473 25.Musical Musical Syntax I: Theoretical Synt Perspectives a Martin Rohrmeier, Marcus Pearce

25.1 Outline...... 473 The understanding of musical syntax is a topic of fundamental importance for systematic musicol- 25.2 Theories of Musical Syntax...... 474 ogy and lies at the core intersection of music theory 25.2.1The Concept of Musical Syntax ...... 474 and analysis, music psychology, and computa- 25.2.2Foundations of Musical Syntax...... 475 tional modeling. This chapter discusses the notion 25.3 Models of Musical Syntax...... 477 of musical syntax and its potential foundations 25.3.1Building Blocks ...... 477 based on notions such as sequence grammat- 25.3.2Structure Building ...... 478 icality, expressive unboundedness, generative capacity, sequence compression and stability. Sub- 25.4 Syntactic Models of Diferent Complexity ...... 478 sequently, it discusses problems concerning the 25.4.1Finite-Context Models...... 478 choice of musical building blocks to be modeled 25.4.2Finite-State Models ...... 479 as well as the underlying principles of sequential 25.4.3Context-Free or Equivalent Models...... 479 structure building. The remainder of the chapter 25.4.4Beyond Context-Free Complexity ...... 481 reviews the main theoretical proposals that can be characterized under diferent mechanisms of 25.5 Discussion ...... 482 structure building, in particular approaches using 25.A Appendix: The Chomsky Hierarchy ..... 483 fnite-context or fnite-state models as well as 25.A.1Type 3(Regular)...... 483 tree-based models of context-free complexity (in-

25.A.2Type 2(Context Free) ...... 483 1 . 25 | C Part cluding the Generative Theory of Tonal Music) and 25.A.3Type 1(Context Sensitive)...... 483 beyond. The chapter concludes with a discussion 25.A.4Type 0(Unrestricted) ...... 483 of the main issues and questions driving current References...... 483 research and a preparation for the subsequent empirical chapter Musical Syntax II.

The idea that there is a grammar of music is prob- ably as old as the idea of a grammar itself. Mark Steedman [25.1,p.1]

25.1Outline What distinguishes music from other sounds? One an- tional, and psychological/neuroscientific approaches, swer is to be found in the manner in which the elements questions about musical structure and the perception are organized and related within a structural frame- of it facilitate a close link across traditional divisions work and, most importantly, the apprehension of this between disciplines [25.2]. Note that we use the term structure by a listener, so that the sound is experienced computational to describe a theory that is expressed in as music by that listener. Therefore, discovering the computational terms, whether or not it is actually im- principles of musical structure building is one of the plemented as a computer program. central questions for theoretical and empirical music Exploring the principles of musical structure build- research. Despite the strong historical (and method- ing naturally requires us to distinguish between the goal ological) divide between music-theoretical, computa- of uncovering rules governing the structure of music (an

©Springer-VerlagGmbHGermany2018 R. Bader (Ed.), Springer Handbook of Systematic Musicology, https://doi.org/10.1007/978-3-662-55004-5_25 to Icon- the internal struc- harmonic syntax Musical Syntax II. happens in a musical it happens. Conversely, is used to refer to the ar- is not, as you can hear if modeling what when syntax [. . . ] is not. Similarly, I-VII6-I6-II6-V7-I is an English sentence, whereas is a coherent progression of chords, whereas )stronglyimplies The disciplines involved in research on musical syn- the concert cert went the to [. . . ] I-I6-VII6-II6-I-V7 refer to the arrangementsions; the order of of chords chords withinis these to at progressions form least as progres- important asguage. the (Other order components of of words inthe harmonic lan- position syntax of are chords withintion and phrases, resolution the of prepara- dissonances,of and chord the progressions to relation melody and bass lines.) you play through thelanguage, two the examples. word In therangement study of of words tois form a sentences; very word importanting order component music, of we syntax. can In use study- the term heard syntax is examinedcomputational by modeling, triangulating and between cognitivewe research. theory, focus Here on theoreticalwhile approaches empirical to research musical usingpsychological syntax computational experimentation, and models, neuroimagingcovered are in the companion chapter, with computational modelingoptimal since structural the description (that search relatesas to for structures an ture of the musica part (whether thereof, or it a is corpus).psychological phenomenon, Since a we music often single is use an psychological composition, inherently understanding to guide themodels of development of music, structural justmusic to as guide we the use developmentlogical and structural theories testing of models of the psycho- of perception of musical structure. tax range from musicologycomputational and modeling, music to psychology theory, through andogy. Although neurobiol- the disciplinary perspectives(e.g., are it distinct is possibleoptimal to according develop to a some structural criterion, such theorybut as that simplicity, not is accordingperception to and cognition), the in criterioncus this on contribution, of a we converging matching picture fo- that human emerges when musical Asyntactictheorymightbeappliedtoanyaspectof musical structure – melody,grouping structure, harmony, form, or rhythm, even aspects metre, suchand as timbre dynamics. Intypically practice, been syntactic applied approachessequence to have – e.g.,and predicting chords (combinations – of) rather pitches than ]) 4 and Iwentto Aldwell ]and,ultimately,re- 6 , 5 ,p.139]: 8 ,p.89]giveabriefaccountofsyn- 7 [25. write the following in order to characterize et al. [25. ]) and, together with other constraints (e.g., 3 One way thatthe music order resembles of things language is is crucial that in both. Finding a formal characterization of musical struc- binations in a language. the rules for arranging itemsparts, phrases) (sounds, into words, their word possible permissible com- acquire the syntactic structure of a musical style. ture brings traditional music theory in close connection produced in compositional practiceare listeners (since before composers they becomethere composers). However, can beand no therefore learning theoretical without modelsespecially a those of grounded hypothesis musical in structure, space provide a computational useful modeling, approach to understandingesis the space hypoth- that human learners are faced with when they give rise to thesic. musical In structures turn, that musical welisteners structure find is from in acquired mu- mere implicitlyand exposure by represented and internally musical [25. interaction harmonic syntax Schachter timbral qualities andapproaches constitute so informal, verbal on. accounts oftactic Many syn- models music-theoretical of music.well-defined Although formalisms the is use not oftheory, (yet) there strict common are and in some music accountsof that syntax employ the in notion music theory. For instance, In human language thebols) set may of items bebirdsong (alphabet they words may of be sym- and pitches,In slides morphosyntactic music and other the units, sounds. voice-leading symbols in may patterns be or melodic relationships notes, between chords, voices, tax as: those imposed by cultural factorserties or of the physical instruments, prop- body, constraints constraints of of the the performance and hands so on or [25. the 125. 2. The Concept of Musical Syntax Berwick 2. 25 Theories of Musical Syntax ception and production ofpsychological these goal). structures Yet (an bothof internal aspects the form same coin: twoman the sides perception capacities and and limitations cognitivepossible of processes structures hu- influence that the composersilar can use argument (for about ato sim- language, [25. the reader is directed external goal) and the cognitive principles of the per- Music Psychology – Physiology Part C

474 Part C | 25.2 Musical Syntax I: Theoretical Perspectives 25.2 Theories of Musical Syntax 475 theories of rhythm and metre often do not take an ex- grammaticality judgements), the fact that the space of plicitly syntactic approach. By analogy, metrical and possible musical compositions is theoretically infinite rhythmic features of language are often studied from (or unbounded), the idea that we want to be able to de- the perspective of phonology rather than syntax. A well- scribe structural relationships within musical sequences formed harmonic sequence, for instance, may be as- (i. e., focus on strong generative capacity compared to signed to metrical structure in a regular or irregular weak generative capacity). Syntax is also relevant to way. It is important to note that despite the predom- tasks such as compression, identifying the stability of inance of Western music in theoretical and cognitive events at different levels within music, and measuring research [25.9], the general notion of musical syntax musical similarity. We investigate these issues further is not limited to Western tonal music – and there are in the following sections. approaches addressing non-Western music [25.10–12]. Different aspects of musical structure may be more Grammaticality or less important in different musical styles and cul- One core foundation for the concept of musical syn- tures. tax is the notion of regularity, permissibility, well- Several models for musical syntax have been pro- formedness or grammaticality, i. e., the characterization posed based on different levels of structural representa- of structures that are regular or irregular with respect to tion (melodic structure, harmony and chords, bass lines, aparticularsystem(representing,forexample,amusi- outer voices, voice-leading, other types of categorized cal style). If such a distinction were irrelevant, the char- sound, polyphonic pitch structure and so on). Here, we acterization of musical syntax would be unnecessary reserve the term syntax for approaches presenting a for- since every sequence would be equally plausible. How- mal system characterizing the sequential structure of ever, musical styles and idioms are implicitly character- such building blocks, in contrast to the more general ized by regular and irregular sequences. Although cat- term musical structure which captures the rich interac- egorical grammaticality decisions are often made, the tion of different musical features such as rhythm, metre, distinction may be one of degree (compare [25.13–15], timbre, counterpoint, dynamics, phrasing, instrumenta- in linguistics). For instance, not every chord sequence tion, agogics, and so on. The precise identity of these or every musical form is regular in the 18th century building blocks is one of the central ongoing research Classical idiom [25.16]. Another example illustrates questions in musical syntax. aregularandirregularcommon-practiceharmonicse- 2 . 25 | C Part The general term musical structure refers to the way quence (Fig. 25.1 adapted from [25.17]). Musical reg- in which one or more pieces of music may be repre- ularity can be characterized empirically through (com- sented in terms of their constituent parts, potentially putational or hand-conducted) corpus analysis, which reflecting a wide range of different musical features in- can provide information about frequent and less fre- cluding rhythm, metre, timbre, counterpoint, dynamics, quent regular patterns and indirectly about irregularity phrasing, instrumentation, agogics, and so on. by the discovery of absent and low-probability patterns Musical syntax is a formal characterization of the (although absence does not necessarily constitute evi- principles governing permissible sequential structure dence for ungrammaticality). Grammaticality can also in music. It characterizes sequences of musical events be experimentally established through psychological generated from a lexicon of building blocks and a set of experiments. Furthermore, introspective analyses by in- rules governing how the building blocks are combined. dividual experts may be regarded as single-participant The lexicon (the set of building blocks) may consist experiments, with some extrapolation to wider groups of single events or patterns (schemata) of notes, glis- (whether expert or otherwise) assumed. In this context sandi, rests, chords, voice-leading patterns, timbres, or it is essential to understand the importance of negative other noises. The rules may constitute any formal sys- evidence in the form of explicit instruction about im- tem that characterizes how sequences may (and may plausible or irregular structures (note that this is differ- not) be formed by combining elements from the lexicon. ent from the absence of positive evidence). While some regularities and rules may be inferred from positive data 25.2.2Foundations of Musical Syntax alone (i. e., the presence of well-formed structures), it is negative evidence that makes the strongest conclusions Why do we need a syntax of music? When characteriz- possible with regard to range, scope of generalization, ing musical structure, and the cognitive representation and mutual interaction of grammatical systems. There is and processing of that structure, several issues arise continuing debate about whether and how people might which motivate the development of a formal syntactical receive negative evidence in the development of lan- understanding of music. These include distinguishing guage, but this topic has received little attention in the regular and irregular musical structures (i. e., making domain of music. ], ]. and 22 21 , ]and 20 23 Mavroma- Aldwell ]. See ,Chap.28]for 27 22 , 26 ]hasusedgeneral-purpose . 24 ]) and Bayesian model comparison pro- 25 ]foranexampleoftheseprinciplesappliedto V7 28 harmonic progression as discussed by Compression Compression-based evaluation of a model of musi- [25. ) b ( Characterizing a set of (musical) sequenceserative using theory a gen- allows us toset capture of a sequences potentially using a infinite finitewe set of can rules. In think thisextent sense, of to generative whichficient theories they representations in enable of termsefficient, compression a of sparse through set the encoding ef- oftutes of a sequences. the core Highly principle environmentand of there consti- cognitive is systems a close [25. compression relationship between because prediction we and onlymation need that to is store notResearch the predictable in infor- using music amusic model information psychology [25. retrieval [25. [25. comparing different candidatecount models differences taking in into theirof ac- complexity free parameters and and the so on numbers tis [25. music. cal syntax is independent from other questions such as compression algorithms as models of musicality. complex- A model thatwith better general captures coverage structural in regularities pected a to given be musical capable idiomtherefore, of is compression. more ex- accurate Conversely, we predictionpressibility and, (of can unseen use data com- as to a ensure measure of generalizability) the powertheory and (and efficiency the of latent a structure generative ever, that it a postulates). more How- space, complex meaning model that will increasedthe level itself data of consume must compression of exceedin more order the to increased be sizelowing efficient. of In the this the paradigms respect, model of(MDL; approaches minimum [25. fol- description length vide closely related methods [25. pression (see below). This broader conceptstrong is generative capacity known as ] 19 , weak gener- 18 VII6 and a poor ) a I6 II6 ( Poor II b) is poor because the dependencies between the chords are disrupted – for instance, the ) b ( However, for most languages there ,p.140]. 8 of formalism that embodies this principle, The contrast between a good I [25. VII6 I6 II6 V7 Good II a) does not refer to (human) music generation but one kind Unboundedness Weak and Strong Generative Capacity are blocks using the rules. Generative grammars [25. we can further imagine(such a as composition ideal thatto airport never construct ends an music). exhaustive Hence, listTherefore, of it all the musical is only sequences. impossible wayture to characterize is musical by struc- and employing recursive (or a iterative)ical finite rules sequences to set based generate on of grammat- the building recombination of blocks building The set of possiblemusic, in musical Humboldt’s famous structures words, makes is infiniteof use finite unbounded means. It – is simple toedness demonstrate the of unbound- musical structures: for everyimagine sequence we a can longeranother one element or (tone, a chord, variation etc.) of into it the that sequence; inserts Fig. 251. a,b Schachter plausible. For this reason ittheory is desirable matches that a theoretical syntactic tively insights relevant (or as adequate) well structures,and provides as testable useful generalizations, cogni- and achieves optimal com- der to reproduce exactlylanguage those (i. sequences. e., For a set amay of given strings), be such accurate a ingenerate characterization the terms set of of strings). coveragegenerative This capacity. (i. e., is referred they tois can as an infinite numberdescribe of formal the models language, that many adequately of which are highly im- by formal rulesby that a well-defined are formal capable mechanismgrammar). (such of as a generating formal them Asyntacticmodelofasetof(musical)sequencesmay focus on the description of the surface sequences in or- often used inmusic theoretical (and approaches other to auditory the sequencesor such birdsong). syntax as Note of language thatative in this context theto term the description and analysis of a set of sequences II6 chord is not functionally wellin connected the to analysis of its the context good (evenIharmony though example, it the authors features propose good a voice-leading). hierarchical Further analysis note of that I VII6 I6 as a prolongation of a single Music Psychology – Physiology Part C

476 Part C | 25.2 Musical Syntax I: Theoretical Perspectives 25.3 Models of Musical Syntax 477 grammaticality or weak/strong generative capacity. In tion of events with respect to a common core struc- particular, the criterion of optimal compression makes ture (for instance, differences between different cover it possible to evaluate and compare syntactic models in- versions of a song). In this context, it is important dependently of the grammaticality distinction as well as that such operations respect (hierarchical) structural independently of tests (such as pumping lemmata) that boundaries of constituents (e.g., a tonic expansion) require grammaticality distinctions over sequences that rather than comparisons between unstructured surface are extremely improbable and do not generalize over sequences. For example, De Haas et al. [25.32]im- corpora (such as n-th level center embeddings). In this plemented a similarity measure that is closely related context, compression provides a better way to provide to structural stability in terms of the largest common afoundationforstronggenerativecapacityandalso embeddable subtree between two compositions. This assess the cognitive relevance of a proposed syntactic approach outperformed edit distance (a structure-free account of a (musical) language. surface comparison between sequences) in predict- ing harmonic similarity between music sharing similar Stability, Similarity, and Semantics melodies. Similarity is also closely related to the con- as Underpinnings of Syntax cept of compression since we can train a syntactic There are several other ways to motivate syntactic struc- model on one piece of music and use that model to ture in music. One of these is the proposal that we predict another piece of music – greater degrees of pre- need an account of syntactic structure in music to be dictability (and hence compressibility) indicate greater able to predict the relative stability of musical events. degrees of structural overlap between the pieces [25.21, Many music theorists observe that in harmonic, melodic 23]. or voice-leading sequences, some events may be con- Finally, semantics may constrain syntactic struc- sidered ornamental or accidental whereas others are tures, particularly in linguistics. Whereas linguistic structurally fundamental [25.8, 29, 30]. If this notion of syntactic structures to a large extent serve the tempo- relative structural stability – not to be confused with ralization/linearization of semantic structure (in terms tonal stability and the tonal hierarchy [25.31]–isex- of form/meaning pairs), there is no immediate anal- tended to a fully recursive structure (i. e., not just to ogy in music. Although music may express meaning in individual notes or chords but also to motifs, phrases, terms of illocutionary acts like warnings, or aggression, and other larger scale components of musical form), it or in terms of symbolic associations, it is agreed that 3 . 25 | C Part can be accounted for using a hierarchical syntactic for- music, in general, lacks complex, explicit propositional malism. Whether or not this type of structure is in turn semantic forms ([25.33]anditsdiscussion[25.33– coextensive with the above forms of establishing hier- 36]). However, the patterns of relative stability outlined archical structure remains a question open for further above (which are themselves related to syntactic struc- theoretical investigation. ture) lead to perception and experience of tension and Another, related avenue for establishing hierar- release by the listener, which can be viewed as a kind chical structure is similarity. From a theoretical and of semantic interpretation [25.37–40]. However, further psychological perspective musical similarity may be research is required to examine these potential relation- construed in terms of operations of omission or inser- ships between syntax and semantics in music.

25.3Models of Musical Syntax Amodelofmusicalsyntaxconsistsoftwocorecom- different levels of representation (or abstraction), such ponents: first, a choice of the underlying representation as: harmony and chord sequence, bass line, melodic line for musical building blocks and how they relate to the (diastematics), outer voices and voice-leading, or poly- musical surface; second, a formalism used to generate phonic pitch structure. Every choice involves selecting musical structure based on the set of building blocks. adistinctionbetweenstructuralandnonstructuralitems with respect to the underlying model.Forinstance, 25.3.1Building Blocks amodelofharmonicsyntaxmayregarddifferentsur- face and melodic realizations of a chord sequence as The choice of building block is fundamentalfor the syn- equivalent; similarly, a theory of voice-leading would tactic model. In contrast to language, where the set of regard certain note repetitions, trills, or ornaments as morphosyntactic features is largely accepted, syntactic nonstructural. Given the divergence of representations, models of music have made different choices of build- styles, and level of abstraction adopted by different ing block. This entails modeling musical structure at approaches in the literature, there is no consensus at k- of not ]. 52 without ]do is part of one way 63 k – 61 ). A sequence is gram- k ]constitutejust ]. k-factor languages 51 is formally defined by a set 55 ]. It is important to note here , ]). Furthermore, computational 60 54 – , 53 5 58 models [25. k-factor language A These characterizations of structure, however, only The languages generated by each class of gram- It is fundamental to note that the Chomsky hierar- -gram models. every possible sequencepurposes equally. this For istures many insufficient occur theoretical as very someare frequently rare, of whereas less these common, otherquirement or struc- transitions demands unlikely. a This characterization that theoreticalonly is re- based on not grammaticalitystraightforward but to also expand the on above definition probability. to It incor- is factor languages) and aren also referred to as Markov or matical iff everythe subsequence set of of length factors.in Several music models theory have and beenfactor cognition proposed that contain,that in part, schema-theoretic approachesnaturally correspond [25. with modification (since theypatterns, involve reductions, and nonlocal thestructurally ability important from to those that distinguish are not). notes that are draw a distinctionquences, between yet regular and within irregular those se- categories, they consider of factors (strings of length the expressive (and compressive) power of thetation represen- and corresponding processing requirements. mar form proper subsets ofclasses the of languages grammar generated higher by up inas the we hierarchy. However, move up the hierarchy,tion the complexity and of parsing recogni- increases inexpressive tandem with power the of increased ular, each while class of context-freein grammar. grammars the In (and hierarchy) partic- those aresuch capable higher as of embedded capturing structure, which phenomena,by cannot be finite-state captured grammars,many they problems of also intractability and bringpecially undecidability, in es- with the context them of grammar induction [25. chy and its extensions [25. strings, except in historical terms. Thereto are characterize many ways sequentialof structure, formal as languages demonstrates any (e.g.,mal Handbook handbook for- languages, [25. models are, fundamentally, in nohand-crafted respect models distinct by from theoristspressive in power [25. terms of their ex- characterizing (musical) sequential structure.in They no are particular way primaryapproaches or more that natural characterize than other classes of infinite sets of - k ], but funda- 50 1ofthe ! ]. In finite n 57 , 56 ]andmetricalstruc- ponding probabilistic 48 – 41 , 37 ]. Characterizations with mod- grammars [25. 51 of building blocks for a musical syntax nguages and corres ]. The core idea of these very local mod- 57 ]. These theories define how these aspects finite context 49 We should mention here that some very interest- assumption that there are no (unbounded)between dependencies events longer than themodel. relevant In context general, of finite-context the the models formal correspond to subcategory of strictly local languages ( tion formally amounts torelationships a between table each that possible lists combinationements grammatical of (such el- as chord,account is note, easily or extended to rootnext larger transitions). context-lengths: element the This may besor, related but also not to only theelements. to sequence its What of 2, predeces- such 3, accounts or have more preceding in common is the end of thequence already [25. processed portionels of is the to characterize input sequentialpossible se- structure element-to-element by transitions identifying (how elements may follow or precede each other). This characteriza- since they characterize the formaljects space rather of than musical explicitly specifying ob- elements how may sequences be of combined, we dotheories not of consider syntax them proper. as context automata, themined next by state testing a is finite portion completely of deter- length .425 Syntactic Models oferent Di f 125.4. Complexity Finite-Context Models There is an interestingwithin subclass the of grammar classknown contained as of finite-state grammars which are models. Many theories oftypes structure of have used formal explicit and languages its in extensions [25. the Chomsky hierarchy .225.3 Structure Building Traditionally thereattempts to have characterize been thements sequential a structure in of number a ele- sequence,context-free of ranging la from theoretical Markov models to ture [25. of music may betentially expressed represented in by algebraic cognitive ways systems and [25. po- mental domain could be establishedgoals. independently of the modeling ing work has beenvarious done on aspects representational spacesnotably, for of pitch spaces musical [25. elements including, most present how (and based on which principles) a els of different complexity involve a trade-off between Music Psychology – Physiology Part C

478 Part C | 25.4 Musical Syntax I: Theoretical Perspectives 25.4 Syntactic ModelsofDiferentComplexity 479 porate probabilities: every entry in a transition matrix cies, nested structures, and cross-serial dependencies. is associated with a probability. Probabilistic instantia- To some extent these limitations can be addressed by tions of the approach, therefore form a superset of the using sophisticated representation schemes such as the nonprobabilistic versions, which only allow the proba- multiple-viewpoint formalism [25.64, 65]thatextends bilities 0 (nongrammatical) and 1(grammatical).Often the range of context that a Markov model can take these probabilities are estimated through analysis of fre- into account by combining several Markov models over quency counts of events in a corpus [25.64, 65]. Such different feature spaces and (possibly) time scales, in- probabilistic extensions of k-factor languages are re- cluding nonadjacent events. ferred to as Markov models or n-gram models (for which n-grams correspond to probabilistic versions of 25.4.2Finite-State Models j k-factors). In an n-gram model, the sequence e.j n/ 1 is called an n-gram (note that the subscript and! su-C Several theoretical approaches can be viewed as having perscript symbols denote the beginning and ending of equivalent representational power to finite-state or reg- asubsequenceinthestring;inthepreviouscaseit ular grammars in Chomsky’s terminology. In contrast refers to the subsequence, from index .j n/ 1tothe to k-factor languages,suchmodelsinvolvegrammars index j)whichconsists,conceptually,ofaninitialsub-! C that distinguish between (hidden) variables (nontermi- j 1 sequence, e.!j n/ 1,oflengthn 1knownasthecontext nal symbols) and surface symbols (terminals). Accord- ! C ! and a single symbol extension, ej,calledtheprediction. ingly, regular grammars (i. e., grammars that only have The quantity n 1istheorder of the n-gram rewrite rules of the form A aB;inwhicha refers to a termi- rule. ! nal and A; B to nonterminals;! see the appendix below) Such models are frequently used in computational characterize sequential structure by building up a string models of music (see below), and also some music the- from left to right. They form a true superset of k-factor oretical accounts (e.g., Piston’s table of common root languages. The formal machine that recognizes the set progressions, shown in Table 25.1). of strings generated by such a grammar is a finite-state By definition all types of strictly local or Markov automaton (informally, a flow-chart). The probabilistic models share the Markov assumption (25.1)and(25.2): counterpart to a regular grammar is the Hidden Markov the grammaticality (gr)ofasubsequenceortheproba- Model (HMM; [25.66]). bility (p)ofasymbolappearinginasequencedepends 4 . 25 | C Part only on its immediate preceding context of length k. 25.4.3Context-Free or Equivalent Models This assumption means that these models cannot repre- sent any nonconsecutive dependencies between musical There are several accounts of structure in music the- elements beyond a fixed finite length. ory which go beyond the expressive power of finite- context and finite-state grammars (for further discus- i i gr e1 gr ei n 1 (25.1) sion [25.38]): D ! C i i p !e1" p e!i n 1 " (25.2) " ! C Differences of structural importance Markov! " models! provide" powerful approximations # Dependency structure, preparation, and ornamenta- to sequential structure for numerous practical applica- # tion tions independently of whether those sequences obey Headedness the Markov assumption. Nonetheless, such models are # Nested structures theoretically as well as practically limited in the ex- # Functional categories. tent to which they can capture and represent more # complex structural features such as nonlocal dependen- Ausefulstartingpointistheinsightthattheelements in a sequence may differ in structural importance, i. e., Table 25.1 Table of common root chord progressions (af- some can be left out without impairing grammatical- ter [25.60]) ity whereas others cannot. An early account by Kostka Is often followed by Sometimes by Less often by and Payne [25.67]referstothisaslevels of harmony I IV or V VI II or III (note, however, that the observation is not restricted II V IV or VI IorIII to harmony). Second, musical structure expresses de- III VI IV I, II or V pendencies: e.g., in a I II V or I III IV progression, IV V IorII III or VI the II or III chord may be understood as preparation V I VI or IV III or II for V or IV and not simply a sequential succession of VI II or V III or IV I I; accordingly, it is dependent on V or IV, not on I. VII III I This is expressed by the rules V II V or IV III IV ! ! trans- which cor- that specify prolongational preference rules ,Fig.10.6]) 76 ’s Generative Theory of The overall structure of grouping structure which corresponds to the pattern well-formedness rules ]proposedatheoreticalaccount which represents the relative struc- ](GTTM)providesanaccountthat Fig. 25.2 GTTM (after [25. While these rules are abstract in that 75 Jackendoff ): first, 71 [25. 25.2 and reflecting patterns of tension and relaxation metrical structure reduction Schenker Lerdahl According to GTTM, a listener unconsciously in- Prolongational which hierarchical structures are permissible andthemselves which may be modifiedformational in rules. limited ways by they define only formalselect which possibilities, well-formed or transformed structures ac- of music baseddifferent layers on of reductional musical structureface analysis ranging to from that foreground, sur- reveals construed, middle his ground account and entailsterpoint Ursatz. (such that as Briefly principles neighbor notes) ofguish may the coun- be structural used importance to of distin- musical events. position. fers four typessurface of (Fig. hierarchical structureresponds in to a theinto musical segmentation units of (e.g.,ond, the motives, musical phrases, and surface of sections); periodically recurring sec- strongtime-span and reduction weak beats; third, tural importance ofestablished pitch rhythmic units; events andreduction within finally, contextually amongst pitch events at variouscording levels to of structure. the Ac- theory,are grouping largely and derived metricaland directly structure these from structures are the usedreduction musical in which is, generating surface in a turn, time-span gational used reduction. in Each generating a of prolon-nization the is subject four to domains of orga- Tonal Musicbrings [25. the ideasbased expressed theoretical framework, inspired by by the Schenkerapproach generative into to a grammarple, rule- in founded linguistics. on the Itcan assumption is, be that for partitioned aments exam- piece into which of hierarchically music mayapplication organized be seg- of derived thethe through same hierarchy. the Specifically, rules recursive yield the a at theory hierarchical, different structural iscognitive levels state description intended of of of to an the experienced final listener to that com- , , , ]; 1 ]). Stability conditions 38 38 and 79 , 68 1 [25. ], and [25. 74 reduction Time-span Lerdahl Tidhar Steedman ], ]; ]; 75 37 ]introducedtheno- 78 , [25. 73 ,namelythatintheII [25. 77 [25. [25. Time-span Lerdahl ]. Various partial or full com- segmentation ]developedacontext-freeformal- ]; Schenker nguages and hierarchical tree- headedness Keiler 68 68 , 30 ]; Riemann , 38 71 38 [25. [25. [25. [25. ]; the reader is also directed to [25. 69 ]). Finally, ]; for nested structure in music see [25. Metrical structure structure Narmour Grouping Context-free la Another central formal concept concerns nested 72 70 ]; , , 72 Rohrmeier putational implementations of thesediscussed below. theories exist, as Jackendoff tion that chordstional can categories be (such classifieddominants) as into that different tonic, may func- dominants, besuch and functionally as sub- interchangeable, IIharmonic and sequences IV leading to to be V. hierarchical Riemann [25. considered Anumberoftheoriesaccountformusicinsuch theoretical terms: ism for theformalism, functional tree structures approach representsequences to different that harmonic fulfill harmony. identical harmonic In functionssame in this way the in higher parts of the tree. based accountsthese kinds are of structural well-suited dependencies in for sequences. representing Rohrmeier 71 68 may be furtherThis elaborated, leads ornamented, to orrecursion recursive prepared. (chains) structures and inasequence).Oneprominentexampleofnestedstruc- nested the recursion form (sequenceture of in in tonal tail music is modulationby (e.g., an [25. early account above). structures: the notion of dependencymay introduced above lead toquences within the a formation dependentquence subsequence (and of within so on). a dependent Forthe se- instance, subsubse- preparation the II of chord the (which V is chord) in the above sequence Vprogression,Visthefundamentalchord,i.e.,the head (as expressed in the left-hand side of the rules This further entails that goalsdamental are structurally than more their fun- preparationsornamentation and and, variation conversely, adds that new materialstructure. to basic This notion ofentails dependency a structure further notion of (for further details the reader is directed to [25. Music Psychology – Physiology

Part C

Part C | 25.4 480 Musical Syntax I: Theoretical Perspectives 25.4 Syntactic ModelsofDiferentComplexity 481 tually apply to particular aspects of the musical surface. other way, we can add degrees of context-free character Time-span and prolongational reduction additionally to regular grammars. depend on tonal-harmonic stability conditions which are internal schemata induced from previously heard 25.4.4Beyond Context-Free Complexity musical surfaces. When individual preference rules reinforce one an- Are there aspects of musical structure that require other, the analysis is stable and the passage is regarded greater than context-free power to be modeled? De- as stereotypical whilst conflicting preference rules lead bates in theoretical linguistics of the past 25 years to an unstable analysis causing the passage to be per- have reached a fairly consensical view that human lan- ceived as ambiguous and vague. In this way, according guage is mildly context sensitive [25.86, 87]. It requires to GTTM, the listener unconsciously attempts to ar- syntactic power that is stronger than context-free but rive at the most stable overall structural description of considerably less strong than the immense computa- the musical surface. Experimental studies of human lis- tional power of full context-sensitive grammars. One teners have found support for some of the preliminary example of this context-sensitive complexity is given components of the theory including the grouping struc- by cross-serial dependencies (as in Dutch or Swiss ture [25.80]andthemetricalstructure[25.31]. German relative clauses [25.86, 88]) that cannot be ex- The theory constitutes a formal predecessor to Jack- pressed by context-free grammars. In the Chomskian endoff ’s later parallel architecture framework of lan- tradition, minimalist grammars, that are equally mildly guage [25.81, 82]. It is important to observe that the context-sensitive [25.89], adopted two mechanisms of GTTM is not a grammar or a syntax of music: it pro- external merge (similar to a context-free tree building vides a model of parsing but contains no generative operation) and internal merge (combining an already rules or mechanisms to derive the musical surface, fur- derived branch of a tree with different free positions ther it does not model a distinction between regular and in the tree). Internal merge may express features such irregular sequences. Rather than generating the musical as movement (Sue wondered which book Peter read?). surface, the GTTM is a theory of musical processing Katz and Pesetsky [25.90]arguethatmusicalandlin- with only limited applicability as a theory of structural guistic structure are formally equivalent in the sense syntax per se. that both require structure-building operations based on It is highly challenging to develop formal context- external and internal merge. 4 . 25 | C Part free grammars that account for musical surface struc- What about music? In his review, Roads [25.83] ture but several efforts have been made (e.g., [25.83, argues that the strict hierarchy characteristic of context- 84]forreviews).Johnson-Laird [25.85]usedgrammat- free grammars is difficult to reconcile with the ambigu- ical formalisms to investigate what has to be computed ity inherent in music. Faced with the need to consider to produce acceptable rhythmic structure, chord pro- multiple attributes occurring in multiple overlapping gressions, and melodies in jazz improvisation. While contexts at multiple hierarchical levels, even adding afinite-stategrammar(orequivalentprocedure)can ambiguity to a grammar is unlikely to yield a sat- adequately compute the melodic contour, onset, and isfactory representation of musical context. The use duration of the next note in a set of Charlie Parker of context-sensitive grammars can address these prob- improvisations, its pitch is determined by harmonic lems to some extent but this also brings considerable constraints derived from a context-free grammar model- additional difficulties in terms of efficiency and com- ing harmonic progressions. In a more recent approach, plexity. There are several attempts to model music Rohrmeier [25.38, 68]introducesasetofcontext-free using grammatical formalisms which add some de- rules modeling the main features of tonal harmony from gree of context sensitivity to context-free grammars the common-practice period. without adding significantly to the complexity of the Context-free languages (and more complex for- rewrite rules. An example is the Augmented Transition malisms) constitute supersets of regular and suprareg- Network (ATN) which extends a recursive transition ular languages. In fact, the latter constitute local bound- network (formally equivalent to a context-free gram- aries of context-free languages (i. e., substrings that do mar) by associating state transition arcs (rewrite rules) not use the features of nested embedding are regular; with procedures which perform the necessary contex- it is further possible to derive precise models of lo- tual tests. Cope [25.91]describestheuseofATNsto cal transitions from context-free models). Accordingly, rearrange harmonic, melodic, and rhythmic structures the distinction between these types of languages does in EMI (experiments in musical intelligence). Another not imply a forced alternative – rather, context-free lan- example is provided by the pattern grammars developed guage models can result from the addition of the above by Kippen and Bel [25.10]formodelingimprovisation structural features to regular language accounts. Put an- in North Indian tabla drumming. ]andre- 12 , 10 ]). Different musical 92 -gram models are easily [25. n . Mavromatis Musical Syntax II Many of these questions are best addressed by While Markov and on substitution intoHowever, in a using previously thedescriptions prepared of grammar blues chord skeleton. to progressions, Steedman generate had to structural introduce implicit meta-level conventions not explicit in the production rulescontext-sensitivity of required the to grammar. adequatelycal The model structure musi- extent requires of further investigation. is also possibleof to the conduct behavior a of ahavior quantitative of computational comparison listeners, model allowing a with rigorous thethe empirical be- test theory of as a psychologicallynitive plausible representation model and of processing cog- ofWe musical syntax. address theseter, points in detail in the following chap- cent work by styles or traditions maybuilding emphasize block different or kindscomplexity show of in different their degreescomparisons syntactic and structure. may kinds Cross-cultural havetheories of implications of for music.requires evolutionary While predetermined each (innate)least assumptions process the about search of space at it inference and must the be noted structuremeans that of implies cross-cultural the innateness of universality model, tions. more by than no these assump- implementing a computationalmodel theory that embodies as a amusical particular syntax theoretical computer stance and on testing thebehavior model with by human comparing behavior. its theory Modeling to requires make the all itsthe assumptions analysis explicit of and complex permits examples and large corpora. It above that areture, essential stability, tension, in and musical similarity. Conversely,powerful more implicative types struc- ofably syntactic more formalisms difficult to are inferclear consider- from data. that It we is not canstance currently develop that one generalizes overarchingtures. across theoretical As musical in stylesmajority other and cul- areas of of researchon empirical on musicology, Western musical the musicafewnotableexceptionsincluding[25. syntax and has harmony focused in particular (with we noted above degreesbe of added context-free to character regular maysensitivity grammars added and to degrees context-free grammars. of context- learned and arecapable useful of for modeling prediction, morecal they complex and are structures, hierarchical barely nonlo- dependencies in music described , 45 [25. Johnson- -gram models; regu- n Longuet-Higgins ies, single or multiple center- ]hasdevelopedacategorial 72 , 1 [25. ], this allows a more elegant description 85 [25. Steedman sicians or nonmusicians) sensitive to? learned, and if so,positions how need and to be which assumed kinds as of innate? predis- The power of a particular syntactic formalism is in- of musical structure such as rhythm, metre,ing? and Or tim- are thesesyntactic aspects formalisms? also best explained using ception ofnonlocal music, dependenc exhibit featuresembedding? of recursion, the relationships present inthere musical examples structure? of Are syntacticrequire musical (mild) structures context that sensitivity?ple, How polyphonic can streams multi- beapproaches? represented by formal ]. Although it is less ambitious than that of of improvisational competence since it does not rely Laird 46 grammar (augmented context-free) toharmonic account structure for of the ory 12-bar of blues, tonal harmony based due on to a the- tant to note here thatway the Chomsky of hierarchy is characterizing justmalisms one but the it does power notto necessarily of lend every itself grammatical naturally aspect for- of musical structure. Furthermore, as erarchy probabilistic counterparts have(e.g., been finite-context proposed grammars: lar grammars: Hiddengrammars: probabilistic context-free Markov grammars). These models;developments suggest context-free as a generalbe strategy beneficial to that move from it deterministic to may probabilistic models for implementation and evaluation. It is impor- dependent of whether it is probabilistic orProbabilistic deterministic. models have distinctof advantages the in subtlety terms withdependencies which for they application can in capturetion, structural prediction, parsing, classifica- andterms learnability/inference of as well robustnesseach as of and the in model graded classes of grammaticality. the extended For Chomsky hi- 4. Which kinds of formal structures are listeners5. (mu- Can such syntactic structures and relationships be 3. To what extent does real music, and listeners’ per- 1. How powerful a grammar do we need to represent 2. How does musical syntax interact with other aspects 5. 25 Discussion This discussion of theoreticaltax accounts raises of several issues musical and syn- questionscurrent which are research: driving Music Psychology – Physiology Part C

482 Part C | 25.5 Musical Syntax I: Theoretical Perspectives References 483

25.A Appendix: The Chomsky Hierarchy Noam Chomsky introduced a containment hierarchy of 25.A.2Type 2(Context Free) four classes of formal grammar in terms of increas- ing restrictions placed on the form of valid rewrite Grammars in this class only restrict the left-hand side rules [25.52]. A formal grammar consists of a set of their rewrite rules to a single nonterminal symbol – of nonterminal symbols (variables), terminal sym- i. e., the right-hand side can be any string of nonterminal bols (elements of the surface), production rules, and symbols astartingsymboltoderiveproductions.Inthefol- A ˛: lowing description, a T" denotes a (possibly empty) ! sequence of terminal2 symbols, A; B V denote non- The equivalent automata characterization of a context- 2 terminal symbols, ˛ .V T/C denotes a nonempty free language is a nondeterministic pushdown automa- sequence of terminal2 and[ nonterminal symbols, and ton, which is an extension of finite-state automata that ˇ; ˇ0 .V T/" denote (possibly empty) sequences employs memory using a stack and state transitions may of terminal2 [ and nonterminal symbols. The difference depend on the current state as well as the top symbol in between different formal grammars in the Chom- the stack. sky hierarchy relates to different possible production rules. 25.A.3Type 1(Context Sensitive) Every grammar in the Chomsky hierarchy corre- sponds with an associated automaton: while formal Grammars in this class are restricted only in that there grammars generate the string language, formal au- must be at least one nonterminal symbol on the left- tomata specify constraints on processing or generating hand side of the rewrite rule and the right-hand side mechanisms that characterize the formal language. Au- must contain at least as many symbols as the left-hand tomata provide an equivalent characterization of formal side, i. e., string length increases monotonically in the languages as formal grammars. production sequence. ˇAˇ ˛; ˇAˇ ˛ 25.A.1Type 3(Regular) 0 ! 0 Ä Context-sensitiveˇ languagesˇ are equivalently char- 25 | C Part Grammars in this class feature restricted rules allowing acterized by a linearˇ boundedˇ automaton, that is a state- only a single terminal symbol, optionally accompanied machine extended by a linear bounded random access by a single nonterminal, on the right-hand side of their memory band, whose transitions depend on the state, productions the symbol on the memory band.

A a ! 25.A.4Type 0(Unrestricted) A aB (right linear grammar) or ! A Ba (left linear grammar) : Grammars in this class place no restrictions on their ! rewrite rules Regular grammars generate all languages which can ˛ ˇ be recognized by a finite-state automaton, which re- ! quires no memory other than a representation of its and generate all languages which can be equivalently current state. characterized by a universal Turing machine (the re- It is essential to note that regular grammars are not cursively enumerable languages), which is the same as equivalent to Markov models or k-factor languages (see the linear bounded automaton for context-sensitive lan- Sect. 25.4.1 above). guages without bounds on the memory tape.

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486 Part C | 25