854439

Set-Class Salience and Forte's Theory of GeneraAuthor(s): John F. DoerksenSource: Music Analysis, Vol. 17, No. 2 (Jul., 1998), pp. 195-205Published by: WileyStable URL: http://www.jstor.org/stable/854439 .

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JOHN F. DOERKSEN

SET_CLASS SALIENCE AND FORTE S THEORY OF GENERA

In the conclusion to his ground-breaking article on pitch-class set genera, Allen Forte suggests that one might further investigate the relationship of'levelled linear structures' to genera (Forte 1988, p. 264). He draws attention in particular to the question of generic relationships between high and low structural levels. In this essay I take up Forte's suggestion, but I recast the problem. Rather than examine the results of a generic analysis for relationships between levels, I explore an analytical approach that lets levels shape the generic analysis: this I call the set-class salience theory. While this approach integrates well with Forte's pitch-class set genera theory, it also stands independently and makes analytical claims that do not bear directly on genera (see Doerksen 1994). Given the focus of the symposium in which this essay appears, however, I shall limit my comments to those aspects of the salience theory that are necessary for understanding its relationship to Forte's genera theory. What fol- lows, then, is a sketch of the main features of the salience theory and a demonstration of its effect on generic analysis.

The SalienceTheory

The definitive characteristic of the salience theory is its attempt to draw from the musical surface the basis for the structural differentiation of pitch materials in a post-tonal work. What this means for analysis, in part, is that referential collections do not hold an a priori privileged status; prior to an assessment of salience, each musical event within a work has the potential to achieve structural distinction. Only an event's involvement at the musical surface will determine its structural import. The implications of this contextualist stance for generic analysis will become more apparent in due course, but for now it is worth noting that, in a sense, the salience theory prescribes a methodology for applying the genera theory.

This analytical stance is built upon a central assumption, namely, that contextual salience equals structural significance. Let me clarify this statement with an example. Take, for instance, the appoggiatura in tonal music. The appoggiatura has contextual salience on its side: it is higher, metrically stronger, and often longer and louder than the note it resolves to, but we hold it structurally less significant than its resolution. The appoggiatura is not structural; it is a non-chord tone. This syntax is at work in tonal music. But post-tonal music

MusicAnalysis, 1 7/ii (1998) 195 o Blackwell Publishers Ltd. 1998. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK



196 JOHN F. DOERKSEN

Ex. 1 The segmentation strategy (Berg, Op. 2 No. 4, bars 14)

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does not generally exhibit such a syntax, so in the absence of such conditions, I claim that contextual salience assumes a more important role.1 In the salience theory, then, an event exhibiting a high degree of involvement at the musical surface will be tagged as being important. Another, closely related, assumption implicit in the contextualist stance is that, at least at the first stage of analysis, the unit of analysis should reflect boundaries created at the musical surface.2 In the case of a texted work, the text, too, creates boundaries, and therefore has a part to play in defining the unit of analysis. The salience theory will call these units events.

Guided by these assumptions, my analysis of Berg's Op. 2 No. 4 demon- strates the main framework of the salience theory.To get a sense of the segmentation process, consider the vocal line of the opening phrase of the song (Ex. 1 ) . Here musical gestures corroborate boundaries suggested by the text. Rests break the line into two primary gestures. The extended duration of the C," on 'Gras' and the quasi-sequential repetition of the ascending FFGeC," suggest a further internal boundary within the second primary gesture. Notice that set- class identity has not entered the discussion at this point. I have tried to iden- tify events on the basis of their gestural qualities, for lack of a better description.

Fig. 1 Event-classes

Repetition Complementarity l

Pitch-Class Set-Class Pitch-ClassExchange Overlapping Embedded l l

Pitch-Class Set Literal I

Pitch-Set l

Literal Formally Significant

Form Inclusion Usage

Phrase I

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MusicAnalysis, 17/ii (1998) o Blackwell Publishers Ltd. 1998



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SET_CLASS SALIENCE AND FORTE S THEORY OF GENERA

Ex. 2 Berg, Op. 2 No. 4: segmentation

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If this is done for the rest of the piece, quite a range of events is generated: 59 set-class types (see Ex. 2). The procedure seeks to affirm the gestural integrity of the musical surface and tries not to limit the range of events artificially. But it does leave the problem of what to do with the range of set-class types resulting from this strategy. One way to solve this problem is to compare events on the basis of properties they possess. These properties might be abstract - set theory describes these well - or contextual. In general we could say that an event's contextual properties include all of its non-pitch parameters. For the purpose of this analysis I have specified several abstract and contextual properties as a basis for comparing events. These are grouped into event-classes (Fig. 1).


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There are five main families of event-classes: repetition, complementarity, form, inclusion and usage. The family named repetition, for instance, offers a way to relate events to each other based on set-theoretic equivalence. The repetition of an event may be very abstract, as indicated by the label Pitch-Class Set-Class, where the repetition need only keep the interval content of the events in common, or it may be quite contextual, as suggested by event-class Literal. Here the repetition would keep not just the pitch content in common, but also the manner of presentation: for example, two events that both comprise the same pitches and sound as chords would meet the requirements of this event- class. For the most part, the meanings of these event-classes are evident from




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SET_CLASS SALIENCE AND FORTE S THEORY OF GENERA 199

Ex. 2 (cont. )

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their names, but this is probably not so for event-class Usage. This category simply ensures that each event is recognised by the salience theory; it is the first category that any event instantiates, and it becomes crucial when tracing set- class salience in more than one work.

The relationships that each event bears to other events in the piece is tracked and represented by the salience matrix (Table 1). In essence, this matrix tallies up the relationships of a given event with all other events in the work, and the most connected events rise to the top. Events are identified in the matrix by their set-class names, and these appear along the y-axis. Event-classes are listed along the x-axis. An integer at the intersection of a set-class name and an event-class indicates the number of times that the set-class has instanti- ated the event-class. Rows of the matrix are summed by event-class family to form the basis for the Set-Class Salience Index (or sst.3 This index appears in the right-most column, and as the term suggests, it reflects the relative salience of a given set-class. It ranges in value from near zero to a possible maxi- mum of one: the greater the value, the more salient that set-class is deemed to be. This index mediates between the set-class salience theory and the genera theory.

The matrix also includes two other types of information-the Event-Class Sa-

Music Analysis, 1 7/ii ( 1998) c Blackwell Publishers Ltd. 1998



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lience Indices (or ESIs) in the bottom row, and a list of numbers in the second column. The values in the second column are necessary for integrating the data with the genera theory, and I will touch on them shortly. Let me take a moment now, though, to comment on the indices in the bottom row.These indices simply reflect relative activity within each of the event-classes. While they are not central to the relationship between the salience and genera theories, the ESIs do underscore a fact that has long shaped set theory, namely, that inclusion relations are, to border on the tautological, exceptionally inclusive. Compare, for instance, the number of set-classes that instantiate this event-class with the number that instantiate the next most prominent event-class (excepting event- class Usage, which by definition marks every event in the work): less than half of all the pc sets instantiate event-class Phrase. Note also that the ESI of event- class Inclusion is more than ten times greater than its nearest competitor, which, in this matrix, is event-class Usage. As an indicator of connectedness, inclusion relations exclude very few sets. Hence, that theorists should have turned first to inclusion as a means of assessing set-theoretic relations should not be surprising: inclusion relations hold the greatest potential of any single- parameter model for yielding rich analytical results. And the genera theory, too, reflects this richness.

To demonstrate how an event moves from compositional surface to matrix, let us look briefly at what happens to set-class 5-32 (Ex. 3), which appears six times in the song. As the matrix shows, its pitch-class content is repeated twice, and its pitch content together with contextual details three times. From the entry under event-class Inclusion on the matrix, we note that 5-32 is not a particularly common subset of other events in the piece. What is especially notable about this set-class, however, is the formal role it plays, and the values under the family of Form event-classes reflect this observation. The phrases it closes also mark the close of the piece as a whole. All set-classes within a work undergo a similar assessment. The salience matrix, then, attempts to reflect the nature and degree of involvement that a given set-class has at the musical surface, and in this way it establishes a hierarchy among pitch materials.

MusicAnalysis, 17/ii (1998) c Blackwell Publishers Ltd. 1998




Fig. 2 The Genus Salience Index (GSI)

sum of SSIs of the entire 1 set-class inventory

GSI(GA)= ((XIY)IZ) x 10

sum of SSIs of sets intersecting | GA and the set-class inventory

Integration of the salience and genera theories The impact of the genera theory on the way the salience theory presents its results is clear, and it is a simple matter to integrate the two through a slight revision of the Status Quotient in the genera theory.4 Fig. 2 shows the formula for the classical Status Quotient, although because of the revision, I have given it a new label. To keep the two distinct, the revised index is called the Genus Salience Index (or GSI). The distinction centres on the two operands in the expression that invoke the set-class inventory of a composition, namely X and Y. Instead of summing the number of sets to determine the values of X andY, the GSI sums the salience indices (SSIs) of sets. Given the inventory of sets that have been identified in the Berg song (see the salience matrix, Table 1), the impact on the GSI of the most salient sets is over three times that of the least salient. (Compare, for instance, the SSIs of set-classes at the top of the list with those at the bottom; the former will have the greater impact.)

All that remains is to work out a sample GSI, but before we can take that step we must revisit the values in the second column of the salience matrix (Table 1). These numbers reflect another type of weighting I have introduced to the genera theory, a weighting that addresses set-class representation in the genera. I have called these numbers exclusivity indices. In a nutshell, these indices give precedence to set-classes that hold membership in few genera. If a set- class holds membership in only one genus, for instance, I claim that it is highly representative of that genus. If, on the other hand, a set-class holds membership in ten genera, I claim that that set-class is not especially representative of any one of the many genera to which it belongs.5 The exclusivity indices play an important role in calculating GSIs.

To illustrate the calculation of the GSI, I turn to Genus 9. Twenty-one set- classes intersect Genus 9 and the set-class inventory (see Table 2). Instead of passing this value to X, however, the GSI sums the salience indices and exclusivity indices of these sets: the indices sum to 8.9758 instead of 21. (The second column in the genera matrix of Table 2 lists the arithmetic mean of the salience index and exclusivity index for each set-class.) Similarly, instead of assigning 59 to variableY, which is the total number of set-classes in the inventory, the GSI

Music A n alysis, 1 7/ii ( 1 9 9 8 ) c Blackwell Publishers Ltd. 1998




G 1 G2 G3 G4 G5 G6 G7 G8

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3-1 [.6547] 3-2 [.6397] 3-3 [.6946] 3_4 [.7198] 3-5 [.8200] 3-6 [.7586] 3-7 [.6073] 3-8 [.6551] 9-9 [.6243] 9-11 [.6237] 4-1 [.8359] 4-2 [.6154] 4-5 [.6284] 4-7 [.6108] 8-8 [.6243] 4-9 [.7093] 4-11 [.6712] 4-14 [.6589] 4-Z15 [.7259] 4-16 [.5762] 4-17 [.6901] 4-19 [.5736] 4-20 [.6670] 4-21 [.6135] 4-22 [.5654] 4-24 [.5904] 4-25 [.8301] 4-Z29 [.6243] 5-1 [.8109] 7-2 [.4995] 7-3 [.4799] 5-6 [.5476] 7-15 [.5843] 5-Z17 [.4686] 7-Z18 [.3658] 7-19 [.4209] 5-20 [.67S9] 5-21 [.5811] 5-22 [.4313] 5-23 [.4691] 7-26 [.2S26] 5-28 [.4740] 5-29 [.3781] 5-30 [.2953] 7-31 [.3577] 5-32 [.6813] 5-33 [.6100] 7_34 [.474S] 5-35 [.6901] 6-Z3 [.3408] 6-5 [.2418] 6-9 [.2472] 6-Z10 [.2159] 6-Z13 [.4059] 6-15 [.1536] 6-16 [.3333] 6-Z17 [.3511] 6-20 [.5080] 6-27 [.3300]

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Music Analysis, 1 7/ii ( 1998) ¢ Blackwell Publishers Ltd. 1998

Table 2 Berg, Op. 2 No. 4: genera matrix




Table 2 (cont.)

squos GS s squos GS s G4 .089 G9 .153 G6 .075 G2 .118

G8 .087 G10 .149 G1 .073 G1 .111 G9 .087 G4 .147 G12 .068 G 12 .109

G10 .087 G8 .144 G7 .064 Gl l .101 G2 .078 G5 .131 G3 .060 G7 .095 G5 .076 G6 .121 Gl l .058 G3 .082

again sums the salience indices and exclusivity indices of all the set-classes in the inventory, and these sum to 32.2842 instead of 59. In this way the GSI admits set-class precedence to the genera theory: squo(G9) = ((21159)14 l)xl 0 = .0868 GSI(G9) = ((8.9758132.2842)118. l 8 l 8)xl 0 = . l 529

It is not meaningful, of course, to compare these two results with each other: like must be compared with like, so that the Status Quotient's meaning comes from comparing it with other Status Quotients just as the meaning of a GSI resides in comparison with other GSIs (Status Quotients and GSIs are listed for all genera in Table 2). One can then see what effect musical context has on generic interpretation. In the case of the Berg song the GSI displaces few genera, which means that set-classes which the Status Quotient interprets to be important are also salient in the compositional context. In summary, let me reiterate a few main points about the salience theory and its relationship to the genera theory. The salience theory is bound up with a contextualist methodology: it takes as its object the entire pc set inventory of a given composition and makes no pre-analytical claims of preference for any one set or collection of sets. By evaluating each pc set according to a standard set of criteria, it seeks to avoid arbitrariness in assigning structural value. The criteria are, of course, the event-classes, and they include classical set-theoretic relations. I should add that one could use different criteria, define different event-classes, depending upon the task at hand. Nevertheless, the model would still reflect the relational richness of a particular node within a network. At the outset of this article I noted that the salience theory recasts the question of 'levelled linear structures'. Two implications flow from this recasting. Firstly, the theory makes no claims for linearity, at least not explicitly, but it does form the basis for analytical statements about linearity, and holds the potential to provide a contextual justification for projected set-classes. Sec- ondly, the theory constructs 'levels' in post-tonal music, but not in the tradi- tional sense. It does not produce a determinate number of structural levels (it identifies no middleground or backgroun4: the number of'levels' is limited only by the number of events in a composition and the interconnections between them. Furthermore, the theory does not construct nested structures, and in this sense it does not recognise 'well-formed regions of analysis'.6 Neverthe- o Blackwell Publishers Ltd. 1998

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less, when integrated with the genera theory, the salience theory allows the compositional context to bear more directly on an interpretation of genus precedence while the essential role of the genera theory remains unchanged. In short, the salience theory simply offers a way to blend into the genera theory another layer of analytical observation.

REFERENCES

Doerksen, John F., 1994: 'ATheory of Set-Class Salience for Post-Tonal Music, with Analyses of Selected Lieder by Anton Webern' (PhD diss., University of Western Ontario).

Forte, Allen, 1988: 'Pitch-Class Set Genera and the Origin of Modern Harmonic Species',3rournal of Music Theory, 32/ii, pp. l 87-271.

Hasty, Christopher,1981: 'Segmentation and Process in Post-Tonal Music', Music Theory Spectrum, 3, pp.5F73.

Lerdahl, Fred, 1989: 'Atonal Prolongational Structure', Contemporary Music Re- view, 4, pp.65-87.

NOTES

1. Other theorists make this claim. See especially Lerdahl (1989) and Hasty (1981).

2. A rather regimented segmentation strategy, which includes a series of well- formedness and preference rules, informs this analytic approach. It is outlined in Doerksen (1994), pp.76-84.

3. For this analysis, each event-class family bears equally on the SSI. One could eas- ily change the weighting so that one event-class family would have a greater impact on the SSI than another.

4. Forte introduces the Status Quotient in Forte (1988), p.232.

5. For present purposes I can limit discussion of genus representation to this basic point, but I might add that exclusivity indices by themselves revise generic analyses of pieces comprising many exclusive sets.

6. Fred Lerdahl uses this expression in the context of post-tonal music (Lerdahl 1989).




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