorithms

This appendix proposes some algorithms related to concepts and methodological procedures developed along the book, especially in Chap. 3. They are written in pseudo-code notation, and are intended to be primordially adopted as basic suggestions for automatizing, and optimizing the process of derivative analysis. Algorithm 1 Melodic contour Description: Given a pitch sequence, the algorithm returns the corresponding melodic contour Input: P (midi-pitch sequence, expressed as a numeric vector. Ex: ) Output: M (melodic contour, expressed as a numeric vector, according to MORRIS (1987). Ex: ) P1 = P // Create a copy of P n = 0; // Create a counter for i = 1 : length(P) do z = find(P==min(P)) // Find where is the minimal value of P P1(z) = n // substitute value of P1 in same position P(z) = 1000 // set value of P in same position to 1000 n = n+1 // update counter end M = P1 // melodic contour

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. d. L. Almada, Musical Variation, Computational Music Science, https://doi.org/10.1007/978-3-031-31451-3

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Algorithm 2 Metric contour Description: Given a sequence of time points (onsets), and information about time signature, the algorithm returns the metric contour Input: T (sequence of time points, measured as multiples of a durational unit. Ex: ); S (time signature, expressed as a two-element vector. Ex: ) Output: M (metric contour expressed as a numeric vector, according to MAYR; ALMADA (2017). Ex: ) u=16*S(1)/S(2) // units per measure G=(repmat(1,u,1))’ // surface level, assign “1” to any element v = u // create a new variable “v” equal to u n = 1 // create a counter, for levels of the grid while v >= 1 do div=u/v // time span for the current level L=(zeros(u,1))’ // create current level, initially zeroed k=1 // create an internal counter for i = 1 : div: length(L) do if mod(k,2) =0 then // activate onsets at level L L(i)=G(end,i); // copy corresponding value of last level G(end,1:i)=G(end,1:i)+0.01; // add a “bonus” to strong events end k=k+1 // update counter k end n=n+1 // update counter n G=[G;L] // incorporate level L to the grid G v=v/2 // update v end T=mod(T,u) // contextualize T in modulo-“unit” T=T+1 // add “1” to each onset (operational finality) score=[ ] // create a empty vector for scoring values of T for i = 1 : length(T) do x=T(i) // pick an onset y=sum(G(:,x)) // sum up matches in G score=[score;y] // update score end sc=score // create a copy of score q=0 // create an internal counter for i=1:length(score) do z=find(sc==min(sc)) // find the minimal value in sc score(z)=q; // assign q to same position in score sc(z)=1000; // set the same position of sc to 1000 q=q+1; // update counter q end M=score; // metric contour of T in function of time signature S

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Algorithm 3 Measurement of similarity Description: Given two related musical ideas P (parent) and C (child) with same cardinality, returns a penalty k, that expresses dissimilarity between them. Input: MP (matrix of attributes of P) MC (matrix of attributes of C). MP and Mc have same dimensions (i,j), such that i = 12 and j corresponds to the number of events present in P and C. Output: k (penalty for dissimilarity between P and C) // define sections of the matrices Pp = MP (1 : 4, :) // pitch section of matrix P Pt = MP (5 : 7, :) // temporal section of matrix P Ph = MP (8 : 12, :) // harmonic section of matrix P Cp = MC(1 : 4, :) // pitch section of matrix C Ct = MC(5 : 7, :) // temporal section of matrix C Ch = MC(8 : 12, :) // harmonic section of matrix C // 1. Calculation of pitch penalty kp // define attributes of P and C Pp 1 = Pp (1, :) // attribute p1 of P Pp 2 = Pp (2, :) // attribute p2 of P Pp 3 = Pp (3, 1 : end − 1) // attribute p3 or d-p3 of P Pp 4 = Pp (4, :) // attribute p4 of P Pp 5 = Pp (3, end) // attribute p5 of P Cp 1 = Cp (1, :) // attribute p1 of C Cp 2 = Cp (2, :) // attribute p2 of C Cp 3 = Cp (3, 1 : end − 1) // attribute p3 or d-p3 of C Cp 4 = Cp (4, :) // attribute p4 of C Cp 5 = Cp (3, end) // attribute p5 of C // formation of the pitch vector v1 = abs(Cp 1 − Pp 1) v2 = abs(Cp 2 − Pp 2) v3 = abs(Cp 3 − Pp 3) v4 = abs(Cp 4 − Pp 4) v5 = abs(Cp 5 − Pp 5) // verification of applicability of adjustment rules // rule 1 - octave equivalence z=find(v3==12 | v3==-12) // search for change of register if sum(z)>0 then // case affirmative v3(z) = v3(z)/3 end

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// rule 2 - transpositional equivalence if length(unique(v1))==1 then // all entries are equal in v1 v1=(repmat(2,length(v1),1))’ // assign 2 to the entries of v1 v2=v1 // the same to v2 // rule 3 - inversional equivalence if v3==2*abs(Cp 3) then // v3 is double of absolute value of Cp 3 v3=(repmat(3,length(v3),1))’ // assign 3 to the entries of v3 // rule 4 - inversional equivalence of contours if length(unique(Pt 4 + Ct 4)) == 1 then // the sum of both contours equals 1, so they are invert each other v4 = (repmat(1,length(v4),1))’ // assign 1 to the entries of v4 // formation of the pitch-vector vp = [sum(v1) sum(v2) sum(v3) sum(v4) v5] // weight vector for pitch domain wp = [15 15 40 25 5] // (N.B.: the user may edit the weights if necessary). // calculation of pitch penalty kp 0 = vp .wp // provisional pitch penalty kp = kp 0/4060 // normalized pitch penalty // 2. Calculation of temporal penalty kt // define attributes of P and C Pt 1 = Pt (1, :) // attribute t1 of P Pt 2 = Pt (1, 1 : end − 1) // attribute t2 of P Pt 3 = Pt (1, :) // attribute t3 of P Pt 4 = Pt (2, end) // attribute t4 of P Ct 1 = Ct (1, :) // attribute t1 of C Ct 2 = Ct (1, 1 : end − 1) // attribute t2 of C Ct 3 = Ct (1, :) // attribute t3 of C Ct 4 = Ct (2, end) // attribute t4 of C // formation of the temporal vector v1 = abs(Ct 1 − Pt 1) v2 = abs(Ct 2 − Pt 2) v3 = abs(Ct 3 − Pt 3) v4 = abs(Ct 4 − Pt 4)

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// verification of applicability of adjustment rule 5 equivalence of durational proportions div = Ct 2/Pt 2 // ratio between IOIs if length(unique(div))==1 then // a unique integer is obtained v1=(repmat(1,size(v1,2),1))’ v2=v1(1:end-1) v3=(repmat(0,size(v1,2),1))’ v4=.5 // formation of the time-vector vt = [sum(v1) sum(v2) sum(v3) v4] // weight vector for temporal domain wt = [15 45 30 10] // (N.B.: the user may edit the weights if necessary). // calculation of temporal penalty kt 0 = vt .wt // provisional temporal penalty kt = kt 0/1140 // normalized temporal penalty // 3. Calculation of harmonic penalty kh // define attributes of P and C Ph 1 = Ph (1, :) // attribute t1 of P Ph 2 = Ph (2, :) // attribute t2 of P Ph 3 = Ph (3, :) // attribute t3 of P Ph 4 = Ph (4, end) // attribute t4 of P Ph 5 = Ph (5, end) // attribute t4 of P Ch 1 = Ch (1, :) // attribute t1 of C Ch 2 = Ch (2, :) // attribute t2 of C Ch 3 = Ch (3, :) // attribute t3 of C Ch 4 = Ch (4, end) // attribute t4 of C Ch 5 = Ch (5, end) // attribute t4 of C // set vh as a zeroed five-entry vector vh =[0 0 0 0 0] for i=1:5 do a=Ph (i, :) b=Ch (i, :) if a==b then // vectors in P and C are identical vh (i) = 1 // weight vector for harmonic domain = [45 25 15 10 5] // (N.B.: the user may edit the weights if necessary). // calculation of harmonic penalty kh = vh .wh // harmonic penalty wt

// 4. Calculation of the global penalty k k=(3.5*kp +5*kt +1.5*kh )/10