toward a formal theory of tonal music.pdf

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Yale University Department of Music

Toward a Formal Theory of Tonal MusicAuthor(s): Fred Lerdahl and Ray JackendoffSource: Journal of Music Theory, Vol. 21, No. 1 (Spring, 1977), pp. 111-171Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843480Accessed: 07/04/2009 04:42

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TOWARD A FORMAL THEORY

OF TONAL MUSIC

Fred Lerdahl and Ray Jackendoff

INTRODUCTORY REMARKS

We take the goal of a theory of music to be a formal de-scription of the musical intuitions of an educated listener. By"musical intuitions" we mean the largely unconscious knowl-edge which a listener brings to music and which allows him toorganize musical sounds into coherent patterns. By "educatedlistener" we mean not necessarily a trained musician but alistener who is aurally familiar with the musical idiom inquestion. Such a listener is able to identify a previouslyunknown piece as an example of the idiom, to recognize ele-ments of a piece as anomalous within the idiom, and gener-ally, to comprehend a piece within the idiom.

The "educated listener" is an idealization. Rarely do twopeople hear the same piece in precisely the same way or withthe same degree of richness. Nonetheless, while it is conceiv-able to hear a piece any way one wants to, there is normallysubstantial agreement on what are the most natural ways tohear a piece. Our theory is concerned not with particular in-stances of hearing, which are always subject to a degree of

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variability, but with the idealized underlying competencewhich the educated listener brings to bear in understandingtonal pieces.

Atheory

about aparticular type

of music is,ideally,

a sub-set of a theory of all music. We are constructing our theory oftonal music with this larger perspective in mind. While manyof our specific rules are applicable only to tonal music, thebasic components of the theory are designed to accommodatemusic of different traditions and historical periods.

A reader who is at all acquainted with contemporary lin-guistics will observe that the goals we have set for ourselvesare in some ways parallel to the goals of transformationalgenerative grammar, which seeks to describe the linguisticintuitions of a native speaker of a language and to discoverthose aspects of particular grammars which are common toall languages. Indeed, our way of thinking about music is pat-terned after the methodology of linguistics in that we demandstrong motivation, formal rigor, and predictive power forevery part of the theory. On the other hand, we do not ap-proach music with any preconceptions that the substance ofour theory will look at all like linguistic theory, since languageand music are on the face of it different manifestations ofhuman cognitive capacity.

Previous theories of tonal music have not met such de-mands of rigor and prediction. Even Schenker's theory, whichcan be construed as having much in common with the genera-tive approach to linguistics, is at bottom inexplicit. One ofthe virtues of a formal theory is not that it is necessarily more

"true," but that, even where incorrect or inadequate, it clari-fies issues precisely.To elucidate in what sense our theory is modeled after lin-

guistic theory, we must mention a common misconceptionabout transformational grammar. It is often thought that aChomskian generative grammar is an algorithm that grindsout grammatical sentences; this view suggests that a genera-tive music theory should be a device which composes pieces.

There are two errors in this view. First, the sense of "gener-ate" in the term "generative grammar" s not that of an elec-trical generator which produces electricity, but rather themathematical sense, in which it means to describe a (usuallyinfinite) set by formal means. Second, Chomsky distinguishes

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the "weak generative capacity" of a theory from its "stronggenerative capacity."' A grammar weakly generates a set ofsentences and strongly generates a set of structural descrip-tions, where each structural

description uniquely specifiesa

sentence, but not necessarily conversely. It is the notion ofstrong generation which is overwhelmingly of interest inlinguistic theory. The same holds for music theory, since atheory which weakly generates "grammatical" tonal pieceswould tell us nothing interesting about their structures.

A strongly generative theory of tonal music would notmerely give a description of what pieces are grammatical.Rather, it would have to specify each tonal piece togetherwith its structural description, i.e., a specification of all thestructure which the educated listener infers in his perceptionof the piece. If a given piece cannot be heard as tonal, thetheory should be unable to give it a structural description; ifa piece can convincingly be heard in several ways, the theoryshould give it a different structural description for each wayof hearing it.

We have found that a generative music theory must notonly assign structural descriptions to pieces, but must differ-entiate the structural descriptions along a scale of coherence,weighting them as more or less "preferred" ways of hearing apiece. Thus, the theory is divided into two distinct parts: well-formedness conditions, which specify possible structural de-scriptions; and preference rules, which designate, out of thepossible structural descriptions, those that correspond to theeducated listener's hearing of any particular piece.

There are various criteria for determining which structuraldescriptions of a piece are "preferred." Among these, ofcourse, are our own intuitions. Furthermore, beyond allseemingly self-evident intuitions, a "preferred" structuraldescription will tend to relate otherwise disparate elements ina satisfying way and to reveal surprising analytic insights. Inaddition, we are aware of relevant research in experimentalpsychology and are concerned that its findings be in agree-

ment with our theoretical constructions. Criteria within thetheory itself include the internal consistency of the rules andtheir generalization from particular instances to the entirebody of classical tonal music. We utilize within the frame-work of the theory such psychologically primary notions as

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parallelism, articulation, and stability. Finally, the nature ofour structures must be psychologically plausible in that theircomplexity must result from the interaction of a fairly smallnumber of processes, each of which taken in isolation is rela-tively simple.

Although it would be possible within the theory to gener-ate new tonal pieces, we have chosen only to generate struc-tural descriptions for existing pieces. It would seem to beinherently less rewarding to specify normative but dull piecesthan to develop structural descriptions for works of lastinginterest. Moreover, in the former case, there would always bethe danger, through excessive limitation of the possibilities in

the interest of conceptual manageability, of oversimplifyingand thereby establishing shallow or incorrect principles withrespect to music in general.

For the common conception of a transformational genera-tive grammar as merely a sentence-generating device is mis-taken in a further respect. Linguistic theory is not simplyconcerned with the analysis of a limited set of sentences;rather it considers itself a branch of psychology, concerned

with making empirically verifiable claims about one complexaspect of human mental life, namely language. Similarly, byputting our emphasis on the musical intuitions of the edu-cated listener, and by taking as our sample a highly complexbody of music, we are asserting that the analysis of pieces ofmusic, though not without a great deal of intrinsic interest, isnot an end in itself. Rather the goal is an understanding ofthe mental process of musical perception, a psychological

phenomenon. From this viewpoint, our theory of music is notjust an analytic system, but makes strong claims about thedelimitation of possible theories of musical cognition.

By regarding music only as apprehended structure, we aredeliberately avoiding the difficult issue of musical meaning.Whatever music may "mean," it is in no sense comparable tothe semantic component in language; there are no substantiveparallels to sense and reference in language, or to such seman-tic

judgmentsas

synonymy, analyticity,and entailment. It is

in the domain of syntax that the linguistic approach has rele-vance to music theory. Yet even here there are no substantiveparallels between musical structure and such grammaticalcategories in language as noun, verb, adjective, noun phrase,

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verb phrase, and so forth. The concepts of musical structuremust be developed in terms of music itself.

We likewise avoid the issue of aesthetic value. Nevertheless,that we are dealing with works of art rather than "mentalobjects" from the everyday world, such as sentences, is to adegree problematic. Whereas a sentence normally has a defi-nite meaning, structural ambiguity in a work of art is com-mon. And whereas a sentence normally has a definite function,it is in the nature of a work of art that it is appreciated andcontemplated from various points of view and with variouspurposes in mind, not only by different people but by thesame person on different occasions. These differences, how-ever, do not mean that the understanding of a work of art cantake any arbitrary form whatsoever; rather, they mean that,to an extent, multiple understandings are possible, desirable,and even inevitable. In constructing our music theory, wehave accounted for this state of affairs by building a systemof interactive components and by emphasizing the "pre-ferred" nature of the resulting structural descriptions.

Under our conception of music theory, then, the under-

standing of a piece of music by the idealized listener consistsin his finding the maximally coherent structural descriptionor descriptions which can be associated with the piece's se-quence of pitch-time events.

Maximizing coherence involves the interaction of a numberof different domains of analysis, each of which must be repre-sented in the structural description. There are four with whichwe will be concerned here, to be termed grouping analysis,

metrical analysis, time-span reduction, and prolongationalreduction. As an initial overview, we may say that the group-ing analysis assigns group boundaries to the music in a hier-archic fashion at every level of a piece. The metrical analysisassigns a hierarchy of strong and weak beats. The time-spanreduction designates "structural beginnings" and "structuralendings" of groups, and assigns to the pitches a hierarchywhich relates them to the grouping and metrical structures.The

prolongationalreduction

assigns to the pitches a hier-archy which expresses harmonic and melodic continuity andprogression; it is the closest equivalent in our theory toSchenkerian analysis.

There are some abstract properties common to these four

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domains of analysis. First, each domain partitions a piece intodiscrete regions, organized hierarchically in such a way thatone region may contain other regions, but may not partiallyoverlap with other regions. For example, Figure 1a representsa possible kind of organization. Figure lb represents an im-possible organization: at j, two regions overlap within level 2;and at k, a boundary in level 3 overlaps a region in level 2.

Another property common to these domains is that theprocesses of organization are essentially the same at all hier-archic levels. A related point is that the processes of organiza-tion of these domains are recursive, i.e., capable of indefiniteelaboration by the same rules. Other aspects of musical struc-ture, however, are not hierarchic in nature. For the presentwe shall ignore these dimensions.2

As mentioned above, the rules which assign structural de-scriptions are categorized as well-formedness rules, whichassign possible structures, and preference rules, which selectcoherent structures from possible structures. In addition,transformational rules, which convert structures into otherstructures, are needed for special cases (such as elisions) not

generated by the well-formedness rules. Although transforma-tional rules have been central to linguistic theory, they play aperipheral role in our music theory, at least at its current stageof development.

The following discussion of the organization of the theorywill be informal. Emphasis will be placed on how the compo-nents work in principle. To give a complete account wouldexceed our present purpose, which is to convey in general the

goals, operations, and implications of the theory as a whole.We begin with a discussion of rhythmic structure in terms ofgrouping analysis and metrical analysis. Then we develop thetwo modes of pitch hierarchization, time-span reduction andprolongational reduction. Finally, we apply the completedsystem to a piece of some complexity.

RHYTHMIC STRUCTURE

When hearing a tonal piece, the listener naturally organizesthe sound signals into units such as motives, themes, phrases,periods, theme-groups, sections, and the piece itself. Ourgeneric term for these units is "group." The grouping analysis

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Figurela

lb

3) .. . J

4)k

a

Figure 2a

2b

:

6 o : o

: :

: */ 0,~?

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picks out the groups and indicates them in the structural de-scription by slurs beneath the musical notation. The groupingwell-formedness rules restrict the possible grouping structuresto the kind of hierarchic organization discussed above, whereFigure 1a was possible and Figure lb was not possible. Underthese conditions, both Example 1a and Example Ib are possi-ble groupings.

However, while Example la would appear to show the cor-rect grouping, Example lb is absurd. In order to select theactually heard grouping or groupings (such as Example la),as against all the merely possible groupings (such as Examplelb), we develop the grouping preference rules. These providethe criteria for picking out groups, and are classified accordingto principles of (a) articulation of boundaries, (b) parallelismin structure, and (c) symmetry.

Group boundaries are articulated by such factors as dis-tance between attack points, rests, slurs written into themusic, change in register, change in texture, change in dy-namics, and change in timbre. A further articulatory device isthe harmonic cadence, which from the phrase level upwardnormally signifies the ending of groups; this will be discussedlater.

Parallelism in structure involves some kind of repetition orsimilarity in the music, such as a motive, a sequence, a section,and so forth. The similarity is particularly crucial at the begin-ning of groups; for even if they diverge later on, they are stillperceived as parallel. In tonal music, parallelism is the majorfactor in all large-scale grouping.

Related to parallelism is the principle of symmetry, whichstates that the ideal subdivision of any group is into equalparts. In Example la, all the groupings are assigned by theparallelism rule (reinforced by various articulatory criteria),except for the groupings marked s, which are due to the sym-metry rule; thus, each 4-measure group subdivides not onlyinto 1 + 1 + 2, but, at the next level, into 2 + 2.

A grouping transformational rule is required to account

for grouping overlaps, which as such do not meet the well-formedness conditions of hierarchic organization. The trans-formational rule relates well-formed underlying groupings tothe musical surface, thereby preserving the sense that overlapsare variations on normal hierarchic grouping. Thus, in Exam-

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Example la. Beethoven: Sonata Op. 2, No. 2, Scherzo

^^f $rf 1 ffr>TI p

lb

P A- $ T ^-zC $% 4^v.?t^ m J j J^

v$^ .^^t^ v^rf^^' ^


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pie 2, the events at u and v function as both endings andbeginnings of groups; these overlaps are disentangled at under-lying levels.

We turn now to metrical structure, which is independent ofgrouping structure but interactive with it. Given appropriatemusical cues, the listener will instinctively infer a regular,hierarchic pattern of beats to which he relates the actualmusical sounds. The metrical analysis assigns to a piece such apattern of beats and indicates them in the structural descrip-tion by dots beneath the musical notation and above thegrouping slurs, as in Example 3.

Each level of dots represents a marking-off of the musicinto equal time-spans. A dot at a particular level represents ajudgment that that moment in the music is a beat at thatlevel. If a beat at a particular level is felt to be "strong," or"down," it is a beat at the next larger level and receives anadditional dot. (Thus, metrical structure does not exist with-out at least two levels of dots.)3 The process is the samewhether at the level of the smallest note value or at a hyper-measure level. The notated meter is usually an intermediate

metrical level.Theoretically, the dots could be built up to the level of a

whole piece. However, the perception of relative metricalstress fades over long timespans. In addition, at large levels,metrical structure is heard within the context of groupingstructure, which is rarely regular at such levels; and withoutregularity there can be no metrical structure. Therefore,metrical structure is a comparatively local phenomenon.

The metrical well-formedness rules assure the hierarchiccondition that a beat at a particular level must also be a beatat all smaller levels. Characteristics of metrical well-formed-ness in classical tonal music include the equal spacing of beatsand the provision that at each successive level the distancebetween beats must be either two or three times that of theimmediately lower level. Musical styles of other cultures andhistorical periods often require more complicated rules of

metrical well-formedness; the rhythmic complexities of tonalmusic arise from the interaction of metrical structure withgrouping structure and pitch structure.

It hardly needs emphasizing that bar lines and beams be-tween notes are notational devices and not part of the physi-

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Example 2. Mozart: Sonata K. 279, I

Example 3. Bach: "O Haupt voll blut und Wunden"

r.,N I i II,"/t4 J LJ i . LLt rbrr r r C rCJ

. I n I j j J u i J j*0 It*

0.

.

' I. '

*

r? ?

e

*0

0)..

. . I. - - I, .., T14 _- I ...

dr I - II II-- I I I I I _I I I I

I

? ? ? ? ?* ? ?

e


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cal signal, as pitches, durations, dynamics, and timbre are.From the physical .signal the metrical preference rules assignto a piece the actually heard metrical structure (or structures)instead of the myriad well-formed but inappropriate metricalstructures applicable to it. For example, they select the metri-cal structure in Example 3 rather than one in a triple meter,or one which places the strongest downbeat of the passage onthe opening event.

The metrical preference rules can be classified according toprinciples of (a) cues for strong beats, (b) parallelism withgrouping structure, and (c) regularity of pattern. The cues inthe music for relatively strong beats include such factors as

attack, accent, change of dynamic, register of pitch, harmonicchange, and suspensions. Added to these is the listener'stendency to ascribe parallel metrical structures to parallelgrouping structures (especially rhythmic patterns which formgroups). If there is any regularity to these various cues, thelistener extrapolates an entire metrical hierarchy, which hewill renounce only in the face of strongly contradictory cues.Syncopation takes place when there are strong contradictory

cues which yet are not strong enough to override the inferredpattern.

A metrical transformational rule is needed for metricaloverlaps, in which a shift in the metrical structure occurs insuch a way that the same moment in a piece serves a doublemetrical function. The transformational rule deletes one setof dots in favor of the other at the musical surface. In Figure2a, the weaker metrical function is deleted; in Figure 2b, the

stronger is deleted. (It may be helpful to think of the largestlevel of dots as representing the measure or half-measurelevel.)

Although it is conceivable for metrical overlaps to ariseindependently, they generally happen as a consequence ofgrouping overlaps which take place on relatively weak metri-cal stresses. On the other hand, if a grouping overlap takesplace on a relatively strong metrical stress (as in Example 2),no metrical

overlapresults.

Usingfamiliar

terminology,we

call a combined grouping and metrical overlap an elision.We further distinguish between elisions in which the weakermetrical function has been deleted (as in Figure 2a), andelisions in which the stronger metrical function has been

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deleted (as in Figure 2b). Example 4 is typical of the formerkind; the effect at s is one of jarring reorientation. In thelatter kind, the effect is rather one of retrospective awarenessthat a shift has taken

place;there is an

exampleof this in the

Schumann song analyzed later in this paper (pp. 149ff).Some general reflections are in order concerning the organ-

ization of the grouping and metrical components. It seems tous essential that the two not be confused inadvertently. It istempting, for example, perhaps on the basis of a misleadingspatial analogy, to attribute duration to the concept of metri-cal stress, so that first a beat, then a measure or motivicgrouping, and finally a phrase receives "strong" or "weak"markings as the analysis proceeds to higher levels. In our view,however, the notion of "beat" or "metrical stress" can onlybe understood as a point in time, without duration;4 henceour use of dots to signify metrical structure. Beats are correct-ly analogous to equidistant geometric points rather than tothe lines drawn between them; while rhythms and groupshave duration, then, beats do not. The metrical componentassigns metrical stress not to groups but to beats withingroups.

Of course the listener senses that a group has a certainweight if within it there is a strong beat. But this does notmean that the group as a whole receives metrical weight; forthe weak beats are all equally weak. In Example 5a, for in-stance, the E-flats are metrically equal, a fact obscured inExample 5b, in which metrical properties appear to be in-cluded within a grouping notation.5

Just as groups as such do not receive metrical stress, metri-cal structure as such does not possess any inherent grouping.Whether a weak beat is heard as an afterbeat or as an upbeatis entirely a matter of the grouping associated with it. Similar-ly, at a larger level of grouping, say the paradigmatic ante-cedent-consequent pattern in Figure 3, there is nothing in themetrical structure which prevents the half cadence in m. 4from resolving as a full cadence to the tonic in m. 5. It does

not so resolve only because of the intervening group bound-ary.6Figure 3 poses a question of a different sort. A phrase can

be roughly characterized as the lowest level of grouping whichhas a structural beginning, a middle, and a structural ending

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Example 4. Haydn: Symphony no. 104, I, mm. 17ff.

i$f r f r 0 1 8 ij^ . a r4^ r

Y)r r J8 -. . . ?

Example 6. Beethoven: "Hammerklavier" Sonata op. 106, I

j f a - t ' t* * ; 'r

'fr r rI

"""=~~''(t i cedent)~ _ ~ ntec de nt

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: ** : :r'' I

? ':, '

: . . .I j[

fr'f TS ' rvmf= u=ro . -- L' @* \

'F rr rJ JJ j 'r r ir J J

(extended consequenT)

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Example 5a. Mozart: Symphony no. 40, I

ibbt

n , n n? . ? . ? . . . ? ? ? ? ? ? ? ? ?

Sb

;9 kr f, r'i v i 1, /,Example 7. Bach

Example 7. Bach: "0 Haupt"

126

* .II

._ ....,

-,

- I'..,? , ,, .? , ? ? , ?

1 1 t1 1 ? J


18/62

Figure 3

I w%%V I-measure: 1 2 3 4 5 6 7 8

Figure 4a

(a'I

pitch events: W z

4b

(b)

(C)

w y z

4c

w x y z 127


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(a cadence). Thus, a phrase normally has a kind of structuralgravity at or near its group boundaries. If groups as such haveno metrical stress, and if metrical structure is an extrapolationof

multi-leveled, regularly spaced beats,how are these

pointsof gravity accounted for in the structural description? InFigure 3 there is no conflict between metrical stress andstructural weight in mm. 1 and 5; but mm. 4 and 8, wherethe cadences occur, are metrically relatively weak (dependingon the example, the cadences might even happen in the secondhalves of these measures). One solution might be to move thesecond strong metrical stress of each phrase from the down-beats of the third and seventh measures to the points at whichthe cadences actually occur.7 However, there are two strongreasons against such a revision of the metrical component.First, it would entail surrendering one of the formal proper-ties common to all the domains under discussion, namely thatthe processes of organization are essentially the same at allhierarchic levels. Secondly, it would mean giving up the tradi-tional distinction between cadences which take place at weakmetrical points and cadences which take place at strong metri-cal points. The latter are particularly important for achievinglarge-scale arrival and-in conjunction with grouping overlaps-continuation. Thus, in Example 68 the consequent phrase isextended so that the cadence at q arrives on a strong beat andoverlaps with the succeeding phrase. According to the pro-posed revision, the cadence of the antecedent phrase at p alsowould have to be metrically strong, with the result that themetrical distinction between p and q would be lost. This is

plainly not acceptable.The points of gravity in a phrase, then, are not to be inter-

preted as metrical phenomena. Rather, they are hierarchicallyimportant elements produced by the interaction of pitchstructure and grouping structure; and they stand with metricalstructure in a contrapuntal relation, so to speak, which under-lies much of the rhythmic richness of tonal music. The metri-cal component therefore remains as originally set forth. As

will be seen below, the proper distribution of structuralweight in a phrase emerges in the analysis as part of the time-span reduction, which treats a phrase as an elaboration of itsbeginning and ending.

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PITCH REDUCTION

Both the time-span reduction and the prolongational re-duction

assignhierarchic structures to all the

pitchevents in a

piece.9 To represent these two kinds of reduction, we haveinvented two somewhat different "tree" notations; these visu-ally resemble, but are substantially different from, the treenotation utilized in linguistics. Linguistic trees represent "is-a" relations: a noun phrase followed by a verb phrase is asentence; a verb followed by a noun phrase is a verb phrase;and so forth. Our musical trees, however, do not involve gram-matical categories. The fundamental relationship which theyexpress is that of a sequence of pitch events as being an elabo-ration of a single pitch event. The dominating event, that ofwhich a sequence of events is an elaboration, is always one ofthe events in the sequence; the remaining, subordinate eventsin the sequence are heard as relatively ornamental. "Reduc-tion"-the process of recursively substituting single events forsequences of events-can be thought of as the inverse of elab-oration.

In the following exposition we begin with the time-spancomponent and then turn to the prolongational component.In both cases, after outlining the notations and the basicprinciples of reduction, we apply the components first to anabstract antecedent-consequent pattern, then to the openingeight measures of Mozart's Sonata, K. 331.

In the tree notation for the time-span reduction, a "rightbranch" ( X), in which a line to the right attaches to a line to

the left, denotes the subordination of an event to the preced-ing event within that region at that level; a "left branch"( X), in which a line to the left attaches to a line to the right,denotes the subordination of an event to the following eventwithin that region at that level. The well-formedness condi-tions for these trees prohibit both the crossing of branches, asin Figure 4a, and the assignment of more than one line of thetree to the same event, as in Figure 4b. (The letters in paren-

theses signify reductional levels.)The relevant notion of elaboration in the time-span reduc-tion is elaboration at successive levels within more or lessequally spaced, discrete time-spans. Within each time-span, orregion, a dominating event must be found; that is, all other

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events in the region are elaborations of that event. Thus, atlevel (c) in the well-formed tree in Figure 4c, x is in the regionof w and is an elaboration of w; y is in the region of z and isan elaboration of z. At level (b), all four events are in thesame region, and z (together with x and y) is an elaborationof w.

Before the rules which establish the tree structure can beapplied, it is necessary to select the regions of application atevery hierarchic level within a piece. These regions for all butthe most local levels of analysis consist of the groupingsassigned by the grouping preference rules. Within the lowestgrouping level, smaller regions are chosen in terms of levels ofmetrical structure. In these smaller regions, a given weak beatis bracketed with the preceding strong beat, unless the pre-ceding strong beat is separated from the weak beat by a groupboundary; in this latter case, the weak beat is bracketed withthe following strong beat. In musical terminology, this meansthat a weak beat is an afterbeat unless it is situated at thebeginning of a group, in which case it is an upbeat. See Exam-ple 7; the brackets indicate the sub-group regions of applica-tion. In a given tree, each level of branching corresponds to aregion of application for the preference rules.

Given these regions of application, the preference rules forthe time-span reduction choose the syntactically most co-herent reduction (or reductions) from all the possible butmostly implausible reductions of a set of pitch events. Syn-tactic coherence in this domain can be thought of in terms ofstability. These preference rules are classified as (a) those

which ascertain the most stable structure (the tonic), and (b)those which establish the hierarchy of relative stability in rela-tion to the most stable structure.

The tonic-at any level, local or global-is selected withreference to the available pitch collection and cadential struc-ture at the appropriate level.

The rules of relative stability or instability are, in broadmusical terms, the principles of relative consonance or disso-

nance. For example, a local consonance is more stable than alocal dissonance; a triad in root position is more stable thanits inversions; a chord is more stable if its melodic note is thesame pitch-class as its root; the relative stability of two chordscan be determined by the relative closeness to the local tonic

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Figure 5

(h) (i) ( (

A .XX

*

Figure 6

(9)

level (h) = [b4 [c [ [?c

level(g) =[b8]

[Cs]

measure: 1 2 3 4 5 6 7 8*

0* *

. .

4 4

8133


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pie 9; h, i, /, and k in the music stand for Figure 5 (h), (i), (j),and (k), respectively. (The sub-phrase bracketing is the sameas in Example 7.)

To carry the reductional process any further than we havein Example 9, it is necessary first to develop conceptions ofthe "structural beginning" and the "structural ending" of agroup. By structural ending, we mean either a V-I progressionwhen it appears at the end of a group, i.e., the full cadence, orthose variations on the full cadence known as the half cadenceand the deceptive cadence. The cadence is designated as asyntactic unit, both elements of which are retained at theappropriate levels in the time-span reduction, with the firstelement subordinate to the last.l1

By structural beginning we mean the most stable eventearly in a group in which there is a structural ending. Theremust normally be at least one intervening event (the "mid-dle") between this stable event and the cadence if the formeris to be designated as a structural beginning. Thus the smallestgrouping levels, such as those specifying motives, usually donot have structural beginnings and endings; all groups from

the phrase level on up do have them.The structural beginning and the cadence of a group arespecially labeled in the reduction, with b standing for "struc-tural beginning" and c for "cadence." They dominate hier-archically all other events in a group. As a visual aid, we placea number, signifying the number of measures spanned, withineach grouping slur beneath the music; the same number ap-pears as a subscript to the b and the c for each group. Thus in

Figure 6 each b and each c receives the subscript "4" at level(h) in the reduction, since they begin and end four-measuregroups. However, the first b and the last c are also the struc-tural beginning and the cadence for the entire eight-measuregroup; therefore, at this next region of application, they re-ceive the subscript "8" and are retained at level (g).

In effect this labeling creates a double-layered pitch hier-archy between those events which are b's and c's and those

which are not. At local levels, all otherevents in a

groupare

subordinate to the group's structural beginning and ending.At more global levels, structural beginnings and endings aresubordinate or dominating by virtue of the hierarchic struc-ture of the groups for which they function. At the end of the

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Example 9. Bach: "O Haupt"

/ \

/ \/

o)(\

Ab b (a)(a)

(b) (b)

C; c1

I I I I F r r r r ' ' r, , ,,,

r J J J J J J

((# i iX J-rj j I

w f, r PfrfI ij jI I

(b)i

(a)

135

? ? ? ?* ? ? ? ? ? ? ? ?- ?*

?? * : *J ?.

,P44 j j j j - j .( WI----wj 6 - p - p tp FF -1-

I


27/62

reductional process, there remains only the structural begin-ning and the structural ending for the piece as a whole.

We are now in a position to investigate a complete time-span reduction. Example 10 gives the music12 and Example11 represents its analysis. By this point, Example 11 shouldon the whole be self-explanatory. The following remarks arein the nature of annotations.

1. The best way to "read" Example 11 is first to examinethe grouping and metrical analyses, then to hear the levels inrhythm, as notated beneath the music, in their reductionalorder. If the analysis is correct, each level should sound like anatural simplification of the previous level. Any difficultieswhich the reader initially has in deciphering the tree can becleared up by a step-by-step comparison with the notationunderneath.

2. Level (f) in Example 11 has already been reduced fromthe actual music (Example 10) to what is felt to be its small-est beat level. This is a convention which we always follow inconstructing the time-span and prolongational reductions for

a piece, partly because syntactically unimportant detail isthereby eliminated, and also because the two reductions donot significantly diverge beneath this level.

3. It would be possible to assign further low-level groupingslurs within these eight measures. However, since the indica-tions for these groupings are somewhat conflicting, and sincesuch groupings would not affect the analysis as a whole, wechoose not to assign them. Therefore, the weak beats within

the lowest grouping slurs are simply bracketed as afterbeats.4. The structural beginnings and cadences are labeled ateach level, but they do not receive subscripts until the reduc-tional process has reached the grouping levels for which theyfunction.

5. The selection of which events are dominating is straight-forward except in mm. 3 and 7 at level (d). In m. 3 there is aconflict in the preference rules between the metrically strong-

er position of the F-sharp-E-Achord and the more stable

structure of the V6 chord. The F-sharp-E-A chord is chosenfor reasons having to do with the phrase as a whole: the regu-larity of harmonic rhythm is preserved, and a descending linein the bass from the tonic to the dominant is created. The

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Example 10. Mozart: Sonata K. 331, I

Andante grazioso. A I

(sa rr %p-r f*-*a

(-)


29/62

K, , I k I K . . I K I . f,

P rr rr r r G rPtd r Gr r 'r rr, rP^r2i 3,r rr r r P r' r pr p pC r cJ

4 8

(e) on> I I iii i, I i. i I i1[) g ~ J 2_ 2'

tb 2 2 2 2tc

J

(C)*'Io |I, 1, ;B

114) (C 1^^ C4 i4 4 4

ca,

aYI

8c c -

~~~~~~~(a) ,a>ll DI b 3 r4 C

138

Example 11

n., A I k I K

I I I


30/62

same chord is chosen in m. 7 because of the parallelism withm. 3.

6. Note that if the cadences were not labeled and retained,in m. 4 the I chord instead of the V chord would have beenselected at level (d). The labeling also causes the retention atlevels (d), (c), and (b) of both elements of the full cadence inm. 8.

7. Levels (c) and (b) can be construed as Schenkerian"background" evels. Schenker's own analysis of this passage13gives the melodic structural weight to the E in the first mea-sure rather than to the opening C-sharp. In our theory, itwould be possible to achieve his result by applying the "arpeg-giation rule," i.e., by regarding the first measure as a brokenchord and therefore compressing it into one event placed onthe downbeat of the measure. However, such a decision pro-duces difficulties at intermediate structural levels (especiallylevel (d)).

8. In the large levels (b) and (a), the sense of metrical struc-ture has faded to the point that it is largely inoperative. Theevents at these levels in the notation beneath the music do

not receive rhythmic values because there are no more dots inthe metrical analysis to which such values could be related. Itis at this point in the reductional process that pitch eventscan be thought of as rhythmically free-floating.

9. It would perhaps be sufficient to stop the reductionalprocess at level (b). Level (a) simply selects the most stableevent from those available at level (b). As is often the case,the most stable event here is the last chord; thus, the rest of

the piece is a left branch to it. This situation can be inter-preted to mean that, in a sense, all the events of a pieceexcept the last constitute a delay of the moment of completerepose, which is the ending. If the point of maximal stabilityhappened in the middle of a piece, there might be no reasonfor the piece to continue.

10. We have considered these eight measures as if they werea complete piece. If the entire theme, which is 18 measures

long,were

analyzed,the

opening event would eventually belabeled b18 and thus would dominate all other events in thefirst eight measures.

11. Observe how the geometry of the tree, while accuratelyconveying the hierarchy of pitch events, also mirrors visually

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the partitioning of the piece into time-spans (groupings andbracketings). Moreover, the tree notation makes visually clearthe interaction of the pitch structure with the metrical struc-ture at all pertinent levels.

There are important hierarchic aspects to the pitch struc-ture in Example 10 which do not emerge in the time-spanreduction in Example 11. To capture these aspects, we mustdevelop the domain of analysis called the prolongational re-duction. In the parameter of rhythm in this domain, eventscome before or after other events, but they are not measuredaccording to some metrical conception. The relevant notionof elaboration is elaboration by harmonic and melodic con-nection.

There are two kinds of elaboration in the prolongationalreduction: prolongation, in which a pitch event is elaboratedinto two or more copies of itself;'4 and contrast, in which adifferent, relatively ornamental, pitch event is introduced. Aprolongation is represented by two branches extending froma small circular node, as in Figure 7a. Neither event takes hier-

archic priority; rather, both events are thought of as an exten-sion over time of what at a deeper level is the same event. Incontrast, hierarchic subordination is designated by right andleft branching, as in the time-span reduction. However, where-as in the time-span reduction these branchings only indicatethe subordination of one event to another, in the prolonga-tional reduction they receive a further interpretation. A rightbranch signifies progression in a piece, whether at a local level

as in Figure 7b or at a large structural level. A left branch isutilized only at local levels and signifies delay in relation tothe bass, as at level (c) in Figure 7c. (Since metrical structuredoes not play a role in the prolongational reduction, Figure7c represents a suspension considered only with respect to itspitch structure.)

The well-formedness conditions and the preference rules forthe prolongational reduction are similar to those for the time-

span reduction.The well-formedness conditions

precludethe

crossing of branches and the assignment of more than onebranch to the same event. Given a sequence of pitch events, thepreference rules select a hierarchy according to principles of

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Figure 7a

(a)

(IM

--_ - U *--

(a)

(b)(b)

J>I^vwv-Y I - I ||

[ba] [C4] [b,] [cs]measure: 1 2 3 4 5 6 7 8

time-span reduction

time-span reduction

(a) > (a)

(b) ) (b)(b) >

(c) > (c(C) )(C

I- s- ^-y I- ^^ --1[

1 2 3 4 5 6 7 8

prolongational reduction

7b 7c

(a)

(b

O 1

(a)

(b)

I

*?

Figure 8

141

i . 1i= -I1dn .f Ii 11


33/62

stability largely resembling those for the time-span reduction.Meter is disregarded; prolongations are maximized.

At any given level down to the phrase level, the sequenceof events available to the

preferencerules for the

prolonga-tional reduction is precisely the same sequence as at theequivalent level in the time-span reduction. In other words,the hierarchy of b's and c's as designated in the time-spanreduction heavily determines the prolongational reduction.Again, we illustrate with a typical antecedent-consequentpattern. In the time-span reduction in Figure 8, levels (a), (b),and (c) refer to events labeled as structural beginnings andendings; these events at equivalent levels become the materialfor the prolongational reduction in Figure 8.

Even when precisely the same events are available at a givenlevel for the two kinds of reduction, the reductions drawradically different connections among these events. For in-stance, in Figure 8 the V chord in the full cadence in m. 8 is aleft branch to the ensuing I chord in the time-span reduction,but is a right branch from a prolonged I chord in the prolon-gational reduction. In the former case, it is a left branch be-cause it is within the time span of the final tonic; in the lattercase, it is a right branch because it progresses to the finaltonic. (In both cases, it is subordinate to the tonic accordingto principles of stability.) Large-scale right branching in theprolongational reduction always indicates significant syntactic"progress" n a piece.

Note, too, that at level (c) in Figure 8 the prolongationalreduction brings out connections of harmonic identity not

captured in the time-span reduction. The tree for the time-span reduction would look the same even if the b for theconsequent phrase were an entirely different chord; in theprolongational reduction, however, such a change would pro-duce a right branch at level (c) instead of the prolongationrepresented there. On the other hand, the tree for the time-span reduction expresses grouping structure, something notconveyed in the prolongational reduction itself. Thus, for

example, the tree for the prolongational reduction cannotdifferentiate between the full cadence in m. 8, and the phrase-ending half cadence followed by the phrase-beginning tonicchord in mm. 4-5.

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From the phrase level down to the smallest beat level-thatis, for all events which are not b's and c's except for the mostlocal details-the prolongational reduction is constructedwithout reference to the time-span reduction. As a result,within this region the two reductions usually differ not onlyin the connections made but in the actual sequences of eventsat corresponding levels.

To facilitate reading the prolongational reduction, we give,beneath the music in a non-rhythmic notation, the actualsequence of events for each level from the phrase on down.(See Example 12.) Unlike the tree, however, this secondarynotation does not convey the types of elaboration within asequence of events.15 For each level from the phrase on up, itis sufficient to mark the events which, as b's and c's in thecorresponding time-span reduction, have caused the equiva-lent sequence of events in the prolongational reduction; thismarking is accomplished by placing the corresponding lettersin parentheses at the appropriate places just below the music.

Example 12 represents the prolongational reduction forthe first eight measures of Mozart's K. 331. Remarks:

1. The most local level in the prolongational reduction(level (h)) corresponds with the "lowest beat level" as deter-mined in the time-span reduction (Example 11, level (f)). Inplacing this lower boundary on the prolongational reduction,we are in effect claiming that beyond this level local detail isnot of prolongational significance.

2. Level (a) represents the level of abstraction at which theA major root position triad is totally unelaborated. The high-est level of the prolongational reduction for any classical tonalpiece always results in an undifferentiated root-position triad;it is a way of saying that a piece is in a certain key.

3. Levels (a), (b), and (c) derive from the equivalent levelsin Example 11. The situation is exactly as set forth in Figure8 and in the discussion of Figure 8.

4. At levels more detailed than level (c), all remaining eventsare connected without reference to the other domains ofanalysis. Let us paraphrase the reductional process for mm.1-4 at each successively lower level (similar considerationshold for mm. 5-8). At level (d) the as yet unelaborated tonicchord progresses through the ii6 chord to the cadential V

143


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L- - il i*, " " 11,L_ *ri,II mI4 rl -1 e l,,C_a C rlll H i115* l111* llll llH

-- L _ I l ' I ll '

2- _ L- * '* * * ?I-J. ?"lb l l "

-I. -- *1*|L **. 1 .l i '1 '| 1 I

Z L I * 1_

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