What Is a CadenCe? - TU Dresden [PDF]

What Is a Cadence? Theoretical and Analytical Perspectives on Cadences in the Classical Repertoire

Markus Neuwirth Pieter Bergé (eds)

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What Is a Cadence? Theoretical and Analytical Perspectives on Cadences in the Classical Repertoire

Reprint from What is a Cadence? - ISBN 978 94 6270 015 4 - © Leuven University Press, 2015

Reprint from What is a Cadence? - ISBN 978 94 6270 015 4 - © Leuven University Press, 2015

What Is a Cadence? Theoretical and Analytical Perspectives on Cadences in the Classical Repertoire

Markus Neuwirth and Pieter Bergé (eds)

Leuven University Press

Reprint from What is a Cadence? - ISBN 978 94 6270 015 4 - © Leuven University Press, 2015

© 2015 by Leuven University Press / Presses Universitaires de Louvain / Universitaire Pers Leuven. Minderbroedersstraat 4, B-3000 Leuven (Belgium) All rights reserved. Except in those cases expressly determined by law, no part of this publication may be multiplied, saved in an automated datafile or made public in any way whatsoever without the express prior written consent of the publishers. ISBN 9789462700154 D / 2015 / 1869 / 19 NUR: 664 Cover and layout: Jurgen Leemans Cover illustration: ‘Cadence #1 (a short span of time), Robert Owen, 2003’, CC-BYNC-ND Matthew Perkins 2009.

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C o n t e n t s 5

Contents

Introduction: What is a Cadence?

7

Nine Perspectives Markus Neuwirth and Pieter Bergé

Harmony and Cadence in Gjerdingen’s “Prinner”

17

William E. Caplin

Beyond ‘Harmony’

59

The Cadence in the Partitura Tradition Felix Diergarten

The Half Cadence and Related Analytic Fictions

85

Poundie Burstein

Fuggir la Cadenza, or The Art of Avoiding Cadential Closure

117

Physiognomy and Functions of Deceptive Cadences in the Classical Repertoire Markus Neuwirth

The Mystery of the Cadential Six-Four

157

Danuta Mirka

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6 Contents

The Mozartean Half Cadence

185

Nathan John Martin and Julie Pedneault-Deslauriers

“Hauptruhepuncte des Geistes”

215

Punctuation Schemas and the Late-Eighteenth-Century Sonata Vasili Byros

The Perception of Cadential Closure

253

David Sears

Towards a Syntax of the Classical Cadence

287

Martin Rohrmeier and Markus Neuwirth

List of Contributors

339

Index

343

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Towards a Syntax of the Classical Cadence 287

Towards a Syntax of the Classical Cadence* Martin Rohrmeier and Markus Neuwirth

“The idea that there is a grammar of music is probably as old as the idea of a grammar itself”1

I. Theoretical foundations 1. The cadence: Essence or family resemblance?

A

crucial aspect of our listening experience is the formation of expectations and predictions.2 No matter what kind of music we are listening to, we have a predisposition towards expecting certain continuations of what we have heard before. In Leonard B. Meyer’s words, a given musical event “implies” another and at the same time may “realize” the implications set up by earlier events.3 The cadence is often cited as one of the most prototypical patterns in Western tonal music, creating and ultimately fulfilling highly specific expectations.4 It has frequently been suggested that cadential contexts differ systematically from non-cadential ones with respect to their expectancy profile. Cadences articulate phrase boundaries at which there is a highly contrastive change in predictability from high to low.



Financial support for the research presented in this chapter has been generously provided to the first author by the MIT I2 Intelligence Initiative, the MIT Department of Linguistics and Philosophy, as well as the Zukunftskonzept at TU Dresden funded by the Exzellenzinitiative of the Deutsche Forschungsgemeinschaft. The work by M.R. was conducted under the affiliation of MIT as well as TU Dresden. M.N.’s research has been funded by The Research Foundation – Flanders. We would like to thank Taiga Abe and Sophia Stuhr for their kind assistance in preparing the figures included in this chapter. 1. Steedman, “The Blues and the Abstract Truth” (1996), 1. 2. Rohrmeier and Koelsch, “Predictive Information Processing” (2012); Rohrmeier, “Musical Expectancy: Bridging Music Theory, Cognitive and Computational Approaches” (2013). 3. Meyer, Explaining Music (1973). 4. See Meyer, Emotion and Meaning in Music (1956). *

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Although cadences are often described as a stock pattern, the variety of cadential structures is in fact enormous, exceeding by far the simple characterizations in terms of “I–IV–V–I” or “I–ii(6)–V–I” found in numerous modern music-theory textbooks and used in typical experimental settings in music psychology.5 This variability of cadences clearly defies music-theoretical attempts to provide an unequivocal and all-encompassing definition. As Robert Gjerdingen polemically points out, a chordcentered definition of the cadence is dubious, as it “highlights only what Locatelli has in common with Rimsky-Korsakov.”6 This type of “lowest common denominator” approach adopted by many theorists may single out V–I as the essence of an authentic cadence,7 although this feature is hardly sufficient to allow a clear-cut distinction between cadential and non-cadential (prolongational or sequential) progressions.8 In other words, the bare harmonic essence may not be capable of providing the differentia specifica to other (non-cadential) types of progressions. Because non-cadential phrases may likewise feature V–I, this element (the alleged “essence”) is too unspecific a requirement for a cadential progression. As shown elsewhere in this volume, the characterization of half cadences is even less straightforward, especially due to the variety of possible predominant chords.9 Conversely, the four-stage prototype of the cadence,10 embracing an initial tonic, predominant, dominant, and final tonic, does not cover all possible instances of the cadence concept, as not all of these stages must necessarily be present: Not only might the initial tonic be omitted, but the predominant as well.11 Even a dominant in root position, sometimes considered an absolute requirement for “the authentic cadential progression” to have “sufficient harmonic strength to confirm tonality,”12 may not be strictly necessary, as (pre-classical) cadences featuring an inverted dominant demonstrate.13 This might suggest that one of the crucial problems in defining cadences seems to be the lack of an “essence”: It appears to be difficult, if not downright impossible, to identify the necessary and sufficient criteria underlying the definition of a cadence,

5. This is hardly surprising, as cadences have evolved over a remarkably long period of time; thus, the ways in which cadences materialize depend to a large extent on the historical style in which they appear. 6. Gjerdingen, Music in the Galant Style (2007), 140. 7. See Temperley, The Cognition of Basic Musical Structures (2001), 336ff. 8. On these types of harmonic progressions, see Caplin, Classical Form (1998), 23ff. 9. Specifically on the problems surrounding a robust definition of half cadences, see, for instance, the contributions by Martin and Pedneault-Deslauriers as well as by Burstein in this volume. 10. Caplin, “The Classical Cadence” (2004). 11. E.g., Mozart, K. 330/i, mm. 5–8. In his discussion of such “incomplete cadential progressions,” William Caplin hypothesizes that “the initial tonic is left out more often than the pre-dominant is, for eliminating the latter results in the loss of a fundamental harmonic function. Excluding both of these harmonies occurs infrequently in the literature” (Caplin, Classical Form [1998], 27). 12. Caplin, Classical Form (1998), 27; see also Caplin, “The Classical Cadence” (2004). 13. See Caplin in this volume.

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Towards a Syntax of the Classical Cadence 289

a fact that might indicate that there is no such thing as “the” cadence. Rather, one could argue, the various forms of cadences are related to one another by way of a Wittgensteinian “family resemblance.”14 Nonetheless, considering other disciplines such as evolutionary biology, linguistics, and economics, the complexity and diversity of real-world phenomena should not prevent us from developing concise analytic characterizations with the aid of formal methods. In this chapter, we seek to account for the combinatorial complexity inherent in the classical cadence by taking advantage of the flexibility of a generative grammar approach.

2. Formal modeling and music theory Music theory seeks to provide a concise description of the principles governing a musical system or a specific, historically bound style. Despite the field’s ambitions with regard to accuracy and conciseness, at present formal models of music description are the exception rather than the rule. Although the notion of (linguistic) syntax has been repeatedly invoked in theoretical writings since the eighteenth century,15 only a few theorists have taken this analogy seriously, among them Aldwell and Schachter: One way that music resembles language is that the order of things is crucial in both. “I went to the concert” is an English sentence, whereas “I concert went the to” is not. Similarly, I-VII6-I6-II6-V7-I [...] is a coherent progression of chords, whereas I-I6-VII6-II6-I-V7 [...] is not, as you can hear if you play through the two examples. In the study of language the word syntax is used to refer to the arrangement of words to form sentences; word order is a very important component of syntax. In studying music, we can use the term harmonic syntax to refer to the arrangement of chords to form progressions; the order of chords within these progressions is at least as important as the order of words in language.16

Despite their explicit reference to linguistic syntax, Aldwell and Schachter do not go on to employ formal tools and their powerful potential in music-theoretical descriptions, although most of the rules in their textbook would lend themselves to formal modeling. However, in the 1980s and over the last decade, a number of theorists have taken advantage of the power of formal grammars for the characterization of music and computational music analysis.17 14. On the notion of “family resemblance,” see Wittgenstein, Philosophical Investigations (1953/2001), 65–68. 15. Most famously by Koch, Versuch I–III (1782–1793). For modern sources, see Albersheim, “Die Tonsprache” (1980) and Kostka and Payne, Tonal Harmony (1984). 16. Aldwell and Schachter, Harmony and Voice Leading (2003), 139 (emphasis in original). 17. Winograd, “Linguistics and the Computer Analysis of Tonal Harmony” (1968); Keiler, “Bernstein’s ‘The Unanswered Question’ and the Problem of Musical Competence” (1978); Steedman, “A Generative Grammar for Jazz Chord Sequences” (1984) and “The Blues and the Abstract Truth” (1996); Lerdahl and Jackendoff, A Generative Theory of Tonal Music (1983); Rohrmeier, “A Generative Grammar Approach to Diatonic Harmonic Structure” (2007) and “Towards a Generative Syntax of

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Formal models are particularly well-suited to music-theoretical purposes, as they (1) contribute to the specificity and conceptual clarity of a given theory, (2) offer precise evaluation criteria, truth conditions, and empirically testable predictions, and (3) allow differentiation between regular and irregular (or grammatical and non-grammatical) musical utterances as well as between general and style-specific features. In addition, it should be noted that a generative grammar approach not only describes the structure implied in the musical pseudo-surface (e.g., chord representation), but also captures functional and dependency relationships between elements in the deep structure (weak vs. strong generativity). Such a formal characterization additionally allows us to make predictions with respect to the corresponding cognitive processes and constraints. More specifically, it makes it possible to model the generalized competence of a well-informed expert listener/composer/performer with regard to the tonal language, deliberately sidelining style-specific, piece-specific, composerspecific, and idiosyncratic aspects.18 Another important (and cognitively informed) theoretical approach addressing the classical cadence embraces the families of schema and prototype theories.19 From a formal perspective, schema or prototype theories and grammar approaches may be construed as the opposite ends of a complexity spectrum. Their primary difference involves a tradeoff in terms of compression: If a pattern evinces both regularity and combinatorial freedom, grammars will be more suitable to describe it; however, if a musical structure exhibits more standardization and less variability, schema-theoretical (and exemplar-based) approaches will be more appropriate. However, the boundaries between schemata and generative grammars may be fuzzy, since the ways in which prototypes can be modified can approximate the ways in which generative rules specify the modification of strings; conversely, building blocks and rules may incorporate fixed block sequences.20 Although, as noted above, the classical cadence appears to be relatively formulaic, closer inspection reveals a high degree of variety, flexibility, and freedom. This flexibility may lend itself in particular to the use of a generative grammar focusing on small building blocks and rule-based generative mechanisms, rather than the schemata employed in music theory. Tonal Harmony” (2011); Katz and Pesetsky, “The Identity Thesis for Language and Music” (2010); De Haas et al., “Modeling Harmonic Similarity Using a Generative Grammar of Tonal Harmony” (2009); De Haas, “Music Information Retrieval Based on Tonal Harmony” (2012); and De Haas et al. “Automatic Functional Harmonic Analysis” (2013). 18. For an earlier attempt, see Lerdahl and Jackendoff, A Generative Theory of Tonal Music (1983), 3; Jackendoff and Lerdahl, “The Capacity for Music” (2006). 19. E.g., Gjerdingen, Music in the Galant Style (2007). For a schema-theoretical approach to cadences, see Byros in this volume. 20. See also Temperley, The Cognition of Basic Musical Structures (2001), 336ff. If a schema-based description employs rules to recursively adapt, modify, and recombine schemata, this converges to a syntactic approach with building blocks of different sizes. For an attempt to transform schemata into a grammar, see Lerdahl, Tonal Pitch Space (2001), 233–248.

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3. Formal grammars: A brief introduction The concept of formal grammars and rewrite systems dates back to Chomsky’s early approaches in the 1950s.21 A crucial motivation for such systems is the insight that infinite or very large sets of sequences can be efficiently and concisely characterized by a small number of rules operating in conjunction with combinatorial principles, rather than listing each of these sequences individually. The formalism requires the following definitions: A language is a finite or infinite set of sequences (or strings) over a predefined repertory of symbols that are commonly referred to as terminal symbols. In the case of music, one may, for instance, decide to choose tones, drumbeats, chords, or bass notes for the set of terminal symbols. A language distinguishes grammatical (regular, licit) sequences from ungrammatical (irregular, illicit) ones generated by the same terminal symbols. A finite number of sequences could simply be described by listing all of the sequences, whereas an infinite set of sequences requires an indirect definition to characterize it, a definition that specifies the generating mechanism.22 A formal grammar (generative grammar) is a way of characterizing an (infinite) set of sequences by means of construction (generation), defined through its terminal symbols, nonterminal symbols (variables that represent a deep structure not observable in the surface sequence; for instance, relationships between grammatical categories), rewrite rules for grammatical production, and one special nonterminal start symbol defining the beginning of production.23 The production process defines strings by beginning with the start symbol and iteratively replacing nonterminals in the current string until the string consists only of terminals. The set of all strings generated by the start symbol using all possible combinations of rule applications defines the language expressed by the grammar. The set of productions derived from one nonterminal is called the yield of the nonterminal; in particular, the language expressed by the grammar corresponds to the yield of the start symbol. The derivation process results in a parse tree that expresses the derivation steps that produce a given sequence. If there is more than one nonequivalent parse tree (the order of the single derivations does not matter), the sequence is referred to as ambiguous. For instance, a toy grammar G = (Σ, V, R, S) may be defined with the terminals Σ = {a,b,c,d}, the nonterminals V = {U,V,W,X}, the start symbol S = {U}, and the following rules in R: U → bV V → cW

21. See, e.g., Chomsky, “Three Models for the Description of Language” (1956). 22. Note also that a finite set of sequences might be more easily or comprehensively described by characterizing it by its underlying structure rather than listing all instances without taking into account a generalizing structure; in other words, a grammatical description is a means of compression by generation. 23. Note that “production” here does not refer to a temporal process or a cognitive model; rather, it characterizes a mathematical (atemporal) construction principle for sequences.

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V → aW W → dV W → bW W → dX X → b

Our simple grammar G now generates strings by successive rewrite steps, beginning from the start sequence “U” until any sequence is reached that lacks a nonterminal symbol. One sample generation process could involve the following steps: U ⇒ b V ⇒baW⇒badV⇒badcW⇒badcbW⇒badcbbW⇒badcbbbW⇒ b a d c b b b d V ⇒ b a d c b b b d a W ⇒ b a d c b b b d a d X ⇒ b a d c b b b d a d b. The final string contains no more nonterminals and therefore the generation stops, resulting in a terminal string. This grammar is recursive because the yield of the symbol W (and V) generates another instance of this symbol, resulting in strings of unbounded lengths. Based on the way the rules for this toy grammar are defined, it is straightforward to conclude that (1) each sequence must begin with a single ‘b’, (2) each sequence must end with ‘d b’, and (3) the possible middle sections include repetitions of the sequence ‘c’ or ‘a’ followed by any number of ‘b’s (including no ‘b’) and ‘d’. Furthermore, one can easily see that no string with a different structure can be produced. Note that the definition of the start symbol is crucial here, since defining ‘V’ or ‘W’ as the start symbol would result in somewhat different structures. The process of reconstructing the possible underlying generation paths that resulted in an observed terminal sequence (and deciding whether a sequence is grammatical) is referred to as parsing (structural listening or language perception would involve a form of parsing). Each class of formal grammars (see below) requires a specific associated parsing process (or automaton) to parse strings. The automata associated with formal grammars differ in terms of the types of memory representation they use—a distinction that is cognitively highly relevant for processing and learning.24 Chomsky’s work on formal languages led to what has come to be known as the Chomsky Hierarchy,25 which differentiates various complexity classes of grammars. In its most well-known form, it encompasses four different types of formal grammars. A grammar in which all rules take the abstract forms of either A → b C or A → C b (not both!) and A → b is referred to as a regular grammar (sometimes also a finite-state grammar). Grammars of this complexity (type 3 in the Chomsky Hierarchy) correspond to a sequence model analogous to a flow-diagram. Relaxing the restrictions of regular 24. See Hauser, Chomsky, and Fitch, “The Faculty of Language: What is It, Who Has It, and How Did It Evolve?” (2002); Fitch, Hauser, and Chomsky, “The Evolution of the Language Faculty: Clarifications and Implications” (2005); Rohrmeier and Rebuschat, “Implicit Learning and Acquisition of Music” (2012); Rohrmeier et al., “Implicit Learning and Recursion” (2014). 25. Chomsky and Schützenberger, “The Algebraic Theory of Context Free Languages” (1963).

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grammars, rules of the more general form A → b1 b2 b3 … bn, in which each bi is either a terminal or a nonterminal, characterize context-free grammars (type 2 in the Chomsky Hierarchy). Such grammars famously have as one of their distinctive features the power to generate sequences with center-embedding, such as “seq1 – seq2 – continuation of seq1,” or “seq1 – seq2 – seq3 – continuation of seq2 – continuation of seq1” as the simplest examples. Such center-embedding structures cannot be expressed by regular grammars (or lower complexity subregular grammars). Grammars in which all rules are of the general form c A → b1 b2 b3 … bn (a nonterminal A rewrites to any sequence of terminals and nonterminals b1, b2, b3 … bn only when it occurs in the context of c; the left-hand side is not longer than the right-hand side) are referred to as context-sensitive grammars (type 1 in the Chomsky Hierarchy). They can model centerembedding recursion, cross-serial dependencies, and other varieties of complex structure. The set of unrestricted grammars has no restrictions on rewrite rules (type 0). The four types of grammar are ordered in terms of increasing complexity in such a way that a higher-order language contains as a subset all languages of lower complexity (e.g., due to its less restrictive rules, every context-free grammar includes the expressive power of regular grammars). The types of formal languages in increasing order of complexity are: regular languages (type 3), context-free languages (type 2), contextsensitive languages (type 1), and unrestricted languages (type 0). It is important to note that there are other types of formal models apart from the traditional Chomsky Hierarchy (e.g., subregular, multiple context-free, and mildly context-sensitive grammars) as well as Hidden Markov Models and Dynamic Bayesian Networks that have been employed in both linguistics26 and music theory27.

26. See, e.g., Clark, “An Introduction to Multiple Context Free Grammars for Linguists” (2014); Seki et al., “On Multiple Context-free Grammars” (1991); and Jäger and Rogers, “Formal Language Theory: Refining the Chomsky Hierarchy” (2012). Furthermore, it is important to note that the Chomsky Hierarchy does not represent the only way to characterize formal languages of different complexity. 27. There are ongoing debates over the type of complexity required for the characterization of music. Some theorists argue for context-free complexity: see Lerdahl and Jackendoff, A Generative Theory of Tonal Music (1983); Rohrmeier, “Towards a Generative Syntax of Tonal Harmony” (2011); Steedman, “A Generative Grammar for Jazz Chord Sequences” (1984) and “The Blues and the Abstract Truth” (1996). Other theorists argue for lower complexity: Temperley, “Composition, Perception, and Schenkerian Theory” (2011); Tymoczko, “Function Theories: A Statistical Approach” (2003). Still other theorists argue for higher mildly context-sensitive complexity: Katz and Pesetsky, “The Identity Thesis for Music and Language” (submitted draft). Note that it is crucial to bear in mind for this debate that, due to the fact that higher-order models by definition include lower complexity models, converging evidence of regular grammar structure in music cannot decide the debate, since the existence of one non-regular (context-free or context-sensitive) feature would obligate a requirement of higher grammatical complexity.

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II. Towards a grammatical model of the cadence 1. Dependency relations Following the music-theoretical consensus, we conceive of a cadence as a stagebased process. We understand cadential progressions not as a linear or (Markovian) forward process, in the manner of “B follows A,” “C follows B,”28 but instead propose to represent the logical structure through implicative dependencies: A prepares (= implies) B, B in turn implies C. This view is consistent with the one outlined in Rohrmeier’s Generative Syntax Model (GSM),29 according to which tonal language can to a large degree be expressed in terms of a series of nested implication-realization relationships.30 Accordingly, the hypothesized principles that motivate our tree analyses of cadences are twofold:31 (A) Two elements can have an implication-realization relationship. For example, the straightforward implications of tonal functions are: Dominants imply (or prepare) the tonic, predominants imply the dominant, embellishments imply their goal event (and only that event). (B) Two elements can establish a prolongational relationship: Two instances with the same tonal function can form a prolonged higher-order region or unit.

Cadential patterns involve a recursive, left-branching, and right-headed structure. The syntactic organization of cadential phenomena can be represented in tree structures that depict this type of implicative and prolongational relationships. The dependency structures that govern such syntactic trees may also be expressed in terms of a dependency graph (see Fig. 1).

28. See, e.g., the approaches by Temperley, “Composition, Perception, and Schenkerian Theory” (2011); Tymoczko, “Function Theories: A Statistical Approach” (2003); Tymoczko, A Geometry of Music (2011). 29. Rohrmeier, “Towards a Generative Syntax of Tonal Harmony” (2011). 30. The core difference between the present approach and a Markovian approach is that preparations and embellishments may be nested in multiple (hierarchical) ways, rather than being restricted to a linear order. 31. We assume that cadences are constituents in tonal phrases that may be modelled using the standard set of generative rules of Western tonality. In other words, cadences are nothing special in the sense that there are no special cadential rules. Rather, cadences are special constituents in which common tonal rules that hold for any other part of a given piece are applied in a particular configuration to achieve a powerful drive towards closure. While cadences may be conceptualized in terms of different stages (predominant, dominant, final tonic), these stages are neither structural parts of the music itself (they have no ontological status) nor of the grammatical model. These stages are reflected in the trees in terms of subordination relationships. Lerdahl proposes a similar view in Tonal Pitch Space (2001).

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Figure 1: An example of a syntactic parse tree and the corresponding dependency graph

The dependency graph visualizes all dependency relationships between the harmonies involved in the cadential process in terms of implication (imp) or prolongation (prl). The dependency structures as implied by the tree and visualized in the dependency graph are identical in the sense that one can easily be transformed into the other.32 Moreover, the syntax tree inherently represents the rewrite rules that were used to generate the structure; for instance, I → V I (second highest branch) or IV → V65/IV IV in the example given above. Every parent node in any branch of the tree or subtree corresponds to the left-hand side of a rewrite rule, and every immediate child of that node corresponds to the right-hand side of the rewrite rule. The syntax (or parse) tree expresses the structure generated by specific applications of rewrite rules to the nonterminals.

32. It is important to note that there are significant differences between dependency grammars and phrase-structure grammars, for instance, with regard to the notion of constituency (see also Kuhlmann, Dependency Structures and Lexicalized Grammars [2010]). The complex issues concerning the evaluation of the use of constituency in musical structures (cf. Rohrmeier, “Towards a Generative Syntax of Tonal Music” [2011]) will be considered in future work.

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2. Level of representation While the syntactic formalism and its complexity are independent of the building blocks involved, the right choice of the level of representation is crucial for the formalism’s expressive power, specificity, and generality. However, in the case of tonal music, the right choice of building blocks is not self-evident; prima facie, there are several candidates on which the syntactic model could be built: the bass line, harmony on scale degrees, harmony with inversion, functional harmony, outer voices, and full four-part blocks. To begin with, a model of the cadence (and of tonal music in general) grounded in the bass line alone is clearly insufficient. Consider, for instance, the following harmonization: 3–4–6–©4–5–1 :=: iii–vii°64–iv6–V/VII–iii6–vi6. This example illustrates the fact that there are possible harmonic realizations of a well-formed bass line that are clearly irregular in common-practice tonal language. Perhaps more importantly, the bass line cited above demonstrates that categorizing cadential events based only on the bass gives rise to representations of harmonically non-interchangeable events (e.g., i6, iii, vi64; IV, IV64/IV, vii°64, etc.). In other words, although a well-formed bass line is fundamental, crucial constraints stem from factors other than the bass line. Another approach, one based on the outer voices, is more constrained, since already two notes of a given harmony are specified. However, choosing outer voices as the building blocks is only useful when a difference in the top voice makes a categorical difference for the syntactic progression. Generally speaking, two chordal tones are still too unspecific (or ambiguous) with respect to the harmony they serve to express. Because the interchangeability of surface elements falling under the same building-block category is crucial for the choice of building blocks, it is necessary to consider a third approach here, namely modeling the harmonic structure.33 As powerful as such a fully generalized approach may be, a model based solely on harmony may result in irregular bass lines and hence be incapable of distinguishing cadential from non-cadential progressions.34 In opposition to an account of harmonic building blocks (as proposed by Keiler), Lerdahl and Jackendoff in their Generative Theory of Tonal Music (GTTM) argue that such harmonic syntax trees would not be able to (1) assign structure to non-harmonic tones or (2) account for Schenkerian interruption.35 This

33. Alan Keiler proposes an elegant generative syntax analysis based on scale degree harmony that expresses generalized musical structure in a manner that is formally more precise and efficient than the one put forth by the “Generative Theory of Tonal Music” (GTTM). See Keiler, “Bernstein’s ‘The Unanswered Question’ and the Problem of Musical Competence” (1978); Keiler, “Two Views of Musical Semiotics” (1981). 34. This discussion is based on the implicit assumption that the core structure has to be modelled within a single system. It may also be viable, however, to model harmonic structure and the bass line separately and relate the sequences by means of constraints or interfaces. 35. Lerdahl and Jackendoff, A Generative Theory of  Tonal Music (1983), 338 (n. 4).

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criticism appears to be misguided: First, a syntactic theory is indeed able to account for interruption. Second, including additional detail is only useful as long as it does not undermine the expressive power and compression by abstraction that is achieved by a generative grammar approach. Again, the question is whether it is worthwhile to enrich the representation to account for voice leading, outer voices, or full texture, despite the fact that this implies eliminating generalization over elements that should be grouped together because they are related in analogous ways. On the other hand, if the building blocks were full four-part harmony textures, it would be possible to distinguish a large number of different cases, but generalization, compression, and abstraction (all of which are the purpose of describing a structure by rules) would no longer be possible. For example, two instances of a ii6–V7 progression should be regarded as identical with respect to syntactic structure, regardless of whether or not the inner voices are exchanged. This aspect may lend itself to a critique of the level of representation chosen by the GTTM. Despite the problems of defining “the” musical “surface”36 by modeling at the note-level, Lerdahl and Jackendoff pay the price for giving up concrete generative rules37 and end up proposing a generative theory without any generative rules. Thus, their proposal is not, and cannot easily be amended to be, a theory of competence or of (tonal) syntax.38 Although models improve with increasing levels of detail, the generalizations drawn from these models may grow weaker. This makes it clear that generality is an important criterion with regard to the choice of building blocks. Functional categories provide a further tool for generalization, allowing us to subsume V7, V, vii° and ii, IV, (vi) under the same categories, and suggesting that categorial constituent heads be employed rather than encoding the bare dependency structure. To be sure, it is not immediately clear whether modeling cadences in terms of functional regions (such as tonic, dominant, and subdominant) is useful; however, if one intends to draw generalizations from categorization by functions, it is certainly possible to express the rules we are outlining here in terms of the functional regions used by the GSM.39 36. For various problems of defining and identifying the “musical surface,” see Cambouropoulos, “The Musical Surface” (2010). 37. Designing a model that generates the entire musical surface is very difficult and is equivalent to creating a generative model of the entire composition (which is unlikely to simultaneously be a good, cognitively adequate model of processing or listening). 38. The GTTM is a theory of parsing that excludes core processing aspects of parsers, such as the use of a stack, online backtracking, and revision. Rather than being a model of musical syntactic competence, GTTM turns out to provide an (incomplete) theory of performative aspects of musical parsing, whereas Keiler proposes a view of modeling tonal syntactic structure at the right level of abstraction. 39. A purely functional approach such as the one advocated by Riemann and his followers might prove insufficient here because it enforces the interchangeability of certain chords belonging to the same functional category, a restriction that our approach does not endorse. Rather, we would argue that there are certain limitations to the arbitrariness with which seemingly functionally equivalent chords can be used to replace one another. For instance, a ii6 chord has a stronger implication of moving to V than a IV chord, as the latter chord may also be used as a neighboring sonority and hence as a means

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Although we are entirely aware of the problem that arises from abstracting from voice-leading characteristics, we opt to concentrate almost exclusively on functional harmony in order to considerably reduce the complexity inherent in cadential patterns. For the reasons outlined in this section, we build our model on scale-degree harmonic representations, including inversions and some common features of voiceleading patterns.40 However, generally speaking, it is important to bear in mind that the chosen level of representation can never be perfect.41

3. The structure of the perfect authentic cadence 3.1. The head of the cadence constituent: The initial tonic, the final tonic, and its preparation (and extension)

Constituents feature a head element upon which the other parts of the constituent depend. The head of a cadence is the final tonic, which at the same time functions as the head of the entire overarching phrase. Furthermore, the tonic chords involved in the cadence are all connected in specific ways. The phrase-initiating tonic and the final tonic constitute the overarching head for the whole phrase, whereas the I(6) chord initiating the cadence may best be modelled as subordinate to the final tonic. Alternatively, the I(6) can be analyzed as preparatory with regard to the (adjacent) predominant chord over 4^. This would imply, however, that the I(6) could not appear without this chord over 4^, which is not necessarily the case: In fact, both elements can appear independently of one another (e.g., I6–V64–V7–I, ii6–V64–V7–I).42 Figure 2 illustrates both interpretive possibilities.

of prolonging a tonic harmony. ii6 and IV are more likely to occur in a cadential context than a rootposition ii chord, due to the fact that both share scale degree 4 in the bass. 40. It is important to note that the challenge in this musical modeling endeavor results from the fact that the formalism is fundamentally required to account for both adequate harmonic dependency relationships and a coherent bass line. This formal correspondence is far from trivial, since a coherent bass line involves certain linear aspects that run counter to harmonic hierarchical dependency relationships. Ultimately, this may require a specifically amended type of formalism. 41. As Temperley cogently argues, “[a]n immediate problem with this model is that its predictions do not always hold: sometimes predominants move to tonics, for example, as in a plagal cadence. Admittedly, such exceptions show that functional harmonic theory is imperfect as a model of tonal harmony; but they do not show that it is useless. A theory whose predictions hold true most of the time can still be of great value; we use such theories all the time in our daily lives. Imperfect though it may be, functional harmonic theory represents a powerful and valid generalization about tonal harmony, better than many conceivable alternatives—for example, a theory that posited that chords are chosen at random without regard for the previous chord, or that predominants move to tonics and dominants move to predominants. On this basis, I would argue, we are justified in positing functional harmony as part of the knowledge that common-practice composers brought to bear in their compositional process” (Temperley, “Composition, Perception, and Schenkerian Theory” [2011], 148). 42. Another way of resolving this problem is to derive the I6 chord in both ways when it precedes a 4^ and to analyze it in terms of a double function. Cf. double functions of pivot elements in the GSM; Rohrmeier, “Towards a Generative Syntax of Tonal Harmony” (2011).

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Figure 2: The tonic frame. The cadence is marked by the box. The overarching frame of the cadence constituent is headed by the tonic (I), which in turn merges with the phraseinitiating tonic to constitute the overarching head of the phrase. The I(6) chord that initiates the cadence could either be modelled as subordinate to the final tonic or to the predominant function.

With regard to the first solution (I(6) subordinate to the final tonic), the following rules establish the tonic frame (bear in mind that the rules are represented as branches in the tree diagram): (1) I → I(6) I (2) I → V I

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These rules define both the head of the entire cadential constituent and the tonic seed.43 In this abstract form, rules (1) and (2) hold for both major and minor modes. It is important to note that we construct our syntactic model of the cadence largely independent of mode. Most rules and generalizations apply to both modes, and most diatonic scale degrees can be employed with the respective mode-specific adaptations. Rule (1) generates the final tonic as well as the initial I(6) chord and, in so doing, defines the overarching link between the I(6) and the final tonic. Note that the cadenceinitiating I(6) chord is optional and not necessary for the formation of a cadence constituent. As for the final tonic, a perfect authentic cadence requires that tonic to appear in root position, whereas an evaded cadence typically (but not exclusively) uses I6.44 The way in which the final tonic is realized in terms of voice leading and grouping defines in part the type of cadence (PAC, IAC, or deceptive/evaded cadence). Crucially, this implies that part of the cadential structure is not definable purely by syntactic tree structure (using harmonic building blocks). The penultimate dominant (V(7)) is necessary for the structure to be a cadence. This is expressed by rule (2). If there were no dominant, a given phrase might be regular in the sense of being in accordance with the rules of tonal harmony, but it would not be considered a cadence. For instance, a dominant harmony cannot be replaced by a diminished (seventh) chord (vii°(7)) on 7^ without the progression losing its cadential capacity. In other words, if the root of a V chord is missing, the resulting harmony cannot express a genuine cadence.45 Within the tonic frame, rule (2) defines the seed for the dominant group and its further recursive elaborations. It should be clear that, in a cadential context, there are naturally no options other than the dominant to immediately precede (and be subordinate to) the final tonic.46 In other words, rule (2) alone demands, and is sufficient to ensure, that the dominant is string-adjacent to I. 3.1.1 The plagal progression as a means of tonic prolongation Opposing a widespread view, we follow the assertion of Caplin and other theorists that plagal progressions (IV–I) do not act as genuine cadences in the classical style. Rather, they are better understood as post-cadential codettas, serving to prolong and consolidate a preceding cadential tonic (e.g., I–IV–I).47 In other words, by the time the 43. Our general analysis (see Fig. 2) is similar to the normative prolongational structure proposed in the GTTM; it differs, however, with respect to the intent to avoid right-branching derivations. 44. Methods of modeling cadences that avoid tonic closure are discussed below. 45. One exception may be the Prinner cadence (IV–I6–ii7–vii°6–I or IV–I6–vii°(7–6)–I), which, however, might also be described as projecting a prolongational progression in Caplin’s sense. See the discussion in Caplin’s contribution to this volume. 46. Compare the formalization by Rohrmeier, “Towards a Generative Syntax of Tonal Harmony” (2011). 47. See Aldwell and Schachter, Harmony and Voice Leading (2003), 193; Lerdahl and Jackendoff, A Generative Theory of Tonal Music (1983), 134; Caplin, Classical Form (1998), 43; and Caplin, “Conceptions and

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plagal progression enters in a given piece, a sense of closure has already been imparted to the listener by means of an authentic cadential progression. This view underpins the approach employed by the GSM, which does not treat the plagal cadence as a strong constituent at a functional level, but rather as a mere appendix at the surface level.48 However, this interpretation is not meant to preclude the possibility that plagal progressions functioned as cadences in historical periods prior to the “classical” era (e.g., the Amen cadence in a Baroque composition) or in contemporary popular music.49 3.2. Preparing the preparation (of the preparation of the preparation)

Having discussed the tonic frame and its corresponding dominant seed, we now consider the remaining elements of the cadence, which essentially function as preparations of V and as potential recursive preparations of such preparations. We distinguish between generalizable and non-generalizable preparations, i.e., those generalizing across different harmonies and those that apply only to specific harmonies. 3.2.1. Preparations of the dominant Non-generalizable preparations of the dominant include versions of IV (IV, IV6), ii (ii, ii43, ii64, ii6, ii6(5)), vi, V64, and N6. Each of these preparations may be modelled using a rule in the following form: (3) V → V-Prep V

One example would be V → ii6(5) V. There are as many specific V-preparation rules as there are chords that can prepare a cadential dominant. These rules are largely identical for major and minor modes (potentially adapted to match the diatonic counterpart, e.g., ii° instead of ii, iv instead of IV). Note that the chords involved in these rules imply specific bass notes: ii implies 2^; ii6(5), IV, and N6 imply 4^; IV6, ii43, ii64, and vi imply 6^. On the surface, V may be realized as either V or V7, or as a combination of the two. Some dominant preparations, in particular the Neapolitan chord and the various augmented sixth chords on ¨6 (see below) hint at the minor mode. 3.2.2. Embellishing the dominant Let us now briefly consider the cadential 64 as the primary means of embellishing the penultimate dominant. After having reached a dominant 64 in the context of a cadential progression, the most frequently used option is to resolve the double-suspension to a V53 (or V7) chord. However, the composer may alternatively choose to add chords at the expected point of resolution that are normally used to lead to (or prepare) the

Misconceptions” (2004), 71f. 48. See Rohrmeier, “Towards a Generative Syntax of Tonal Harmony” (2011). 49. See also the discussion in De Clercq and Temperley, “A Corpus Analysis of Rock Harmony” (2011).

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cadential 64, namely either vii°7/V, vii°6/V, or a German augmented sixth chord, as though stepping backwards in time within the multi-stage process. Consequently, these chords cannot be understood as mere insertions between a cadential 64 and its resolution, but may instead be analyzed as a means of embellishing (and thus, in a way, prolonging) the cadential 64 itself. In other words, they seem to act as a means of embellishing the embellishment. Two examples from Mozart’s oeuvre can be cited in support of this interpretation. Towards the end of the reprise in the first movement of his D-major Piano Sonata, K. 576 (see Figure 3), Mozart launches a normative cadential progression, starting with a tonic harmony in m. 148, proceeding to a ii6 (m. 149), and finally arriving at a cadential 64, which is extended for two measures (mm. 151f.). At the moment one expects the resolution of the 64 suspension, a fully diminished seventh chord suddenly enters, likewise sustained for two measures. But rather than entirely abandon the attempt to complete the cadence, the progression reverts to the cadential 64 (m. 154), this time resolving the suspension properly, ultimately arriving at the final tonic (m. 155).

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Figure 3: Mozart, Piano Sonata in D major, K. 576/i, mm. 148–155

In Mozart’s F-major Piano Sonata, K. 533/i, the exposition is about to close with a PAC but avoids closure by means of a deceptive harmony (vii°6/V) at the progression’s ultimate stage (m. 82), the cadential resolution being delayed until m. 89. However, before the cadential 64 is resolved, it is prolonged by various embellishing chords, including vii°7/V (m. 84) and an augmented German sixth (m. 86), both of which subsequently return to the cadential 64, which ultimately gives way to a root-position V in m. 88. This suggests that chords that are normally employed to prolong a predominant harmony, connecting it to the cadential dominant, can also be used (even within the same temporal sequence) to extend in time a particular dominant embellishment, namely the cadential 64.

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Figure 4: Mozart, Piano Sonata in F major, K. 533/i, mm. 80–89

Alternatively, the situation described in the two Mozart sonatas may also be interpreted as a form of dominant prolongation. This would help avoid an increase in complexity resulting from additional rules. In addition, it would explain the fact that both of the embellishing chords (vii°7/V and the augmented sixth chord) can easily be followed by a plain dominant sonority. The fact that another cadential 64 is inserted, thus framing the appearance of these embellishing chords, might simply be regarded as a coincidental surface event, one that lacks any deeper-level structural significance. 3.2.3. Predominant preparations There are several non-generalizable preparations for different predominant chords. (4) IV → iii IV (5) ii6 → vi ii6 (6) vi → iii(6) vi

Rule (4) expresses the somewhat less frequent case in which IV is prepared by iii in the major mode. Rules (5) and (6) express diatonic fifth relationships that apply in the context of both major and minor modes (given diatonic adaptations of the respective harmonies). 3.2.4. General preparations: Applied dominants Several preparations of dominants, predominants, and other chords appearing in a cadential context result from the general flexibility with regard to preparing chords

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with applied implicative chords. Analogous to the formalization of dominant preparations, all the general preparations discussed here follow the same abstract rule pattern (note the similarity to the general rule (3) above): (7) X → X-Prep X

In this context, V6(5) and vii°(7) may function as chord preparations from a semitone below (for instance, in a rule such as X → V6(5)/X X). Ger6, Fr6, It6, and V42/X6 can function as preparations from a semitone above, and V/X as a preparation from a fourth below. In particular, chords on IV and ii are often approached via their own secondary dominants (either V6(5) or vii°(7)).50 IV6 may be prepared by its applied V42. Specifically, the cadence-initiating tonic (I or I6) may also be embellished from above by an applied dominant (e.g., V43, V42, vii°6). Augmented sixth chords on ¨6^ constitute a common way to prepare the dominant in minor cadences. It is important to note that it is possible (although rare) for some of these preparatory embellishments to themselves be recursively embellished—for instance, a prepared Neapolitan, as found in Mozart’s Piano Concerto in D minor, K. 466 (mm. 347f.; see the discussion below). However, this list of embellishments is not exhaustive, as further examples may be discovered (e.g., the harmonically flexible diminished chord and its variants). Further note that for the sake of simplicity, this list of rules does not imply an explicit ranking of different preparations with respect to their order.51 3.2.5. Sample analysis A sample analysis will illustrate the formalism of the cadential structure developed in the previous sections (see Fig. 5). This example shows that the multiple predominants are characterized by the fact that they are all dependent on V. For instance, there is no rule that vii°7/V may follow ii6; rather, both chords are dependent on V. Further, it is important to note that V65/IV (or vii°/IV) is very similar to I6, yet it fulfills a different function: Instead of serving as an initial tonic, it acts as the subordinate applied dominant preparation of IV (compare the discussion above regarding the tonic frame). Note that this tree diagram (like all syntax trees) contains all rules that were applied to generate or parse the tree structure. From the top, some of the rules involved are I → I I, I → V I, V → IV V, IV → V65/IV IV, etc. The tree represents the (atemporal) dependency relationships between the elements and does not have direct implications regarding the order of rule applications when generating or parsing such a sequence.

50. Note that in Caplin’s theory, V65/IV is considered an initial tonic embellishment, rather than a predominant embellishment; see Caplin, Classical Form (1998), 29. 51. Such an encoding could be achieved by using different symbols for V that represent different stages of preparations (such that preparations on 4^ would precede preparations on ©4^).

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Figure 5: Analysis of an elaborated PAC with the sequence I–V65/IV–IV(5–6)–ii6–vii°7/V–V64–V7–I

3.3. The order of dominant and predominant embellishments

There is considerable variety in the ways in which dominants and predominants may be prepared in a cadential environment. It is by no means a simple task to define the specific constraints governing the temporal order of these elements, as there is a remarkable combinatorial power inherent in cadences and their underlying syntax. However, it is possible to state several general constraints. First, predominant and dominant elements establish their own order in straightforward ways: One or more dominants may be used to prepare the tonic; none, one, or more diatonic predominant elements may precede each dominant. Within diatonic predominants, IV (when present) usually precedes ii or ii6, whereas the order that chords over 6^ take is more flexible. Second, within this diatonic frame, chromatic embellishments may enter the cadential pattern, creating additional implications. For instance, embellishing chords on ©4^ (such as V65/V or vii°7/V) may be inserted between 4^ and 5^. The same holds for embellishing chords on (¨)6^ (either V43/V or vii°6/V) or ¨6^ in minor mode (e.g., Ger6). Both of these elements— chords based on ©4^ and on 6^—may be present, the order being interchangeable. Once scale degree 5^ is reached, a cadential V64 may be introduced to delay the entrance of V(7). These rules indirectly control the sequential order of the cadential constituents. The table below summarizes the different components that affect the order of the elements involved. In this context, it is important to bear in mind, as mentioned above, that the difficulties arising from the attempt to bring cadential elements into a linear order stem from the fact that two structures, linearity in the bass line and hierarchical harmonic dependency, are closely linked, and linear order is sought where the structure arises from hierarchical organization. Note that because the generation process

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is recursive, a number of complex chord progressions can be generated as a result of coordination and recursive expansion (see Section 5.1 below). Table 1: Overview of the order of different dominant embellishments and dominant or predominant preparations

V64, V54

Dominant embellishments Order of dominant preparations Order of predominant preparations

diatonic

ii6(5), IV, ii, IV6, VI, ii43, ii64

other

V/V, V/V6(5), vii°/V(7), Ger6/V, It6/V, Fr6/V, N6

diatonic

IV: I, iii, V6(5)/IV, I6, vii°6/IV ii(6): vi, V42/ii, vii°6/ii vi: iii(6)

other

X: V/X, V6(5)/X, vii°(7)/X, Ger6/X, It6/X, Fr6/X, N6/X, V42/X6

4. Ways of avoiding (perfect) cadential closure In the history of music, composers have used numerous devices to play with the strong patterns of expectancy conveyed by the cadence, manipulating them by means of delaying, replacing, or omitting the final tonic. Naturally, these cases are far more difficult to systematize than authentic cadential closure, since breaking an established formulaic structure is a creative act that is open to all kinds of possibilities. Nonetheless, there are some underlying syntactic structures common to all of these cases, in particular the usage of operations such as replacement, coordination, and elision, all elements known from linguistics. 4.1. The imperfect authentic cadence

The so-called “imperfect authentic cadence” (IAC) does not differ from the PAC in terms of its syntactic parse or its harmonic structure; rather, it differs with respect to voice leading. An IAC is generated such that the final tonic does not support the first scale degree in the soprano (as is the case with a PAC), but rather scale degree 3^, which may represent the genuine goal of a descending linear motion, as is the case with the Prinner cadence (see the example given in Fig. 6).52 A sense of closure is imparted to the listener because the final chord is a tonic (a fact also reflected in the tree); only the voice-leading treatment weakens the sense of closure. For this reason, a purely harmonic interpretation may prove insufficient for a proper char-

52. See Caplin in this volume.

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acterization of the IAC, as it does not include the treatment of the upper voice—the very criterion that sets the IAC apart from the PAC. Although this difference is not reflected in the tree at the chosen level of representation, it informs the interface between the syntactic dependency, constituency structure, and voice leading and thus disambiguates the parsing process.

Allegro moderato                                  mf  p                    

Figure 6: Cadential closure reached by a Prinner cadence, see Haydn, Keyboard Sonata in D major, Hob. XVI:14/i, mm. 1–4

While in some situations, scale degree 3^ may indeed be the proper melodic goal, in other situations perfect cadential closure is purposefully avoided by disappointing the expected voice-leading motion: Rather than completing the melodic line by moving down to 1^ (as shown in (a)), this pattern is interrupted, leading back to 3^ (potentially with a ©2 preparation, as shown in (b)) and thus creating a deceptive effect. (a) 5–4–3–2–1 (b) 5–4–3–2–[©2]–3

This structure ensures that the phrase context remains open and implies further continuation. In other words, the imperfect authentic cadence may trigger a subsequent phrase structure (most typically by using a “one-more-time” strategy53) upon which the preceding phrase is syntactically dependent. In this case, it is advantageous to model the syntactic structure of the overarching phrase structure in terms of a recursively embedded phrase that is inserted between the tonic (supporting 1^) bringing about perfect cadential closure and the original V (supporting 2^). The following example by Mozart illustrates this (see Figure 7).

53. Schmalfeldt, “Cadential Processes” (1992).

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                          f p                     p

                           f

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            

     

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    

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vii /vi

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Towards a Syntax of the Classical Cadence 309

Figure 7: Mozart, Piano Sonata in F major, K. 332/i, mm. 71–93. IAC as a deferral of perfect authentic closure through2the recursive embedding of another phrase. This structure is combined with a one-more-time pattern.

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4.2. The half cadence

4.2.1. Distinctive features of half cadences Half cadences are often described as incomplete forms of authentic cadences, lacking the final tonic and hence the necessary requirement for an authentic cadence. However, half cadences also seem to be special in the way they function. Several contextual features distinguish half cadences from authentic cadences. Some of these features help to weaken the tonic-implications of the dominant, stabilizing the latter in such a way that the degree of closure conveyed by a given phrase is partial rather than perfect. One of these features concerns the surface realization of the dominant: Whereas authentic cadences often use V7 to approach the final tonic, a dissonant V chord has been regarded as irregular when it functions as the final moment in a half cadence.54 However, Poundie Burstein argues that half cadences can likewise end on a V7 sonority, a notion previously rejected by theorists, as this chord had been considered too unstable a harmony to act as a true (local) goal.55 In addition, half cadences seem to make occasional use of a dominant sonority in first inversion (e.g., Mozart, Piano Sonata in C major, K. 309/i, mm. 106–108), an option that is irregular in the context of authentic cadences. Irrespective of these distinctions, it is important to note that a (phrase-initiating) I immediately following the halfcadential V is merely a coincidental surface adjacency resulting from the juxtaposition of two phrases, rather than a structural V–I relation. Another difference between half and authentic cadences is derived from metrical criteria: Whereas the concluding V in a half cadence typically arrives on a strong beat, in an authentic cadence, V more frequently occurs on a relatively weak beat. This suffices to make it clear that the generation of half cadences also differs from the generation of authentic cadences in the ways non-harmonic factors are affected. This concerns in particular factors that help determine grouping boundaries (such as pitch proximity, rests, articulation, texture, register, dynamics, and relative lengthening, factors that have been formalized in the grouping preference rules 2 and 3 in the GTTM). The generation of such grouping boundaries, in conjunction with the cadence type, strongly contributes to the recognition of the half cadence in the reverse-engineering (i.e., parsing/listening) process, since the mere harmonic structure generated by the syntactic core can frequently result in ambiguities that may, e.g., arise from an unrelated phrase-initiating tonic following a half-cadential dominant.56

54. See Burstein in this volume. 55. Ibid. 56. Lerdahl and Jackendoff note: “If a grouping boundary intervenes between the two chords, the V does not resolve into the I; instead the V ends a group and is heard as a half cadence, and the I is heard as launching a new phrase. Metrical structure alone cannot account for these discriminations, precisely

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Grouping boundaries are an essential factor in disambiguating parses that would be ambiguous based on the harmonic structure alone, and they make it possible to determine whether the surface has been generated by the underlying form (phrase type) of a half cadence or an authentic cadence. In cases in which harmonies other than the tonic follow a half cadence, such ambiguities are clearly avoided, reinforcing the half cadence parse. Half cadences and authentic cadences might also differ statistically from each other with respect to both the functional sonorities used as predominants and the bassline. Phrygian half cadences often approach their goal by a descending bass melody, unlike the majority of authentic cadences. Furthermore, in authentic cadences, it is generally rare to have a 6–5 tenor clausula in the bass before arriving at the final tonic (e.g., Beethoven, Piano Sonata op. 7/ii, mm. 22–24). Consequently, authentic cadences seldom use an augmented-sixth chord as a predominant (they may rarely use this chord prior to a V64), whereas this is the defining feature of a Phrygian half cadence. In half cadences, by contrast, the final dominant is frequently approached by a 6–5 motion in the bass. For instance, the half-cadential progression over a (¨)6–5 line in the bass (clausula tenorizans) can best be understood as achieving a sense of local stability with the arrival on V. This V represents a genuine goal harmony, rather than interrupting an authentic cadential progression. 4.2.2. Revisiting the syntax of the half cadence: Where on earth is the missing tonic? The points outlined above highlight the fact that there are striking differences between half cadences and authentic cadences in terms of stability, metrical structure, and voice leading, all features that help disambiguate grouping structure. Notably, in all forms of half cadences, the chord progression lacks a tonic implied by V within the confines of the phrase. One challenge for a generative approach is to explain how to reconcile the missing tonic with the core of the cadential tonic frame. The reason why the missing tonic creates a problem for the generative approach is that the dominant can only be generated with reference to its implied tonic (or as the goal of a tonicization), otherwise it would be “left hanging” in empty space, violating the dependency principle. One core test for the investigation of the syntactic structure of the missing tonic consists of tonic completion: It is possible for almost all types of half cadences to transform their harmonic sequence into an authentic cadence by adding the missing tonic and by adjusting nonsyntactic parameters such as the meter, rhythm, melody, and other features that are affected by the syntactic structure when generating the surface. The example of Mozart’s Piano Sonata in A major, K. 331 (see Figure 9, below), may serve as a useful illustration of a composed version of such a tonic because it has no inherent grouping. Both components are needed” (Lerdahl and Jackendoff, A Generative Theory of  Tonal Music [1983], 29).

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completion test. The fact that tonic completion is possible in a large number of cases motivates solutions that explain half cadences in terms of manipulated authentic cadential progressions. Such solutions are likely to end up being more parsimonious than transformation-free approaches because they avoid adding inelegant rules that generate V as the stable, reference-free end of a phrase and might interfere in problematic ways in non-cadential contexts. The fact that there are some forms of half cadences (such as Phrygian half cadences or simple half cadences such as V65– I–V) that cannot easily be completed (or transformed) in such a way could suggest that there may be additional factors determining the type of half cadence. However, this does not necessarily diminish the general power of our proposed analysis or the usefulness of the completion test for a significant number of cases. Overall, there are several options for dealing with the problem of the missing tonic: (a) proposing a different tree analysis in which V is not subordinated to the final I (but instead, for instance, to the initial tonic; see Fig. 8a), (b) using an empty (hidden) element of a I that is present in the analysis but does appear on the musical surface (see Fig. 8b), (c) modeling V as dependent on the final I of the subsequent phrase context (see Fig. 8c), and (d) assuming that the I upon which V depends was moved away, either to an earlier or a later position (see Fig. 8d). Options (c) and (d) are similar; one of the main differences is that (d) maintains the parallelism and coordination between the two analogous phrases, whereas (c) models the second phrase as subordinate to the first phrase at the final tonic position.

Figure 8a: Departure analysis of V as subordinate to the preceding I (tonic departure). The GTTM prefers this analysis. This solution models the V as not having any implications of the phrase-final I.

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Figure 8b: Empty element analysis. The tonic I first generates V and is then rendered empty. The epsilon symbol represents the empty element.

Figure 8c: Subordination analysis of the half cadence. The consequent (or one-more-time) phrase is subordinate to the antecedent phrase, and the final tonic of the consequent phrase terminates the tonic implication of the V of the initial half cadence. Note that because the initial and final tonic nodes of the consequent phrase are joined at a higher level, both the string-adjacent and the sequence-final tonic relate to the open V of the half cadence. This analysis models the open structure of the underlying (Schenkerian) interruption as well as its continuity.

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Figure 8d: Movement analysis of the half cadence. This analysis is similar to 6b (empty element) and 6c (subordination). In contrast to the previous analyses, however, both the antecedent and consequent phrases are arranged in a paratactic (not hypotactic) way. The tonic implication regarding the phrase-final tonic is achieved through a rightwards movement operation: Beginning from a perfect cadence in the antecedent phrase, the final tonic I is moved to the right and merged with the tonic seed of the consequent phrase. In this way, the analysis manages to retain the paratactic order of both sequences while still maintaining the implication of the half cadence to the final V.57

It is important to bear in mind that previous syntactic approaches have faced considerable difficulties in making sense of the half cadence, which may perhaps best be exemplified by Lerdahl and Jackendoff’s account (analogous to the option depicted in Figure 8a): In order to explain the counterintuitive result that the phrase-concluding V is subordinate to the preceding tonic in their prolongational reduction, the authors propose to think of “cadences as signs, or conventional formulas, that mark and articulate the ends of groups from phrase levels to the most global levels of musical structure.”58 Rather than invoking the heavy baggage of a concept that is external to one’s theory, such as the semiotic notion of a musical sign, the present approach addresses the issue of cadence with the help of syntactic methods only. The final analysis, depicted in Figure 8d, involves a structure that expresses a dependency relationship between the two tonic elements that exceeds what 57. It is important to note that this movement operation cannot be achieved by means of a contextfree grammar (i.e., a tree-structured analysis); it requires a more complex mildly context-sensitive grammar that in turn entails remarkably greater complexity in terms of parsing, processing, and learning the grammatical structure. 58. Lerdahl and Jackendoff, A Generative Theory of  Tonal Music (1983), 134.

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a context-free grammar is capable of expressing. While there are historical precedents,59 the analysis of the half cadence constitutes a promising candidate for analogies with complex syntactic structure in linguistics.60 The open dominant at the end of the first phrase may be analyzed such that the final tonic node of the first phrase is associated with the final tonic of the second (or final) phrase. This can be understood as the first tonic “moving” to the end of the second phrase or the first dominant selecting the final tonic of the second phrase. The final tonic of the first authentic cadence form is not moved to a position before the V but rather to the end of the following phrase, forming the overarching structural head for both phrases (for instance, within the antecedent-consequent structure of a period). Thus, the second analysis emphasizes the view that the V of the half cadence may be locally stable (despite the lack of a directly adjacent tonic and without any immediate need for resolution), yet in an overarching hierarchical context, it is closely linked to the higher-order superordinate tonic. This view is also reflected in the Schenkerian concept of interruption: As Cadwallader and Gagné note, the I immediately following a half cadence may not necessarily be regarded as providing a moment of tonic closure.61 According to a Schenkerian analysis, periods are viewed as a bipartite structure resulting from an interrupted (or divided) structural motion from the primary note (3, 5, or 8) to the first scale degree. Since the first attempt at completing the fundamental line fails because of the interruption on 2^ (supported by V), this motivates a repetition of the

59. One historical example of an explanation by transformation accounts for the half cadence by means of a V–I inversion thus: x V I → x I V. This analysis makes use of an operation that reverses the sequence V–I from an authentic cadence to I–V. This view is found in Koch’s Handbuch bey dem Studium der Harmonie, in which the half cadence is regarded as the inverted form of the authentic cadence in harmonic as well as metrical terms: “With regard to the underlying harmony, the original form of the half cadence is the reversed form of the Kadenz [authentic cadence], that is, in the half cadence, the tonic triad precedes the dominant triad on a weak beat [...]” (Koch, Handbuch [1811], 378 [our translation]). Today, this explanation is commonly considered incorrect, and the argument can be countered in the following way: Half-cadential subtypes such as the converging or the Phrygian half cadences do not feature a tonic preceding the concluding dominant sonority. In these progressions that feature a I at their very beginning, this I is structurally identical to those found at the beginning of authentic cadences. Accordingly, there are cases in which there are no positions where the I could move if it were required to move towards the left. Hence, the reversal explanation is only partial and requires another significant addition to the formalism to explain other variants of the half cadence (that may ultimately render this form of reversal unnecessary). If one assumed a combination of reversal and empty element (i.e., moving to an empty element position), this solution would be inelegant because the empty element could be directly positioned after the V without requiring any movement at all. 60. At present, it remains an open question whether or not forms of transformations and movements known from linguistics occur in music or are meaningfully necessary in music analysis. For a discussion of cadential V–I locality as another potential phenomenon requiring movement, see Katz and Pesetsky, “The Identity Thesis for Language and Music” (submitted manuscript). 61. Cadwallader and Gagné, Analysis of  Tonal Music (2011), 119f.

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whole phrase,62 this time completing the line by moving further to 1^/I: I–V || I–V–I. According to the Schenkerian interpretation, the half cadential V is a higher-order event that remains active across the phrase boundary between antecedent and consequent, aiming at the theme-concluding tonic. However, the structural dominant that resolves the tension is that of the consequent phrase, not the half-cadential V concluding the antecedent.63 In this respect, there is an important structural similarity between the half cadence and the deceptive (or evaded) cadence, both of which imply the use a one-more-time technique. From a syntactic perspective, the analysis of interruption outlined in Fig. 8d may be regarded as analogous to the linguistic phenomenon of right node raising as seen in the following example: Mary likes ___ and John hates tomatoes.

In brief, the two verbs “likes” and “hates” can be analyzed as being coordinated and jointly selecting the noun “tomatoes.” Alternatively, one may wish to employ the concept of rightward across-the-board movement, analyzing the noun “tomato” as having been moved to the end of the sentence.64 Which of the pre-existing linguistic approaches (e.g., rightward across-the-board movement or type-raising, as is typical in categorial grammar65) the musical analysis adopts need not concern us at this (early) stage in building a syntactic theory of the cadence. What matters is that the movement analysis outlined above indeed proposes a structure that is capable of reflecting the unrealized forward implication of the interruption, the sense of local completion, and the parallelism and paratactic arrangement of the two phrases. This movement-based solution must be compared to other possible accounts that do not use movement. First, an empty element solution (lacking the first tonic; see Fig. 8b) cannot explain the open implication of an interruption and hence the syntactic motivation for the second phrase. It would further predict that phrases in general could end on a missing empty tonic, which is problematic in cases in which there must be a final tonic (e.g., phrases that conclude entire sections or pieces). Second, a solution in which V is dependent on the initial tonic (see Fig. 8a), as proposed by the GTTM, faces the same problem as the empty element solution and is inconsistent with the interpretation of the trees and strong generativity (V as a forward implication to a final tonic, not as a departure from the initial tonic), as well as the other rules for V (it would require a specific rule on its own). Third, a solution such as that proposed 62. Note that the present syntactic formalism does not enforce the parallelism that both phrases are built identically. 63. Lerdahl and Jackendoff, A Generative Theory of Tonal Music (1983), 140. 64. Across-the-board movement constitutes a problematic example in generative linguistics, since one of the core tenets requires movement to be only left-directed (i.e., against the direction of speech). 65. See Steedman, The Syntactic Process (2000).

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by Fig. 8c is consistent with the unrealized V-implication and the one-more-time pattern. The only issue is that the two phrases are not paratactic, but instead hypotactically arranged. It is clearly possible to model interruptions in this way.66 One final problem with considering the half cadence as an instance of movement is that the dissonant V7 chord very rarely (if at all) occurs in half cadences, whereas it frequently occurs in perfect cadences (see above). Pure movement of the tonic to a different position may not account for this phenomenon, and this may mean that only the late surface realization of the subsequent tonic in fact licenses the appearance of the seventh in the subsequent generation process. Overall, whether the lack of paratactic analogous phrase structure truly necessitates the introduction of a new syntactic mechanism that massively increases the formal complexity of the syntactic mechanism (beyond context-free) is unclear and remains questionable. Future research may shed more light on this point. Nonetheless, two syntactic mechanisms manage to account for interruption, which is impossible under Lerdahl and Jackendoff’s system, even in their analysis of Schenkerian interruption. An excerpt from Mozart’s A-major Piano Sonata, K. 331 (see Fig. 9), illustrates the above point, showing that it is possible to model interruption with the help of a syntactic approach. Andante grazioso                              p                                  

                         

                   

          

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66. This has been previously proposed by Lerdahl and Jackendoff, A Generative Theory of  Tonal Music (1983), and Rohrmeier, “Towards a Generative Syntax of Tonal Harmony” (2011), and employed in Koelsch et al., “Processing of Hierarchical Syntactic Structure in Music” (2013).

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I

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Figure 9: Analysis of the theme (mm. 1–8) from Mozart’s Piano Sonata in A major, K. 331, using the movement analysis of half cadences

4.3. The deceptive cadence as coordination and recursive embedding

The deceptive cadence constitutes a more radical way of avoiding cadential closure than the half cadence.67 Rather than leaving the V-implication unrealized and establishing the V chord as the ultimate goal, the deceptive cadence builds up a “normal” cadential framework, one that raises but ultimately frustrates strong tonic expectations by introducing a non-tonic chord (or a first-inversion tonic) at the final stage of the cadential progression. In the classical style, a variety of non-tonic chords have been employed to break the expected pattern. Some of these chords may be viewed as acting as tonic replacements (such as vi or ¨VI), others less so (e.g., vii°6/V or viio7/vi). The latter situation can be illustrated by the following two examples: (a) I–viio6–viio/IV–IV–iv–V–It6–V64–V–viio7/vi–vi (b) I6–IV–V64–V7–viio7/vi–vi

In both (a) and (b), a strong cadential context sets up a drive towards I that is (temporarily) suspended by vii°7/vi, which itself leads to vi (thus acting as an embellishment of the submediant). While music theorists have primarily devoted attention to the local surprise effect conveyed by the deceptive cadence, linguistically minded attempts at modeling the syntactic structure of the deceptive cadence must take the overarching phrase context into account. If we understand the vi chord as a tonic substitute, as has often been proposed in the literature,68 the tree structure of the cadence would be almost 1 67. See also Neuwirth in this volume. 68. Aldwell and Schachter, Harmony and Voice Leading (2003), 197; Caplin, Classical Form (1998), 25.

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the same as for an authentic cadence: The tonic head that governs the entire phrase would be rewritten as a tonic replacement with the unary rule “I → vi” after the V chord is derived (maintaining that V does not prepare vi, but rather I). The tree resulting from this interpretation would suggest a structure that is completely closed at the moment of arrival on the tonic replacement. While this reading may be adequate for some cases (especially in which there is a sense of redirected closure), in many cases the overarching phrase context is different: The deceptive tonic replacement vi does not act as a completion of the preceding phrase, although it may convey a local sense of rest; instead, it functions as the setup to a subsequent phrase that promises cadential closure. Often this second attempt at closure is a repetition of the previous phrase, in the manner of a one-more-time pattern. This yields a different interpretation of the syntactic structure: Rather than being a tonic replacement, the non-tonic chord at the end of the deceptive cadence functions as a preparation of another harmonic sequence that finally leads to the previously suspended tonic. Expressed in formal terms, the deceptive cadence effects the recursive embedding of another, longer phrase into a cadential context. Therefore, the deceptive chord itself represents a far left-branching subordinate dependency of a second final tonic phrase. This understanding further predicts that other chords (or almost any chord) may occur in a deceptive cadential context, or more precisely: Any chord that occurs in the left-hand corner of a V constituent may function as a deceptive sonority. The following examples illustrate this point as well as the empirical occurrence of the predicted flexibility. The first example is by Haydn (see Fig. 10) and exemplifies several quite complex aspects: (c) vii°6/V–V7–I7 == V7/IV–IV–I6–ii6–V–vii°6/V–V7–I7 == V7/IV–IV–I6–ii6–V–I– I6–ii6–I6–ii6–V–I

This example combines the avoidance of cadential closure through deceptive strategies with several nested one-more-time patterns. What is remarkable about this excerpt is that cadential closure is denied no fewer than three times, with each deceptive cadence followed by a further attempt at a complete cadence. The first cadential progression replaces the expected tonic not by a stable submediant sonority, but by a much more active secondary dominant function, vii°6/V. The deceptive harmony sets up an expectation for another attempt at completing the cadence.69 This expectation is indeed fulfilled: A second cadential goal is approached in mm. 47–48. Here, again, the soprano carries out its expected, typical concluding formula (1^–2^–1^, featuring a trill on the penultimate 2^), but once more the bass denies closure by moving from 69. Note that here we indeed have a context in which I6 is subordinate to ii6, since otherwise the tree lines would cross if ii6 were linked to its related I (the same holds for the two subsequent one-moretime patterns). The phrase continues with a repetition of the same deceptive patterns before cadential closure is reached via another two (simpler) embedded one-more-time patterns.

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3–4–5 to 6, the latter again harmonized as vii°6/V. The preceding unit is now repeated in its entirety (mm. 48–52 ~ 43–47). However, this time, the roles of the soprano and bass are exchanged: Whereas the bass provides its expected clausula 5–1, the soprano violates expectations by moving from 2^ to 3^ (rather than repeating its previous 1^–2^–1^ pattern). This cadence may be heard as an imperfect authentic cadence, but because a perfect authentic cadence is truly expected at this point, and this expectation is disappointed, we can refer to this pattern as a melodically deceptive cadence. The unsatisfactory character of the IAC is also revealed by the fact that a two-bar unit is attached to this cadence, closing the exposition as a whole by completing the previously denied 1^–2^–1^ pattern (mm. 54–55).

        p                   

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Figure 10: Haydn, Keyboard Sonata in G major, Hob. XVI:27/i, mm. 38–57. Modeling the 2 deceptive cadence as a pattern of recursive embedding. The deceptive context vii°/ii illustrates the fact that the deceptive cadence forms part of a one-more-time pattern in which the deceptive chord initiates the phrase. The parse tree shows that the second phrase constitutes a copy of the first phrase that is recursively embedded within the V–I concluding the first phrase.

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Figure 11: Mozart, Piano Sonata in F major, K. 279/ii, mm. 1–6: An example of a deceptive cadence based on a IV6 chord within the context of a one-more-time pattern

Another example by Mozart further illustrates the proposed analysis of the deceptive cadence (see Fig. 11). In this example, which contains another one-more-time pattern, it is a IV6 chord that denies cadential closure—that is, a chord that cannot function as a tonic replacement.70 In what follows, this chord initiates a second subsequent embedded cadential preparation. This embedded deferral ensures that a level of tension is maintained from the previous phrase and increased with the new subsequent cadential context. This analysis shows that the syntactic tree can express the fact that the tonic chords in mm. 2 and 4 serve a double function: They close the preceding constituent and initiate the subsequent constituent (analogous to an elision; see below). It further represents both the disruption of the expected cadential closure by IV6 as well as its overarching embeddedness in the surrounding dependency relation-

70. At times, however, IV6 is characterized as a variant of vi produced by a 5–6 motion over a sustained bass note (6^).

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ships. This example, like the previous one, illustrates how closely connected methods of deferring cadential closure are to the general syntactic principle of coordination and embedding (as discussed in detail in Section 5.1. below). A third example exhibits a context in which a diminished chord fulfills the deceptive function (see Fig. 12). This example further shows that in terms of an embedded tonic deferral, the deceptive cadence is independent of the use of a one-more-time pattern; it may or may not be found in conjunction with it.

Figure 12: Haydn, Keyboard Sonata in F major, Hob. XVI:21/ii, mm. 13–26: An example of a deceptive cadence based on a diminished chord

A fourth example (see Fig. 13) demonstrates that even a Neapolitan chord can follow V in a deceptive cadential context and may nonetheless give rise to a regular, syntactically well-formed structure.71 The syntactic analysis shows that V and N6, although 71. Note that this situation could also be analyzed as involving a cadence evasion (see below), since the Neapolitan sonority does not articulate the goal of the preceding phrase, but rather launches a new one. This relates to the discussion on interpreting the neurocognitive ERAN patterns of Neapolitans after a cadential context (as presented by Koelsch, Brain and Music [2012]). The interpretation of such

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string-adjacent on the surface, are in fact not dependent in the deep structure; the two chords belong to different branches of the tree.

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Figure 13: Haydn, Keyboard Sonata in E-flat major, Hob. XVI:28/i, mm. 82–89. An example of a deceptive cadence based on a Neapolitan chord that follows V and initiates a subsequent phrase with a tonic continuation (no one-more-time pattern in this case). Note that the sequence is syntactically regular and well-formed despite the Neapolitan following V.

Any attempt at understanding the V–N6 chord succession as a direct progression would render the structure ungrammatical, as this succession itself is highly unlikely (or implausible), especially with the Neapolitan occurring on a strong beat. Rules such as ii prepares V, V prepares I, etc., are hierarchical rules (V → ii V) rather than simple Markovian progressions. Therefore, a context-free grammar approach need not consider rules such as V → V N6 or N6 → V N6, whereas a strictly local or Markovian approach would be obliged to. In contrast, V(7) → N6 V(7) would be an appropriate rule that would license the example as grammatical, with reference to a (remote) second V chord. After all, merely sequential or strictly local rules (such as those used in a Markov model or a Piston progression table) cannot both adequately describe this harmonic structure and at the same time characterize irregular V–N6 progressions. structures should not be based on the distinction between regular and irregular sequences but instead on “completion vs. revision” in musical processing (as suggested by Koelsch, ibid.).

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However, not all deceptive cadences are explicable in terms of embedded phrases ultimately leading to the previously withheld tonic (“delayed realization”). Although it is crucial to understand the syntax of the deceptive cadence from the perspective of the overarching musical context, this context may not necessarily be the preceding prolonged tonic but also the subsequent (novel) musical segment. An example by Haydn, the first movement from his D-major Keyboard Sonata, Hob. XVI:37/i, illustrates this case (see Fig. 14). In this example, the deceptive chord (V7/iv in the key of vi) serves a double function, not only concluding the development section but also initiating the modulation back to the home key by means of a brief descending-fifths sequence, with the return to the tonic coinciding with the entrance of the primary theme that launches the recapitulatory rotation (i.e., an instance of denied realization). From a syntactic perspective, this example can be modelled by two different analyses: If one prefers the consistency of syntactic embeddings, this example forces an analysis that treats the entire B-minor section as a subordinate preparation of the subsequent tonic; that is, (part of) the development section would be syntactically subordinate to (and preparatory of) the subsequent head tonic of the recapitulation. However, there is a second interpretive option, one that is analogous to the (linguistic) notion of “elision” as used by Lerdahl and Jackendoff.72 As the example shows, it is not a remote non-tonic harmony that replaces the expected B-minor tonic, but rather the right tonic root (B) upon which a major seventh (dominant) rather than a minor tonic chord is built, thus transforming the expected minor tonic into a secondary dominant. In other words, we hear the expected tonic root at the end of the cadence, but its function has been transformed. This interpretation can be reframed using the linguistic concept of elision: Instead of saying that the tonic is transformed, the example can be analyzed such that the final tonic and the subsequent dominant-seventh chord that share the same root and fifth are merged into one sonority. Instead of two chords that link different ends of branches in the tree (as in the Neapolitan example above), the surface adjacency and similarity of the two chords permits them to be merged into one item that fulfils two different functions in the tree. This analysis does not require (as the first subordination analysis does) that the development section as a whole be understood as subordinate to the recapitulation. In addition, it emphasizes the implicative density inherent in the classical style and links it to the notion of elision.

72. In linguistics, elision refers to the omission of an expected sound in such a way that the resulting sequence is easier to pronounce or to produce. In English, for instance, “vegetable” is pronounced like “vegtable,” and in German “teuerer” becomes “teurer.” Lerdahl and Jackendoff (A Generative Theory of  Tonal Music [1983], Chapters 3 and 4) discuss elision in the context of two overlapping groups in a grouping structure and beat overlaps in a metrical structure.

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Figure 14: Deceptive cadence with denied realization rather than delayed realization, from Haydn, Keyboard Sonata in D major, Hob. XVI:37/i, mm. 54–64

4.4. Evaded cadence

Another important strategy of avoiding cadential closure must also be mentioned here: the evaded cadence.73 Unlike a deceptive cadence, in which the deceptive harmony groups backwards, the evaded cadence is characterized by the fact that the harmony following the penultimate cadential dominant is cut off from the preceding phrase, initiating a new formal unit. Whereas the deceptive cadence features a goal harmony (although a deceptive one) on the level of mere surface relationships of metrical and phrase structure, the evaded cadence does not. Cadential evasion is typically generated by means of secondary parameters (most importantly, texture and register). The harmony most commonly used to launch a new phrase is I6, but other harmonies can be chosen here as well (e.g., V7/IV); the decisive criterion for differentiating an evaded cadence from a deceptive one is not the identity of the chord following the dominant (since almost any chord can initiate the one-more-time pattern), but the location of the grouping boundary. A special (and frequently used) variant of the evaded cadence 73. Schmalfeldt, “Cadential Processes” (1992).

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even allows for the inversion of the penultimate dominant74: At the moment when the proper resolution of a cadential 64 is expected, the bass starts moving downwards (5–4), thus giving rise to a V42 sonority moving to I6 (typically initiating a new group). In this sense, one can conceive of this variant as a possible means of embellishing the I6 chord. Despite the surface differences with respect to the deceptive cadence, it can be argued that the deep structural relationships underlying the evaded cadence are fundamentally analogous to those modeling the deceptive cadence. For both types of cadences, cadential deferral is achieved via a recursive insertion of a chord progression between V and the final I.

5. The generative power of syntactic formalisms 5.1. Coordination

The formalism outlined above characterizes the different elements of the cadential progression and their embellishments. However, another aspect must be addressed here as well, namely the coordination and expansion of cadential elements, as this will allow us to account for some more extravagant examples that occur in actual compositional practice. The following list displays some of these examples: (a) V64–V–V64–V–I (b) I–V64–V7–V64–V7–V64–V7–I (c) I–V7–vi–ii6–V64–V–I (d) i–iv65–ii65–V7–I65–IV7–VI7–ii°6–V64–V7–i (e) I–IV–ii–V–V7–V64–viio43/ii–ii6–V64–V7–I (f ) I–ii6–V–viio7/vi–vi–ii6–V64–V7–I (g) V64–V7–V64–V7/V–V–V7–V64–V7–vii°7/vi–vii°7/V–V64–V7–i (h) I6–ii6–ii65–V7–vi–ii6–V64–V–I

In each of these cases, either there are multiple V harmonies, or the V harmony is extended through an insertion that sets up a sequence of implications to another subsequent V. However, in the latter case, the inserted second sequence is not necessarily implied by the preceding V (e.g., V–viio7/vi in example (f )). This pattern is structurally similar to, and may indeed be combined with, a deceptive cadence. To account for this phenomenon, one must find a way to make sense of these multiple instances of V. There are several ways to achieve this. First, one could regard these different instances of V as separate left-branching derivations from I, or as coordinated instances of an overarching V group. In the case of coordination, a second possible distinction concerns whether more than two instances are simultaneously derived 74. Note that this is the only form of cadence that allows a V42 chord.

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from the same parent node (referred to as “n-ary branching”), or whether branching is restricted to binary branching, forcing a binary hierarchical structure in different V nodes.75 Moreover, coordination may be applied at the highest level or below, depending on whether predominants such as IV prepare a single instance of V within the dominant group or the entire group. Although there is a practical limit imposed on the number of coordination instances of V, this limit is not a theoretical one and should not be incorporated at the level of this rule (since we are able to correctly perceive or generate an even larger number of coordinations). Figure 15 illustrates the different ways of modeling multiple V harmonies.

Figure 15: Different methods of accounting for multiple V nodes

The derivation of the first example in Fig. 15 (left) does not require any additional rules, since it follows from multiple applications of the I → V I rule. The second case (middle) requires a rule such as (8), which coordinates multiple instances of V in a single derivation from their V parent. In contrast, the third option (right) enforces binary branching, employing a rule such as (9) multiple times to create internal structure in the different V nodes. (8) V → V+ (that is: V → V | V V | V V V | …) (9) V → V V

Of course, the coordination principle is not restricted to the V chord; there are other relevant instances that do not concern V. For example, the very end of the first movement of Mozart’s D-minor Piano Concerto, K. 466, features a repeated alternation between V6/iv and iv (mm. 390–393) that prolongs the subdominant harmony and

75. This point is also discussed in detail in the GTTM; see Lerdahl and Jackendoff, A Generative Theory of Tonal Music (1983), e.g. 326–328.

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thus represents an instance of subdominant coordination. Other situations of alternating harmonies cannot be easily explained in terms of coordination and must be analyzed more carefully. Consider, for example, the following harmonic sequence: (i) I6–ii65–V64–iv6–V64–iv6–V64–vii°7/V–V

In this case, we do not want to model the multiple instances of V64 and/or iv6 via coordination, since this would only make sense if, for instance, iv6 were subordinated to V64 or vice versa. Otherwise, the derivation would involve crossing branches, which cannot be expressed with context-free grammars (instead requiring a context-sensitive grammar). Because both iv6 and V64 are in fact themselves subordinate to V, the preferable analysis in this case is to regard the entire harmonic sequence (except for I6) as multiple subordinate derivations from the final V at the end of the sequence. In contrast, the case exemplified in (j–l) may be better understood in terms of coordination of several instances of predominant chords. However, as above, it is equally possible and plausible to analyze this example in terms of derivation (iv being derived twice from V). It is again a matter of choice on the part of the analyst whether such cases should be regarded as one extended predominant subtree or as several individual predominant derivations. (j) i–V43/iv–iv–V65/iv–iv–iio6–V64–V–i (k) I–V7/vi–IV–vii°43/ii–ii6–vii°65/ii–ii6–vii°65/ii–ii6–V65/V–V (l) I–vii°64/IV–IV6–IV–V64–V7–I

In summary, it can be seen that from a structural syntactic perspective, coordination and methods of avoiding or delaying cadential closure employ very similar structures of recursive embedding. 5.2. Generalizations regarding the order of implicative elements

A second generalization concerns the order of elements with the same tonal implication. Multiple instances of the same tonal function (1) each carry their own tonal implication (multiple implications instead of multiple replications of a single implicative function) and (2) typically appear in increasing order of implicative strength (see (a) and (b*)). For instance, IV has a weaker implication of V than ii. Therefore, the coordination of IV–V and ii–V results in the order IV ii(6) V rather than ii IV V. (a) IV–ii(6)–V (b*) ii(6)–IV–V

V has a weaker implication of I than V7. By analogy, this predicts a predominance of the order V–V7–I (as opposed to V7–V–I). In general, the dependency structure of these examples is (IV (ii–V)) and (V (V7–I)) and not ((IV–ii) V) and ((V–V7) I), since the latter parallelization of “ii, IV” and “V, V7” would not predict the restriction in the order.

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5.3. Tonicization, key borrowing, and modulation as forms of recursion

A fundamental strength of the combinatorial complexity of generative grammars involves recursion as well as the power to express multiply embedded structures. This recursive combinatorial power is particularly useful for modeling multiple applied embellishments, coordination, and prolongation, as well as the derivation of multiple embellishments from other keys and tonicizations.76 The GSM formalizes the idea of modulation and tonicization as center-embedding recursion within a concise framework. (a) I–It6/ii–V/ii–vii°43/ii–ii6–V64–V7–vii°7/vi–vi (b) i [...] ii6–V64/iv–iv–V6/iv–iv–V6/iv–iv–V6/iv–iv–V6/iv–i6–V7/V–V7–I–V/N6– N6–vii°7/V–V64 [...] V7–i (c) I6–vii°65/ii–VI6/ii–vii°43/ii–ii6–V64–V7–I (d) I6–ii65–ii43–V64–V42/IV–IV6–IV–V65/IV–V7/IV–IV–ii6–V64–V (e) I6–vii°65/V–V43/V–V–IV6–V7–I

Example (a) illustrates the possibility of temporarily borrowing harmonic elements from keys other than the tonic, especially those keys in which the constituent elements of a cadential progression act as a new tonic. This allows us to derive one or even multiple subordinate implicative harmonies. This change in our point of reference precisely defines an instance of recursion in tonal music. The slash “/” symbol in standard notation represents a functional switch to a different tonal space and can be logically applied in a recursive manner, e.g., V/V/ii. However, instead of notating the sequence given above as It6/ii–V/ii–vii°43/ii–ii6, it is more parsimonious to reflect the underlying regular structure by choosing ii as the new point of reference or as a local point of tonicization: (It6–V–vii°43–i6)/ii. Otherwise, the rules of tonality would be represented in a very redundant and unsatisfactory manner (see, for instance, It6/ ii–V/ii, It6/iii–V/iii, etc.). Recursion and coordination challenge the adequacy of local grammars, Markov models, and schema-based approaches, all of which lack the power to express these aspects (see also 5.5.). A generative hierarchical approach, by contrast, is capable of representing the underlying structure in a parsimonious fashion. Figure 16 provides an example of how such a recursive embedding analysis of key borrowing/tonicization/modulation is modelled following the formalism of the GSM. Note that such recursive generation generalizes to any part of a phrase (or, here, a cadence), not only to key relationships between phrases and modulations.

76. The use of recursion to capture tonicization and key embedding has already been suggested by Douglas Hofstadter in his Gödel, Escher, Bach (1979).

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Figure 16: Tree analysis of the sequence I–It6/ii–V/ii–vii°43/ii–ii6–V64–V7–I. The tree illustrates recursive embedding using a local change of key reference based on the modulation mechanism proposed by the GSM.

Another example of the power of recursion in the generative mechanism is given above in example (b) from the end of the first movement of Mozart’s Piano Concerto in D minor, K. 466 (mm. 338–356). The cadential pattern considered here starts with an initial tonic (after a preceding cadence) and features a tonic extension by means of a onemore-time pattern. After recursively extending the area around iv via coordination, the sequence reaches the major tonic via the secondary and the home-key dominant recursively borrowed from the corresponding major key. It then diverges to the Neapolitan, which is itself prepared by its applied dominant. The “grand cadence” ends with an extended V64 (preceded by a V preparation with a diminished chord based on ©4^), inviting the soloist to deliver a cadenza ending with a trill on 2^ over V7 and the final tonic. A final example, the end of the second movement of Mozart’s Piano Sonata in F major, K. 280, demonstrates the generative power of recursion in music and the way it is modelled in our grammar (see Fig. 17). The dramatic effect conveyed by this ending results from the various steps employed to extend and delay cadential closure. First, a simple conventional cadential context is established by the first four chords, using the standard pattern i6–IV–V64–V7. The concluding sonority (i) is avoided by means of a i6 chord, which provides only partial closure. At the same time, i6 serves two further functions: It embellishes the subsequent IV and initiates a (subordinate) one-moretime repetition of the cadence in an attempt to complete the phrase. In other words, i6 represents an example of elision here. In the following measures, the cadence-final dominant is delayed until after the cadential V64 with several subsequent embellishments that each imply the cadential

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V: Ger6, vii°43/V, and again Ger6. The first Ger6 is in turn recursively embellished with a relative diminished chord (preparing VI, iv, and the Ger6). This delay of the cadencefinal V creates a strong momentum towards V, such that V is extended and split over an entire bar into V and V7. What follows is a brief and harmonically incomplete tonic (represented by its root only), immediately followed by a V7/iv in first inversion, in a manner that almost provides another case of elision (although here the two chords are separate). This again recursively triggers a one-more-time pattern, this time serving to prolong tonal closure for the cadence and the entire movement. This one-more time pattern is repeated once more (initiated by a stronger subdominant preparation by means of a diminished chord) before the final closure is finally achieved. These one-more-time patterns establish a tonic extension via overarching coordination. The tonic concluding this movement is itself delayed by a dominant overhang. Altogether, all of these recursive embellishments and extensions create a cadential progression that is longer than 13 chords.

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Figure 17: Mozart, Piano Sonata in F major, K. 280/ii, mm. 51–60. The end of this piece provides an example of an extensively elaborated and embellished cadence using multiple parallel recursive embellishments and several applications of coordination (one-more-time technique).

5.4. The combinatorial power of a recursive grammar

Another powerful characteristic of a generative grammar is that it models strong assumptions about conditional independence and is effective at modeling combinatorial flexibility. For instance, it formalizes the intuition that the application of one or more secondary dominants is independent of whether or not there is coordination and whether or not the cadence is deceptive or perfect authentic. Example (a) illustrates this point: It combines several preparatory implicative dependents of V (one with a recursive secondary dominant, viio), coordination of dominants, and a deceptive cadence framework in which vi is itself recursively prepared by viio7. Note in this context that predominant chords are themselves combined with dominant preparations, as seen in examples (a)–(d). (a) I–viio6–viio/IV–IV–iv–V–It6–V64–V–viio7/vi–vi […] (b) I6–IV6–V64–IV–viio65/ii–ii6–V64–V7–I (c) I6–ii65–V64–iv6–V64–iv6–V64–viio7/V–viio64–viio65/ii–ii6–V7–I (d) I–V7–V43/V–V7–vii°7/vi–vi–V64–IV–V7–I

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Cadences can potentially be enriched and expanded to a very large extent. One example of this is found in the famous C-major Prelude from J. S. Bach’s Well-Tempered Clavier I, the second half of which can be analyzed as one greatly extended cadence-like pattern.77 The strong combinatorial power of a generative grammatical approach such as the one proposed here makes it possible to plausibly express complex cases such as (a)–(d) as well as the examples given in the previous sections; however, if the rules are stated in a probabilistic form, they would predict that such complex cases will be rare and will appear much less frequently than simple, (seemingly) prototypical cases. 5.5. Inadequacy of local grammars and forward or Markov models

One may further note that the combination of different elements in a generative grammar is enormously flexible. However, this means that it is difficult to express the syntax of the cadence purely in terms of a forward model (such as a Markov model). As mentioned repeatedly in this chapter, if one modelled cadential progressions solely in terms of what can be followed by what (i.e., I–vi, vi–ii6, ii6–V, V–I), each forward step would have to incorporate all possible forms of recursive insertions (e.g., I–It6/ ii, I–V/ii, IV–It6/ii, IV–V/ii, etc.), which would result in a model containing hundreds of redundant multiple representations of the same structure. Furthermore, recursive insertions that generate and permit seemingly irregular surface progressions (such as V–N6 or many of the forms found in deceptive cadences or transitions at boundaries between separate subtrees) cannot be modelled as regular or irregular by strictly local grammars that neglect the hierarchical context. Because a syntactic approach aims at the characterization of structure in the sense of strong rather than weak generative capacity (capturing the internal logic of relationships between elements rather than merely their surface sequence), a Markovian representation would be a possible and computationally powerful model of the musical surface, but would fail to meet the criterion of a parsimonious and concise representation of the internal logic of a given sequence.

Conclusion As the title of this chapter indicates, we do not claim to provide a comprehensive or exhaustive theory of the syntax of the classical cadence; rather, our aim is to demonstrate the manifold advantages of adopting a generative grammar approach (such as those devised more than half a century ago in the field of linguistics by Noam Chomsky 77. See Schoenberg’s assertion that “[i]n a general way every piece of music resembles a cadence, of which each phrase will be a more or less elaborate part” (Schoenberg, Musical Composition [1967], 16).

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and his followers) in music theory. Although generative linguistics inspired Lerdahl and Jackendoff to develop an enormously powerful generative theory of tonal music, we believe that adhering more closely to formal language theory and well-defined generative rules than the GTTM did could be rewarding, allowing us to account for both the linearity and the hierarchical (and recursive) nature of the classical cadence and tonal music in general. As pointed out in the introduction to this chapter, one of the main challenges in addressing the classical cadence is the almost infinite variety of cadential patterns, even when we consider the harmonic level in isolation. This variety seems to be difficult to capture using an exemplar-based or chunk-based approach. Rather, the representation of the internal logic of the tonal dependency structures inherent in cadences requires a hierarchical approach, as opposed to a linear or Markovian model. A syntax approach such as the one proposed in this chapter (1) accounts for the underlying deep structural dependencies in the main cadence types found in the classical repertoire and described by music-theoretical approaches, and (2) models sequential structures based on a small set of rules that generate temporal sequences through their multiple independent combinations. Such an approach allows us to generate embellishments, insertions, coordination, and functional relationships independently for each event in the sequence. In other words, it provides combinatorial power and predicts a great variety of cadential phenomena (with divergent frequencies of occurrence). We suggest that the empirical variety of cadences found in the classical repertoire, along with the very frequent occurrence of a few very common cases, can be successfully predicted by (potentially probabilistic) generative mechanisms. The family of formal approaches subsumed under the umbrella of formal and generative syntax provides precise, explicit, empirically testable, and computationally implementable ways to characterize hierarchically organized sequential structures in music. At present, our proposed formalization is limited by the selected level of representation (scale-degree harmony and bass line); it does not yet address the interface between metrical and grouping structure on the one hand and the cadential progression on the other. The mathematical formalization of such relationships remains to be addressed in future work. Furthermore, we regard the rules proposed in this chapter as merely a (potentially suitable) starting point for further studies. The development of a more refined apparatus of generative rules and their empirical examination by means of corpus research must await future studies.

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