composition, perception, and schenkerian theory

8/10/2019 Composition, Perception, and Schenkerian Theory

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Composition, Perception, and Schenkerian Theory

david temperley

In this essay I consider how Schenkerian theory might be evaluated as a theory of composition

(describing composers mental representations) and as a theory of perception (describing listenersmental representations). I propose to evaluate the theory in the usual way: by examining its predic-tions and seeing if they are true. The first problem is simply to interpret and formulate the theoryin such a way that substantive, testable predictions can be made. While I consider some empiricalevidence that bears on these predictions, my approach is, for the most part, informal and intuitive:I simply present my own thoughts as to which of the theorys possible predictions seem mostpromisingthat is, which ones seem from informal observation to be borne out in ways that supportthe theory.

Keywords: Schenkerian theory, music perception, music composition, harmony, counterpoint

1. the question of purpose*

More than seventy years after schenkers death,Schenkerian theory remains the dominant approachto the analysis of tonal music in the English-speakingworld. Schenkerian analyses seem as plentiful as ever in musictheory journals and conference presentations. Recent biblio-graphical works,1translations of Schenkers writings,2and ex-tensions of the theory to non-canonical repertories3testify tothe theorys continuing vitality. In music theory pedagogy,Schenker remains a powerful and pervasive force. Several recenttextbooks either focus on the theory directly4or present basicharmony and counterpoint from a Schenkerian perspective.5

Schenkerian analysis remains the standard next step in tonalanalysis after the rudiments of the undergraduate core; at manyinstitutions, including my own, graduate or upper-level under-graduate courses on tonal analysis are exclusively devoted toSchenkerian analysis.

No theory should be exempt from critical scrutiny; suchscrutiny is especially important if a theory is highly influentialand widely accepted. Schenkerian theory (hereafter ST) hascertainly not escaped criticism over the years. Numerous cri-tiques of the theory have been offered,6to which its advocateshave responded in vigorous defense.7I will argue here, however,that some basic questions about ST have not yet been askedor, at least, have not been given the careful and sustained atten-

tion that they deserve.

A theory is (normally, at least) designed to make accuratepredictions about something; we evaluate it by examining how

well its predictions are confirmed. Thus, the first step towardevaluating ST is to determine what exactly its predictions areaboutthat is, what it is a theory of. This is, in fact, an ex-tremely difficult and controversial issue. In some respects, thereis general agreement about the basic tenets of ST. By all ac-counts, the theory posits a hierarchical structure in which musi-cal events are elaborated by other events in a recursive fashion:Event X is elaborated by event Y, which in turn may be elabo-rated by event Z. (This process of elaboration is sometimes alsocalled prolongation, diminution, or transformation.) The elabo-rating and elaborated events may be pitches, although in some

cases they may also be more abstract entities, notably structuralharmonies or Stufen. Viewed generatively, the process beginswith the Ursatza descending stepwise melody 31, 51, or81, over a IVI harmonic progressionand ends with thepitch structure of the actual piece. A Schenkerian analysis thusforms a kind of tree structure, similar to the syntactic treesposited in theoretical linguistics. There is widespread agreement,also, that ST is primarily a theory of common-practice tonalmusiccertainly this was Schenkers intentionthough somehave argued that it is applicable to other kinds of music as well,such as pre-tonal Western music, post-tonal music, jazz, and rock.

ST, then, posits a certain kind of musical structurehereafterI will call these Schenkerian structures. The difficulty lies in de-

termining what, or perhaps more precisely where, these struc-tures purport to behow they map onto the real world. Torepeat my earlier question: What exactly is ST a theory of ?8Twogeneral possibilities are that it could be a theory of compositionor a theory of perception. As a theory of composition, the theorystates that Schenkerian structures are mental representationsformed in the minds of composers as part of the process of creat-ing a piece. This view does not necessarily imply that pieces are

* Thanks to Nicholas Temperley for helpful comments on an earlier versionof this essay.

1 E.g., Ayotte (2004); Berry (2004).2 Schenker ([1924] 2004); ([1930] 1994).3 E.g., Judd (1992); Covach and Boone (1997); Larson (1998); and Maisel

(1999).4 Cadwallader and Gagn (2007).5 Roig-Francoli (2003); Gauldin (2004).6 Rosen (1971); Narmour (1977); Smith (1986).7 Schachter (1976); Keiler (1978); Beach (1987).

8 For a more thorough discussion of this issuewhich I treat only very brieflyheresee Temperley (1999).


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composition, perception, and schenkerian theory 1

generated in a top-down orderstarting with the Ursatzandproceeding downwardbut simply that the entire Schenkerianstructure is generated at some point during the compositionalprocess. The compositional view of ST also does not imply thatcomposers were necessarily consciously aware of Schenkerianstructures; it is a basic tenet of cognitive science that many men-

tal processes and representations are not consciously accessible.Schenker himself clearly advocated ST as a theory of the com-positional process (though he believed that only great compos-ers of tonal music, not mediocre ones, had Schenkerianstructures in mind).9Today, one rarely finds explicit endorse-ments of ST as a theory of composition, but it is certainly im-plicit in much Schenkerian writing. For example, a statementsuch as Beethoven recomposes the untransposedfalling sixth . . .over the span of 13 bars in the development section10surelyconstitutes a claim about the compositional process. And someauthors, notably Brown,11have claimed quite forthrightly thatST should be construed as a theory of expert tonal composition.

The second possibility is that ST could be construed as a

theory of perception. By this view, in hearing a piece, listenersinfer a Schenkerian structure for it. As with the compositionalview, the perceptual view of ST does not necessarily imply thatSchenkerian structures are available to consciousness; they may

well be present at an unconscious level. The perceptual view ofST, like the compositional view, is rarely explicitly embraced;indeed, some authors have argued against it.12But again, oneoften finds it implied in Schenkerian writings, and I suspect itis actually quite widely held (which is probably why some peo-ple bother to argue against it!). In particular, as I have arguedelsewhere,13claims that Schenkerian structures explaincertainaspects of our perception and experiencethe sense of coher-ence we may experience in a piece, for exampleseem to imply

that such structures are being mentally represented.The compositional and perceptual views of ST are not mutu-

ally exclusive. A parallel here comes to mind with linguistics(one of many that will be drawn in this essay). When linguistsposit syntactic tree structures, for example, the usual assumptionis that such structures play a role in both production and com-prehension: speakers form them in creating a sentence, and lis-teners form them in understanding the sentence. Similarly, onemight argue that Schenkerian structures characterize both com-position and perception. In support of this view, one mightargue that the purpose of music, after all, is communication.

Why would the great composers have bothered to create suchelaborate mental structures if they thought that these structures

would never be shared by listeners? Thus, the possibility of con-struing ST as a theory of both composition and perceptionshould be borne in mind.

The two possible views of ST presented aboveas a theoof perception or a theory of composition (or both)are not thonly ways in which the theory might be construed. Anothpossibility is that a Schenkerian analysis could simply be rgarded as an interesting or satisfying way of hearing a piece, nentailing any claim either about listeners existing perceptions

about the compositional process. Indeed, I would argue that thviewwhich I have elsewhere called the suggestive viewin fact the dominant construal of the theory in contemporaSchenkerian discourse, or at least the one that is most ofteexplicitly advocated.14This construal of ST is perfectly coheent, and the goal it envisions for the theory is a very worthwhone. Indeed, it has a direct benefit that the perceptual and compositional goals do not: it enhances our enjoyment and apprciation of a piece of music. But this is a very different goal frothe compositional or perceptual construals of the theory; indeed, one might question whether, under the suggestive construal, ST is a theory at all. Construed suggestively, the aim the theory is not to generate true statements or correct predi

tions about anything, but simply to give rise to a satisfying eperience for the listener. Perhaps the theory could, in some wabe systematically evaluated as to how well it achieved the gobut that is not my intent here. For now, I will give no furthattention to the suggestive view of ST and will focus on thperceptual and compositional construals.15

In what follows I will considerin a very preliminary anspeculative wayhow Schenkerian theory might be tested boas a theory of composition and a theory of perception. I propoto evaluate the theory as we would evaluate any theory: by examining its predictions and seeing if they are true. The firproblemand really the primary concern of this essayis simply to interpret and formulate the theory in such a way th

substantive, testable predictions can be made. I will considsome evidence from corpus and experimental research thseems to bear on these predictions (at present, little such evdence is available). But for the most part, my approach in thessay will be quite informal and intuitive: I simply present mown thoughts as to which of the theorys possible predictioseem most promisingthat is, which ones seem from informobservation to be borne out in ways that support the theor

9 See, for example, Schenkers remarks in Free Composition([1935] 1979, xxii,xxiii, 6, 12829).

10 Cadwallader and Gagn (2007, 7).11 Brown (2005, 22233).12 See for example Schachter (1976, 28586) and (1981, 12223); and Ben-

jamin (1981, 160, 165).13 Temperley (1999, 7071).

14 See Ibid. (6975) for discussion of the suggestive view of analysis anexamples from the Schenkerian literature.

15 One other possibility is that ST could be regarded as an objective theory musical structure, without entailing any claim about either perception (eith

descriptively or suggestively) or composition. This brings to mind the semotic approach of Nattiez (1990, 1112), which distinguishes between tpoietic level of musical structure (in the mind of the producer), the esthelevel (in the mind of the perceiver), and the neutral level, which is simply tobjective structure of the piece itself: the latter would seem to correspond the goal that I have just articulated. I do not believe that this view of STwidespread in current thinking, though it is difficult to be sure.

There is certainly more to be said about the relationshipsconflicts aconvergencesbetween these various goals. For example, the suggestiand compositional goals of analysis may often converge: the composeconception of a piece is likely to be an interesting and satisfying way hearing it, though perhaps not always the most satisfying way.


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principles of functional harmonic theory, these principles arenot held to be inviolable laws, but, rather, generalizations thatare mostly true; and on that basis, they arequite legiti-matelywidely accepted to have been part of compositionalthinking. Of course, such claims are generally based not on sys-tematic data analysis, but on informal observationpassed onand reconfirmed over many generations. In recent years, how-

ever, a number of these principles have been subjected to em-pirical testing, generally with positive results.18

Once appropriate data have been gathered, the questionarises: how can we judge whether, or to what degree, the dataconfirm the theorys predictions? One can of course judge thisintuitivelyas I did in saying that the data in Example 2 giveconsiderable support to harmonic theorybut clearly suchstatements are vague and informal; one might wonder if the fitbetween theory and data could be measured more objectively.

There is in fact an extremely powerful method for doing this,namely, Bayesian probabilistic modeling. Under the probabilis-tic approach, the validity of a theory of a body of data dependson the probability that it assigns to the data. A finite-state

model, for example, can be made probabilistic by assigningprobabilities to the transitions; the probability of a series of out-put symbols is then the product of the probabilities of all thetransitions. By this reasoning, the best model of the dataandthus the one most likely to characterize the process of their cre-ationis the one that assigns the data highest probability.Elsewhere I have explored the possibility of using this approach

to evaluate claims about compositional processes.19Conceivabthis approach could be applied to ST as well. I do not attempthis here, and I believe it would be premature to do so, for resons that will become apparent below. But the probabilistic approach is still worth bearing in mind, as it suggests a way

which even informal and qualitative empirical statementse.dominants usually go to tonicsmight ultimately be sub

jected to quantification and rigorous testing.To summarize the previous discussion: A theory can be rgarded as a valid theory of compositional practice to the extethat it makes true predictionsnot necessarily perfectly true, bmostly trueabout compositions. We now apply this approato ST. What exactly are the predictions of ST? And are they tru

2.2 preliminary issues

Before proceeding, we must consider several basic issues the testing of ST. One is the issue of musical domain. If ST istheory of the compositional process, whose compositional prcess does it purport to describe? While there is some diversity

opinion on this point, the usual view seems to be that ST istheory of tonal music20or some more narrowly defined subcaegory such as functional monotonality21or triadic tonality.As a theory of compositional processes, then, it presumably aplies to composers of such music. Those interested in testing S

18 The work of David Huron is especially notable in this regard. See, for ex-ample, Huron (2001) with regard to chord spacing (1718) and the avoid-ance of perfect parallel intervals (31).

19 Temperley (2007, 15979); Temperley (2010). See also Conklin and Wten (1995); and Mavromatis (2005).

20 Cadwallader and Gagn (2007, xi).21 Brown (2005, 22, 24).22 Forte (1977, 25).

example 2. Harmonic progressions in the Kostka-Payne corpusThe data represent both major-key and minor-key sections. Antecedent chords are on the vertical axis, consequent chords on the horizontal;

each cell in the table shows the number of occurrences of the antecedent-consequent pair. All chords were categorized as labeled in the Romannumeral analyses (i.e., in relation to the local key), with the assumption of enharmonic equivalence (e.g.,I =II); applied chords were

relabeled as immediate chords of the local key (e.g., V/V as II). Cadential six-four chords are notated in the analyses in a two-level fashion,with a I 64 V within a larger V; the latter analysis was used here.


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150 music theory spectrum 33 (2011)

have recognized the importance of defining the domain towhich the theory applies. In Matthew Browns words:

[S]ince laws and theories delimit classes of phenomena, they musthave boundaries. . . . [A] valid theory of tonality must be able topredict specific pitch relationships that occur in tonal pieces and notin non-tonal ones.23

Here Brown voices a valid and important concern. A theorymust be judged not only on its ability to generate all the ob-served data, but on its ability not to generate other possible (butnot observed) data. To return to our harmonic finite-statemodel: If we added transitions from every node in the model toevery other node (for example, connecting D to PD and con-necting PD to T), the model would then generate every possibleharmonic progression. It would still of course generate all of theprogressions that were generated before, but it would now allowmany progressions that are uncharacteristic of common-practiceharmony (or, at least, rare). Such a model clearly does not pre-dict the data well, and therefore has little plausibility as a modelof the compositional process. The crucial question to ask of ST,then, is not so much whether it can generate the observed data,but whether it can discriminate between what is observed and

what is not.24

How, then, should the domain of ST be defined? Definingthis domain as tonal music, or something similar, is certainly apossibility; this may not be the best approach, however. First ofall, whether a piece is tonal is highly subjective and, I wouldsuggest, quite ambiguous. Surely we cannot draw a sharp linebetween music that is in this category and music that is not. 25Secondly, it is not crucial to define the domain of the theory inthe largest possible way. Suppose, for example, that we confinedourselves at least initially to music written in Europe between

1770 and 1800. This is surely a well-defined domain; there is nodoubt (barring a few pieces of uncertain date or composer) as towhich pieces it contains. It certainly does not contain everything

to which ST would normally be considered applicable, but it isa start. If one had a theory about dogs, and it later turned outthat the theory was true of all mammals, this would not invali-date the theory as a theory of dogs (though it would mean thatthe domain of the theory could be more accurately character-ized). One might argue, indeed, that it is more prudent to begin

by trying to model a relatively small and unproblematic domainrather than a large and ambitious one.There is no need to resolve the question of domain defini-

tively here, given the preliminary nature of this investigation.My aim has only been to consider some of the issues that arise.Rigorous, quantitative testing of ST will require resolution ofthese issues, but this is a project for the future. For now, I willfollow convention and simply refer to the domain of ST astonal music.

Another issue to consider is the terms in which the theoryspredictions will be expressed. The important point here is that,for a theory to be convincing, it must make predictions aboutentities that are themselves not in questionthat is to say, it

must predict accepted facts. To use the language of the philoso-phy of science, the theorys predictions must concern observa-tion termsthings that are directly observable and henceuncontroversial.26Return again to the finite-state model inExample 1, which makes predictions about harmonic progres-sions. Such a model presupposes that the sequence of chords ina piece is a matter of fact, about which all qualified observers

will agree. Clearly this assumption is not completely truethere is sometimes disagreement as to how to assign Romannumerals to a piece (for example, what is a chord and what ismerely an ornamental or linear event). This is a potential prob-lem for harmonic theory as a model of compositional practice.

To address this problem, the theory would have to be supple-

mented with additional principles that relate harmonies to theobservable facts of music: first and foremost, actual patterns ofnotes. And this points to an important virtue of ST: clearly, thetheory doesaddress patterns of notes, and the relationship be-tween note patterns and higher-level structures, in a way thatfunctional harmonic theory does not. Any formalization of STas a predictive theory should surely take advantage of this aspectof the theory. By contrast, a model of ST that makes predictionsonly about entities posited by the theory itself will have littlepersuasive force. For example, the prediction that every piece isbuilt on either a 31, 51, or 81Ursatz will cut little ice withanyone who is skeptical about the theory, since they will ques-tion the very facts the theory is predicting.

This issue of observation terms arises in a study by Brown,Douglas Dempster, and Dave Headlam.27These authors sug-gest that ST makes a prediction regarding Stufen(structuralharmonies), namely, that a I Stufeis never directly elaborated bya IV/ V. For those who take for granted the presence of Stufenand the relations of elaboration between them, this is indeed atestable prediction. For someone who is skeptical about ST,

3 Brown (1989, 29).4 Probabilistic methods offer an elegant solution to this problem. As noted

earlier, a probabilistic model assigns a probability to a body of data, andmodels can be evaluated and compared using these probabilities: the higherthe probability, the better the model. By definition, the probabilities as-signed to all possible events by a probabilistic model (where events mightbe pieces, harmonic progressions, or anything else) must sum to 1. Thus ifa model assigns relatively high probability to events that occur (or occurfrequently) in a musical corpus, it must assign relatively low probability toother events. For this reason, there is no need to actually test a model on its

ability to assign low probability to events not occurring in the corpus; oneneed only test it on the corpus itself.

5 In effect, defining the domain as tonal music appeals to our intuitionsabout what is tonal and what is not. But the basis for these intuitions is aninteresting perceptual problem in itself; what is it that makes a piece soundor seem tonal? One could certainly devise a model that predicted thesejudgments; this would be an interesting project. (Indeed, ST itself might beused for this purpose, as I will discuss.) But I would argue that modelingcompositional data, and predicting judgments of tonality, are two differentproblems and should be kept separate. The domain of a compositionaltheory might roughly correspond to music that seems tonal, but to definethe domain in those terms seems ill-advised.

26 Fodor (1984, 25, 38).27 Brown, Dempster, and Headlam (1997).


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however, the reality of Stufenis exactly what is at issue. To con-vince such a skeptic, it would need to be shown how the IV/ Vhypothesis led to predictions about actual note patternsor,perhaps, about surface harmonies or other entities whose valid-ity was not in question.

2.3 schenkerian theory as a context-free grammar

As stated at the end of Section 2.1, my aim is to formulateSchenkerian theory in a way that yields testable predictionsabout what occurs and does not occur in tonal music. I begin byconsidering the idea of formulating the theory as a context-freegrammar, a kind of hierarchical model widely used in linguis-tics. After reaching skeptical conclusions about this approach, Iconsider some other approaches that might be used in testingthe compositional validity of ST.

As mentioned above, an essential part of ST is its hierarchi-cal, recursive nature, and it seems appropriate to try to formu-late the theory in a way that reflects this aspect. Let us suppose

we begin with the Ursatza 321linear progression over a151bass line (other forms of the Ursatzwill not be consid-ered here). Each element of the Ursatzcan be expanded or

transformed to introduce new elements; then these elements inturn can be elaborated (see Example 3). For example, the initial3of the melody might expand into a 35arpeggiation, as shownin Example 3(b), or into a neighbor motion 343. (The pos-sibility of a single event undergoing multiple elaborations wouldalso have to be considered.) In some cases, an elaboration mightbe generated from two superordinate events; passing-tones andother linear progressionswhere a consonant interval is filledin with stepwise motioncould be generated in this way, asshown in Example 3(c).28There might also be elaborations of a

vertical nature: for example, building a triad under an existingmelody note or over a bass note (the Schenkerian idea of con-sonant support).

What I have just described is a context-free grammar (CFG),a widely used formalism in the modeling of natural language.29

The grammar is context-free because the possible expansioof an element depend only on the element itself, and not on anlarger context. The construal of ST as a context-free grammhas antecedents in important attempts to formalize the theoby Michael Kassler and Brown.30Kassler proposes a model thgenerates Schenkerian structures through the recursive appliction of vertical and horizontal elaborations, and shows howtonal pieceor more specifically a fairly foreground-levSchenkerian analysis of that piececan be generated in thmanner. Brown, similarly, formulates ST as a set of transformtional rules which can be recursively applied to an Ursatzgenerate tonal surfaces. Both Kassler and Brown state as thegoal the development of a grammar that can generate all anonly tonal pieces.

In considering how to formalize Schenkerian theory asCFG, we immediately encounter a number of difficulties. Onconcerns the interaction between the vertical and the horizontal. In Example 3, suppose one generated a 345pattern frothe initial E5, and then wished to give consonant support the 5by adding a C3 in the bass, as shown in Example 3(d

Would this be generated horizontally (from the previous Cvertically (from the simultaneous G5), or both? Probably bot

as the generation of such notes seems to depend on both vertcal and horizontal connections. This is a major complicatiothat cannot easily be addressed within the framework of context-free grammars.31This is, however, not a problem that unique to Schenkerian analysis; even under a more convention

view of harmony and counterpoint, the placement of notes conditional on both horizontal and vertical considerations, anthese complex dependencies would certainly have to be adressed in some way. Another issue is how the generation Stufenshould be integrated with the generation of notes. Opossibilityadopted by Kassler32is that Stufen could b

viewed simply as emergent features of the note-generation prcess: for example, the bass note of a Stufecould first be gene

ated horizontally, and the upper notes of the chord generatefrom the bass note. In that case, no explicit recognition of Stufwould be needed in the generative process. This does not seevery satisfactory, however; one might argue that, once a Stufe28 Allowing an event to be generated from two superordinate events might be

considered to violate the requirement of context-freeness (as definedbelow). However, this does not seem like a serious problem, as the generat-ing context in this case is still relatively local. Alternatively, one could beginwith a single pitch, forming one endpoint of the progression, and thengenerate the other endpoint and the internal pitches with a single ruleforexample, generating Examples 3(b) and 3(c) in one step.

29 This term is usually attributed to Chomsky (1957), though he used theterm phrase structure grammar.

30 Kassler (1977); Brown (1989) and (2005). Two other interesting computtional implementations of ST, which came to my attention too late to bdiscussed here, are Gilbert and Conklin (2007) and Marsden (2010).

31 Brown (2005, 21617) discusses other phenomena of tonal music that canot easily be modeled within a context-free grammar.

32 Kassler (1977).

A. B. C. D.

?

example 3. Generating a Schenkerian reduction with a context-free grammar


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generated, subsequent elaborations (such as linear progressions)depend on the Stufeitself rather than on its individual pitches.

Another issue that arises here (and in the modeling of STgenerally) is the role of counterpoint. Schenker and his follow-ers maintain that strict counterpointfor example, as codifiedby Fux33has a profound connection to common-practicecomposition. From the point of view of modeling composition,this is an interesting connection, since the rules of strict coun-terpoint are quite well defined and restrictive; it is natural to

wonder whether these rules might be incorporated into a gen-erative Schenkerian grammar. However, the exact nature ofthe relationship between strict counterpoint and ST is unclear.Some, such as John Peel and Wayne Slawson, have gone so faras to claim that the higher levels of a Schenkerian analysis arestrictly governed by traditional contrapuntal rules.34Most otherauthors have argued for a more indirect relationship betweencounterpoint and free compositionobserving, for example,that both idioms are constructed from fluent, largely stepwiselines, and that they share certain basic patterns such as passingtones.35

Clearly, the formalization of ST as a CFG would entail anumber of difficult challenges, but this should hardly surprise

us, and is itself not a decisive argument against the approach. Afurther problem that arises is, in my view, more fundamental.

The problem is that a Schenkerian CFG seems in serious dan-ger of being able to generate everything. That is, it seems thatalmost any possible pitch pattern could be generated using somecombination of basic Schenkerian transformations. In a recentexperiment of my own, I showed that a completely randompitch pattern could be generated using a fairly small set ofSchenkerian transformationsjust single-note elaborations(incomplete neighbors) or consonant skips (see Example 4).36Admittedly the generative model assumed in this experiment is

very far removed from actual ST, mainly because it only allowsmonophonic pitch sequences, thus completely ignoring the ver-tical dimension that (as noted earlier) is a crucial part of thetheory.37But the essential point remains: Even if polyphonicstructures were assumed, it is unclear how a Schenkerian CFG

would impose substantive constraints on what surface note pat-terns could be generated. One possible way to address this prob-lem would be to add probabilities to the transformations; by theusual logic of probability, the probability of a note pattern wouldbe given by the sum of the probabilities of all of its deriva-tions.38A piece would then be judged as ungrammatical ifthere was no way of generating it that was high in probability.

The prospects of success for such a project are uncertain; it isunclear what kinds of regularities the grammar would be ex-

pected to capture.This leads us to what is really the most fundamental ques-tion of all: What is it about tonal music that a Schenkerian CFG issupposed to predict? In what way will the model make substan-tive, testable predictions about tonal music, over and above whatcould be done with more conventional principles of harmonyand voice leading? To answer this question, it is perhaps usefulto consider what the motivation is for using CFGs with naturallanguages. This issue has been much discussed in linguistics.Noam Chomsky famously asserted that natural languages mustinvolve something more powerful than finite-state machines,because they permit infinite recursions that cannot possibly becaptured by a finite-state machinesentences such as the fol-

lowing: The man who said that [the man who said that [the manwho said that [. . .] is arriving today] is arriving today] is arrivingtoday.39Sentences such as this (and other similar constructionsproblematic for finite-state models) involve long-distance de-pendenciesin which an element seems to license or requiresome element much later in the sentence. If there were analo-gous cases in music, this might warrant the kind of context-freegrammar suggested by Schenkerian theory. Do such long-distance dependencies arise in music? Perhaps the first examplethat comes to mind is the Ursatzitself. The Ursatzreflects theprinciple that a tonal piece or tonally closed section typicallybegins with a I chord and ends with VI, though the initial andfinal events may be separated by an arbitrarily long span ofmusic. However, this dependency might also be captured in

3 Fux ([1725] 1971).4 Peel and Slawson (1984, 278, 287).5 Forte and Gilbert (1980, 4149); Cadwallader and Gagn (2007, 2339).6 Temperley (2007, 176).7 Schenkerian theory is really polyphonic in its very essence; even mono-

phonic pieces such as the Bach violin and cello suites are generally under-stood in Schenkerian analysis to have an implied polyphonic structure.

example 4. A random fifteen-note melody generated from an even distribution of the notes C4B4(from Temperley [2007]):The analysis shown is one of 324,977 generated by a Schenkerian parser which finds all possible analyses using just one-note elaborations(incomplete neighbors and consonant skips). The beamed notes indicate the highest-level line or Urlinie; elaborations (shown with slurs)

can then be traced recursively from these notes (for example, the first Bis elaborated by the Ctwo notes earlier, which is in turnelaborated by the Es on either side).

38 Probabilistic context-free grammars are widely used in computational lin-guistics (Manning and Schtze [2000, 381405]). Following the usualmethodology of probabilistic modeling, the probabilities for various differ-ent kinds of elaborations could be based (at least as a first approximation)on counts of them in an analyzed corpusfor example, a set of publishedSchenkerian analyses. A study by Larson (199798) offers an interestingfirst step in this direction. Gilbert and Conklin (2007) also attempt tomodel hierarchical pitch structure with a probabilistic CFG.

39 Chomsky (1957, 22).


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simpler ways, without requiring all the apparatus of a CFG. Atlocal levels (for example, a tonally closed phrase or period), itcould be modeled quite well with a finite-state model of har-monic progressions; indeed, it appears to be captured well bythe model in Example 1. At larger levels, too, it is not clear whatthe Ursatzadds, in predictive power, to a more conventional

model representing a piece as a series of key areas with a pro-gression of chords in each key, beginning and ending in thesame key, and ending with a perfect cadence in the home key.

In short, it is unclear what is to be gained by modelingSchenkerian theory as a CFG. To justify this approach, one wouldneed to find long-distance dependencies that are not predictedby more conventional principles of tonal theory. I am not con-

vinced that such long-distance dependencies exist. Rather thanpursuing this approach any further, I will argue that there are other

ways of testing Schenkerian theory (as a theory of compositionalpractice) that seem more likely to lead to successful results.

2.4 ZGEandSTUFEN

The linear progression or Zuga stepwise line connectingtwo structural tonesis a ubiquitous element in Schenkeriananalysis and is given great importance in Schenkers own writ-ings. One might wonder if this concept could be used to maketestable predictions about tonal music. The simple fact thatstepwise melodic intervals are very prevalent in tonal melodiescould be predicted by a simple note-to-note model, and doesnot requireZge. However, the Zugconcept suggests a furtherprediction: If most stepwise motions are part of a larger Zug,this suggests that a step (whole step or half step) is likely to befollowed by another step in the same direction. The idea that asmall melodic interval tends to be followed by another same-

direction intervalsometimes called process or inertia40has been widely discussed in connection with melodicexpectation (I will return to this below). As a prediction aboutcompositional practice, the inertia idea can be tested quite eas-ily, using computationally encoded corpora. Example 5 showsthe results for two corpora, one of Bach chorales and one ofHaydn and Mozart string quartets. In both cases, the predictionis strongly confirmed: an ascending step is much more likely tobe followed by another ascending step than by a descendingstep, whereas for descending steps, the reverse is true.41

An extension of the Zugideaand one that is perhaps moredistinctively Schenkerianis that linear progressions operatenot just at the surface of melodies but at higher levels. In ST, amelody is typically analyzed as a long-range linear progression,recursively elaborated by other linear progressions (along withother elaborations such as arpeggiations). However, this doesnot appear to constitute a very strong prediction. It is true that

one usually can find a long-range stepwise linear progressioacross a melody; but this may largely be due to the fact thmelodies are usually confined to a fairly limited pitch range. many cases, the predictive power of linear progressions seems lie not so much in pitch patterns themselves, but rather in th

way that pitch patterns are realized rhythmically and metricalAs an especially obvious example, consider the melody Un bd from PuccinisMadama Butterfly(Example 6). Consideresimply as a sequence of pitches, this melody could be generatin many ways; generating it hierarchically from the linear pro

gression G FE D seems no more compelling than othalternatives. But when rhythmic context is added, we see theach of the notes G FE D falls on the downbeat and agogically accented. This seems like more than mere coincdence and suggests to me that the composer was aware of thlinear pattern in choosing a rhythmic setting for the melodone might argue on this basis that he had the same linear fram

work in mind in generating the pitches as well.Another way in which linear progressions might yield tes

able predictions concerns harmonic progressions. Consider thprogression shown in Example 7, from the first movement Mozarts Symphony No. 40 in G Minor, K. 550a progressiodiscussed by John Rothgeb.42As Rothgeb notes, viewed in co

ventional harmonic terms, this progression seems extremeodd: it contains root motions from V to IV and from ii to both of which are usually considered incorrect. And yet the prgression does not sound wrong and is not uncommon in tonmusic. (Two other examples are the opening of HandelConcerto Grosso, Op. 6, No. 4, and the second theme Brahmss Violin Sonata in G Major, Op. 78, I, mm. 4042

The Schenkerian perspective provides an explanation: the logof the progression is built on a stepwise descending bass line. B

40 See Narmour (1990, 2-4) for process and Larson (1997, 102) for inertia.41 Huron (2006, 7778) reports a similar study using a large corpus of both

Western and non-Western melodies; the data reflect step inertia for de-scending steps only, not ascending steps. This intriguing result suggeststhat the bi-directional step inertia found in common-practice music may bepeculiar to that style, or at least not universal. 42 Rothgeb (1975, 27779).

example 5. Stepwise intervals in Bach chorales and in Mozarand Haydn string quartets

The table shows, for all cases of two successive steps within amelodic line, the proportion of cases in which an initial ascendingor descending step (half-step or whole-step) was followed by an

ascending or descending step. Two corpora were used: ninety-eighBach chorales (in their entirety) and the first eight measures of al

of Mozart s and Haydns string-quartet movements. In both

corpora, all four melodic lines were analyzed.


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this view, the chords are not to be understood as constituting aconventional harmonic progression, but rather as a kind of elab-oration of the underlying bass notes. Thus a linear progressioncan license harmonic successions (not really progressions atall) which would not normally be allowed.

A possible generative model of this process could be con-structed in the following way: We could begin with a harmonicfinite-state model such as that shown in Example 1, which gen-erates a series of harmonies and then generates pitches fromthese harmonies. This could be augmented by an additionalnetwork that generates bass-line linear progressions and thenforms triads on the bass notes. Upon entering a functional har-mony node such as T, the machine would have the option of

jumping into a linear-progression network and generating aZugbased on that harmony.43A fragment of such a modelshowing just the part needed for the Zug in Example 7isshown in Example 8. Many questions then arise. What are theconstraints on when linear progressions can occur? By generalagreement, linear progressions usually connect two notes of anunderlying harmony; a strict view might hold that they can onlyarise this way. But one often sees linear patterns in tonal musicthat cannot easily be explained as prolongations of an underly-ing harmonyfor example, a linear bass progression 17654in the first part of a phrase under what would normally be con-

sidered a prolongation of I (certainly not a prolongation ofIV).44There are also constraints on the triads formed from thebass notes, which appear to be quite complex. In Example 7, forexample, why does Mozart choose a I 64triad over the F ratherthan (for example) a iii6triad?

A further situation in which Schenkerian principles maymake testable predictions is illustrated by Example 9, a passagefrom Mozarts Clarinet Concerto in A Major, K. 622, I. The pro-gression of local chords, shown below the score, is mostly quite

conventional, and could perhaps be accommodated by a finite-state model of harmony such as Example 1 (suitably expanded toinclude chromatic harmonies and applied chords). But in somerespects it seems rather arbitrary: Why the alternations betweenIV and V/IV in mm. 14344; why the diminished seventhchords in mm. 14647? It can be seen that there is a large-scalelogic governing this passage: the first part of the progressioncenters around IV and the second part around V, leading finallyto a I in m. 154, thus forming a large-scale IVVI cadence.Rather than generating the progression as a series of local har-monic moves, it might be generated more convincingly by a

3 This is essentially an Augmented Transition Networka formalism oncefashionable in psycholinguistics (see Garman [1990, 34555]).

4 Cadwallader and Gagn (2007, 7475) suggest that linear progressionssometimes serve to connect one harmony to another, rather than prolong-ing a single harmony; they explain 17654progressions in this way.

34

34

I64B :

I V6 IV6 ii6 I6

1

START END

3

4

5

6

7V6

IV6(vi?)

I6/4

(iii6?)

ii6

(IV?)

I6

I

example 6. Puccini, Un bel d from Madama Butterfly

example 7. Mozart, Symphony No. 40 in G Minor, K. 550, I, mm. 2833

example 8. A finite-state model for generating a Zug


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more hierarchical process, which first constructed an underlyingprogression of Stufen, IVVI (in itself a very normative pro-gression), and then generated surface chords as elaborations ofthese structural harmonies. Once mm. 14245 are identified asan expanded IV, the surface progression of those measures (in-

terpreted as IVIV . . . in relation to IV) becomes extremelynormative; the diminished-seventh chords in mm. 14647could be treated as linear connectives between the IV and the V(using the logic of linearly generated harmony discussed ear-lier). In this way, a hierarchical view of the passages harmonymakes it seem more normative, and thus predicts it morestrongly than a purely finite-state model.

2.5 predicting other phenomena of tonal music

So far my approach to evaluating ST as a theory of tonalcomposition has been to examine the theory and consider whatits predictions might be. Another way of approaching this prob-

lem is to examine phenomena of tonal musicespecially phe-nomena that have not been satisfactorily explained by othermeansand consider how ST might predict them. Here I willconsider just two aspects of the common-practice musical lan-guage for which, as far as I know, no adequate explanation hasbeen provided.

As is well known, certain key schemes are especially com-mon in tonal pieces. For one thing, given a particular main key(e.g., C major) certain secondary keys are especially common(such as G major, F major, and A minor). Various theories ofkey relationsoften employing spatial representationshave

been proposed that explain these facts fairly well.45What hnot been explained so well is the temporal arrangement of key

which also reflects consistent patterns. The key scheme IVviI is extremely common; it is frequently seen, for example, Baroque suite movements and classical-period minuets an

slow movements. Other patterns such as IVIV(ii)I or IVviIV(ii)I are also common. What is especially puzzling abothese patterns is that they do not follow conventional rules harmonic progression; indeed, they are virtually mirror imagof conventional progressions. IVIVI is a common kescheme but an uncommon harmonic progression; IIVVI ubiquitous as a harmonic progression, but very rare as a kscheme.

Could ST be of any value to us in explaining these patternHere we encounter the difficult issue of the relationship Schenkerian structure to what might be called key structurethe arrangement of key sections within a piece. Schenkeriaanalysis does of course feature tonal entities prolonged over lar

portions of a piece, and a nave encounter with a Schenkeriaanalysis might suggest that these entities corresponded to ksections. It should be clear, however, that this is not the casSometimes, high-level events in a Schenkerian analysis do represent the tonic triads of key sections; but very often they dnot.46 If one considers the deepest structural level ofSchenkerian reductionthe Ursatzthe V triad almost nev

45 Krumhansl (1990, 4049); Lerdahl (2001, 4188).46 For an insightful discussion of this point, see Schachter (1987, 29296)

141

144

147

V65 IV

V64V

IE: IV V IV

IV VIV

IV VIV

IV VIV

IV

4vii3

7vii

53 I

)(

example 9. Mozart, Clarinet Concerto in A Major, K. 622, I, mm. 14154


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represents an extended section in the dominant key, and indeedis often not tonicized at all. Conversely, a key may be implied

without its tonic triad being represented as a high-level reduc-tional event; and when the tonic triad is a high-level event, thespan of music over which it is prolonged may be quite differentfrom that of the corresponding key section.

In short, Schenkerian analysis does not appear to representkey sections in any direct way. Even if key sections did consis-tently correspond to Stufen, the prediction that would seem tofollow from thisif anyis that key schemes follow the samelogic as harmonic progressions; and as we have seen, this is farfrom the case. It is true that key areas do sometimes correspondto Stufen, and that treating key sections as expanded harmoniessometimes helps to make sense of them. Example 9 offers a casein point: the tonicization of IV is nicely explained by treating it

as an expanded predominant of a large structural cadence. Butfor the most part, the correspondence between Stufenand keysections does not hold. It remains possible, however, that STcould be used to generate predictions about key structure in lessdirect wayfor example, by specifying which Stufen permit (orrequire) tonicization and which ones do not. This possibility hasnot been much explored, but it might offer an interesting way oftesting ST as a model of compositional practice.

Another aspect of the common-practice language that hasnot been fully explained is sequencesmelodic patterns that re-peat at changing pitch levels. While sequences could conceivablyoccur in all kinds of different harmonic contexts, it is well knownthat certain harmonic patterns are particularly common and ac-

count for nearly all sequences in the common-practice repertory.The patterns most often presented in theory textbooks are thefirst four shown in Example 10. Patterns A, B, and C are all

widely used; pattern D, the ascending-fifth sequence, sometimesoccurs but is much less common than the descending-fifthpattern (A).47In some respects, sequences are well-suited to

Schenkerian analysis. Since the melodic patterns of sequencesgenerally shift up or down by a step at each repetition (with the

exception of pattern C, which shifts by a third), they can oftenbe analyzed as elaborations of linear progressions (though it issometimes uncertain which notes of the pattern are structural).Some authors have argued that sequences are best understood asbeing essentially contrapuntal, rather than harmonic, in origin.Allen Cadwallader and David Gagn, for example, refer to se-quences as linear intervallic patterns and write:

In textures of more than two voices, chords naturally arise in con-junction with the repeated pattern, thereby forming chordal se-quences. The chords in the pattern . . . therefore result fromcontrapuntal motion.48

This view is also reflected in contrapuntal names for sequencessuch as ascending 56 for pattern B and descending 56 forpattern C.49

From the point of view of explaining compositional practice,the essential question posed by sequences is this: Why is it thatcertain patterns are widely used and others are not? One couldtry to explain these regularities in contrapuntal terms. The popu-larity of outer-voice patterns such as 106 or 1010 reflects thepreference for imperfect consonances in species counterpoint; thealternating 56 pattern is also familiar from species and can beseen as a strategy to avoid parallel fifths. However, the contrapun-tal view of sequences only takes us so far. In particular, it fails toexplain why patterns A, B, and C are all much preferred over theirmirror images, patterns D, E, and F. These mirror-image patterns

seem contrapuntally just as good as those of A, B, and C; why arethey so rarely used? (As noted earlier, the ascending-fifth patternis generally described as much less common than the descending-fifth pattern.) A more promising explanation for these facts maylie elsewhere. Traditional harmonic theory holds that descendingfifth motions are preferred over ascending fifths, and descendingthirds over ascending thirds;50this nicely predicts the preference

7 Aldwell and Schachter note that most diatonic chordal sequences fall intoone of [these four] categories (2003, 265); see also Laitz (2008, Chapter22). On the rarity of the ascending-fifth sequence, see Aldwell andSchachter (2003, 248); and Laitz (2008, 501).

48 Cadwallader and Gagn (2007, 8182).49 Aldwell and Schachter (2003, 27378).50 See, for example, Schoenberg ([1954] 1969, 68).

A. B. C.

D. E. F.

example 10. Common and uncommon sequential patterns


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for pattern A (all descending fifths) over D (all ascending fifths),

and the preference for B (descending fifths and thirds) over E(ascending thirds and fifths). The predictions of harmonic theoryregarding C (ascending fifth/ascending step) and F (descendingfifth/descending step) are less successful. But harmonic theory

would seem to present at least a partial explanation for the pre-ferred sequential patterns; the contrapuntal view of sequencesdoes not. Here again, then, is a puzzling set of facts that a theoryof tonal music might reasonably be expected to explain, yet STseems to be of little help to us.

Overall, the view I have presented of ST as a theory of com-position is fairly pessimistic. Formulating the theory as a CFGseems to offer little hope of making substantive predictionsabout tonal music. I also considered certain phenomena abouttonal music that one might like a theory to predict, such asregularities of key structure and common sequential patterns;ST seems to have little to say about these phenomena (thoughperhaps future extensions of the theory will shed light on theseissues). However, some aspects of Schenkerian structure do

yield clear and plausible predictions, notably certain fairly localuses of Zgeand Stufen; and these aspects of the theory mightmake an important contribution to the modeling of tonalmusic. I will return at the end of the article to what all thismeans for our view of ST. I now turn to the second issue posedat the beginning of this essay: the testing of ST as a theory ofperception.

3. schenkerian theory as a theory of perception

The hypothesis under consideration here is that Schenkerianstructures are formed in the minds of listeners as they listen totonal music. One basic question for this construal of the theoryis: Which listeners does the theory purport to describe? A theoryof listening might claim to apply, for example, to all those havingsome familiarity with tonal music, or perhaps only to those witha high degree of musical training. Given the speculative natureof this discussion, there is no point in trying to be very precise

about this. My general focus hereand the focus of most of t

relevant experimental research as wellis on listeners who hasome musical training and familiarity with common-practimusic, but who have not had extensive training in ST.51Thisnot in any way to deny the validity of perception informed badvanced training in music theory; but to claim something astheory of perception implies that it characterizes peoples peception beyond what is due to explicit study of the theory.52

A natural approach to testing ST as a theory of perceptionto use the methods of experimental music psychology. Raththan simply asking listeners to report their mental representtions of a piece, the usual approach of music perception researis to access these representations in an indirect fashion, usualby having subjects make judgments about some aspect of muscal passages such as their similarity, tension, or expectedness.

what follows I will review some of this research; I will also adsome of my own thoughts as to the aspects of ST that seemost promising with regard to the modeling of perception.

3.1 similarity

One way of testing the perceptual validity of ST is througjudgments of similarity. If people form reductions of pieces they listen, it stands to reason that passages sharing the samunderlying structure should seem more similar than those thdo not. This reasoning has sometimes been used to argue for th

perceptual reality of pitch reduction. For example, Fred Lerdaand Ray Jackendoff point to the two melodic passages froBeethovens PastoralSymphony shown in Example 11, and nothat listeners have no trouble hearing Example 11(b) as an elaboration of Example 11(a);they argue that such judgments aevidence that the second melody is understood as an elaborate

68

68

68

68

A.

B.

C.

D.

example 11. (a) Beethoven, Symphony No. 6 in F Major, Op. 68, I, mm. 912; (b) Measures 11720 from the same movement;(c) A possible reduction of (b); (d) Measures 1316 from the same movement

51 Most of the experimental studies described below, such as Serafine et (1989) and Bigand and Parncutt (1999), used undergraduate students wivarying amounts of musical training.

52 For discussion of this point, see Temperley (1999, 7576).


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version of the first (or, perhaps, that both are heard as elabora-tions of the same underlying structure).53

The possibility of testing the psychological reality of pitchreduction through similarity judgments was explored in a studyby Mary Louise Serafine et al.54In one experiment, the authorstook passages from Bachs unaccompanied violin and cello

suites, generated correct and incorrect reductions for them, andhad subjects judge which of the two reductions more closelyresembled the original. Subjects judged the correct reductions asmore similar at levels slightly better than chance. As Serafineand her collaborators discuss, one problem with this methodol-ogy is that there are many criteria by which passages might be

judged as similar, some of which are confounded withSchenkerian structure. Given two reductions of a passage, if onereduction is inherently more musically coherent than the other,it may be judged more similar to the original passage simplybecause it is more coherent; this in itself does not prove theperceptual reality of reduction. Another possible confound isharmony. Two passages based on the same surface harmonic

progression are likely to seem similar, and a passage and itsSchenkerian reduction are also likely to share the same impliedharmonic progression; thus any perceived similarity betweenthem may simply be due to their shared harmony.

Serafine et al. made great efforts to control for these con-founds. With regard to the coherence problem, they had sub-

jects indicate their degree of preference for correct andincorrect reductions heard in isolation; these judgments werenot found to correlate strongly with the perceived similarity be-tween the reductions and surface patterns. The authors also triedto tease out the effects of surface harmony from Schenkerianstructure. In one experiment, listeners compared passages withharmony foils that were different from the original passage in

surface harmony but alike in Schenkerian structure, and coun-terpoint foils that were different in Schenkerian structure butalike in harmony. After a single hearing, subjects tended to

judge the harmony foils as more similar to the originals than thecounterpoint foils; after repeated listening, however, they tendedto judge the counterpoint foils as more similar, suggesting(somewhat surprisingly, as the authors note) that Schenkerianstructure is a more important factor in similarity on an initialhearing but that surface harmony becomes more important withrepeated hearings.55

The work of Serafine et al. is a noble attempt to explore theperceptual reality of ST using experimental methods. Themethodological pitfalls of using similarity judgments have

already been discussed; I wish to add a further note of cautionabout this approach. Let us return to the Pastoralexcerpts inExample 11. If someone unfamiliar with the piece were simplygiven Example 11(b) and asked to indicate (by singing, forexample) what underlying melody it was an elaboration of,I wonder if they would produce Example 11(a). They might

well judge Example 11(a) as a plausible reduction of Example11(b), if this were suggested; but other melodies might also beconsidered plausible reductions, such as Example 11(c). Onemight argue that the ease of hearing the connection betweenExamples 11(a) and 11(b) in context demonstrates the remark-able ability of listeners to find connections between two melo-dies that are presented side by side (or presented in formallyparallel positionsthat is, in positions in which one expects atheme to be repeated). No doubt, this process often involves akind of reduction. The question is whether listeners performsuch reductions spontaneously even when such a juxtapositionis not involved. Comparing one musical pattern to another, andextracting their similarities and differences, may often involve a

simplification or reduction of the more complex pattern. Butthis does not mean we spontaneously reduce complex patternsto simpler ones. And the fact that many different simplificationsof a complex patternnot all of which, presumably, are gener-ated spontaneouslyoften seem like plausible reductions of itsuggests that such intuitions are not particularly good evidenceof spontaneous reduction.

Notwithstanding all this, it seems clear that in many cases,listeners do perform reductionas evidenced by their ability torecognize the similarity between a melody and an elaborated

version of it. Such situations are of course commonplace incommon-practice music; they include not only actual theme-and-variations pieces, but the countless other situations in

which a theme is followed by an elaborated repeat, such as thePastoralmelody in Examples 11(a) and (b). (Romantic pianomusicespecially Chopinsalso offers innumerable examplesof this.) The reductive process involved here seems to be highlysusceptible to the power of suggestion; that is, our reduction ofa passage in listening may be steered toward whatever has beenpresented as the simplified version of it. And yet, not everythingis a plausible reduction; for example, the second half of theBeethoven theme, shown in Example 11(d), is not a plausiblereduction of the variation melody in Example 11(b). An inter-esting question is then, what are the factors that make one pat-tern a plausible reduction of anotheror, to put it differently,

what are the criteria whereby reduction is done? A number offactors come to mind here, all well known from tutorials onSchenkerian reduction56and related work.57In general, a plausi-ble reduction of a passage retains the notes that are rhythmicallyemphasized (through duration and metrical placement) and har-monically stable (chord-tones rather than non-chord-tones), andalso makes some kind of inherent musical sense, with the notesof the reduction forming coherent, mostly stepwise lines.

3 Lerdahl and Jackendoff (1983, 105).4 Serafine et al. (1989).5 A similar study by Dibben (1994) tested the perceived similarity between

note patterns and correct or incorrect reductions, and found that correctreductions were perceived as more similar than incorrect ones. This studyused Lerdahl and Jackendoffs (1983) time-span reduction rather thanSchenkerian reduction, so it is less relevant for present purposes; time-spanreductions are rather different from Schenkerian ones, as Dibben acknowl-edges.

56 Beach (1983); Cadwallader and Gagn (2007).57 E.g., Lerdahl and Jackendoff (1983).


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To explore these criteria experimentally would seem to be aninteresting project for music psychologya project alreadybegun by the work of Serafine et al. cited above.

One might also use similarity judgments as evidence of re-duction in a more indirect way. If two passages are reduced insimilar ways, then one might expect them to have similar effects

or functions with respect to other aspects of music cognition.Example 9, the Mozart concerto passage discussed earlier, is acase in point. I argued that this passage can be seen to express alarge IVVI cadence; it seems reasonable to suppose that anexperienced listener of tonal music would comprehend this, andthus would experience a sense of large-scale closure at thispoint. But this large-scale progression cannot be understoodunless one reduces the surface harmonies. If listeners did indeedcomprehend the closural effect of this passage, this could betaken as evidence that they perceived the underlying cadenceand thus were performing reduction; this might be an interest-ing area for experiment.

3.2 typicality and expectation

Listeners can, to some extent at least, judge the typicality ornormality of a piece of music, in relation to the norms of a style

with which they are familiar. A classically trained listener canjudge that a string quartet by Mozart is more normative withinthe classical style than one by Elliott Carter. Such judgmentsform a kind of psychological data that can be used to test theo-ries of music perception. With regard to ST, the obvious predic-tion is that passages judged as normative by the theory will alsobe judged as such by listeners. This raises the question of whatkinds of patterns the theory judges as normative. One obviouscandidate is the Ursatzitself, which is claimed to be the back-

ground structure not only for complete pieces but also for smallertonally closed formal units such as periods and sections. Thissuggests the prediction that a short tonal piece constructed on anUrsatzshould be judged as more normative than one that isnot.58As with similarity judgments, the challenge in construct-ing such passages would be to avoid other confounding factorsin particular, tonal closure itself. If a conventional Ursatz-basedpassage were compared to one that was not tonally closed at all(i.e., not ending on VI in tonic), no doubt listeners would judgethe first as more normative, but that could be easily explained inmore conventional harmonic terms (e.g., using finite-state har-monic models such as Example 1). One could focus instead onthe melodic aspect of the Ursatz, comparing two passages basedon the same normative harmonic structure but with differentmelodies, one that featured a clear Urlinie(such as 321) andone that did not. Of course, many actual pieces and tonallyclosed sections do not feature clear Urlinie; such passages are

typically analyzed in ST either by locating the Urliniein an innvoice or by indicating one or more of its notes as implied.59Bone might argue that the theory judges such structures as lenormative than those with explicit, top-voice Urlinien, and thlisteners should therefore do so as well.

A related aspect of perception deserving consideration he

is expectation. A great deal of work in music perception hfocused on modeling listeners expectations.60 In most sucstudies, subjects are played some kind of context followed b

various possible continuations, and must then indicate the dgree to which the continuation seems good or expecteAgain, the prediction of ST is that the continuations expecteby listeners will be those judged as normative by the theorOne important body of expectation data is shown in Examp12, taken from a study by Lola Cuddy and Carol Lunney.61this study subjects were played a single context interval (twnotes in succession) followed by a continuation tone, and hto judge the expectedness of the continuation tone given thcontext interval. The figure shows Cuddy and Lunneys resul

for context intervals of descending major sixth and ascendinmajor second. To a large extent, these results can be explaineby basic factors of scale membership (pitches within the key keys implied by the context interval are more expected) anpitch proximity (pitches close to the second pitch of the conteinterval are more expected);the principle of reversal62is alapparent, as the most expected pitches after the major sixth i

volve a change of direction.63

One interesting feature of Cuddy and Lunneys data concerns the phenomenon of inertia. As noted earlier, the centraity of linear progressions in ST suggests that melodic stepshould tend to be followed by further steps in the same diretion; if the theory applies to perception, then melodic expectan

cies should reflect this tendency as well. Inspecting Cuddy anLunneys data for the ascending-second context, we see thcontinuation intervals of +1 and +2an ascending half-steand whole-stepare indeed highly expected, more so than dscending steps. Notably, this is not predicted by any of the principles discussed in the previous paragraph: key membershipitch proximity, or reversal. The fact that +3 (an ascendinminor third) is also highly expectedeven more than +1 an+2does not accord so well with ST. But there seems to be least some evidence here of inertia in melodic expectatio

judgments, suggesting an awareness of linear progressions.Schenkerian theory then suggests a further question: Do li

teners also project expectations of linear progressions at largstructural levels? Consider the progression in Example 7. It wnoted that harmonic motions such as V6IV6are generally ra

58 It seems reasonable to assume that the psychological validity of Schenker-ian structures, if any, declines over longer time spans. Indeed, this has beenfound to be true of tonal closure itself; Cook (1987b) found that, with allbut the shortest pieces, listeners (even those with considerable musicaltraining) were unable to tell whether the piece ended in the same key inwhich it began.

59 See, for example, Schenker ([1935] 1979, 51); and Cadwallader and Gag(2007, 2123).

60 See, for example, Schmuckler (1989); Narmour (1990); Cuddy and Lunn(1995); and Schellenberg (1996).

61 Cuddy and Lunney (1995, 46061).62 Narmour (1990, 56).63 However, see Von Hippel and Huron (2000) for an alternative explanatio

of this phenomenon.


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in tonal music, but in a context such as Example 7, they seem tobe justified by the linear progression in the bass. This could betested perceptually: does a IV6following a V6seem more ex-pected or appropriate in a linear-progression context, such as

Example 13(a), as opposed to another usage of V6

such asExample 13(b), where no linear progression is involved? To myears at least, there is no question that V6IV6seems righter ina linear progression context. This is a kind of testable predictionthat might allow us to evaluate ST, or at least one aspect of it, asa theory of music perception.

3.3 motion and tension

Another way of exploring the role of ST in perception isthrough its effect on the experience of musical motion. A num-ber of authors have extolled the theorys power to explain thisaspect of musical experience. Almost sixty years ago, Milton

Babbitt heralded ST as a body of analytical procedures whichreflect the perception of a musical work as a dynamictotality.64Felix Salzer, in critiquing a Roman-numeral analysis of a Bachprelude, laments, What has this analysis revealed of thephrases motion, and of the function of the chords and sequences

within that motion?;65the clear implication is that ST fulfillsthis need. Similarly, in Nicholas Cooks introduction to ST inAGuide to Musical Analysis, he argues that the theoryunlike

Roman numeral analysisshows how each chord in a piece isexperienced as a part of a larger motion towardssome futureharmonic goal.66But how does a Schenkerian analysis maponto experiences of motion; what exactly are the theorys pre-dictions in this regard? Very few concrete discussions of thisissue can be found. One of them is Cooks analysis (in the essaycited above) of Bachs Prelude in C Major from TheWell-Tempered

Clavier, Book I; thus we will focus on it at some length.Cooks discussion of the prelude is framed very much as an

investigation of where its essential motion resides. (Example 14shows a slight reduction of the piece, with rhythmic figurationremoved.) Cook notes first that the beginning of the end ofthe piece is the move to V at m. 24, and that the end of thebeginning is m. 19the end of a large-scale prolongation of I.(The tonicization of V in mm. 511 is not structural; thus the

4 Babbitt (1952, 262); italics mine.5 Salzer (1952, 1: 11). 66 Cook (1987a, 29), italics in original.

Descending major sixth

0

1

2

3

4

5

6

7

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

Ascending major second

0

1

2

3

4

5

6

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

example 12. Melodic expectation data from Cuddy and Lunney (1995)The data show average expectedness ratings (on a scale of 1 to 7) for continuation tones given context intervals of a descendingmajor sixth (A4 C4) and an ascending major second (B3C4). The continuation tones are shown in relation to the second tone

of the context interval; for example, 1 represents a continuation tone of B3.

V6 IV6I V V6 IV6

A. B.

example 13. V 6 IV 6progressions in two different contexts


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I at m. 19 is continuous with the initial I and not a return.)It is not until m. 21 that we experience a structural departurefrom the initial tonic. Thus mm. 20 23 constitute the essentialmotion of the piece: Harmonically, the entire evolution of thepiece . . . lies in these four bars.67Cook then goes on to showhow Schenkers analysis of the piece (shown in Example 15)brings out the contrast between the tonally enclosed begin-ning and ending sections and the goal-directed passage in be-tween. The general point, then, seems to be that the Ursatz

implies a motion from its initial I to its final I.68This does not,of course, imply that the tonally enclosed portions of the pieceare entirely devoid of motion; indeed, Cook also discusses linear

motions within the initial prolonged I. Presumably the patteof stasis-motion-stasis within the Ursatzcan be replicated lower levelswherever a Stufeis expanded with points of stbility at either end and instability in between. But the impliction is that the main motion of a tonal piecethe primaharmonic evolutionis that of the Ursatzitself, in the regio

just preceding the final cadential I, and that other motions asubsidiary.

Cooks account presents a hierarchical structure of motio

of varying levels of importance, corresponding to levels ofSchenkerian reduction. This is a perfectly coherent proposand in the case of the Bach prelude it is not unreasonable. Bas a general prediction about motion in tonal music, it seemrather counterintuitive. In most Schenkerian analyses (incluing Schenkers own), the V of the Ursatzand the immediatepreceding predominant chordscorresponding to mm. 20 2of the Bach preludeoccur at the very end of the piece (perhaps just before the coda), so that almost the entire worka prolongation of the initial I. For example, consider Schenkeanalysis of Chopins Etude Op. 10, No. 12, shown in Example 1

67 Ibid. (3234).68 The logic here is actually not completely clear. According to Schenkers

analysis, the IVII harmonies of mm. 2023 are not literally part of theUrsatz; but they are (perhaps) transitional between the initial I and the Vof the Ursatzand therefore might be considered part of the top-level mo-tion of the piece. Another question is whether the V of the Ursatzis partof this top-level motion as well; Cook seems to assume not, but it is notclear why.

example 14. Bach, Prelude in C Major, fromTheWell-Tempered Clavier,Book I

example 15. Schenkers middleground and background analysis ([1933] 1969) of Bachs Prelude in C Major, fromThe Well-Tempered Clavier, Book I


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According to the view that Cook seems to advocate, the firstseventy-one of the eighty-four measures of this piece are char-acterized by relative stasis; it is only in m. 72 (or perhaps mm.7276, if the V of the Ursatzis included) that the main motion

of the piece occurs. I would argue, rather, that the primary tonalmotion of the piece is the motion away from the tonic key at m.24, through the keys of B major, F minor, and other fleetingtonalities, and back to the tonic at the recapitulation inm. 49.69The function of the IIVI progression in mm. 7277is not to create large-scale tonal motion but rather to confirmthe tonal destination at which we arrived some measures earlier.Intuitions about musical motion are obviously very subjective,and we should be cautious about projecting our own experienceson to other listeners. I can only say that, by myintuitions, the

view of tonal motion advocated by Cook does not ring true.The pattern of musical motion I have asserted for the

Chopin etude might be predicted much more straightforwardly

by a conventional key-based view of motion, in which the es-sential motion of a piece begins when the tonic key is left andends when it is regained. I would not claim, however, that sucha view is completely satisfactory either. One serious problem

with the key-based account concerns large-scale tonal returns insonata movements and other large forms. I would argue that thebig moment of return in a sonata-form movement generally

coincides with the return of the opening theme at the beginningof the recapitulation; importantly, this is not only the momentof thematicreturn (that much seems obvious) but of tonalreturnas well. But the return to the tonic keyas represented by a

conventional Roman-numeral analysisdoes not usually occurimmediately at the return of the main theme, but rather some-what earlier ; the recapitulation is almost always preceded by aprolonged dominant which is clearly V/tonic (or V7/tonic), notI/dominant. (In the case of the Chopin etude, the return to Cminor occurs at m. 41 at the latestthe harmony here is V/C,not I/G; it might even be said to occur a bit earlier.) One couldargue that the moment of return coincides with the first tonicharmony in the tonic key, but this argument, too, encountersproblems. In many cases, the return to the tonic key even pre-cedes the retransitional dominant, and may involve fleetingtonic harmonies.70A Schenkerian view could be of help here; insuch an analysis, the return to I at the beginning of the reca-

pitulation is usually quite a high-level event. But it is not usuallya top-level eventit might be at the second or third level, as itis in Example 15and it remains to be explained why the es-sential tonal motion of a piece should be conveyed at this levelrather than at the top level.

9 The piece also contains a shorter tonal excursion in mm. 6570 of thecodashort but dramatic, as it arguably moves further from the tonic thananything in the works main tonal journey.

70 Beethovens Piano Sonata in C Major, Op. 53, I, offers a typical example.The thematic returnand surely the tonal return as welloccurs atm. 156. But this is preceded by a passage of some twenty-six measures inC (though mostly minor rather than major), including a very prominentC-minor harmony at mm. 13233.

example 16. Schenkers middleground and background analysis ([1933] 1969) of Chopin, Etude in C Minor, Op. 10, No. 12


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Another area of musical experience in which ST may maketestable predictions is patterns of tension and relaxation (orsimilarlyperhaps equivalentlystability and instability). Ofparticular interest here is the work of Lerdahl.71Lerdahls ap-proach to musical tension builds on Lerdahl and Jackendoff stheory of prolongational reduction, in which the notes of a

piece are assigned a hierarchical structure quite similar toSchenkerian structure.72 Lerdahls theory assigns a tensionvalue to each event in a prolongational reduction. The theoryrelies heavily on a spatial representation theory of chords andkeys (which ST of course does not). Oversimplifying somewhat,the tension value of an event in a reduction is the pitch spacedistance between the event and its immediately superordinateevent in the tree, plus the distance between the parent event andits superordinate event, and so on, up to the highest-level eventin the piece. As a result, events that are more embedded in thetree carry higher tension, as do events that are less closely re-lated to their contexts (such as chromatic notes and chords).

Lerdahls tension theory has been the focus of several ex-

perimental studies. In a study by Emmanuel Bigand andRichard Parncutt,73subjects were instructed to judge the ten-sion of chords at each point in a harmonic progression, and theresults were compared to the predictions of Lerdahls theory. Atglobal levels, the match between the theorys predictions andtension judgments was poor; a phrase in a non-tonic key is ratedhigh in tension by Lerdahls theory, but was not judged to be soby listeners.74At more local levels, the theory predicted tension

judgments fairly well, although the authors observe that thismay be largely due to the effect of cadences, which both thesubjects and the theory judged to be low in tension. Experimentspresented by Lerdahl and Carol L. Krumhansl,75in which thehierarchical tension model was applied along with melodic (at-

traction) and surface dissonance factors, yielded more positiveresults; in a wide variety of musical excerpts, tension judgments

were found to correlate strongly with the theorys predictions ofhierarchical tension.

Altogether, the evidence gives at least partial support toLerdahls model as a predictor of perceived tension; there seemsreason to hope that ST might yield successful results in thisregard as well. An important question here concerns the roleplayed by pitch space in Lerdahls theory. Consider a chromatic

chord in the middle of a phrase, which presumably would nomally be perceived as high in tension. Lerdahls theory assighigh tension to such a chord for two reasons: first, because it deeply embedded in the reduction; secondly, because it is hamonically remote from neighboring events. The first of thefactors can be captured by ST; the second cannot. A skept

might argue that the perceived tension of a chromatic chordprimarily due to the second factorthe simple fact that it chromatic in relation to the current key; this could be reprsented without any use of reductional structure. Whether peceived tension is due more to pitch-space or to reductionfactors is currently an open question. In any case, the possibiliof testing ST as a perceptual theory through its predictions tension deserves further study.

4. conclusions

In this essay, I have explored the prospects for testing Sboth as a theory of composition and a theory of perception. A

a theory of composition, I suggested that we could evaluate Sby considering how well it predicts what happens and does nhappen in tonal music. I first considered the idea of formulatinthe theory as a context-free grammar, but expressed skepticisas to the prospects for this enterprise. I then considered somspecific concepts of ST, Stufenand Zge, and suggested som

ways in which they might indeed make successful predictioabout tonal music.

With regard to perception I considered several ways that ththeory could be construed to make testable predictions abothe processing of tonal music. ST predicts that passages witsimilar reductions should seem similar; it predicts that even

judged as normative by the theory will be highly expected; an

it may also make predictions about the experience of musicmotion and tension. It seems to me that the prospects for ththeory here are extremely mixed. Listeners ability to perceivthe similarity between a theme and a variation of it seems tpoint to some kind of reduction; but we cannot conclude frothis that listeners spontaneously generate reductions as they liten. The potential for the theory to tell us much about musicmotion also seems doubtful, or at least unproven. On the othhand, melodic expectation data seem to indicate some awarness of linear progressions; reduction may also play a role explaining tension judgments, as suggested by tests of Lerdahtheory.76I also pointed to several specific situationsregardilinear harmonic progressions such as Example 7 and large-scaStufeprogressions such as Example 9where the theory makconcrete predictions about perception, which, in my opiniohave a good chance of being borne out.

In Section 1, I argued that a theory might well be construeas both a theory of composition and a theory of perceptionsimilar to a linguistic grammar. It may be noted that the aspecof ST that I have cited as most convincing with regard to compositional practicenamely, certain uses of Stufenand line

76 Lerdahl (2001).

71 Lerdahl (2001).72 As Lerdahl and Jackendoff readily acknowledge (1983, 116, 27374, 337

38), there are significant differences between prolongational reduction andSchenkerian reduction. Perhaps the biggest difference is that the elementsof a prolongational reduction are verticalities, whereas a Schenkerian re-duction may analyze the voices of a polyphonic texture as independent ele-ments. Another significant difference is that in prolongational reduction, aVI cadence is analyzed as a single indivisible event, in contrast to STwhere the V and I are separate events.

73 Bigand and Parncutt (1999).74 We should note that this mismatch between theory and perception regard-

ing large-scale tonal structure also holds true for more conventional theo-retical structures; see Note 58 above.

75 Lerdahl and Krumhansl (2007).


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progressionsare precisely those that I have argued for withregard to the modeling of perception. Perhaps these aspects ofST constitute a small part of the knowledge about common-practice music that composers and experienced listeners shareits grammar, one might saya body of knowledge alsoentailing principles of harmony, meter, phrase structure, me-

lodic structure, form, and other things.I readily acknowledge the subjective, speculative nature ofthe conclusions presented above. They are really just opinions atthis point, and others may disagree with them. But I wouldargue that these opinions are given substance, and broughtcloser to factualclaims, by the empirical methods I have pro-posed for testing them. That is to say: the claim bass-line linearprogressions are part of the compositional process is very dif-ferent from the claim bass-line linear progressions, when com-bined with a finite-state model of surface harmonic progressions,can make harmonic progressions more normativeand thuspredict them betterthan a finite-state model alone. Whileneither of these claims has yet been tested in any systematic

way, the latter claim is nearly, if not quite, a claim about obser-vation terms, and thus, in principle, objectively testable.77Withregard to perception, too, I have tried to go beyond mere intui-tive guesses about what is perceptually valid and what is not,presenting concrete ways that such hypotheses might be tested.

If the opinions I have expressed above are correct, theysuggest a new way of thinking about Schenkerian theory.Schenkerian structures, by this view, are not basic buildingblocks of tonal music. Rather, they are patterns that were usedoccasionally by tonal composersperhaps quite often, but notall the timeand usually at fairly local levels. Using terms Ihave coined elsewhere,78they are occasional rather than in-frastructural aspects of musical structure. A composer might,

for example, generate the first and second themes of a sonatamovement using traditional harmonic logic, but bring in a bass-line linear progression in the transition, and a large-scale Stufeprogression at the end of the exposition. This view of StufenandZgebrings to mind other similar schemata such as those pos-ited by Leonard B. Meyer and Robert Gjerdingenfor exam-ple, the 1743 pattern and the Prinner (see Example17).79The Zugand the Stufeare of

composition, perception, and schenkerian theory

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