In theory only. [Vol. 13, no. 1]

MUSIC Via Y~.I c ql IAL~ in ti only ieory Unv. of Mich. dpskc /ýr Michael Buchler John Covach Justin London Tiina Koivisto Stephen Peles Janet Schmalfeldt Stephen Smoliar William Thomson volume 13, numbers 1-4 september 1997 UNIV. OF MICH Music LIBRARY

in theory only volume 13, numbers 1-4 september 1997

PAL, OL v \S o-\Z 1-44 Editors John Covach William E. Lake Reviews Editor Dave Headlam Editorial Board Richmond Browne David Damschroder Walter Everett Edwin Hantz Robert Hatten Marianne Kielian-Gilbert Robert Morris William Rothstein Charles Smith Robert Snarrenberg

contents 3 Lerdahl and Jackendoffs Strong Reduction Hypothesis and the Limits of Analytical Description Justin London 29 The Defining Moment: The Thema as Relational Nexus in Webern's Op. 27 Tiina Koivisto 71 Panel: Peter Westergaard's Tonal Theory: A Perspective of the History of Contemporary Theory 73 An Introduction To Westergaard's Tonal Theory Stephen Peles 95 Coming to Terms: Speaking of Phrase, Cadence, and Form Janet Schmalfeldt 117 Forum: Analysis-What Is It Good For? We Won't Get Fooled Again: Rock Music and Musical Analysis John Covach 143 Review The Analysis and Cognition of Melodic Complexity: The Implication-Realization Model by Eugene Narmour Stephen W. Smoliar 155 Response to Buchler William Thomson 179 Rejoinder to Thomson Michael Buchler Copyright @ 1997 by In Theory Only, ISSN 0360-4365, Volume 13, numbers 1-4

In Theory Only publishes eight issues per volume at irregular intervals. Subscriptions are $20.00 per volume for individuals ($28.00 foreign) and $30.00 for institutions ($38.00 foreign). Prices are subject to change without notice. Subscriptions begin with the current issue. Some back issues are available and the price for these is $2.50 ($3.50 foreign) per issue. A list of back issues is available on request. Reprints of all out-of-print back issues, as well as microffims of complete volumes, are available from University Microffims International, 300 North Zeeb Road, Ann Arbor, MI 48106. Foreign subscribers should contact University Microfilms International, 18 Bedford Row, London, WC1R 4EJ, England. In Theory Only welcomes papers of any length on any theory-related topic, including finished papers, reports of work in progress, pedagogical articles, and comments on ideas appearing here and elsewhere. Authors must submit FOUR laser-print-quality copies, double-spaced throughout (including reference list and endnotes), with examples and endnotes appearing after the text. The author's name should appear only in an accompanying cover letter. Style should conform to the current edition of the Chicago Manual of Style. Please note that In Theory Only uses the author-date style of bibliographic citation. Manuscripts will be returned only if sufficient postage is included. Authors whose manuscripts are accepted for publication must provide a copy of the finished text on 3 1/2" computer diskette in Macintosh or PC format. Authors must provide camera-ready copy for musical examples and figures and obtain copyright permission for the reproduction of any copyrighted material. Any fees required by the copyright holder are the responsibility of the author.. All references to specific pitches in In Theory Only should be made according to the notation suggested by the Acoustical Society of America: pitch class is symbolized by an upper-case letter and specific octave placement by a number following the letter. An octave number refers to pitches from a given C through the B above. Middle C = C'. Editorial and Business Office: In Theory Only College of Musical Arts Bowling Green State University Bowling Green, OH 43403-0290 [phone]

from the editors.. The editors are proud to present this quadruple issue, the largest issue of In Theory Only published to date. (The second largest, the "Index of Music Theory in the United States" [vol. 3, nos. 7-11], was a quintuple issue but was less than half a volume, since at that time a volume consisted of twelve issues. It totaled 171 pages, compared to this issue's almost 200 pages.) Among this issue's eight items are stimulating commentaries on tonal theories of Peter Westergaard and Lerdahl & Jackendoff, a penetrating analysis of Webern's Variationen, Op. 27, and a provocative essay on the analysis of rock music. The latter may spark some disagreement, and we welcome responses. If interest proves sufficient, we may establish a contining forum on metatheoretical issues of analysis. This quadruple issue is proof that ITO is alive and moving forward, although its pace to date has been much slower than many (ourselves included) would have preferred. We apologize for the almost three-year hiatus since the last issue. We will not recite the litany of causes, nor will we promise the unattainable. We are an irregular publication, and that will no doubt continue to some degree. Suffice it to say that we intend to continue to produce quadruple issues and that the next, volume 13, nos. 5-8, includes the conclusion of the Westergaard panel begun in this issue, as well as a group of papers on composers of the Second Viennese School. We are still accepting submissions for Volume 14.

Lerdahl and Jackendoffs Strong Reduction Hypothesis and the Limits of Analytical Description Justin London 1.0 Introduction Socrates is taller than Theatetus-true or false? This question is often used to illustrate one of the basic goals of philosophical inquiry, which is to come up with true descriptions of the world, as well as to introduce students to the conventions of philosophical argument. The exegesis of this question runs along the following lines: When Theatetus was just a boy, Socrates was indeed taller than Theatetus, and so the statement is true. But as the years went by, Theatetus grew into a tall man, whereas Socrates was always a rather short fellow, and so at this later point, the statement is false. This parable illustrates the fact that one cannot simply say "Socrates is taller than Theatetus," since this statement, in being both true and false, is incoherent. Now the reader may claim that all one needs to do is simply specify the domain over which the statement applies, and so one can say something like, "When Theatetus was a boy, one could say that 'Socrates is taller than Theatetus.' This approach attempts to define the conditions under which the statement is true. But, as the saying goes, "True under certain conditions... that is to say, false." For one of the desiderata of analytical philosophy is free repeatability, that is, to be able to make tenseless statements that are simply true, such as "2 + 2 = 4." The grammatically astute will have noticed a tense agreement problem in the conditional version of the statement, for if

In Theory Only I am talking about "when Theatetus was a boy" then I also ought to say that at that time "Socrates was taller than Theatetus." But analytical philosophers don't want to make such tense-conditional statements. Now the foregoing example might seem to illustrate very little (other than that analytical philosophers seem to have some very arcane concerns), but the point of the following pages is that music theorists very often have similar analytical urges. Marion Guck has pointed out that in theorists' discussions of tonal function, "we often talk as though such things as harmonic function are unequivocal aspects of a sound-e.g., a chord is or is not V.... [whereas] on the contrary, function can be weakly or strongly projected" (Guck 1993, 50, n. 7). Even if we admit that there are some aspects of musical description and analysis that are or may be equivocal, more often than not we still pursue these aspects as if they could be made unequivocal-that is, while under one set of conditions (perhaps when we first hear a particular chord) we are able to make some claim regarding its function, under another set of conditions (perhaps when we hear a few of the following chords) we can revise the initial claim, and then get it right. Thus, dominant sevenths are transformed into augmented sixths, and that is the end of it. Indeed, this is perhaps the chief theoretical assumption in any final-state analysis: at the end of the piece the structural function of each and every musical element becomes clear, and in this clarity one is able to reconcile any and all doubts or ambiguities regarding those portions of the piece that had posed a problem when they first occurred. David Lewin presents a complex and subtle critique of this kind of final-state analysis in his well-known discussion of a single G6 chord in m. 12 of Schubert's Morgengruss (Lewin 1986). For this one harmony Lewin provides a list of contexts in which this musical object may be considered, along with examples of the sorts of perceptions, relationships, and descriptive statements one could make about the G6 in each context. Even though some perceptions and descriptions contradict others, they are not mutually exclusive: What-p9-perceives includes the perception that p2 does (did) in fact make sense, even though it was (is) "denied" by p3b and "virtually annihilated" by p.... To put the matter more elegantly: P2, P3b' p, and p, are not all cohabiting the same phenomenological place at the same phenomenological time. They are different objects (or acts) in different parts of phenomenological space-time, exercising a variety of interrelationships. (Lewin 1986, 356; "px" is his notation for individuating the various perceptions that attach to this musical object.)

London, Limits of Analytical Description 5 Lewin thus argues that for a single musical event there may be (and perhaps usually are) a number of separately true and valid analytical descriptions.' Moreover,, he urges us not to assume that amongst these various analytical descriptions one can or should choose one as the single, best, "truest" account of the musical object. However, within each separate phenomenological continuum, Lewin suggests that one can make more or less determinate descriptions in some analytical language. This is because he believes that one has but a single perception of a musical object at any given time (1986, 368-70). So, for example, one cannot perceive a chord as being both a dominant and a tonic at the same time, though one might be able to regard the chord as a tonic in one continuum and as a dominant in another. Yet there may be some contexts in which: (a) we can and do perceive a single event as two different things at the same time, as in the case of a phrase overlap where one event is heard simultaneously as the end of one rhythmic group and the beginning of another;2 (b) there are events that are ambiguous, that is to say, indeterminately perceived as either X or Y in some context (but the listener is unable to make any more definite description of the event); and (c) there are contexts in which we are simply unable to perceive (and hence describe) any particular parametric structure (e.g., an extended modulatory passage in which one ceases to make any assignations of scale degree or functional category). This suggests that one can take Lewin's arguments a step further. For if listeners are able to inhabit the various phenomenological continua that Lewin lays out, then it seems plausible that they are also able to inhabit "meta-continua" in which they are aware of the tensions and antinomies between their lower-level perceptions and descriptions. In these latter perceptual contexts one may have many-to-one relationships, ambiguity, and, perhaps most interestingly, outright uncertainty. In A Generative Theory of Tonal Music (1983; henceforth GTTM) Fred Lerdahl and Ray Jackendoff have crafted one of the most comprehensive and carefully thought-out theories of rhythmic and Seealso Jackendoff (1991 & 1992) for discussions of a similar "parallel multiple analysis model." 2Indeed, it may be quite useful and interesting to sort out those parametric domains where such coextensive percepts and descriptions are possible versus those domains where only the singular sorts of perception and description described above occur. For example, while a single rhythmic event can be heard as both a begin~ning and an end, metric events cannot be heard in such an equivocal fashion (and hence the notion of a metric overlap or elision is problematic). A full discussion of this, alasz

6 In Theory Only tonal structure. GTTM is the final-state model par excellence of the discipline. Through the interrelated processes of grouping analysis, metric analysis, timespan reduction, and prolongational reduction, Lerdahl and Jackendoff are able to provide not merely a comprehensive analysis of a musical structure, but have framed this analysis as an account of the listener's mental representation of his or her final-state knowledge of that musical structure. More precisely, GTTM produces a representation cast in terms of a structural description written under their cognitively based grammar (GTTM, 2; see also Jackendoff 1987 and 1991). Consider their analysis (GTTM, 35) of the grouping structure of the opening measures of Beethoven's Hammerklavier sonata (ex. 1). Example 1. Analysis of Beethoven, Sonata No. 29, Op. 106, mvt. I, mm. 1-17 from Lerdahl and Jackendoff (1983, 35) Reprinted by permission of the MIT Press ri. iP I ~ Jl ~ -(urbrrrku~ This analysis shows all parts neatly nested and with each structural level in a clear subordinate or superordinate relationship to the other parts of the rhythmic hierarchy. However, a reading more sensitive to ambiguities of grouping structure within these measures might look something like example 2.

London, Limits of Analytical Description 7 Example 2. Alternative analysis of Beethoven, Sonata No. 29, Op. 106, mvt. I, mm. 1-17 mismatches, though its broad hierarchic outlines remain intact. If in 9L R,------------A- X SC.,. analytical overconfidence. GTTM, as a result of its Strong Reduction focus of this paper is a critique of the SRH through the exegesis of a Sg -------analytical domai This grouping structure contains overlaps, gaps, conflicts, and mismatches, though its broad hierarchic outlines remain intact. If in listening to this music we cannot sort it all out, then any analysis that overstates the confidence and precision with which group boundaries are marked is in some sense misleading. More problematic than individual analyses is a theoretical framework that engenders such analytical overconfidence. GTTM, as a result of its Strong Reduction Hypothesis (henceforth SRH), explicitly requires resolute analytical decisions, whether or not the music supports such decisions. The focus of this paper is a critique of the SRH through the exegesis of a number of problematic grouping structures.3 In a more general sense this critique could be mounted against any number (if not most) analytical methods, to the extent to which they operate by assigning 3Though the focus is upon the analysis of rhythmic grouping structures, the more general points made herein can be applied to other analytical domains. Furthermore, grouping structure is often regarded as an unequivocal aspect of musical structure (or at least more unequivocal than, for instance, tonal relationships). One reason for focusing upon grouping structure is to show how even the most mundane aspects of analytical description may have both covert and overt difficultie s.

In Theory Only a single analytical descriptor to each individual musical event. In other words, some version of the SRH is usually operative whenever we embark upon an analytical journey. Lerdahl's and Jackendoff's work is the focus here because: (a) they are forthright in overtly stating their SRH, why they hold to it, and, most notably, under what conditions they might be forced to abandon it; (b) in their discussion of grouping analysis in particular they make especially clear the way in which the presence of the SRH delimits the range of permissible grouping structures within their analytical system; and (c) as will be shown, even without the SRH, much (if not most) of the GTTM analytical system can still operate effectively. In the second part of this article a number of problematic grouping structures will be discussed. First, a normative case (the opening measures of Mozart's Piano Sonata K. 283) will be presented, followed by an instance of a parenthetical group (in Haydn's String Quartet Op. 50, No. 3). Instances of ambiguous relationships among subgroups, including the opening measures of Mozart's G-minor Symphony, K. 550, as well as a case of grouping overlap (in Schubert's B-flat Piano Trio) will next be explored. Lastly an example of conflict between prospective versus retrospective locations for group boundaries (in the finale of Haydn's Symphony No. 92) will be considered. Along the way in presenting these examples various particulars of the GTTM grouping theory will be examined when appropriate. In the third part of the article, the SRH will be gone over in some detail and an alternative hypothesis, a weak reduction hypothesis, will then be proposed. In the closing section of the article a number of possible motivations for the SRH, both in the particular case of GTTM and in music theory in general, will be considered, concluding with a few remarks on music theory as a scientific endeavor. 2.0 Some Problematic Examples of Grouping Structure 2.1 A Normative Instance: Mozart's Piano Sonata K. 283 If a piece of music is a more-or-less contiguous stream of sound events, then in large part music analysis is concerned with the segmentation of this stream into parts, subparts, and so on. Musical analysis is also concerned with the ways in which these parts are internally and externally bound together. Furthermore, clarity and precision are what we value in analysis-we generally do not prefer an analysis that, after looking at the interplay of rhythm, meter, pitch processes, and harmony, concludes merely that the phrasing in a given passage is hopelessly murky. Therefore, it is most pleasing when the outcome of one's analytical efforts looks something like example 3.

London, Limits of Analytical Description 9 Example 3. Analysis of Mozart, Sonata K. 283, mvt. I, mm. 1-4 An analysis of the grouping structure of the right-hand part of the first four measures of Mozart's Piano Sonata in G, K. 283 is given below the staff; it shows three levels of structure whose boundaries are recognized by the onset and offset of pitch/durational patterns. This sort of grouping structure, with clear boundaries for each unit and a clear sense of intralevel nesting, passes as a textbook example for grouping well-formedness according to GTTM. But the opening melody of K. 283 is, in fact, a rather exceptional passage: the boundaries of each group are marked by rests, melodic and rhythmic patterning, explicit articulation, harmonic change, register, and meter. Most group boundaries do not contain anything close to this degree of parametric redundancy.4 So this example is given as a cautionary aside, even though the account of its grouping structure given above is perfectly right and proper. It will be useful to contrast this analysis with other analyses that, though similar in form, attach to rather different musical structures. 2.2 A Parenthetical Group Leonard Meyer defines musical parenthesis as "internal prolongations which, while not affecting established implications, interrupt the musical structure, usually after arrival at some point of provisional stability. Because they do not really 'belong' to the preceding and following patternings, such internal interruptions have been called parentheses" (1973, 239). Example 4 shows Meyer's parenthetical illustration, using Haydn's String Quartet Op. 50, No. 3, mvt. IV. 4All too often the opening measures and/or themes of classical movements are used as models for rhythmic, metric, and tonal structure. Yet opening measures have special functions: they must establish key, pulse, meter, principal motive, principal register and/or timbre, and so forth. In these passages, where durational patterns and groupings cue the metric organization, the overwhelming redundancy such as is found in K. 283 is both useful and necessary. It does not follow, however, that the rhythmic hierarchies one finds in the first eight measures will serve as archetypes for the rest of the piece (in particular) or for musical hierarchies (in general).

10 In Theory Only Example 4. Analysis of Haydn, Quartet Op. 50, No. 3. mvt. IV, mm. 1-12 from Meyer (241) From Explaining Music: Essays and Explorations by Leonard B. Meyer. Copyright 1973. Used by permission of the University of California Press After an extensive analysis of the melodic structure of the passage, Meyer concludes that "we recognize at once that measures 5-8 are not part of the 'real' melody.... [T]he real melody is characterized by goal-directed motion, but the parenthesis is static. It is as though a person purposefully striding toward some objective should suddenly pause, perform a dancelike caper, and then continue to his objective" (1973, 241). The parenthetical nature of mm. 5-8 becomes readily apparent when one attempts to perform a grouping analysis and/or prolongational reduction following Lerdahl and Jackendoffs analytical method. Example 5 lists a series of possible groupings for this passage. Example 5a presents the grouping structure of the passage through m. 8. At that moment we can hear mm. 5-8 as a prolongation of the initial tonic harmony, specifically a prolongation of ý, which functions as the goal of the melodic sequence in the first four measures. As the passage continues, however, mm. 5-8 may also be regarded as an antecedent to the articulation of 5 in m. 9. Thus one is tempted to revise the grouping analysis by bracketing mm. 5-8 with mm. 9-12, as indicated in example 5b. Yet example 5b is not really adequate either, as it ignores our initial understanding of the passage as given in example 5a. In order to include both our prospective and retrospective understanding of mm. 5-8, one might indicate some sort of grouping overlap, as given in example 5c.5 Nor is this sufficient, for the overlap would seem to indicate that mm. 5-8 give rise to an interlocking and SLerdahl and Jackendoffs grouping well-formedness rules are discussed in detail below (section 3.1). Here it will be sufficient to note that examples 5c, 5d, and 5e are not well-formed groups according to GTTM (though the grouping overlap in example 5c is permitted via a transformational rule).

London, Limits of Analytical Description 11 hence tightly bound structure for the entire passage, which, at least according to Meyer, is precisely the opposite of their parenthetical function. Perhaps, then, we should analyze the grouping either as example 5d, where mm. 5-8 are simply a gap in the larger group, or as example 5e, where the three contiguous phrases do not form a continuous group on higher level. Of course, one could just ignore the problems created by the parenthesis and simply claim that there are three phrases that form one large group, as in example 5f; however, this would seem to be an oversimplification of the grouping relationships in the passage. Moreover, this oversimplification constitutes a misrepresentation of the grouping structure of the passage in that it ignores the discontinuity created by mm. 5-8. When we attempt to represent the prolongational structure of this passage, other problems arise. Example 6 contains a series of possible prolongational trees for this passage. The task here is to decide how to connect the node representing mm. 5-8 to the surrounding timespans-is it a left branch or right branch? If we want the timespan reduction to reflect the sense of parenthesis, we could just refuse to construct the branch and leave the node unattached (ex. 6a). Alternatively, as a neutral prolongation of 3 we could connect the node with two branches, one in each direction (ex. 6b). However, these two timespan reductions are forbidden by GTTM's well-formedness constraints on tree-structures (113-14, as well as ch. 7 passim). A third choice (and one that follows the GTTM prolongational preference rules) would be to default to a right branch (ex. 6c). However, this forced choice does not accurately reflect the above understanding of these measures since it implies a rather cut-and-dried (though weak) sense of prolongation of the initial tonic. The larger issue here is how to deal with discontinuous musical gestures and the noncontiguous grouping structures they create. If a measure or a phrase (or even a formal section) is understood as an interruption within the context of a larger gesture, then we are faced with a problem when such structures are to be described under a theory which demands that all groups exhaustively and recursively nest and that all events form a continuous left-to-right stream.

12 In Theory Only Example 5. Possible grouping structures for example 4 a mm. 1-4 ''5-8 I b C d e f I I Imm. 1-4 ''195-8 9-12 'mm. 1-4 15-8 I 9-12 I Imm. 1-4 1 19-12 II I 'mm. 1-4 115-8 119-12 f. I I mm. 1-4 I 15-8 1i 19-12 1 Example 6. Alternative branching structures (after Lerdahl and Jackendoff)

London, Limits of Analytical Description 13 2.3 Ambiguous Relationships Among Subgroups 2.3.1 Grouping the Elements of an Extended Anacrusis Mozart's G-minor Symphony, K. 550 begins with an often-discussed extended anacrusis (ex. 7). My commentary shall focus on the first two measures of the passage and on the groups that occur on the half-bar and full-bar levels. According to the analysis in GTTM, the first two instances of the two-eighth/quarter-note figure (E65-D5-D5) each manifests itself as a group on the half-bar level and the remaining notes (up to and including the rest in the second measure of the example) form a single group. Lerdahl and Jackendoff also claim that the first two iterations of the E 6 -D5-D5 figure form a group in their own right, a group that balances the duration of the following group, thus forming a symmetrical structure (see their discussion of GPR 5, GTTM, 49-50). One could also argue that a grouping pattern, once established, tends to perpetuate itself (if possible), and so one can describe the single group formed by the E 6 -D5-D5-B b6 as the product of a grouping overlap (ex. 8). (Overlaps are discussed in greater detail in the following section.) Example 7. Analysis of Mozart, Symphony No. 40, K. 550, mvt. I, mm. 1-4 from Lerdahl and Jackendoff (1983, 48) Used by permission of the MIT Press A 1 '-\ '" N _ -rt- _ ^ -^ tr * \ ____ -____________ i r i r fin I n n i I ~ Example 8. Grouping overlap in Mozart, Symphony No. 40, K. 550, mvt. I, mm. 1-2, I |JJ., ^,,,^, iT, I '~! I ^ V II / If's11 ^ = Nonetheless we may accept the lowest level of grouping posited by Lerdahl and Jackendoff as a given; my argument lies with their combination of the first two half-bar groups into a larger group on the basis of symmetry. Such (1+1)+2 grouping structures are, of course, quite possible (ex. 9).

14 In Theory Only Example 9. (1+1)+2 grouping in Haydn, Symphony No. 104, mvt. IV, mm. 7-10 I t I L I Ii I In the finale of Haydn's London Symphony, mm. 7-8 (each a group in its own right) form a larger unit that balances mm. 9-10 (note, however, the substantial melodic contrast between mm. 1-2 and mm. 3-4). But the opening of K. 550 is quite different, as its groups collectively form an extended anacrusis, an unstable gesture that leads to the anchoring point of stability on the downbeat of m. 4. It seems counterintuitive to place the first two presentations of the half-bar E b s-Ds-D5 motive into a separate group that supposedly balances the following one-bar timespan. I am especially uncertain how to relate the middle group to its surrounding groups, for while it is a continuation of the first group, it is also an anticipation of the third. Each presentation of the E -Ds-D5 motive is an insistent reiteration of anacrustic function, a refusal to let the phrase begin. If pressed, I might suggest example 10, which shows a series of successively revised groups. At the top of the analysis are the half-bar groupings (including the overlap) as they are clearly manifest in the durational pattern of the music. Example 10. Alternative grouping structure in Mozart, Symphony No. 40, K. 550, mvt. I, mm. 1-2 ---,I II,i I I 1,1 W,. - I I I I_ This passage is not balanced, at least not in terms of its kinetic qualities (and hence in terms of the articulative stability that is a precondition for group boundaries). Instead, it is a continuous sweep through the three iterations of the anacrustic figure to its point of

London, Limits of Analytical Description 15 arrival on the downbeat of m. 3.6 The grouping analysis in example 10 is an attempt at acknowledging this rhythmic sweep and the problems it creates with respect to the grouping structure of the passage. Lerdahl and Jackendoff are aware of the kinetic qualities of this passage, and they argue that "the inner tension of this music is in part a product of the rhythmic conflict between the periodicity of the metrical structure (reinforced by the accompaniment) and the complexity of the time-spans resulting from such out-of-phase conditions" (GTTM, 127). On the contrary, the inner tension in this extended anacrusis comes not from the out-of-phase condition between meter and grouping, for of course these same out-of-phase conditions also obtain with respect to the opening measures of K. 283. Rather, it comes from the instability of the grouping structure itself. 2.3.2 Grouping Overlaps Another problematic grouping structure occurs when a single musical event (which may itself be a group) is held in common by two adjacent groups. Example 11, mm. 59-62 from Schubert's Piano Trio in B b, is taken from Cooper and Meyer (1960, 70).7 The second half of m. 61 is common to two groups, and thus a grouping overlap is created. Example 11. Schubert, Piano Trio D. 898, mvt. I, mm. 59-62 (after Cooper and Meyer, ex. 87) Sm.61; 4 I ~ J-~ - I b ";, t, 3 Lerdahl and Jackendoff also discuss grouping overlaps in their more formalized theory, but they do so in an interesting fashion. On the face of it, an overlap such as that found in example 11 violates their Grouping Well-Formedness Rule #4: "If a group G1 contains part of a group G2, it must contain all of G2" (GTTM, 38). They are quick to add, however, that: 6Even though this downbeat is a likely location for the structural downbeat of the phrase, and hence of the transition from arsis to thesis within the passage, we are not certain of this fact until the B bs is sounded on the second beat of the measure, forming the leap that convincingly and dramatically breaks the motivic repetition. 7 I have included Cooper-and-Meyer-style analytical symbols, marking the strong and weak members of the group as befits their analysis.

16 In Theory Only There are in fact cases in tonal music in which an experienced listener has intuitions that violate GWFR 4. Such grouping overlaps and elisions are inexpressible in the formal grammar given so far. However, since overlaps and elisions occur only under highly specific and limited conditions, it would be inappropriate simply to abandon GWFR 4 and permit unrestricted overlapping of groups. Instead overlaps and elisions receive special treatment within the formal grammar, involving transformational rules that alter structure. (GTTM, 38) Thus an overlap is a surface structure that is formed when an underlying structure, which does not contain an overlap, is changed via a transformational rule. This transformational process can be shown as in example 12. Example 12a shows two groups with separate and distinct boundary elements. In example 12b this passage has been transformed into an overlap, as the G' is now common to both groups. Finally, in example 12c this G' is ornamented, creating the overlap found in example 11 (see the transformational rules for groups, GTTM, 55-62). However, the use of transformational rules is not without problems. As Lerdahl has noted, "If the TRs [transformational rules] could apply arbitrarily, any absurd analysis would be admissible... A set of preference rules (PRs) is needed to model the requisite intuitions and constrain the application of the TRs" (Lerdahl 1991, 279). Accordingly, transformational rules play a fairly limited role in Lerdahl and Jackendoff's theory (GTTM, 11). While it is perhaps impossible to develop a generative theory that, while based on a manageable number of base structures, does not involve some transformational rules, why is it that overlaps cannot be included into the base structure of the grouping component of their theory? Constraints could then be added to delimit the contexts in which overlaps could occur (that is, only in cases of grouping juncture). The answer to this question is that the inclusion of overlaps would then create subsequent problems with the binary logic of the GTTM timespan and prolongational trees. That is, if the hierarchic structure is to be represented by binary trees (or more abstractly, by binary suband superordinate relations), then base structures that contain overlaps will generate unsatisfactory trees. What is intriguing about the original form of the GTTM theory is these constraints do not seem to be motivated by any empirical rationale, since overlaps are quite

London, Limits of Analytical Description 17 common.' Instead, the constraints on overlaps seem to stem from the SRH itself, as shall be shown in section 3.0 below. Examples 12a, b, and c. Transformational process producing overlap in example 11. (a) (b) 7 1". - - % f 2.4 Retrospective Relocation of a Group Boundary In the previous two examples the problem one confronts is one of how to relate the various components of the group to each other-while the boundaries of each lower-level group were clear, their higher-level relationships were unclear. In other instances it is the location of the group boundary itself that is in question (ex. 13). On the second beat of m. 4 there is a clear sense of arrival on tonic harmony following the inflection to the dominant. Had the following phrase been a varied repetition of the first, creating a parallel period, our sense of a group boundary between mm. 4 and 5 would have remained secure (ex. 14). Yet in Haydn's version the cadential arrival at m. 4 is undermined by the motivic repetition in mm. 5-6; when the melodic and harmonic See, for example, Cooper and Meyer (1960) for many examples of overlapping and interlocking groups.

18 In Theory Only Example 13. Haydn, Symphony No. 92, mvt. IV, mm. 1-8 A r.1 e. I I I i m.n. 4 1i 0000 q), - \./ * arrival in m. 4 is heard within the context of a sequential passage, its sense of closure and phrase articulation is retrospectively weakened. How definite are the group and phrase boundaries between mm. 4-5 in the ear and mind of the listener once she has heard the entire passage? The structure of one group (mm. 5-6) directs one to retrospectively recast one's understanding of the grouping structure of that which preceded it. Indeed, are not most of our judgments of group boundaries subject to retrospective confirmation or revision? Even once we make our final-state analysis, these articulations are not yes-or-no propositions, but rather a matter of degree.9 Example 14. Author's recomposition of example 13 lr r:121 i ti O I0P I Tr i 1 l ri I I rJ 1 m. 4 pin 9If not all group boundaries on the same level mark closure to the same degree, it then behooves one to examine one's analytical symbology if it tends to represent all group boundaries in the same fashion (implying more-or-less equal degrees of articulation and closure). On the importance and influence of analytical symbology see Narmour (1984).

London, Limits of Analytical Description 19 3.0 Reduction Hypotheses 3.1 The Strong Reduction Hypothesis of GTTM The bedrock of the GTTM model is the Strong Reduction Hypothesis. Lerdahl and Jackendoff approach the SRH by first positing a general reduction hypothesis (GRH): "The listener attempts to organize all the pitch-events of a piece into a single coherent structure, such that they are heard in a hierarchy of relative importance" (GTTM, 106). Note that this general hypothesis does not say that the listeners will organize the pitch-events, but only that they will attempt to organize those events. With this telling hedge Lerdahl and Jackendoff recognize that the GRH involves an assumption that may not always be warranted, for there are times when the listener's attempts at organization are doomed to failure-and that, perhaps, is often the musical and aesthetic point. Lerdahl and Jackendoff themselves are quick to acknowledge the reductive price they have had to pay in the pursuit of formal rigor. They note that: It might be thought arbitrary to have to attach a subordinate event... either to the preceding... or... ensuing event. One might argue instead that subordinate events should appear simply in between structurally more important events at the next smaller reductional level, and therefore that a "network" notation... is more appropriate. In response, we observe that the sheer geometry of networks creates insuperable notational difficulties once even a moderate number of events are considered together; network notation is simply impracticable for the analysis of real pieces.... A more substantive reason for maintaining both left and right branching is that it enforces the generally pervasive intuition that subordinate events are elaborations of particular dominating events, not just elaborations within a certain context. (GTTM, 114-16) Strict and strong hierarchic reduction, then, is the means by which structural coherence is obtained, as well as the means by which listeners can manage the cognitive complexities of musical structures. Their "pervasive intuition" is that one relates a given event to those that precede and follow it, either as a continuation (which marks a right-branching relationship) or an anticipation (which marks a left-branching relationship), but not both. Lerdahl and Jackendoff then refine the General Reduction Hypothesis by adding the following conditions: a. Pitch-events are heard in a strict hierarchy. b. Structurally less important events are not heard simply as insertions, but in a specified relationship to surrounding more important events. (GTTM, 114-16)

20 In Theory Only The SRH gives their theory the necessary rigor for the formalized apparatus of their well-formedness rules and preference rules. The crux of the matter is the characterization of the hierarchy as "strict" in part (a) by means of the specified relationships in part (b). Thus the well-formedness rules for meter, grouping, and their tree structures all must obey requirements of nonoverlapping, adjacency, and recursion. When Lerdahl and Jackendoff later give the formalization of timespan reduction, these strictly specified hierarchic relationships are expressed in terms of subordination to the head of a given timespan. This subordination is both transitive and recursive: "That is, if pitch-event x is subordinate to pitch-event y, and y is subordinate to [pitch-event] z, then x is subordinate to z" (GTTM, 152). In the case of grouping structure Lerdahl and Jackendoff specify that transitivity would require that "a time-span Ti immediately contains another time-span T. if T, contains T, and if there is no time-span Tk such that Ti contains Tk and Tk contains Tj. Informally, Ti immediately contains T. when Tj is exactly one level smaller than Ti" (GTTM, 152). When the Grouping Well-Formedness Rules are framed, the requirements for transitivity and recursion get cashed out as in example 15. As grouping and metric structure are the inputs to formation of timespan and prolongational reductive trees, one can understand why such grouping strictures are necessary. As Lerdahl and Jackendoff note: "The tree notation is possible only if subordination is transitive. If the Strong Reduction Hypothesis turns out to be false, the notation for reduction will have to be modified accordingly. On the other hand, we find it difficult to envision a theory lacking the Strong Reduction Hypothesis that would be both sufficiently rich and sufficiently constrained to constitute a plausible account of musical cognition" (GTTM, 152; italics added). The principal problem with the SRH is that its high level of constraint too sharply limits the range of well-formed structures the grammar is supposed to cover. It simply is not flexible enough to account for all of the grouping structures one encounters in tonal music, though it is powerful enough (under such constraints) to recursively generate grouping, timespan, and prolongational structures at all structural levels. The SRH is strongly tied to GTTMs stated goal of giving an account of musical cognition, a linkage that will be considered below. But more immediately, an alternative reduction hypothesis will be explored.

London, Limits of Analytical Description Example 15. Results of Lerdahl and Jackendoffs Grouping Well-Formedness Rules 21 Grouping Well-Formedness Rules 1. Any contiguous sequence of pitch-events, drumbeats, or the like can constitute a group, and only contiguous sequences can constitute a group. 2. A piece constitutes a group. 3. A group may contain smaller groups. 5. If a group G1 contains a smaller group G2, then G1 must be exhaustively partitioned into smaller groups. 4. If a group G1 contains part of a group G2 it must contain all of G2. Result/Effect of Rule Prohibits: ( )---- Rules 2-3 establish the following top-down analytic procedure: Prohibits: IF Prohibits: - But allows for: / A V 3.2 A Weak Reduction Hypothesis Now we often attempt to organize the events of a piece into a single coherent whole (or at least at some times and for some pieces). So one would like music structures to follow the well-formedness rules outlined in GTTM. (In this sense, Lerdahl and Jackendoffs pervasive intuitions are correct.) But we cannot always do so, and to encompass the messy multiplicity that one finds in the kinds of grouping structures given above, I propose a loosening of the strictures of the SRH. My motivation for such loosening is based on the following observations: 1. Not all events admit structural descriptions of equal robustness, both on the same level as well as from level to level. 2. Some events admit more than one valid structural description (and let us assume that this set of structural descriptions is neither ambiguous nor contradictory).

22 In Theory Only 3. Given (1) and (2), an analysis should not overstate the case for a particular structural description merely because one's chosen analytical method is biased toward a certain set of relationships. 4. We are able to cope with uncertainty. Not all musical objects one encounters can be given a precise structural description. We simply file certain structures as indeterminate or undecidable, and then move on. In most cases our knowledge of the structure of a piece of music, even at some final state of understanding, is neither comprehensive nor complete. The problem with the SRH is that it tends to generate music analyses that are comprehensive and complete. Let us reconsider the two grouping analyses given at the outset of this paper (exx. 1-2, pp. 6-7). Most of the problem patches in the opening measures of the Hammerklavier can be accounted for in terms of unresolvable conflicts amongst and between GTTM's well-formedness and preference rules. For example, at m. 6 we hear a conflict between symmetry (GPR 5) and motivic continuation (the contrary of GPR 3, which deals with changes in melodic direction, dynamics, articulation, and duration), hence the uncertainties on level (b) of the grouping analysis (ex. 2). In m. 12, there is a conflict between motivic parallelism (GPR 6) that leads us to construe the material on the downbeat as cadential (in a manner analogous to mm. 7-8), versus the contraries of GPRs 2 and 3, whereby the increased melodic motion and registral/melodic continuation (not to mention the Gb 3 on the second beat) collectively subvert the sense of closure and articulation. As a result, what we might have initially thought to be a point of melodic arrival on the downbeat of m. 12 turns out to be only a fleeting instance of melodic similarity.10 The GWFRs and GPRs allow us to precisely describe those aspects of the musical surface that are giving us trouble-what exactly is giving rise to doubt in our mind's ear as to the precise location and/or nature of a group boundary or prolongational branching. The bulk of the GTTM apparatus provides the analyst with an elegant (in its separation of WFRs from PRs) and lucid means of untangling lONot all aspects of the grouping structure in these measures can be accounted for as unresolved clashes between GTTM's current set of GWFRs and GPRs, most notably the parenthesis in mm. 9-12 (marked by the repeat that echoes mm. 5-8 in a higher register). However, it would be possible to account for these sorts of grouping structures through a parsimonious expansion of the GWFRs and GPRs, along with some protocols for the interaction between the GWFRs and the prolongational and timespan reduction rules. Thus a parenthetical grouping structure, for example, could be the product of a particular prolongational structure (e.g., a span of structural stasis coupled with motivic/thematic repetition) and its interaction with GWFRs that impinge on the location of group boundaries and on subgroup organization.

London, Limits of Analytical Description 23 parametric conflicts and the problematic grouping structures such conflicts create.'1 With the SRH standing behind the WFRs and PRs, however, the analyst is forced to banish these conflicts from the final rendering of the musical structure. If one relaxes the strictures of the SRH, however, we will have a place for such conflicts and the sense of doubt they engender in one's analyses. One cannot, however, simply dismiss the SRH from the GTTM model. For the SRH is what maintains the integrity of Lerdahi and Jackendoff's generative hierarchy; it keeps each level separate and distinct and makes the level-to-level relationships clear via transitive subordination. Therefore, in its stead I propose a Weak Reduction Hypothesis (WRH). Like Lerdahl and Jackendoff, I first establish a general reduction hypothesis: Alternative GRH: Listeners will structure the series of musical events that constitute a piece (including but not limited to pitch events) as hierarchically as possible, given the circumstances of a particular listening context. Like Lerdalil and Jackendoffs GRH, this one recognizes that hierarchic structure is not simply an aspect of musical organization; it is also part of the way in which we tend to perceive and structure our musical experiences. We actively pursue hierarchic structures, "listening out" for motives, for phrases, for formal sections, and for large-scale shape and closure. We thus come to the listening experience with a highly developed set of hierarchic expectations. My GRH differs from the original in two important respects. First, it does not specify that reductions must comprise a "single coherent structure" but more 11Indeed, Lerdahl and Jackendoff acknowledge that the GWFRs and GPRs may be used in this fashion: "First, intuitions about grouping are of variable strength, depending on the degree to which individual grouping principles apply. Second, different grouping principles can either reinforce each other (resulting in stronger judgments) or conflict (resulting in weak or ambiguous judgments)" (GTTM, 42). This is all to the good. But they go on to say "Third, one principle may override another when the intuitions they would individually produce are in conflict.... [O]ur hypothesis is that one hears a musical surface in terms of that analysis (or those analyses) that represent the highest degree of overall preference when all preference rules are taken into account. We will call such an analysis the 'most highly preferred' or 'most stable'" (GTTM, 42). Thus, here and elsewhere (e.g., pp. 22-25, 64-66, and especially 266-68) a conceptual tension is evident in Lerdahl and Jackendoff's treatment of ambiguous or indeterminate grouping structures. On the one hand, their analyses often show how particular WFRs and PRs are in conflict. On the other hand, when push comes to shove Lerdahl and Jackendoff seem to feel obligated to posit one particular grouping structure as the truest structural description of the passage in question, precipitating out much, if not all, of the ambiguity implicit in their analyses.

24 In Theory Only modestly claims that listeners will make the best hierarchic sense possible within a given listening context. Second, my GRH is mute with respect to the relative importance of various events. The reason for this lacuna is clarified by the weak reduction hypothesis proper: a. Musical events are heard as comprising hierarchically nested structures. b. Within a given musical style the approximate size and scope of musical structures on each hierarchic level is relatively constant and predictable. c. On different levels different parameters are relatively more or less salient and hence structurally determinate. The WRH is viable because, as item (c) makes explicit, the musical hierarchy has important nonrecursive aspects. As Meyer has noted: The way in which a particular parameter acts in articulating structure may be different on different hierarchic levels. For example, on lower levels dynamics and orchestration tend to contribute to the articulation of rhythmic patterns, but on higher levels they generally serve in the structuring of large-scale formal relationships. Similarly durational relationships are crucial in the shaping of low-level events such as motives and phrases, while tonality and texture are especially important for the organization of high-level structures. Moreover, the role played by a particular parameter depends not only upon hierarchic level, but also upon style. Harmonic relationships play a central role in the structuring of tonal music, but none in the ordering of most serial compositions. Timbre plays a very significant role in defining relationships in Webern's music, but only a minor role in the music of Bach. (1973, 89) It is due to the nonrecursive nature of the musical hierarchy that the revised GRH is silent with respect to the relative importance of hierarchic events. This is not to deny that the distinction between structural versus ornamental tones does not exist on a given level or across adjacent levels. But if different musical parameters play particular roles in organizing various levels of musical structure (and indeed, if the salience of a particular parameter is characteristic of a given structural level), then a number of corollaries follow. First, it is possible for musical hierarchies to emerge that are not dependent on transitively subordinate relationships within any single parametric domain. Second, in many cases the question of relative importance becomes moot-for what does it mean to speak of the "relative importance" of timbral contrast on one level versus cadential strength on another level? Nonrecursion also means that where there are unresolvable conflicts between structural descriptions on a given level, listeners will turn to other structures on adjacent levels, especially

London, Limits of Analytical Description 25 those structures that emphasize other musical parameters, to maintain a global sense of hierarchic integrity. Were this not the case, that is to say if the musical hierarchy were wholly or even primarily recursive, then the WRH would not be powerful enough to serve as an organizing framework for one's musical experience and understanding. Nonrecursion allows for robust structures on one level that are capable of coexisting with structures that are ambiguous or intermittent on other levels. 4.0 Conclusion: Why pursue the RH so Strongly? As stated at the outset of this paper, Lerdahl and Jackendoff's SRH is a particularly clear articulation of a more general (though usually tacit) analytical assumption: for every musical event there is a single, best structural description, and the job of the music analyst is to produce that description. What is instructive about the SRH in the context of GTTM is that the motivations for holding to such a strong reductive hypothesis can be seen and understood with considerable clarity. Lerdahl and Jackendoff are quite overt about construing music theory as a subdiscipline of psychology. They begin their book with this sentence: "We take the goal of a theory of music to be a formal description of the musical intuitions of a listener who is experienced in a musical idiom" (GTTM, 1). But the questions and goals of cognitive science and the questions and goals of music theory (and in particular music analysis) are not the same. The different aims and claims of each discipline can be better understood if we consider a number of different questions or analytical contexts for Beethoven's Hammerkiavier. 1. What kind of analysis is necessary in order to say that this piece is tonal-that is, what is involved in the recognition and comprehension of the basic elements of the tonal syntax of Western art music in general and Viennese classical (and perhaps early romantic) music in particular? 2. Given (1), what kind of analysis is required to identify this particular piece as (and perhaps even distinguish among particular performances of) the Hammerklavier? 3. Given (2), what additional analytical claims can be made about the structural relationships among and between its various elements, as well as the ways in which these elements and their relationships serve as vehicles for aesthetic expression? It seems clear that cognitive psychology is primarily concerned with questions (1) and (2) while music theory and analysis is primarily concerned with (3), though of course there is considerable overlap between the disciplines in the area of question (2). In the first

26 26 In Theory Only analytical context the structural description of the piece must be one that places it into a general class of musical phenomena. Thus one must identify a number of basic elements (e.g., particular scales and their temperaments, characteristic sonorities, characteristic rhythms and meters, etc.).- In order to answer the second question one needs a structural description that individuates specific musical works (and, similarly, that allows one to recognize two separate performances of the same work as being just that-two instances of the same piece).This sort of structural description must identify the various structural elements of the piece more closely than in question (1), as well as attend to their particular combination. The third question/ analytical context assumes that the second has been answered (or at least answered well enough to individuate a given piece). Thus, given that certain scale steps, durational patterns, harmonic complexes, and so forth, are present in a certain arrangement, how then do these elements cohere (or resist coherence) into a work of art.? In large part, the kinds of structural descriptions that GTTM is able to generate are those that serve primarily an identifying function; if two listeners both are able to construct the same structural description, with respect to GTTM, they are then able to say that they have both heard and in some sense now "know" the same piece.- In this context having a SRH that disambiguates and simplifies is good, in that the kinds of structural descriptions it engenders readily and efficiently distinguish the opening measures of the Hammerklcwier, for example, from other pieces of tonal music in general and from Beethoven's other piano sonatas in particular. While this is a useful kind of analysis, music theorists usually want to probe the structure of pieces more deeply, often to try to understand their more idiosyncratic aspects. Psychology is a science (or at least endeavors to be a science); music theory is not a science, nor should it endeavor to become one. Narmour, who would like music theory to become a subbranch of cognitive science as much as Lerdahl and Jackendoff, observes that "It may be beneficial.., to emphasize how different the goals are between the scientific approach to natural facts... and the humanistic approach to artifacts. Science seeks to discover commonality out of dissimilar instances, to reduce the disparate to the uniform, to account for the world in as few universal laws as possible" (1990, 58). This is all well and good, but he goes on to say that- "Music theory has similar goals in attempting on all levels to discover what stylistically ties pieces, genres, cultures, etc. together and in trying to formulate parsimonious rules that will assimilate the dissimilar works of art into a unified field of research. In these things, a music theorist is no less

London, Limits of Analytical Description 27 a scientist than any physicist, chemist, or biologist" (1990, 58). Here I believe Narmour conflates the search for parsimonious rules-that is, the construction of any theory of music, art, literature, or cultural practice that wishes to make general claims-with the particular strictures of rules that operate under the scientific method. For what "stylistically ties pieces together" is not the broad application of natural laws, but rather the confluences of cultural contingencies. Musical works are contingent artifacts, each the product of a set of unique historical and compositional circumstances. They are not replicable in the manner of natural phenomena. A scientific theory of music would strive to parse a musical structure the same way each time it is encountered. We would not accept as valid a theory of molecular structure that tended to come up with different stoichiometric equations every time a chemical reaction was observed in the laboratory. Similarly, we may perhaps be uncomfortable with a method of music analysis (and its overarching theory of musical structure) that would produce one structural description for a piece on one occasion and then another structural description for the same piece on another occasion. Yet that is what listening to music is like. Clearly, this is not an anything-goes situation; we must, at the very least, come to the same sort of question (2) structural description each time we listen to a piece or else we are in serious trouble. Unlike science, however, music theory need not seek to find the structural description for any given piece of music. It is precisely the domain of music theory to adjudicate between the sorts of structural descriptions that might be valid for a piece of music versus those that would not or could not be valid. Music theorists are neither scientists nor philosophers, though of course their work is much enriched by a consideration of these other domains. Rather than searching for scientific certainty, we should be prepared to confront (with careful consistency and rigor) the uncertainty and contingency that is the hallmark of humanistic endeavor.

28 In Theory Only References Cooper, Grosvenor and Leonard B. Meyer. 1960. The Rhythmic Structure of Music. Chicago: Univ. of Chicago Press. Guck, Marion A. 1993. Taking Notice: A Response to Kendall Walton. The Journal ofMusicology 11/1: 45-51. Jackendoff, Ray. 1987. Consciousness and the Computational Mind. Cambridge, Mass.: MIT Press.. 1991. Musical Parsing and Musical Affect. Music Perception 9/2: 199-230.. 1992. Musical Processing and Musical Affect. In Cognitive Bases of Musical Communication, ed. M. R. Jones and S. Holleran, 51-68. Washington, D.C.: American Psychological Association. Lerdahl, Fred. 1991. Underlying Musical Schemata. In Representing Musical Structure, ed. P. Howell, R. West, and I. Cross, 273-90. London: Academic Press. Lerdahl, Fred and Ray Jackendoff. 1983. A Generative Theory of Tonal Music. Cambridge, Mass.: MIT Press. Lewin, David. 1986. Music Theory, Phenomenology, and Modes of Perception. Music Perception 3/4: 327-92. Meyer, Leonard B. 1973. Explaining Music. Chicago: Univ. of Chicago Press. Narmour, Eugene. 1984. Toward an Analytical Symbology: The Melodic, Harmonic, and Durational Functions of Implication and Realization. In Musical Grammars and Computer Analysis, ed. M. Baroni and L. Callegari, 83-114. Florence: Olschki.. 1990. The Analysis and Cognition of Basic Melodic Structures. Chicago: Univ. of Chicago Press.

The Defining Moment: The Thema as Relational Nexus in Webern's Op. 27 Tiina Koivisto While most analysts currently agree that the actual set of variations in Anton Webern's Variations for Piano, Op. 27 is to be found in the third movement, Webern's own comments suggest that the initial eleven measures of that movement serve as a thema not only for the subsequent five variations, but also for the composition as a whole. In a letter to Eduard Steuermann, in which he included a copy of the piece, Webern wrote: [The Variations] are divided into three independent movements. I do not display the thema explicitly (at the top, like before). It is almost my wish that it could stay as such unrecognized. (But if people ask me about it, I would not hide it from them.) Nevertheless it is better that it stay back there. (It is-to you I tell it right away-the first eleven measures of the third movement.)' (Webern 1983, 32-33; author's translation) 1"Ich schicke Dir mit gleicher Post meine 'Variationen' u. bin sehr glfcklich, daL Dich meine Widmung an Dich freut. Wie ich Dir, glaube ich, schon angedeutet habe, sind sie in fir sich abgeschlossene Sitze [drei] aufgeteilt. Ich stelle auch das 'Thema' gar nicht ausdracklich hinaus [etwa in friherem Sinne an die Spitze]. Fast ist es mein Wunsch, es m6ge als solches unerkannt bleiben. [Aber wer mich danach fragt, dem werde ich es nicht verheimlichen]. Doch m6ge es lieber gleichsam dahinter stehen. [Es sind-Dir verrate ich es natirlich gleich-die ersten 11 Takte des 3. Satzes]."

30 In Theory Only This article seeks to explicate Webern's remarks and suggests that the thema emerges as the kernel of the relationships employed in the entire piece. By examining the relationships inherent in the thema, it is possible to develop meaningful ways to approach and describe the underlying structural features not only in the set of variations, but also in the preceding two movements.2 Furthermore, this approach reveals the manner in which the thema lies at the midpoint of a series of nested symmetrical structures, against which one hears the temporal accretion of the music. The Thema Example 1 shows the thema of Op. 27 with Webern's remarks about the performance of the piece, as they appear in the Stadlen edition (Webern 1979).3 The thema contains three phrases (A, B, and A), each of which comprises a single member of the work's row class.4 There are a multitude of ways to hear this musical surface. The specific articulation of the thema phrases invites us to hear several prominent groupings. For example, one may hear strands of long and short notes, in which the long notes can be described as a melodic part and the short notes as an accompanying part.5 Furthermore, one may hear strands of slurred and nonslurred dyads.6 Both these groupings partition the phrases into chromatic hexachords. In addition, the single tritone of the thema's intervallic vocabulary, marking the midpoints of the phrases, parses the phrases yet another way into pairs of chromatic hexachords: in this instance into the actual hexachords of the rows. 2Anton Webem's Op. 27 has been discussed from various points of view; see Babbitt ([1960] 1962, 1987), Bailey (1991), Hasty (1981), Leibowitz ([1949] 1970), Lewin (1962, 1987, 1993), Mead (1992, 1993), Nolan (1989), Schnebel (1984), Stadlen (1958), Travis (1966), Wason (1987), and Westergaard (1963, [1962] 1972). 3These remarks were made by Webem to Stadlen as instructions for the world premiere of the work in 1937. Stadlen's annotations include both verbal recollections and comments written on the score. 4In discussion of structural features, the thema phrases will be referred to as A and B if there is no need to differentiate between the outer phrases. SRobert Morris was the first to point this out in a seminar at Yale University in 1975 (Mead 1992, 123). Also Wason (1987,78) cites Morris as the source of his way of construing the thema. 6An alternative interpretation of the thema that also uses the distinction between slurred and nonslurred dyads is developed by Lewin (1987, 39).

Koivisto, The Defining Moment Example 1. Weber, Variationen, mvt. III, mm. 1-12, thema Reprinted by permission of Universal Edition. Thema phrase A (P),dit zablricben Tempmeracbael uirn jeweils den BgPwfine rets ren Saetr an " Rnhig mflieend Jzso (8 V A _( I i2 -3 "- 1 f g sicb ibeiubnd (^ - -^ - ^'* =- f - 31 A' (RP) Thus, chromatic hexachords are articulated in the music as segmental hexachords, as motivic (that is, as melodic and accompanying) hexachords, and as slurred/nonslurred hexachords, as shown in example 2. In order to explore relationships inherent in the thema more deeply, one needs first to examine some of the most significant hexachordal properties of the work's row class, which yields the types of hexachordal collections so prominently displayed in the thema. The row class of the work can be generated from P: 3et210 647598. The chromatic collection-class [012345] is an all-combinatorial hexachord possessing several compositionally significant properties. First, as the hexachord's interval vector

32 In Theory Only <543210> indicates, each interval class (ic) is represented a unique number of times. This property allows the row class the highest degree of differentiation among its hexachordal areas.7 Second, the exclusion of ic 6 from the hexachords places special significance on tritones as intervals that can only occur between hexachords. Example 2. Segmental, motivic, and slurred/non-slurred hexachordal areas of thema phrases A and B P: 3et210 647598 phrases A B Hexachordal areas (segmental) segmental motivic non/slurred P melody 3 2 6475 {te0123} {234567} {6789te) accomp. et 10 98 {456789} {89te01} {012345} (motivic) 16P melody 3 4 02el {345678} {e01234} {789te0) accomp. 78 56 9t {9te012} {56789t) {123456) As example 2 illustrates, the three criteria for parsing each of the thema phrases yield all possible members of the [012345] collection class exactly once. The presence of all hexachordal areas in the thema exhausts the possibilities for the various degrees of pitch-class (pc) intersection available with this type of collection class. As shall be seen, the discrete degrees of pc intersection introduced in the thema become an important factor in the formal shaping of the music. The actual pcs of the pc intersection have significant roles as well. Among the hexachordal areas of the thema one may define more close and more remote relationships having as their criteria properties arising from the orderings or partial orderings of the hexachords. As will be discussed below, one may define a close relationship between the segmental and motivic hexachordal areas arising from ordering properties. Based on this, comparison between the segmental and motivic hexachordal areas alone throughout the thema reveals the presence of all the possible degrees of pc intersection (ex. 3). Degree of pc intersection is indicated with two digits separated by a slash, the first showing the number of pcs shared between hexachords at like 7This property is shared by only one other hexachordal collection-class, [024579], the diatonic hexachord.

Koivisto, The Defining Moment 33 order positions, and the second showing the number of pcs unique to each hexachord. As comparison between, for example, the segmental hexachords of P and I6P shows, the hexachords at like order positions, that is, the first hexachords of both rows, have pc intersection 1/5 sharing pc 3; the second hexachords of these rows share pc 9. Example 3. Pc intersection between segmental and motivic hexachordal areas of the thema A B A' P6/0 P l6P R segmental {te0123}{456789} {345678}{9te012} {456789}{te0123} hexachords 1/5 4/2 2/4 motivic {234567}{89te01} {e01234}{56789t} {89te01}{234567} hexachords One may now define further relationships within the thema by considering the hexachordal areas in terms of partially ordered hexachords based on the semitone partitioning of the thema. Hence, the hexachords are interpreted as ordered sets of unordered ic 1 dyads. Example 4 shows these ordered sets of dyads of the segmental and motivic hexachords of phrases A and B. The interpretation of the hexachordal areas as partially ordered has significant compositional consequences on various structural levels. Significantly, there will always be two rows that yield the same ordered sets of dyads. Thus, each ordered set of dyads shown in example 4 can be generated from two discrete rows. Example 5 illustrates this by showing the four rows connected with the set of dyads of the segmental and motivic hexachordal areas of phrases A and B. Example 6 then groups these rows into row families. (Since the focus here is on hexachord content, each row in the example also represents its retrograde.) Hence, the relationships articulated in the thema generate row families that closely reflect the thema, most obviously by producing the same ordered sets of dyads as in the thema.

34 In Theory Only Example 4. Ordered sets of dyads of segmental and motivic hexachordal areas of thema phrases A and B A/seg {(23}{te}{01} {67){4589})) Almot {{23}{67}{45} {te}{01}{89}) mel. ac. B/seg {{34){78}{56} {eO}{12}9t}} B/mot {{34}{eO){12} (78}{56}{9t}) mel. ace. Example 5. Rows connected with the ordered sets of dyads of thema phrases A and B A/seg lnn P: 3et210 RT6P:23el t0 LJ L-4J A/mot H RleP: 326 475 ISP: 267345 L-JU r-1f n 647598 {23}{teo}{01}{67}{45}{89})) 678459 LiKY Pt 9 1 >(23}{67}{45}{te}{01H89}} eits089' r-fiI Fl 02e1 9t {{34}{78H{56}He012}{9t}} Oet219 0 9e t 2 19 ){{34}{eOX}{12}{78}{56}{9t} 7586t9 LU U B/seg FTh1Fl 16P: 378456 RIoP: 437586 U L.- -= B/mot R: 7 401 -RT~P: 3402e1 TP: 40e321 LW U The relationships articulated in the thema have further consequences. One may group the rows of the work's row class into six equivalent four-row families (eight-row families if one distinguishes the retrogrades) that represent the relationships between the rows yielding the ordered sets of segmental and motivic dyads of phrases A and B.

Koivisto, The Defining Moment 35 Example 6. Two main four-row families 15P T6P 10P TIP leP ( @ T7P ord. ord. ord. ord. A/mot A/seg B/seg B/mot ord. ord. ord. ord. A/mot A/seg B/seg B/mot Among these six families, the four-row families containing P and I6P (ex. 6) will be referred to as the two main four-row families. One may consider partial orderings of the hexachords yet another way, as unordered sets of unordered ic 1 dyads. This approach yields two additional rows (and their retrogrades) that produce sets of dyads connected with the thema phrases, in this instance the unordered sets of the dyads within the hexachordal areas. Example 7 illustrates this by showing the rows yielding the unordered sets of dyads of the segmental hexachords of thema phrase A. The six four-row families, established above among the rows of the work's row class, comprise rows that yield the ordered and unordered sets of segmental, motivic, and slurred/nonslurred dyads of phrases A and B. Thus, in addition to the two main four-row families, which are closely connected to the thema, one has four additional four-row families that reflect to various degrees the thema through the properties arising from its segmental, motivic, and slurred/nonslurred hexachords. Most significantly, however, these additional four-row families produce the same relationships between them and among their rows as the two main four-row families. In this manner, the relationships arising from all the various hexachordal areas of the thema may be expressed to represent the relationships among the rows of the two main four-row families.

36 In Theory Only Example 7. Rows producing the unordered sets of dyads of the segmental hexachordal areas of thema phrase A nr f=F-i 1FIrF-1 RyIP et2031 7 6 5 9 8 P{{te}23}{01}{67}{89}45}} 11P t23e01 796845 LWL L W IUJU Relationships within the row class may be helpfully illustrated using relational diagrams. Example 8 shows a relational diagram comprising the rows of the row class. In the diagram, the two concentric rings of nodes each represent rows related by T. The aligned nodes represent inversionally related rows whose individual hexachords map onto themselves. Each axis represents one hexachordal area. Any row of the row class may be inserted in any one of the nodes, and the remainder of the nodes would be filled automatically.8 We shall now further explore the relationships among the rows of the work's row class by using this relational diagram and by invoking two ways to view inversional relations. In this composition, Webem deals with relationships between pairs of inversionally related rows both in terms of a preserved index number and in terms of a fixed degree of hexachordal pc intersection. The relational diagram will be used to show the two ways of expressing inversional relations, followed by an illustration of relationships within and between the four-row families, invoking these two ways of construing inversional relationships. In example 9 the various degrees of pc intersection are indicated with dotted lines. A similar chart may be made for any row in the diagram, either by inserting the row in the appropriate node, or by reorienting the net of dotted lines. Doing so groups the rows of the row class into twelve discrete sets of inversionally related row pairs that fulfill two conditions. First, the row pairs have the same degree of pc intersection between hexachords at like order positions, and second, they have the same properties when considered as ordered row pairs.9 8The diagram shows the rows related by T and I; operation R would reverse the number of pcs shared and held unique between hexachords of the rows. 9As an example, by inserting the main row pair P-I6P into the nodes marked by asterisks (P at the top) and by inserting rows into the rest of the nodes according to example 8, the eleven other members fulfilling the two conditions are found by reorienting the net of dotted lines until the initial position is reached. These eleven

Koivisto, The Defining Moment 37 Example 8. Relational diagram of rows in the row class TV (D-( cn I etc. same hexachc areas )rdal i represents the appropriate index number to map segmental hexachords onto themselves other row pairs of the same set are T1P-17P... TP-I5P. (The row pairs also represent their retrogrades, that is RP-RI6P, etc.)

38 In Theory Only Example 9. Relational diagram showing degree of pc intersection between hexachords at like order positions of rows Additionally, the same two conditions are fulfilled in the twelve discrete sets of transpositionally related row-pairs, thus allowing extension of pc intersection to transpositionally related rows.1'0 In example 10 the index number of inversion between the rows of the row class is indicated by showing the two patterns that arise from even and odd indices of inversion. A given index groups the rows into twelve sets of row pairs that share pc pairs determined by the index but do not maintain the same degree of pc intersection.11 10For example, the hexachords of P and T5P have pc intersection 1/5 in the same fashion as the hexachords of P and 16P. 1 ln example 10, by inserting rows into the nodes like in example 9, the net of solid lines indicates all twelve members of one set of inversionally related row pairs sharing the same index of inversion. In example 0la the rows are related by 16. The I6-related row pairs are P-16P, TIP-I5P... TeP-I7P (and their retrogrades). A reorientation of the net of solid lines yields a different set of row pairs sharing another

Koivisto, The Defining Moment 39 Example 10. Relational diagrams of rows related by (a) even and (b) odd index numbers 1Oa. Finally, example 11 shows row relationships within and between the two main four-row families. The most significant relations preserving the degree of pc intersection between inversionally related rows within and between the four-row families are labeled as follows. (1) The inversion relation specifying the degree of pc intersection between the hexachords at like order positions of P and 16P and additional such pairs is labeled an Is relation (s = segmental). Thus, the inversion relation between the rows of the thema can be expressed both as Is and as 16, depending on the analytical orientation; both relations are important in the work. In the diagram one may also see index of inversion.

40 In Theory Only Example 10 (cont.) 10b. how moves between Is and T, related rows are identical in terms of degree of pc intersection, a feature that is significant in the work, as well. (2) The inversion relation specifying pc intersection between hexachords of the rows yielding the segmental and motivic hexachordal areas of a thema phrase will be labeled as an Ism relation (sm = segmental/motivic). The first Ism relation (Ism') denotes pc intersection 4/2 and the second Ism relation (Ism2) denotes pc intersection 2/4 between hexachords at like order positions of these rows and additional such pairs. As was discussed above, this type of inversion relation specifies also the properties of ordered rows. Example 12 illustrates this by showing the ordered rows of the four-row family containing P. Most prominently, in the first Ism relation the melodic hexachord possesses the same order as one of the

Koivisto, The Defining Moment 41 segmental hexachords.12 The second Ism relation yields an invariant linking dyad. Example 11. Relational diagram of pc intersections among the two main four-row families ism 0, / o / / Ism1 M2orris pointed out this property in a seminar at Yale University in 1975 (Andrew Mead, personal communication).

42 In Theory Only Example 11 (cont.) s: 15P T6P 1P T1 P lep ( T7P two main four-row families sm 1 ordered sets of segmental and motivic dyads of A and B, and additional such pairs Ism2: pc intersection 2/4 between hexachords at like order positions of rows yielding the ordered sets of segmental and motivic dyads of A and B, and additional such pairs As will be shown, the relationships inherent in the thema form the basis of the pitch organization of all three movements. The first and second movements employ the two different ways to express the relation between the rows of the thema, the Is and 16 relations, and both movements employ one of the two Ism relations, as well as inversionally related row chains generated by Ts. In the third movement, the variation form is based on the use of the sets of ordered and unordered dyads connected with the discrete hexachordal areas of the thema, and the culminating variation employs both Ism relations. As is often noted, one of the most familiar aspects of this piece, as well as Webern's twelve-tone music in general, is his penchant for symmetrical structures, both in time and in register. Nevertheless, one hears in this music a strong sense of progression, suggested by the plasticity of its phrases. This article pays special attention to the manner in which this sense of progression arises from the underlying structures and, further, to the manner in which the symmetrical, frozen structural aspects of the pitch organization interact with our sense of musical accretion over time. In order to demonstrate the various aspects of the underlying pitch organization, the notion of compositional design, introduced by Morris (1987), will be employed. Furthermore, the form of each movement will be explained as arising from an interaction between the underlying

Koivisto, The Defining Moment 43 design and its surface realization. Each analysis begins by examining the underlying design, continues by illustrating some of the specific properties inherent in the design, and then moves to a demonstration through brief examples of the ways in which the design becomes vivid on the musical surface. The First Movement The first movement is in tripartite form, the sections of which consist, with a few exceptions, of mirror-symmetrical phrases. The underlying design may be characterized by focusing on four different aspects: (1) simultaneous retrograde-related row pairs, (2) consecutive row pairs forming row-pair couples, (3) hexachordal areas of the row pairs, and (4) pc intersection between these hexachordal areas. The retrograde-related row pairs form two inversionally related row-pair chains generated by T. (ex. 13). The beginning retrograderelated row pairs are RT8P/T8P (mm. 1-7 and 11-15) and RI2P/I2P (mm. 8-10 and 15-18). As the diagram indicates, T8P and 12P (and RT8P and RI2P) are related by Is. The middle section of the movement consists solely of interlocked members of the two T5 chains (Lewin 1987, 182), whereas the outer sections are extended with T6-related row pairs. Example 12. Two types of Ism relations in the four-row family containing P [-r--rrr P: 3et210 647598 I I| I | Ism, leP: 801,9te 574623 " ^ lsm 2 15P: 267345 e1t0'981 <1 T6P: 954876 0t1e324

44 In Theory Only Example 13. Underlying design of mvt. I: retrograde-related row pairs T7 5 Ts T6 L111T. T5. T5 An examination of the row-pair design in terms of consecutive row-pair couples reveals that the movement exhibits two discrete inversion relations inherent in the thema (ex. 14). The outer sections (mm. 1-18 and 37-54) exhibit the Is relation, the relation between the rows of the thema, whereas the middle section (mm. 19-36) exhibits the second Ism relation, which arises from the segmental and motivic hexachordal areas of the thema. The Is-related row-pair couples of the outer sections are T8P-IP (with their retrogrades) in the first section and P-16P and kP-TsP (with their retrogrades) in the last section; the Ism2-related row-pair couples of the middle section are I7P-T2P, IoP-T7P, IsP-P (with their retrogrades). In this manner the underlying design manifests a tripartite formal layout. However, within this tripartite shape, the last section combines the relationships of the two previous sections by generating the Is-related row-pair couples by T, in the same fashion as the middle section. Example 14. Underlying design of mvt. I: row-pair couples FA- T, B _ A7 T, I - I 1 I s Ism, 1s Yet another deep structural level emerges if one interprets the row pairs as hexachordal areas (ex. 15). In this interpretation two identical chains of hexachordal areas permeate the movement. In both chains, adjacent hexachordal areas have the same degree of pc intersection, that is, intersection 1/5, between hexachords at like order positions.

Koivisto, The Defining Moment 45 Example 15. Underlying design of mvt. I: pc intersection between hexachordal areas Climax End Epilogue 6/0 i f 116I ~. -^ * Ii I I1/5 115 I Hexachordal areas have several consequences for the formal shape of the movement. First, the identical hexachordal areas at the ends of the chains punctuate two structurally prominent moments: the first chain concludes with the row pair that forms the climax (mm. 32-34), and the second chain concludes with the penultimate row pair (mm. 47-51). As such it leaves the last row pair, whose hexachordal areas are a repetition of the third-to-last row pair, as a separate unit. Thus, the role of the last row pair as an epilogue (mm. 51-54), a role confirmed by Webern's remarks in the Stadlen edition, arises from a deeper level of the pitch organization. Aspects of pc intersection between hexachords also reveal how the row pairs manifest all the possible degrees of pc intersection with regard to the initial one. In the two chains, the maximum degree of remoteness from the initial hexachords, pc intersection 3/3, demarcates structurally significant moments. In the first chain the midpoint of the middle section occurs after these hexachords are stated. At this moment the surface interpretation of the rows changes as well: new configurations are introduced, and deviations from the symmetrical arrangements are made to help shape the climax and a transition to the last section. In the second chain, the last row pair is singled out, since it repeats pc intersection 3/3, thus emphasizing the special role of the epilogue. Lastly, the segmental hexachords of the thema have an important role as points of departure and arrival. Most importantly, the middle section is framed by the segmental hexachordal areas of phrase A, arising from the row pairs IP/RIP and P/RP. In addition, the

46 In Theory Only beginning of the last section is demarcated by entrances of the exact rows of the thema. In light of the above, the form of the movement can be viewed as arising from an interaction among the various aspects of the underlying design. First, the design articulates a tripartite formal layout, within which there is a special emphasis placed on the last section. Second, the aspects arising from the hexachordal areas manifest a continuum stretching throughout the movement, significantly, however, leaving the last row pair, which forms the epilogue, as a separate unit. Finally, the climax as well as the conclusion before the epilogue are demarcated as the last members of the two chains. The properties inherent in the underlying design have manifold compositional consequences offering ways to shape the specifics within the overall flow of the music determined by the hexachordal areas. For instance, example 16 shows the manner in which three discrete segmental ic 1 dyads of the work's row class are related by T,. Consequently, invariant dyads arising from the T5 chains permeate the movement. Through the partitioning schemes and their surface realizations, these invariant dyads become a prominent feature in the shaping of the mirror-symmetrical phrases acting, for example, as their framing and middle dyads. Furthermore, the dyads are typically connected with an additional pc to form collections belonging to the collection class [016], a characteristic sonority of this movement. These aspects may be illustrated with the movement's climactic phrase (mm. 32-34; ex. 16). The dyads E-E6 and A-G t, which arise from the T. chains and form prominent elements in the previous phrases-most often emphasized by the same registers-become the framing and middle ic 1 dyads of the climaxing phrase. As another illustration of ways in which the underlying design offers opportunities to shape the climax, the example shows how the tritone B-F, having its first entrance here as an axis tritone (m. 33), together with the pitch E3 evokes the movement's opening sonority. The opening retrograde-related row pair, TP/RT8P, and the row pair of the climaxing phrase, I5P/RI5P, exchange the first and middle trichords belonging to the set class [016], that is {892} and {45e} (at order numbers {teC} and {456} of I1P). Moreover, the pcs of the work's opening melodic dyad, E-CO, become the framing pcs of the climax's ascending line (mm. 32-33). (It is worth mentioning that the corresponding

Koivisto, The Defining Moment 47 C/) Cý 0 Q. L) 0 -V ar a fl- C)l M.1 U.) a) L R IL LEro C0ICT0 OD iJ ~CV)z c-oI gL) FaLal: IicL

48 In Theory Only gesture in the middle section's opening phrase is framed by the tritone B-F.) The goal of the ascent, D b6, features the first two sections' high registral extreme. The richness with which the design becomes vivid on the musical surface may further be illustrated using one particularly telling passage, the epilogue, in which several trajectories of relationships coincide. The special qualities of the epilogue arise from various structural levels, ranging from the underlying design, as demonstrated above, to its registral isolation in the context of the last section. This register connects it to the climax of the movement. Additionally, as the last member of a Ts chain, the epilogue merges invariances within the chains that have permeated the movement. One of the most intricate ways in which the epilogue brings back the events of the movement is the way in which it invokes the opening. The Stadlen edition reproduces Webem's instructions to emphasize the melodic pitches of the opening phrase (ex. 17). These nonadjacent pitches of the rows form a hexachordal collection that belongs to the same class as one arising between segments of simultaneous row pairs, as Robert Wason has noted (1987, 95-96, 99). Most significantly, however, the pcs of the opening melody are precisely those that form the last trichords of the epilogue (ex. 17). That these last trichords form a hexachord that is a member of this same hexachordal collection-class arises inevitably from Webem's use of trichordal partitioning and retrograde-related row pairs in this movement; that the trichords are formed of the exact same pcs as the opening melody reveals Webem's sensitivity to the possibilities inherent in his compositional design for its surface realization. The Second Movement The second movement has been discussed in great detail by several analysts (Babbitt [1960] 1962, Bailey 1991, Lewin 1962 and 1993, Mead 1993, Nolan 1989, Wason 1987, and Westergaard 1963). As Babbitt has noted ([1960] 1962, 117), the second movement is based on simultaneous row pairs generated by the same index number of inversion as employed for the rows of the thema, and thus the row pairs yield the same simultaneous dyads. As is well known, the bipartite second movement is based on a two-voice canon and the pitches are, for the most part, registrally fixed and arranged around the axis of symmetry, A4

Example 17. Connections between the opening and concluding measures of mvt. I I I I I I I 63 rP. n(D u N.- I. /- r\ A N I hl Ad i I rl k Wl I 44 A I v [in AM it a A4 0 0) (tf 0 (D~ (D P-A 0 (D ci 0 first R rel. row pair @5Wn0 0*0*** e 7 6(9*) * * * last R rel. row pair t. 2 1 _348*

50 In Theory Only Like the first movement, the formal layout of this movement can also be considered as an interaction among various aspects of its underlying design (ex. 18). The 16-related row pairs of the second movement are RP-RI6P and RTP-RIP in the first part, while the second part employs RT2P-RI4P and RT5P-RIP. As example 18a indicates, the 16-related row pairs form Ism'-related row-pair couples, which articulate the bipartite shape of the movement. The T5 chains (ex. 18b), which run in opposite directions, form a continuum over the entire composition connecting the last and first row pairs as adjacent members of the chains. In this movement, the degree of pc intersection proves important both between adjacent and simultaneous hexachordal areas, as it reveals the overall rhythm of change of the music. In hexachords of adjacent rows the degree of pc intersection decreases until the last row pair returns to the hexachordal areas of the very beginning (ex. 18c). Example 18. Underlying design of mvt. II: (a) row pair couples, (b) row pairs,. and (c) pc intersection between hexachordal areas 18a. '6 Sim' sm, 18b. 6 5 i. T TS 16 TZT5T T57 '5

Koivisto, The Defining Moment 51 Example 18 (cont.) 18c. 110 S115 j53 1571 1/5 (100 --> * * * * A/seg B/mot B/sl A/seg B/seg A/mot A/sl B/seg The degree of pc intersection between simultaneous rows indicates how rapidly the dyads change within the hexachords, as noted by Mead (1993, 184) and intimated by Wason (1987, 84-85). The opening and conclusion manifest the same rhythm of change, the most rapid, whereas the third row pair manifests a contrasting rhythm of change, the slowest. Finally, the second movement displays all the hexachordal areas of the thema. A-segmental and B-segmental hexachordal areas frame the movement acting as points of departure and arrival, while the middle row pairs yield the ordered sets of motivic and slurred/ nonslurred dyads of thema phrases A and B. Thus, in the row pairs of the second movement, the intersecting pcs are precisely those between the three pairs of discrete hexachordal areas of the thema phrases, and the pcs A and E b, arising from pc intersection 1/5, become the axes of symmetry. The rhythms of the phrases arise from the rhythm of change inherent in the hexachords of the rows. Within this overall rhythm of change the particularities are determined by the properties inherent in the Ts chains and in the Ismi row-pair couples. The following analytical vignettes, the first showing the opening phrase and the second comparing the opening and concluding phrases, will demonstrate the more general principle of the manner in which the

52 In Theory Only underlying design becomes, with its different rhythms of change, the source of the surface composition. The opening phrase (ex. 19) introduces the basic elements of the phrases. (Lewin [1993] calls the first three dyads of the opening phrase TUNE.) The parts of the phrase have their roles as opening, middle, and concluding elements, associated with the specific dyads, dynamics, articulation, and contour. The remainder of the phrases are based on these same elements, and on their varied and extended forms (Westergaard 1963). In these varied forms the pcs may remain the same while some other characteristic, such as articulation or dynamics, changes; or in some cases the other features remain the same while the pc content is altered. Example 19 further compares the opening and concluding phrases, which manifest the same rhythm of change. The Ts relation between the rows results in the segmental dyads forming the opening phrase occurring at every other order number of the concluding row pair, that is, at order numbers {2468t}. Thus, these dyads maintain the same order while the intervening dyads are employed to vary or extend the phrasal elements. Finally, a closer look at these passages shows that the very end of the last phrase, before the stinger (m. 22), brings back the events of the opening of the movement in a way that echoes by analogy the epilogue of the first movement. Example 20 indicates how the three last dyads of the concluding phrase (mm. 20-21) are RT, of the three initial dyads of the movement. Furthermore, this transposition level differentiates the three initial dyads as those dyads that have only one fixed register all through the movement, from the three final dyads as those that may change their register according to their functions within the phrases (Mead 1993, 186). Example 20. Opening and concluding dyads (mm. 1-2 and 20-21) of mvt. II mm.1-2 {8t} {99} {15} RT6 {7e} {33} {24} mm. 20-21

Koivisto, The Defining Moment Example 19. Elements of the opening and concluding phrases (mm. 1-3 and 18-22) of mvt. II opening phrase 53 opening elements middle element concluding element concluding phrase A_ upbeatf'stinger s s 91i P: 8 9 5 7 4 16P: t 9 1 e 2 II I I I 16 *8697 1 e32 *tO9e 56734 The Third Movement With the entrance of the thema at the beginning of the third movement, the relationships of the two previous movements receive a condensed interpretation, which elucidates the relationships of these movements in a crystallizing moment. This then serves as a basis for the subsequent five variations. The variation form arises from discrete

54 In Theory Only areas of the thema phrases, which comprise those rows that can produce the ordered or unordered dyads of the segmental and motivic hexachordal areas of the thema. Since the third movement is based for the most part on the partitioning scheme of the thema, the use of the discrete A and B areas makes possible an unfolding of the same dyads and tetrachords within one area. In this manner the surface composition of the variations may employ the various possibilities inherent in the dyadic sets of the segmental and motivic hexachordal areas of the thema to form an intricate motivic interplay based on the thema phrases. The first two variations are based on discrete A and B areas, whereas the fourth, the culminating variation, intermingles various relationships of the thema in the same fashion as the first and second movements; the two variations surrounding it, the third and the fifth, initiate and conclude these chains of relationships. The first and second variations' underlying designs are based on discrete A and B areas. In the first variation (ex. 21), the rows of the B area belong to the main four-row family, whereas the rows of the A area are based on P and the two inversionally related rows that extend its hexachordal areas (Mead 1992, 127-28). Example 21. Underlying design of mvt. III, var. 1: A and B areas mm. @ TqP RI6P RI7P RI1P Ri7P RIoP ordered/ ord. ord. ord. unord. unord. unord. ord. unord. sets B/mot B/seg A/seg Alseg A/seg A/seg B/seg of dyads The underlying design of the second variation (ex. 22) employs row pairs from the two main four-row families, in this instance those generated by R, from P and 16P and their Ism1-related row pairs. Hence, the relationships, which in the thema are interpreted with one row, are in the second variation interpreted with two consecutive rows that produce the sets of dyads heard in the corresponding phrase in the thema. Example 23 sketches some of the ways in which the possibilities inherent in the underlying design become vivid on the musical surface. The example shows the variation's three phrases by aligning their corresponding elements. It further shows the main partitioning scheme as it is applied to the first two rows. This scheme follows closely the partitioning of the thema, thus producing the same set of dyads (with one exception). A quick comparison with the thema and the first variation reveals how the motivic material introduced in

Koivisto, The Defining Moment 55 the thema and further developed in the first variation forms the basis of the surface interpretation of this partitioning. In addition to the obvious motivic connections based on the various articulations of the ic 1 dyads, one could mention, for example, the accented four-note forte gesture (mm. 25-26). This gesture echoes, through its contour and pc content, m. 10 of the thema and its various elaborations in the first variation (mm. 13-14 and 18-22). The Ism' relation between the rows offers opportunities to shape the phrases as well. For instance, the Ism'-related row pairs produce an invariant tetrachord at order numbers {2345} and these invariances are employed to connect the soft ritardando figures within the phrases. Example 22. Underlying design of mvt. III, var. 2: A and B areas mm. ( @ R15P R6P RioP RIoP RP ordered/ ord. ord. ord. ord. ord. unord. sets A/mot A/seg B/seg B/seg B/mot of dyads The fourth and culminating variation is permeated by row chains that are initiated in the third variation. These row chains employ rows of two four-row families, the first containing P and the second containing the rows yielding the unordered dyads of the motivic hexachordal areas of the middle thema phrase. These families are connected by the same index number of inversion. In the music, the rows of these families are grouped to form row pairs exhibiting the second Ism relation (P-RIsP, T3P-RIsP, T6P-RIeP, and T9,P-RI2 P) in the same fashion as the row-pair couples of the first movement's middle section. Example 24 shows the manner in which these row pairs form two interlocked row-pair chains (Lewin 1987,182). The seven last hexachords of these chains form the culmination of the third movement as well as the climax of the entire work. One of the ways in which the special qualities of this passage are achieved is the particular manner in which it invokes the thema by unfolding its various dyadic areas (ex. 25).

56 In Theory Only Example 23. Mvt. III, var. 2, mm. 23-33 r4.- - - -, r.4 U r4 IYC I ~~Sf Cjk w u 1 11 smRI5P: Ism1 TeP: 89 Otle 23 e ltO - LW-l 5 43 76 2 6 78 45 9 The beginning of the culminating passage is demarcated first by the ordered accompanying tetrachord of thema phrase A and second by the ordered melodic dyads of the opening phrase of the thema. These ordered melodic dyads arise from the sole row of the row class that yields the ordered melodic hexachord of P as an ordered segmental hexachord (Mead 1992, 130-31). The culminating passage then continues with dyads that combine accompanying dyads of A and B.

Koivisto, The Defining Moment 57 With these dyads the climax reaches the extreme high register of the entire work (A6, mm. 53-54), a structurally significant pc all through the piece. The passage concludes with the dyadic sets of the B area, first with its melodic tetrachord and second with its accompanying and melodic hexachords. Example 24. Underlying design of mvt. III, var. 4: interlocked row-pair chains culminatin9 passage ism, ISm2 The fifth and concluding variation becomes a truly prominent moment in which the symmetrical aspects interact with the accumulation of the relationships that contribute to its sense of arrival. This accumulation takes place on various structural levels. For example, as Lewin has pointed out (1987, 183), there is a return to the five-beat phrasal units, which characterize both the thema and the first movement. The very end of this variation, the passage that follows these five-beat phrasal units, forms a coda based on the rows of the thema, which concludes not only the third movement but also the entire work by representing in a condensed form the basic characteristics of the piece.13 That is, the passage manifests the way in which the phrasal shaping and sense of progression interact with the symmetrical, frozen aspects of the pitch organization. This passage combines the register-symmetrical arrangement of the second movement with the mirror-symmetrical aspects of the first movement: the pcs that in the second movement are arranged symmetrically in register around the pitch A4 are in this concluding passage arranged 13Lewin calls the entire fifth variation a coda (1987, 183). The final passage of the fifth variation may well be described as "a coda within a coda."

Example 25. Dyadic areas of hexachords and tetrachords in the culminating passage of mvt. III (mm. 51-55) 00 94 r 4h s.w wl 14E hI i H 0 ord./unord. sets of dyads 1T6P RIeP ord. ord. tetr. hex. N/acc ANmeI {teX(0ll (23)(67)(45) T9P hepta.: dyads A+B/acc (te)9{O1}(78H(te)9 unord. tetr. B/mel (12)(34) R12P unord. hex. B/acc {56){9t){78} unord. hex. B/mel (1 2}{341(eo}

Koivisto, The Defining Moment 59 symmetrically in time around the second axis of symmetry, the pc E b (ex. 26). As the example further illustrates, registral differentiation in the coda extracts precisely those pitches that change registers in the second movement. Interacting with this symmetrical arrangement is a phrasal shaping that divides the passage into three subphrases. In the context of the fifth variation, the sense of conclusion achieved with these subphrases arises from various features, such as a mirror-symmetrical construction with regard to the very beginning of the variation and a T, transposition of the conclusion of the main body of the variation. On a deeper structural level, the sense of conclusion achieved with the last chords arises from the manner in which they complete the symmetrical arrangement, which is a manifestation of the thema through the first and second movements. Thus, as sketched in example 27a, just as the epilogue in the first movement and the concluding phrase in the second movement bring back the openings of these movements, the coda at the very end becomes a representation of the relationships of the thema and the first and second movements. Example 26. Symmetrical aspects in the coda (mm. 62-66) Eb4 m e3 A4 8 0 t 7 5 9 5 7 6 2 4 0 2 1 9 4 1 e 8 6 e t opening phrasel mvt. II * floating pitch classes/mvt. II

60 In Theory Only cq CD 0> o IIP w a ll. C tId 0 0 ci. 4) N a ~]

Koivisto, The Defining Moment Example 27. Variations, op. 27 (a) temporal accretion and (b) symmetrical structures -/ N ccnud n.\\ phras I Cu x 61 EpBogue a. b. I Thema Coda I___________ Thema l ------ i -- 11 ----- B A' A B A' A ABA' A B A' I II':1: Thema var. 1 2 3 4 5 II III Conclusion As suggested at the outset, this work shows Webern's deep interest in symmetrical structures that interact with through-composed aspects. This has been illustrated with examples from the first and the second movements, as well as with an analysis of the manner in which the coda encapsulates the qualities of the piece as a whole. However, this dialectic exists in the formal layout of the entire composition (ex. 28). The third movement's first two variations reflect the second movement, and the concatenation of variations three, four, and five echo the tripartite structure of the first movement. The structural similarities arise from the use of the same hexachordal areas but motivic connections also occur.

62 In Theory Only Example 28. Parallels between mvt. III and mvt. I-II Mvt. Ill A I 00p Thema var. 1 motivic connections 2 3 4 5 same hexachordal areas 1 ____ 1: Mvt.:II Mvt. II same hexachordal areas motivic connections A Ism2 row pairs rows of the Thema enter 4, 4$' B A' Mvt. I The third movement's first two variations are based on the segmental and motivic hexachordal areas of thema phrases A and B, which are the hexachordal areas of the second movement's first part.14 Example 29 illustrates ways in which the underlying structural similarities are signaled by motivic connections by sketching the manner in which some of the significant moments in the first two variations invoke the second movement. First, both variations open with prominent motivic materials of the second movement: the first variation picks up as its initial motif the exact pitches of the concluding element of the second movement's final phrase (ex. 29a), and the second variation opens with the characteristic repeated staccato As of the second movement (ex. 29b). Moreover, the melody that leads to these ticking As in the first variation's concluding phrase is framed by pitches A b3 and B b, the exact pitches of the second movement's opening phrasal elements. Example 29c illustrates the manner in which the first variation's climactic phrase (mm. 18-21) is 14Wason has noted the structural similarity between the second variation and the second movement by pointing out common inversional-symmetrical aspects of the row organization in these two sections (1987, 70, 86).

Koivisto, The Defining Moment 63 framed by gestures that evoke the middle phrasal elements of the second movement through their pitch contents and melodic contours. Lastly, when trichords belonging to the collection class [016] enter for the first time in the third movement's second variation, they are exactly those chords heard before in the second movement's opening part (ex. 29d). The opening of the third variation signals its similarity to the first movement by being the only variation to employ mirror-symmetrical phrases.is On a deeper structural level, the connections between variations three, four, and five and the first movement arise from aspects in their underlying designs (ex. 28). The symmetrical phrases of the third variation and the first movement's A section are based on the same hexachordal areas yielding the slurred/nonslurred dyads of thema phrase A and the motivic dyads of thema phrase B. The fourth variation and the first movement's middle section are based on row chains formed from Ism2-related row pairs.16 The fifth variation and the first movement's A' section are punctuated by a return of the rows of the thema. Example 30 illustrates the vivid manner in which the associations between the first movement and the third variation's opening phrases are emphasized by registral connections. Example 31 then shows the manner in which the opening of the culminating, the fourth, variation prominently invokes the climax of the first movement's middle section: at these moments the row-pair chains feature the same row, and the associations are emphasized by surface composition. The sense of return in the fifth variation and in the first movement's A' section as well as their structural similarities have been pointed out and discussed by Lewin (1987, 183). 15Wason presents the idea that the third variation "grew eventually into the first movement," by pointing out similarities in the retrograde symmetrical aspects between the third variation and the first movement (1987, 70, 87). 16Lewin has discussed the structural similarities of the row transformations of the two chains in the first movement's middle section and in the third movement's fourth variation (1987, 182).

64 In Theory Only Example 29. Motivic connections between mvt. III, var. 1-2 and mvt. II: (a) var. 1, mm. 12-13 and mvt. II, end of concluding phrase (m. 21); (b) var. 1, concluding phrase (mm. 21-23) and mvt. II, opening (m. 1); (c) var. 1, mm. 18 and 21 and mvt. II middle phrasal element (m. 16); (d) var. 2, trichords and mvt. II trichords 29a j0D 0 0 ý4; ý - @ ME If b ~ 29b. El..,1..i - c~- r

Koivisto, The Defining Moment Example 29 (cont.) 29c. 65 (ii3t1 * I I - h -,ý I A o - LA -Ar____ ____ ____ ___ _ __t-,I LI 1 4 4T-f -*1I I1 1 Ir I. '1 ii I F I 5. v j vw 29d. mvt iii: m. 24 26 30/33 31 myt. ii: m. 8 4 9 3

66 In Theory Only Example 30. Associations between mvt. III, var 3, opening (mm. 33-35) and mvt. I, opening (mm. 1-4) mo-h ri.____ _______ 4orp r 9l- - L-' Example 31. Associations between mvt. III, var. 4, opening (mm. 45-46) and mvt. I, mm. 32-33 i 1 I I * I i kqLi I-I m--I

Koivisto, The Defining Moment 67 The structural similarities between variations one and two and the second movement, and between variations three, four, and five and the first movement, clarify the manner in which the thema lies at the focal point of a large-scale symmetrical structure. This large-scale structure, as well as the series of nested symmetrical structures within it, duplicates the tripartite formal layout of the thema (ex. 27b, p. 63). Hence, as comparison between examples 27a and 27b reveals, the entire work's large-scale formal shaping manifests the dialectic between the balanced, symmetrical structures and the sense of progression, a quality that characterizes the work in various spans of time, as well as the thema itself. In this music, the strong sense of progression, interacting with the symmetries, arises from continuous, multilayered accumulation of relationships and events. In the process of hearing the entire work, the thema emerges as the defining moment: it enters at the focal point of a large-scale symmetrical structure, but at the same time it crystallizes the relationships of the two previous movements. This crystallized interpretation then serves as a basis for the varied elaborations introduced in the subsequent five variations. The coda concludes the piece by capturing the relationships of the thema through their interpretations in the first and second movements. The wealth of relationships inherent in Webern's Variations for Piano, Op. 27, a composition that has fascinated musicians for decades, cannot fully be enjoyed without taking into account the interaction between the surface composition and the underlying structures, just as it cannot fully be appreciated without acknowledging the dialectic between its symmetrical structures and the sense of temporal accretion. By inspecting these dialectics, whether between the surface and deeper levels, or between symmetry and temporal accretion, one may learn more about this music than by inspecting any element alone. It is only through this interaction that in such concise idioms of composition as Webern's a piece may become an intensified moment with depth that penetrates all its structural layers.

68 In Theory Only References Babbitt, Milton. [1960] 1962. Twelve-Tone Invariants as Compositional Determinants. Musical Quarterly 46: 246-59. Reprinted in Problems of Modern Music, ed. P. H. Lang. New York: Norton. _____. 1987. Words About Music. Ed. S. Dembski and J. Straus. Madison: Univ. of Wisconsin Press. Bailey, Kathryn. 1991. The Twelve-Note Music of Anton Webern: Old Forms in a New Language. Cambridge: Cambridge Univ. Press. Hasty, Christopher. 1981. Rhythm in Post-Tonal Music: Preliminary Questions of Duration and Motion. Journal of Music Theory 25/2: 183-216. Leibowitz, Rene. [1949] 1970. Schoenberg and his School. Trans. D. Newlin. Reprint, New York: Da Capo. Lewin, David. 1962. A Metrical Problem in Webemrn's Op. 27. Journal of Music Theory 6/1: 124-32. ___. 1987. Generalized Musical Intervals and Transformations. New Haven, Conn.: Yale Univ. Press. ____. 1993. A Metrical Problem in Webemrn's Op. 27. Music Analysis 12/3: 343-54. Mead, Andrew. 1992. Review of The Twelve-Note Music of Anton Webern: Old Forms in a New Language by Kathryn Bailey. Integral 6: 107-135. _____. 1993. Webemrn, Tradition, and Composing with Twelve Tones. Music Theory Spectrum 15/2: 173-204. Morris, Robert. 1987. Composition with Pitch-Classes: A Theory of Compositional Design. New Haven, Conn.: Yale Univ. Press. Nolan, Catherine. 1989. Hierarchic Linear Structures in Webemrn's Twelve-Tone Music. Ph.D. diss., Yale Univ. Schnebel, Dieter. 1984. Die Variationen fir Klavier Op. 27. Musik-Konzepte: Sonderband Anton Webern 2: 163-217.

Koivisto, The Defining Moment 69 Stadlen, Peter. 1958. Serialism Reconsidered. The Score 22: 12-27. Travis, Roy. 1966. Directed Motion in Schoenberg and Webern. Perspectives of New Music 4/2: 85-89. Wason, Robert. 1987. Webern's Variations for Piano, Op. 27: Musical Structure and the Performance Score. Integral 1: 57-103. Webern, Anton. 1983. Aus dem Briefwechsel Webern-Steuermann. Musik-Konzepte: Sonderband Anton Webern 1: 23-51.. 1979. Variationen fir Klavier, Op. 27. Ed. P. Stadlen. Vienna: Universal Edition. Westergaard, Peter. 1963. Webern and "Total Organization": An Analysis of the Second Movement of the Piano Variations, Op. 27. Perspectives of New Music 1/2: 107-20.. [1962] 1972. Some Problems in Rhythmic Theory and Analysis. Perspectives of New Music 1/1: 180-91. Reprinted in Perspectives on Contemporary Music Theory, ed. B. Boretz and E. T. Cone, 226-37. New York: Norton.

panel Peter Westergaard's Tonal Theory: A Perspective of the History of Contemporary Music Theory In Theory Only is pleased to present in this issue the first installment of a collection of essays addressing Peter Westergaard's contributions to tonal music theory. This collection grew out of a special session entitled "The Tonal Theoretical Work of Peter Westergaard" presented at the fourteenth annual meeting of the Society for Music Theory on 1 November 1991 in Cincinnati, Ohio. The majority of the papers presented at that special session are included here, but several are not. One, Fred Everett Maus's "Teaching With Westergaard's Counterpoint Rules," was published in the Fall 1992 issue of the Journal of Music Theory Pedagogy. Composed especially for this occasion are a bibliography of Westergaard's theoretical writings and a response to the panel by Westergaard himself, both of which will appear in the next issue. Westergaard is a composer, theorist, and conductor on the faculty of Princeton University. He studied composition with Walter Piston, Darius Milhaud, Roger Sessions, and Wolfgang Fortner. Before joining the Princeton faculty in 1968, he taught at Columbia University and Amherst College. He has contributed a number of articles on various topics to Perspectives of New Music. His textbook, An Introduction to Tonal Theory (1974), extends Schenkerian concepts and examines thoroughly the rhythm of tonal music. The 1960s were an exciting time in the development of modem music theory. Composers and theorists at Princeton and elsewhere were subjecting existing theoretical models for music to penetrating critical reexamination. Several of them developed models exhaustively reconstructed from primitives, justifying each assumption and every deduction along the way. Many consider this rigorous brand of music

72 In Theory Only technical inquiry to constitute the most sophisticated kind of musictheoretical discussion. Westergaard's theoretical model is perhaps especially noteworthy, then, for being cast in the form of an undergraduate textbook. The result of this combination ultimately made esoteric theory more accessible to the nontheorist, while students were at the same time exposed to the beneficial exercise of theory-building. The essays presented here provide an interesting perspective of the history of contemporary theory. The first gives an overview of Westergaard's theory. Each successive contribution then deals with an aspect of the theory, interpreting and evaluating it in comparison to other theories. The result is a rich perspective of Westergaard's theory in particular and of music theory in the 1960s and early 1970s in general. We hope you enjoy these essays.

An Introduction To Westergaard's Tonal Theory Stephen Peles Introduction Within the American music-theoretical literature of the postwar era, Peter Westergaard's tonal theory distinguishes itself as one of the most influential yet inadequately acknowledged bodies of work produced by a theorist of his generation. Portions of Westergaard's theory are distributed across various professional journals; portions of it are inferable from certain of Westergaard's writings on nontonal theory; others have been disseminated orally to several generations of students at Columbia and Princeton. The theory receives its most complete and detailed expression, however, in his seminal 1975 book, An Introduction to Tonal Theory (hereafter IT7), and it is that text that this essay primarily addresses. In many ways ITT is emblematic of the profound shift in methodological concerns that-under the influence of such diverse intellectual currents as Schenkerian theory, cognitive psychology, and philosophy of science--characterized American music theory in the 1960s and 1970s and effectively redefined the agenda for subsequent decades as well. Indeed, ITT was one of the texts that prompted that shift, and the book is therefore an important source document not only for practicing theorists but also for those historians concerned with tracing the development of postwar American music theory. Thus,

74 In Theory Only while ITT synthesizes and reconstructs important aspects of the theoretical tradition that precedes it-Westergaard modestly acknowledges (ITT, viii) that most of the book is at least as old as Schenker ([1910/1922] 1987), and much is as old as Fux ([1725] 1943) or even Bernhard (Hilse 1973)-it does so in a selective manner consistent with a larger program that intersects only partially with the concerns and intentions of its antecedents. That larger program is nothing less than a unified theory of classical tonality and entails, among other things, a rational derivation of the tonal collection based upon the acoustical properties of sound waves and human auditory physiology, neurophysiology, and psychology; a psychologically based theory of tonal rhythm-including phrase rhythm-that is sufficiently integrated with the pitch operations proposed by the theory to warrant the claim that this is indeed a pitch/time syntax for tonal music; an account of what performers do-or perhaps ought to do-and why; and a number of extended commentaries on ambiguity in tonal music. As one would expect of an enterprise of such ambitious scope, Westergaard's tonal theory is nothing if not big. By any standards the aspirations of the present essay are notably more modest: it is solely intended to provide readers unfamiliar with ITT with the kind of broad overview that will be most helpful in placing the subsequent essays in this series in a proper perspective. Considerations of space preclude anything approaching a complete exposition of Westergaard's tonal theory; what follows is intended only to sketch its most important contributions. Species-Counterpoint Model of Tonal Music ITT consists in essence of a single complex argument, which both defends and demonstrates the theory's two central theses: 1. We can generate all the notes of any tonal piece [and of no nontonal piece] from the pitches of its tonic triad by successive application of a small set of operations, and moreover 2. the successive stages in the generation process show how we understand the notes of that piece in terms of one another. (ITT, 375) Notice that while these two claims are closely related they are nevertheless quite distinct. The first proposes that the infinite set of all possible tonal pieces can be defined by positing as axiomatic a finite set of possible tonic triads together with a finite set of operations. The theory asserts that the iterative application of the operations, first to some instance of a tonic triad and subsequently to the result of the immediately prior operation, will suffice to generate any tonal piece. The essential property of a tonal piece is its generability in this sense.

Peles, Introduction to Westergaard's Theory 75 The second claim constrains the first and no doubt reflects Westergaard's awareness of the problem that anything that can be axiomatized at all can usually be axiomatized in an indefinite number of nonidentical ways; or, to state the same problem a bit differently, two sets can be identical in their extensions (i.e., have exactly the same members) but can differ in their intensions ("the set of all things that are the planet Venus" and "the set of all things that are the Evening Star" is probably the most overworked example). This fact is a problem for any theory that aspires to more than the most trivial sort of acoustic-level description. Taken alone, Westergaard's first claim leaves unanswered the vexing question of what to do when faced with two different axiomatizations each of which is capable of accounting for (i.e., generating) the same set of data (tonal pieces in the present case). Westergaard's second claim proposes a Chomsky-flavored answer: prefer the one that also accounts for some other set of data, in this instance prefer the one that most accurately models the relevant intuitions of a tonally literate listener. This concern with fundamental methodological questions is one of the most interesting characteristics of Westergaard's work and is one to which this article will return. For now it is sufficient to point out that one practical consequence of this metatheoretical concern is an expansion of the domain of the theory: Westergaard's theory is not just a theory of what tonal pieces do, it is also a theory of what tonal listeners do. As indicated by his second thesis, Westergaard is most concerned with what listeners do when they attribute priority relations to events, that is, when they understand events as instantiating a set of asymmetrical relations that entail thinking of some events in terms of others. An event in terms of which other events are understood is said to have conceptual priority over those other events, and it is a measure of Westergaard's parsimoniousness in such matters that his metalanguage here is neutral with respect to its interpretation in pitch or time: a conceptually prior event can be either a pitch or a timepoint. Westergaard thus begins constructing his model of tonal music by defining the smallest elemental sound-constituent of the system capable of sustaining such relational attributions.' For Westergaard, that irreducibly atomic element of tonal music is the note, a theoretical term that bears a fairly simple correspondence to the actual acoustical observables with which it is correlated (to the "measurable properties of sound waves," as Westergaard puts it). Notes are, in this sense, 1Undergirding this is Westergaard's elaborate reconstruction of the tonal collection, which appears in the Appendix to ITT. For discussion, see Stephen Dembski's contribution to this series of articles (forthcoming).

76 In Theory Only idealizations of actual sounds: they are what remain after (1) the reduction of the sounds of tonal works to a point beyond which they would cease to be sounds and (2) the elimination of all properties of the resulting atomic sounds except the tonal-syntactically relevant ones. What is left is an irreducible package of sonically inseparable attributes: pitch, onset time, and duration. Westergaard maintains, "pitch and time relationships are the primary stuff of the structure of any piece of tonal music" (ITT, 11). From the elemental notes, Westergaard proceeds to recreate the tonal universe from the ground up, the ensuing ontogenesis comprising two steps. First, as the minimal bearers of pitch and time information, notes combine to form temporally deployed strings called lines. What is special about lines is that their temporally adjacent notes are understood as connected, which is to say that "we can conceive of the notes as appearing one by one in a succession, each note replacing the one before it and each note in its turn being replaced by the next note until the last note occurs" (ITT, 29). It is that sense of replacement that is invoked when one speaks informally of a line going from one place to another, or of one pitch moving to another.2 Second, lines combine to form pieces, which is to say that "we can conceive of a piece of [classically tonal] music as being made up of two or more such lines unfolding simultaneously" (ITT, 29). Westergaard's careful choice of language emphasizes that notes may be largely acoustical phenomena, but lines (and even whole pieces) are largely psychological phenomena. They are ways of thinking about acoustical events as related in particular ways. At this stage, then, Westergaard's immediate task is to formulate a theory that constrains the kind of note configurations that obtain both between and within lines to those that best conduce to this sort of thinking. Roughly, his strategy is to break the theory into two parts, corresponding to within- and between-line note relations: (1) a generative component for creating new lines from prior lines (usually moving from the simple to the complex) and (2) a collection of constraints that serve to restrict the output of the generative apparatus by imposing well-formedness conditions on between-line relations. That strategy naturally inclines Westergaard toward a contrapuntally based model, since, on balance, contrapuntal theory is better equipped than harmonic theory (conventionally construed) to 2The sort of line Westergaard's theory is principally concerned with is the structural line, which results when "a series of notes forms a [time]span and pitch structure that gives us a way of understanding other notes in terms of that structure" (ITT, 289).

Peles, Introduction to Westergaard's Theory 77 draw maximumly fine distinctions between note relations of either sort. Thus, for example, where harmonic theory is biased to regard dissonance and consonance as properties of intervals, contrapuntal theory is further disposed to reduce the dissonant interval to its component notes and to attribute the property of dissonance (instability) to one and the property of consonance (stability) to the other. Contrapuntally speaking, the two events marked S1 and S2 in example 1-the two F4/G3 verticalities of a half-note duration-are precisely opposite in their effect and the internal structural status accorded the constituent notes, despite the fact that the two events are acoustically equivalent to within pitch and duration; that is, over timespan S1 the F4 is stable and the G3 is unstable, while over timespan S2 the reverse holds true. Example 1. Differing stability characteristics of identical verticalities S1 S2 As an embodiment of a set of well-formedness conditions on lines in combination, species counterpoint provides Westergaard with a basis upon which to explain such contextually determined phenomena by introducing in their most basic forms the two most important priority relations of tonal music: the subordination of dissonant pitches to consonant ones and the subordination of off-the-beat timepoints to on-the-beat ones. As Westergaard formulates it, species counterpoint is not just a pedagogical device intended solely to introduce the student to "the infinite world of fundamental musical problems" (Schenker [1910/1922] 1987, v. 1, p. 10) by reducing those problems to their simplest cases, although it certainly serves that function. Additionally, Westergaard conceives of his species exercises as actual miniature compositions, which differ from their more elaborately constructed counterparts in the established repertory solely by virtue of the fact that they employ a radically simplified version of the syntax of real-life tonal pieces. Westergaard's set of generative operations that determine allowable within-line relations is small. Indeed, Schenkerian theorists will no doubt be struck by the comparatively parsimonious nature of

78 In Theory Only Westergaard's typology of tonal transformations. A whole battery of Schenkerian operations, such as arpeggiation, unfolding, motion from an inner voice, and the like, collapse into a single operation category in Westergaard's system: namely the arpeggiation operation. The other basic operations are rearticulation, neighbor embellishment, step motion, anticipation, and delay.' The last two are of course time transformations and alter the temporal position of a note, usually displacing it from some timepoint regarded as the note's normative location. Although normative timepoints in this sense come ready-made in species, the rules of species preclude temporal displacements of this sort, and hence, with the obvious exception of the delay operation inherent in fourth species, anticipation and delay are not directly represented in Westergaard's set of rules for constructing species lines. Those rules (given below in condensed form; see ITT, 55-58), moreover, are slightly different depending upon whether the line in question is an "upper line with a basic step motion" (essentially a Schenkerian fundamental line), a bass line (similarly Schenkerian in essence), or an upper line other than the one responsible for the basic step motion. In the last case the rules are a simplified variant of the first case, omitted here for brevity's sake. I. Rules for Constructing an Upper Line with a Basic Step Motion A.The basic step motion 1.The final note must be a tonic. 2.The first note of the basic step motion must be a tonic-triad member a third, fifth, or octave above the final note. 3.These two notes must be connected by all of the pitches of the intervening diatonic degrees to form a descending step motion. B. Secondary structures 1. Any tonic triad pitch may be repeated. 2. A neighbor may be inserted between consecutive notes with the same pitch. 3. Any tonic triad pitch may precede the first pitch or may be inserted between any two consecutive pitches as long as no dissonant skip and no skip larger than an octave results. 3Certain of these terms (e.g., neighbor and step motion) refer both to operations used to generate lines (the sense that is currently under discussion) and also to functional assignments that may be imposed upon a line only when it is combined in counterpoint with other lines. Maus (1992) offers a clear discussion of the distinction in Westergaard's theory between, for example, a passing tone conceived as a stage in the generation of a single line and a passing motion conceived as a way of relating the structures of two simultaneously sounding lines.

Peles, Introduction to Westergaard's Theory 79 4. Any two consecutive notes forming a skip may be connected by a step motion. II. Rules for Constructing a Bass Line A. The basic arpeggiation 1. The final note of the basic arpeggiation must be a tonic. 2. The first note of the basic arpeggiation must be a tonic. 3. The middle pitch of the basic arpeggiation must be a dominant either a fifth above or a fourth below the final tonic. B. Secondary structures 1. Any tonic triad pitch may be repeated. 2. A neighbor may be inserted between consecutive notes with the same pitch. 3. Any tonic triad pitch may be inserted between any two consecutive pitches as long as no dissonant skip and no skip larger than an octave results. 4. Any two consecutive notes forming a skip may be joined by a step motion. The operations authorized by the A-rules are obligatory and ordered, and may be usefully understood as tonality establishing; the operations authorized by the B-rules are optional and unordered, and may be understood as tonality preserving (although Westergaard does not use this term). Application of the A-rules will serve to generate any of the three paradigmatic forms of Schenker's Ursatz, minus, however, Schenker's attribution of Stufen to the resulting note structures. Westergaard calls a structure that results from a complete instantiation of the operations demanded by the A-rules a basic structure; subsequent structures that arise from application of the operations licensed by the B-rules are called secondary structures. Example 2, an example of what I will refer to as a derivation sequence, illustrates the use of Westergaard's operations to generate a simple species line.4 4While the almost complete exclusion of inherited notions of harmony is characteristic of Westergaard's work, he was apparently well aware of the explanatory role played by harmonic entities in theories that relied upon them and was concerned to present an alternative account that required nothing apart from actually instantiated notes. Thus, for example, the A-rules, together with rules B1 and B3, result in something like the attribution of a tonic Stufe to the entire species exercise (though the Stufe in question is an effect of the notes, not their source); more particularly they indicate how an account of Stufen effects might be constructed solely on the basis of differences in observed behavior between actually sounding notes and without recourse to abstract entities. Similarly, in the more complex case of free composition, Westergaard's borrowing operation-which is a kind of doubling of an already generated note in another line-does much of the work in his theory that Stufen do in those theories that employ them explicitly.

80 In Theory Only Example 2. A derivation sequence (IT1, 66) Reprinted by permission of Peter Westergaard. A A] A2 B4 M4 A A3 The structural descriptions of lines provided by the derivation sequences that generate them establish a convenient basis upon which to explicate the notion of structural ambiguity. That term has a precise technical meaning for Westergaard, and the phenomenon to which it refers arises only when both of two conditions are met: (1) the line or lines in question must be properly generable in more than one way and (2) there must be no principled way of isolating as the clearly preferable alternative a single derivation sequence from the set of available candidates. Westergaard states his general position vis-_-vis such ambiguity when he observes that "any but the simplest tonal structure is bound to have a degree of ambiguity somewhere: the greater the number of subsequent operations required to build up a structure, the greater the chance that some other series of operations would result in the same structure" (IT/I, 409). The two most crucial terms in that statement are degree and somewhere. Ambiguity not only comes in varying strengths, but it is also localizable and can be traced to a particular point of origin. The advantage, then, to regarding ambiguity as a function of the degree of nonintersection between two or more distinct derivation sequences that intersect completely at their terminal levels is that a derivation sequence provides an extremely fine-grained view of the tonal structure in question-hence more degrees and more somewheres-which in turn allows correspondingly

Peles, Introduction to Westergaard's Theory 81 greater specificity with respect to assessments of degree and extent of ambiguity. Westergaard proposes a number of disambiguation principles intended to help decide between such alternative derivation sequences, which in this context amount to alternative analyses. The details of those principles vary somewhat depending upon whether the object under analysis is a single species line, more than one species line in counterpoint, or an actual tonal piece (in which event Westergaard's theory of tonal rhythm applies in full force, and a number of transformational operations excluded from his formulation of species may come into play). But in all cases those principles seem to be motivated by two sorts of concerns: considerations based upon assumed or explicitly postulated characteristics of human cognitive processes (essentially empirical concerns) and methodological considerations of explanatory adequacy and form, which to a certain extent can be said to stand above the fray (superempirical concerns). The former are all consistent with a single overriding characteristic of Westergaard's hypothetical listener, which Westergaard summarizes under the maxim "given the choice of two ways of understanding the same notes, he [the listener] will choose the easier way" (ITT, 249). For example, when the set of notes comprises a single species line, Westergaard proposes the following set of principles: prefer the derivation that (1)... uses fewer stages to construct the line and therefore provides a more efficient explanation, or (2)... determines the pitches of the line more closely and therefore provides a tighter explanation, or (3)... corresponds more closely to any regularities in the pattern of the line and therefore provides a more consistent explanation, or (4)... attaches secondary structures to more basic pitches and therefore provides a more connected, unified explanation. (ITT, 67) The first of those principles is self-explanatory: as an empirical claim it follows from the maxim quoted above; as a superempirical methodological constraint it is Ockham's long-familiar razor. The second cautions the analyst against settling for a relatively vague explanation when a more precise one is available (e.g., given two consecutive notes, each an instance of the same tonic-triad pitch, prefer to explain the second note as a repetition of the first rather than as the insertion of another tonic-triad pitch). The third privileges important regularities and parallelisms (and might be thought of as expressing a more general principle: other things being equal, analyze similar events similarly). The fourth suggests that an analysis that construes secondary structures as connected to pitches of the basic

82 In Theory Only structure (in the sense of using those pitches as subglobal points of departure or arrival) is more unified than an analysis that construes secondary structures as "simply ifilling up the time" (ITT, 66). Westergaard's disambiguation principles for lines in combination expand upon the set of principles for individual lines and rely upon a set of categories for describing the degree of structural correspondence between the lines in question. Briefly, two lines are said to be aligned when their "simultaneously sounding notes have the same order of conceptual priority in their respective lines" (ITT, 82), that is, when simultaneously sounding notes are generated at the same stage in their respective derivation sequences. Otherwise they are unaligned. Additionally, when "simultaneously sounding notes serve the same function in their respective lines" (ITT, 82), that is, when each note is generable under the same operation, the structures are functionally parallel. Thus, the pair of lines shown in example 3a are functionally parallel; those in example 3b are aligned (but not parallel); those in example 3c are neither. Westergaard orders these structural categories in terms of what he regards as the degree of correspondence between their respective lines, resulting in a scale of such correspondences ranging from unaligned and nonparallel at the low end to functionally parallel at the high end. Having thus established a set of principles for roughly ranking alternative derivations of the same line and another set of principles for roughly measuring the degree of structural similarity between the respective derivations of two different lines, Westergaard is in a position to elaborate upon a global explanatory strategy of minimizing overall degree of ambiguity while maximizing overall degree of derivational similarity, and he states the following general principles for deciding between alternative analyses of species lines in combination: Given the combination of an ambiguous line with an unambiguous one, we will prefer to understand the ambiguous line in terms of whatever structure corresponds most closely to that of the unambiguous line. Given the combination of two ambiguous lines, we will prefer to understand them as having whatever available structures correspond most closely to one another. (ITT, 86).-.. Given two ways of understanding a combination of lines, we will choose whichever way gives the closer correspondence between linear structures. (I7T, 87)

Peles, Introduction to Westergaard's Theory 83 Examples 3a and b. Disambiguation principles for lines in combination (ITT, 82-83) Reprinted by permission of Peter Westergaard. a. BI BI B2 B2 B2" b. B3 SB2 B2 ___________Ri Y>... 12K_ 1K: n ^

84 In Theory Only Tonal Rhythm With the species model as a point of departure, Westergaard introduces his theory of tonal rhythm by posing the question: "What durations can the composer give his notes and still be reasonably sure that his listener will in fact understand them as segments of a longer span?" (ITT, 228).5 Unlike species rhythm, which is additive (i.e., a new note simply increases the overall duration of the line), Westergaard conceives of tonal rhythm as divisive: a new note segments a conceptually prior timespan into two parts. In the simplest case, then, Westergaard's question is a question of the possible proportions the durations of those two parts may assume such that the listener's sense of the conceptually prior timespan is preserved. Westergaard's answer is based upon two principles, an order principle and a size principle: 1. Given two different points in time, it is easier to think of the second in terms of the first, and 2. Given two unequal periods of time, it is easier to think of the shorter in terms of the longer. (ITT, 228-229) He begins by considering the implications of those principles for individual lines. A simple case is that of a passing motion between notes a third apart. To understand the passing tone as segmenting the prior span of the first note is to think of the onset time of the passing note as an "interior point of articulation within the span of time" initiated by the onset of the first note (ITT, 229). In second species, for example, this is easy, since the principles of species counterpoint ensure that notes with conceptually prior pitches (the notes forming the third) will occur at conceptually prior timepoints (downbeats). Westergaard's claim is that listeners can easily maintain their sense of that priority relation (and by extension their sense of a single, large timespan partitioned into two parts) in the context of more complex durational ratios so long as neither the size principle nor the order principle interferes. In the case of a passing tone the order principle, of course, can not interfere, since the note with the conceptually prior pitch occurs first. The size principle might interfere, however. If the second note (the passing note) is longer than the first note, the two principles will be in conflict (since the note with the conceptually prior SWestergaard's theory of tonal rhythm accounts for almost half of ITT and more than any other component of the theory suffers from the much abbreviated summary that space allows here. Interested readers are urged to consult the original, as well as Walter Everett's contribution to this series of articles (forthcoming).

Peles, Introduction to Westergaard's Theory 85 pitch will be the shorter of the two), and the listener's sense of the prior span will be jeopardized. From these principles Westergaard derives his law ofsegmentation, which, in its most general form, he expresses in the formula "s, s2 = S3... = s," (ITT, 227), to be read as "the timespan of the first note that segments a superordinate timespan is greater than or equal to that of the second note, and all timespans after the first are of the same duration."6 Segmentations that conform to this law are, in Westergaard's terms, stable segmentations; those that do not are unstable. Unstable segmentations are likely to leave the listener uncertain of how to group the notes into superordinate timespans. Example 4a illustrates the principle at work in the case of a single passing tone connecting pitches a third apart; example 4b shows a passing motion connecting pitches a fifth apart. One important claim that is being made is that durations of notes will determine in part the number of stages the listener needs to generate the line; thus, while the line in example 4c consists of the same sequence of pitches as the line in example 4d, the durational patterning of the former allows the listener to generate it in a single step, while the latter requires more stages. It is important to understand that Westergaard's law says little about what might happen in a tonal piece (and less about what must); it is concerned instead with how the durations of notes influence the way a listener will interpret what does happen. To be sure, Westergaard's rhythmic principles attempt to capture certain regularities of tonal behavior. But the sort of normativity Westergaard is talking about is not a matter of simple statistical frequency-of-occurrence defined over some finite corpus of existent pieces; rather, it is a matter of conceptual priority. What makes one instance of a phenomenon rather than another more central to one's conception of the class of phenomena it represents is, Westergaard seems to be saying, roughly its relative simplicity. Explaining what makes one rhythmic configuration simple relative to another is one purpose of his rhythmic theory. For example, in Westergaard's view what makes an unaccented passing tone relatively simple compared to an accented passing tone is that it associates conceptually prior pitches with conceptually prior timepoints. Consequently listeners are 6Westergaard expresses the constraints on delay and anticipation as special cases of this general law. A delay is understandable as such as long as the note in question is delayed by at least half of its original duration; an anticipation is understandable as such as long as its duration is no greater than half the duration of the span it anticipates.

86 In Theory Only Examples 4a, b, c, and d. The law of segmentation (ITT, 230-32) Reprinted by permission of Peter Westergaard. a. s Sl >S2 may become... -- Sl S2 or Sl< S2 but not b. 2 3 S4TN )ecomes r--- 3 ---7 -or3 or C. s PI P2 l p A-C1. l-owI

Peles, Introduction to Westergaard's Theory 87 d. from S Sl S2 from Sl S2= S3 from S' inclined to think of the latter as a more complex instance of the former, a fact reflected in its derivation, which requires an extra stage-usually a delay. To the extent, then, that Westergaard's law of segmentation is in the business of prediction, it is in the business of predicting listeners' responses-what kinds of durational patterns are apt to produce what kinds of effects. This is a nontrivial point since it bears directly upon the question of what counts as evidence for the theory, and what counts as evidence against it.

88 In Theory Only Thus far Westergaard has treated tonal rhythm merely in terms of its raw durations. To understand his account of the way these principles interact with beats, measures, and the like, it is helpful to understand his definitions of the terms involved. Beats are "equally spaced reference points" (IT, 21) that the listener can use to measure durations (the term is also used to denote the timespan between reference points). Downbeats (also called primary beats [ITT, 259]) are "primary reference points" in terms of which other (secondary) beats are understood (ITT, 21-22). Measure refers to "the [time]span between consecutive primary [beats]" (ITT, 22). Meter refers to "the way the secondary reference points divide the spans between primary reference points" (ITT, 22). Notice that in each case the term is defined without reference to any acoustical event whatsoever. All of these terms refer to psychological constructs and taken together describe a hierarchy of conceptually prioritized timepoints and durations.7 (Thus, if accent is understood as a matter of relative loudness, then it is fair to say that accent has nothing to do with Westergaard's notion of meter.) The interaction of this psychological hierarchy with the onset and offset times of actually sounding notes allows Westergaard to extend the application of his segmentation principles. Once beat-level and measure-level timepoints are established, the listener will understand them as implicitly segmenting any notes that are sustained through them; conversely, notes that begin off the beat will tend to be understood as segmenting a conceptual timespan that begins on the beat (even if that beat is not articulated by an attack). Thus Westergaard would explain the fact that example 5a is harder to grasp (more complex) than example 5b by invoking the implicit segmentations involved. While the passing-tone D in example 5a might appear to follow the rule (it segments the span of the E precisely at its midpoint), the D itself is implicitly segmented into two parts by the downbeat, and these two parts stand in an unstable relation to one another. The additional rhythmic resources that Westergaard's nonspecies rhythmic principles make available are accompanied by the introduction of a number of new pitch operations supplementing those available in species. (Some of these, such as anticipatory arpeggiations, are old operations that take advantage of the now-available anticipation operation.) Most will be familiar: chromatically altered neighbor and passing tones, chromatic notes resulting from mixture, 7The properties of a succession of notes that encourage the establishment of one such hierarchy rather than another are discussed by Westergaard in a section entitled "Establishing Beats," (TT, 269-76).

Peles, Introduction to Westergaard's Theory 89 incomplete neighbors (which under Westergaard's definition are incomplete only in that they are approached by skip; in their simplest form these are anticipatory), octave transfer, and others. Example 5. Metric segmentation (ITT, 249-50) Reprinted by permission of Peter Westergaard. a. Si' S2' B1 IB2 SA I fi I P'I AZ b. With these additional operations and with his account of beats and measures complete, Westergaard turns to the next larger span of interest: the phrase.8 Westergaard considers a phrase to be a different kind of thing than a beat or a measure. A measure is a purely temporal structure, a set of reference points in time in terms of which we can understand whatever notes may occur during that measure. A phrase, on the other hand, has a particular kind of pitch structure that implies a particular type of span structure. A phrase 1. establishes one set of pitches and then 2. moves to a second set of pitches in such a way that 8estergaard touches briefly upon hypermeter, which he refers to under the rubric of macro-measures, defined by him as "measure-like structures that are made up of measures the way measures are made up of beats" (ITT, 311, fn.), but does not elaborate upon the phenomenon, regarding it as presenting "no essentially new problems."

90 In Theory Only a. we expect those pitches, b. we have some sense of when they are about to occur, and c. once they have occurred we know the phrase has gotten where it's going and that no further pitches are needed to complete that phrase. (ITT, 311) The two collections of pitches around which a phrase is structured are associated with two timepoints (called the first primary beat of the phrase and the second primary beat of the phrase), and the two timespans associated with these collections are understood as contiguous parts of a single superspan that encompasses them both. Since the point of initiation of the second collection therefore segments that larger span, the listener will expect it to occur no earlier than the larger span's midpoint. Indeed, Westergaard's explanation of phrase structure relies heavily on considerations of the listener's expectations, and he is able to account for those expectations on the basis of principles he has already established. A case in point is the listener's expectation that the goal of the phrase will be a particular set of pitches that arrive at a particular point in time. The theory as developed thus far has already determined the initial conditions: (1) the listener expects pitch structures to be complete (in the sense first developed in species); (2) as a psychological principle, the law of segmentation holds; and (3) a temporal grid of beats and measures has been established (or will be early in the phrase). Given these conditions, Westergaard argues, the expectation can be triggered by "creating pitch structures that are incomplete up to B2 [the second primary beat of the phrase] but are completed at B2" (ITT, 316). Westergaard's simplest example of this process concerns a hypothetical phrase in C minor that begins on E b5 and moves to D5, with a secondary structural detour to Gs via F5, as shown in example 6a (the line above the staff shows the elapsed time in seconds from the onset of the Eb5; the F5 occurs 0.75 seconds into the phrase). The question that concerns Westergaard's hypothetical listener is this: given a phrase that begins with these notes, when and where will the phrase end? Westergaard's answer is simple but ingenious. To begin with, given the tempo and the durational proportions of the notes, it is simplest for the listener to understand the E bs and G5 as occurring on downbeats. To complete the structure, the listener expects either a Cs (i.e., an underlying E b5-Ds -C5 step motion) or an E b (i.e., an

Peles, Introduction to Westergaard's Theory 91 underlying E 6'-D5-E 6' neighbor structure)? In either case the D 5 segments the span of the conceptually prior E 61 that begins the phrase, and thus whichever note completes it is constrained to arrive no later than the three-second mark, as indicated by the shaded area in example 6b (anything later would cause the D' to segment the span initiated by the E 6' in an unstable way). Therefore, since the listener prefers to think of conceptually prior pitches as occurring at conceptually prior timepoints (and if they do not, will regard them as displaced), this leaves only the beats at the two- and three-second marks as most probable locations for the final note of the phrase. One of these is problematic however. If the final note does not arrive until the three-second mark (ex. 6c), then the timespan of the D' will be (implicitly) segmented in an unstable fashion by the beat at the two-second mark. Consequently, with only these four notes to go by, the listener will instead expect that the final note of the phrase will fall on the earlier of the two beats (ex. 6d). Of course the listener could be wrong, and it might not happen that way. But it is the fact of the expectation, not the fact of its fulfillment that Westergaard is concerned to explain. "The listener's sense of what is about to happen in a phrase is rarely a matter of knowing exactly what will happen next. It is usually a matter of having some inkling of the structural features shared by a number of things that might happen next" (ITT, 318). Examples 6a and b. Expectations within a phrase (ITT, 317) a. 10Osec. 11isec. 12 sec. 13 sec. b. 0 sc. sec. sec.14 sec. 'B2? X B2?" B2? 9Phrases can be based upon incomplete structures, of course, but these pose additional problems. See ITT, 327-328 for discussion.

92 In Theory Only Example 6 (cont.) C. O sec. 1 sec. 2 sec. 3 sec. B1 Ib lb IB2 A b60 d. O sec. i sec. 2 sec. BI B IB2 - I I I Westergaard continues at roughly this level of fine detail for the one hundred and twenty-odd pages that remain of ITT, considering cadences, incomplete structures, phrases that combine to form larger spans (and phrases that do not), large-scale tonicizations, structural parallelism, and a host of other complications. To follow him in that pursuit is impossible here; further discussion is offered in subsequent articles in this series. It should be clear, however, on the basis of the general principles already discussed, precisely what issues are at stake. Conclusion No theory is without its critics, and Westergaard's has had its share; any theory as fine-grained in its level of analysis and as detailed in its style of argumentation as ITT is apt to invite disagreement if only by virtue of having more points of substance at which disagreement might arise. Moreover, by casting his net as wide as he does-by considering the proper domain of a tonal theory to include psychological as well as acoustical phenomena-Westergaard has many more ways in which he could go wrong. As an indication of the value of Westergaard's theory, it left those of us who first encountered it in the mid-1970s (including many like myself who were previously acquainted with Schenkerian theory) wondering how we had managed to get along for so long without an explicitly articulated account of tonal rhythm. If nothing

Peles, Introduction to Westergaard's Theory 93 else, Westergaard's work successfully shifted the burden of proof to those still inclined to think they could adequately express the relevant generalizations needed by any workable formulation of tonal syntax without a theory of rhythm or with a theory of rhythm wholly independent of a theory of pitch. Regarding pedagogical issues raised by ITT, it is, of course, a curiosity that a theory of this sophistication should take the form of a textbook (and an introductory undergraduate textbook at that). No doubt many readers will assume from this brief survey of the theory that ITT is not a typical textbook in other respects as well. The assumption is warranted, although the atypicality of the book is, pedagogically speaking, entirely virtuous. ITT does not simply present the student with a ready-made, prefabricated theory; instead, the student is put in the uniquely beneficial position of watching the theory being constructed. A set of problems the theory is meant to address is delineated; initial definitions of theoretical terms are offered; observations are generalized into principles expressed in an unambiguous metalanguage; a simple model that incorporates those principles is constructed as a first approximation and evaluated for its goodness-of-fit with the target domain; further constraints and principles are invoked to account for discrepancies between model and domain; and so forth. In the process, the student is not only taught a theory, but also how to theorize. Like a latter-day Josephus apprenticed to Aloysius, the student learns the craft in the traditional way: by looking over the shoulder of someone who knows how the thing is done.

94 In Theory Only References Fux, Johann Joseph. [1725] 1943. Steps to Parnassus: The Study of Counterpoint. Trans. and ed. A. Mann. New York: Norton. Hilse, Walter, trans. and ed. 1973. The Treatises of Christoph Bernhard. Music Forum 3: 1-196. Maus, Fred. 1992. Teaching with Westergaard's Counterpoint Rules. Journal of Music Theory Pedagogy 6: 83-95. Schenker, Heinrich. [1910/1922] 1987. Counterpoint. 2 vols. Trans. J. Rothgeb and J. Thym. Ed. J. Rothgeb. New York: Schirmer Books. Westergaard, Peter. 1975. An Introduction to Tonal Theory. New York: Norton.

Coming to Terms: Speaking of Phrase, Cadence, and Form Janet Schmalfeldt As the safest way of ensuring an intelligible communication network among composers, performers, and listeners, Peter Westergaard's An Introduction to Tonal Theory (1975; hereafter ITT) advocates that you "develop your own [verbal] metalanguage and teach it to people by applying it to actual statements in the object language--that is, to passages of music, written or performed" (ITT, 9). Westergaard's position might well describe the implicit, if not express, pedagogical goal of music theorists throughout the ages, but it particularly characterizes the American music-theoretical agenda of the 1970s. Those were the days when to be a genuine music theorist most especially meant to create one's own, preferably formalized and systematic, theory about music. The more original one's theory, the better; and the more distinctive one's terminology, in Westergaard's sense of a "small, closely knit vocabulary of key words-words that pertain directly to the ways in which we think of the music we hear" (ITT, 9), the greater the mark of one's accomplishment. Both during and since the 1970s only a few have been as rigorous and successful as Westergaard in achieving the kind of goal he has recommended. But this might in the long run be fortunate: after all, were each of us to develop our own metalanguages for discourse about music, our communication with one another would probably regress to the level

96 In Theory Only depicted in the wonderful cartoon from Westergaard reproduced as example 1. Example 1. Cartoon from ITT (9) Reprinted by permission of Peter Westergaard. bl blah I I blah \blah.} blah' ah ~ ~. blah. Let us consider the vast spectrum of professional musicians who, along with nonprofessionals, aficionados, and music students, might be represented by the listeners shown in Westergaard's cartoon as talking in their own way about the music they hear. First, although the composer and the performer are each given a separate representation here, that they are talking about what they have written and played serves as a reminder that composers and performers are also listeners. Indeed, for the simplest explanation as to how musical styles become established, developed, transmitted, transformed, and eventually abandoned, one need only acknowledge that composers both listen and react-sometimes positively, sometimes with disdain-to the works of other composers, and that performers do the same. Next, the listeners in the cartoon might just as well stand for an impressive array of specialists: music theorists and analysts in general, historians of music theory, scholars of music history, aestheticians, ethnomusicologists, social and cultural music historians, style analysts, specialists in musical perception, experts on historical performance, and music critics, to name some of the broadest categories with which distinct metalanguages tend to be associated. Now, listening to music does not necessarily serve as the central concern for all of these types of scholars. Nor do all of these make a habit of writing about how they conceive of the music they hear; on the contrary, it would seem that until recently in this country, that activity was thought to be the special, if not exclusive,

Schmalfeldt, Coming to Terms 97 domain of the theorist/analyst. But distinctions of this kind have now been queried, at the same time that metalanguages have proliferated. For example, whether or not music critics and historians are themselves talking more today about how they hear music, many of these have insisted that doing this should have historical and cultural as well as purely theoretical relevance. In response, if a new music-theoretical agenda has emerged, perhaps it could be described as integrative, pluralistic, interdisciplinary, and historically critical on the question of the role of music theory within its broader cultural context. Thus the pressure to develop one's own, unique metalanguage has given way to the challenge of arriving at metalanguages that accommodate a more diversified range of listeners and music-historical concerns. In the face of that challenge, much can be said for the metalanguage that traces whatever origins it might have within the music-theoretical tradition and draws when appropriate upon established terminology. For students of Westergaard's text who know nothing about Schenker, Fux, and Bernhard, and who therefore might skim over the author's acknowledgments to those theorists in his preface, many aspects of both Westergaard's metalanguage and his tonal theory might be taken as purely Westergaardian. It is, of course, not unusual for beginning students to think at first that the author of their tonal theory text invented tonal theory. Perhaps in part to ward off that misunderstanding, certain theory texts before and since Westergaard's have made a point of including the occasional reference to earlier theorists and to stages in the development of tonal thought. Westergaard's approach is to use the historically collective we when introducing traditional terms, axioms, basic operations, and constructs; for instance, "We conceive of music as being made up of units of sound called notes" (ITT, 11). Of interest here is that the meaning of the word we implicitly changes when the author proceeds to introduce his own, entirely original admixture of Fuxian and Schenkerian principles. In statements such as "we will make use of a special kind of tonal music... called species counterpoint" (ITT, 53), the we now represents Westergaard himself plus those who are about to follow his procedures, rather than the collective we that might refer to a theoretical tradition. Evident on every page of his text is Westergaard's unfailing capacity for engaging students to think not only about tonality but also about what it means to construct a theory of tonality. Were it not for this pedagogical flair as well as genuine commitment, that is, had the author not chosen to present his theory in the format of an introductory undergraduate text, perhaps he might have documented, for example, which elements of Fux's species

98 In Theory Only counterpoint system remain intact and which recede when the operational rules for composing a species counterpoint are reformulated to produce a structure that composes out a Schenkerian Ursatz-form. In the light of the present-day concern for historical evaluation, it should come as no surprise that this kind of clarification served as a common thread among the papers comprising the special session on Westergaard's work presented at the 1991 conference of the Society for Music Theory. From the same perspective, a renewed examination of Westergaard's metalanguage in general and his terminology in particular seems overdue. More than twenty years have passed since the publication of his text, and a barrage of writings on tonal theory---especially Schenkerian theory-has appeared in the interim. Readers of that literature who turn, or return, to Westergaard's text today will undoubtedly find occasion to note a long-standing tendency within the field: the apparent need within different metalanguages for different terms in reference to similar perceptual phenomena. As Westergaard puts it, how can one be sure that two different people "wouldn't use different words for the same conception?" (ITT, 9). Or, conversely, what can be learned about individual metalanguages when two different people use the same term for different conceptions? Given that the topics of Westergaard's chapter 8, "Phrases, Sections, and Movements," and his chapter 9, "Performance," have not only long held venerable positions in discourses about music but also enjoyed much recent attention, these chapters most especially invite a comparative terminological inquiry. This article shall focus in particular on notions of phrase and cadence as these relate to ideas about performance and form. Let us begin with the term phrase. One would be hard pressed to find a comparable term to which, from the mid-eighteenth century to the present, so many different conceptions have been attached. As a means of describing one organizational level within the hierarchy of formal units perceived as grouping structures in music, the term phrase has become standard and thus unavoidable; but conflicting definitions reflect divergent views concerning both the content and the length, or span of time, of the musical unit to which the term might best be applied. On the matter of phrase lengths, perhaps the only noteworthy consensus among tonal theorists has been an agreement about the tendency toward duple, as opposed to nonduple, construction. In William Rothstein's words, "Most theorists, whatever their conceptual framework and vocabulary, have recognized that phrases of two, four, eight, and sixteen measures enjoy a privileged status in tonal music" (1989, 33). To Rothstein's observation, one

Schmalfeldt, Coming to Terms 99 might add that the eighteenth- and early nineteenth-century pioneers of phrase-structural analysis-notably Riepel, Kirnberger, Koch, and Reicha-all tended to regard the four-bar phrase as a stylistic norm, subject to expansion. Moreover, especially Riepel (1755, 36ff) and Koch ([1787] 1969, 414-15) tended to identify phrases in terms that would clarify their harmonic goals: thus Riepel's and Koch's Grundabsatz-a phrase that ends on the tonic; or Koch's Quintabsatz-a phrase that ends on the dominant, that is, with a half cadence. Such characterizations anticipate Arnold Schoenberg's theory that specific types of phrases fill specific formal functions (see below). Certainly the most comprehensive study of the concept of phrase to date, Rothstein's work (1989) takes Westergaard's (ITT) as well as Roger Sessions's (1950) ideas about phrase as its ostensible point of departure. In particular, Rothstein cites the following definition from Westergaard, and he acknowledges what must surely be celebrated as its ground-breaking feature-an emphasis on pitch structure as well as tonal rhythm. For Westergaard, a phrase 1. establishes one set of pitches and then 2. moves to a second set of pitches in such a way that a. we expect those pitches, b. we have some sense of when they are about to occur, and c. once they have occurred we know the phrase has gotten where it's going and that no further pitches are needed to complete that phrase. (ITT, 311) More specifically, Westergaard's phrase is to be understood as a "span structure" itself divided into "two large segments, the first occupied by the first set of pitches, the second by the second set" (ITT, 311). In the excerpt by Mozart reproduced from Westergaard's text in example 2a, the complete span structure of this phrase has been labeled SS; S, and S2 comprise the two segments of the span, and the downbeats on which these begin-labeled B1 and B2-serve as the two "primary reference points" of the phrase. My overtly Schenkerian graph in example 2b attempts to show how one might interpret the pitch structure of the complete span in accordance with Westergaard's definition of phrase. The graph proposes that the upbeat anticipation labeled a by Westergaard activates a compound line effecting an arpeggiation to the active tone 5 in the descant; thus Westergaard's "first set of pitches" expresses the tonic triad in support of ý, already in effect on the first downbeat but, due to the delay produced by the arpeggiation, literally not achieved until the third beat. The "second set of pitches," that is, the goal of the phrase at B2, takes the form of a i6 -chord supporting i. In Westergaard's terms, "the pitches creating the

100 In Theory Only motion to the second set may be understood [in this case] as segmenting the first large segment [S,]" (ITT, 311); or, in Schenkerian terms, the V-chord at m. 2, creating the secondary beat labeled b, provides consonant support for the passing-tone ý, whose function is to fill in the overall span from ý to 1. Example 2a. Dual segments of a span structure (ITT, 316; Mozart, Symphony No. 40, mvt. III, mm. 1-3) Reprinted by permission of Peter Westergaard. SS Example 2b. Voice-leading graph of Mozart, Symphony No. 40, mvt. III, mm. 1-3 A i

Schmalfeldt, Coming to Terms 101 Example 2c. Voice-leading graph by Heiniich Schenker (1926), with formal annotations added (Mozart, Symphony No. 40, mvt. III, mm. 1-14) Reprinted by permission of Drei Masken Verlag, Munich. Theme type: sentence (after Schoenberg) presentation continuation (basic idea varied repetition),.t.......-....................... ~-..... o PAC in v In contrast, Rothstein would not regard the excerpt in example 2a as a phrase. For clarification, my discussion turns to a term long associated with phrase and just as multivalent: cadence. Within Rothstein's study, Westergaard's work serves as the precedent for an emphasis upon "directed motion in time from one tonal entity to another" (1989, 5); but Sessions's much looser definition of phrase, "a constant movement toward a goal-the cadence" (1950, 13), ultimately provides the stronger basis for Rothstein's position, namely, that the term phrase is best reserved for describing a complete tonal motion, and that completeness can only be fundamentally achieved by cadential means (1989, 7-9). Precisely what do these different theorists, Sessions, Westergaard, and Rothstein, mean by the term cadence? From a glance at example 3, which shows the three excerpts from Beethoven that accompany Sessions's discussion of phrase and cadence, it should be clear that, as Rothstein concedes, Sessions's cadence is primarily a rhythmic phenomenon, rather than, for instance, a type of harmonic-contrapuntal progression. Although Westergaard and Rothstein would identify the goal of Sessions's third excerpt as effecting a half cadence, neither of these two would claim that cadences have been completed at the end of Sessions's first and second examples.

102 In Theory Only Example 3. Sessions's (1950, 13-14) examples of phrase and cadence Reprinted by permission of Princeton University Press, Princeton, NJ. For instance, I have sometimes been distressed to hear the following passages from Beethoven's Quartets played thus: '7- Op. 18,No. 1 and._..-.,.. ~ T, p(>Op.l&No.2 instead of thus: I 114% -. and Or, still more distressingly, I have heard the opening Of the Scherzo of Beethoven's Fifth Symphony played thus: (a) instead of, correctly, thus: (b)

Schmalfeldt, Coming to Terms 103 Like many Schenkerians, Westergaard avoids the terms perfect authentic cadence and imperfect authentic cadence, but his complete or full cadences conform with the traditional harmonic and rhythmic requirements for those types of progressions.' Since Westergaard does not at any point within his text invoke chordal inversion theory, and since the only harmonic values to which he consistently alludes are those represented by the triads built on the dominant and the tonic, one knows and can confirm from his examples that, when he defines the cadence as a motion from a dominant collection to a tonic triad, he describes the progression from root-position V to root-position I. If one now specifies the motion of the descant within such a progression, a fundamentally stepwise descent through ý to 1, one arrives at what Rothstein means by authentic cadence-the only kind of progression that can effectively bring closure to an Ursatz-form or to a replica thereof. It follows that, for both Westergaard and Rothstein, the i-chord that closes the excerpt by Mozart in example 2a rules out the notion of a cadence there, and this in part explains why Rothstein would regard the excerpt as, at the very most, only a subphrase. In short, like many theorists during and since the eighteenth century, Westergaard takes a less restrictive attitude toward phrase endings than does Rothstein, whose position, it can be mentioned, finds its most direct precedent in the second of Carl Schachter's three studies on rhythm and linear analysis (1980, 203). For Westergaard as of 1975, a phrase need not end with a cadence in order to qualify as a phrase; in fact, Westergaard's cadence is just one means-albeit the surest means--of getting the listener to sense the forthcoming arrival of the phrase's second primary beat, B2 (ITT, 318-19). Although Westergaard posits that one senses a phrase ending as the moment where no further pitches are needed to complete that phrase, he also proposes that a phrase need not be a self-contained or complete structure, that it might instead be incomplete. If the complete phrase structure is one that arises from "rearticulation and arpeggiation of the tonic triad" (ITT, 313n), then presumably the phrase by Mozart (ex. 2a) is complete, despite its conclusion on the i6. The same would be true for the additional excerpt identified as a phrase by Westergaard and reproduced at example 4a. By contrast, Westergaard regards the first of the two phrases shown in example 5 as an incomplete structure, since it ends with a half cadence. Finally, the phrase from his text reproduced in example 6a would without a doubt have to be 'On the subject of past and recent views about cadential concepts, terms, and strategies, see Schmalfeldt (1992).

104 In Theory Only considered very incomplete, since it concludes on the passing-tone 4 as supported by the highly unstable ii4 chord. Example 4a. From ITT, 316 (Mozart, Die Entfi2hrung aus dem Serail, Act II, "Wenn der Freude Tranen fliessen," mm. 1-2) Reprinted by permission of Peter Westergaard. 'b [Bi lb IB2 Adagio Example 4b. Mozart, Die EntfUihrung aus dem Serail, Act II, "Wenn der Freude Tranen fliessen," mm. 1-9 Theme type: 8-bar period, featuring interruption antecedent (basic idea Adogio 2 Ohe 2"^iil 2i 7.,d ~JP Aram contrasting idea) PAC

Schmalfeldt, Coming to Terms 105 Example 5. From ITT, 334, with formal annotations added (Haydn, Quartet Op. 3, No. 3, mvt. III, mm. 1-8) Reprinted by permission of Peter Westergaard. Theme-type: 8-bar period, featuringinnterruption antecedent consequent VMLI II A A I h - Example 6a. From ITT 316 (Mozart, Symphony No. 40, mvt. I, mm. 1-4) Reprinted by permission of Peter Westergaard. b IB1 b IB2 Moto AUcgro Ai'., i -.i'.. T,,', Both Westergaard and Rothstein develop hierarchical phrase-structural terminologies, within which subphrases, or small phrases, can be understood as embedded within larger phrases and pairs of phrases create even larger spans. From Westergaard, on the one hand, we have example 7, where the two small four-bar phrases are embedded in the larger eight-bar phrase; and, on the basis of this example, one can imagine that the excerpt shown in example 6a would have become only the first of two small phrases, or perhaps only the first of two "segments of a single large span" (ITT, 333), had Westergaard discussed the passage within its larger context (ex. 6b). From Rothstein, on the other hand, we have the durational reduction shown in example 8a, where each of Johann Strauss's notated four-bar segments within his initial thirty-two-bar theme has been appropriately represented as a single hypermeasure (or macro-measure, to use Westergaard's term from 1975), such that the thirty-two notated measures express an eight-bar structure. As shown with Rothstein's graph (ex. 8b), a complete tonal motion-an Ursatz-replica-is achieved only by means of the cadence at the end of the theme.

106 In Theory Only Example 6b. Voice-leading graph by Heinrich Schenker (1926) with formal annotations added (Mozart, Symphony No. 40, mvt. I, mm. 1-22) Reprinted by permission of Drei Masken Verlag, Munich. Theme-type: sentence presentation continuation (basic idea varied repetition) Allegro moZ o _ -- [ wo T-nrt\i 'HC (standing on te dmi imVordetO mOM At) 4..-L -........-' 0-.. $-IV V ----- Bdur HC (standing on the dominant) Thus Rothstein regards this entire thirty-two-bar excerpt as a single, large phrase, and he concludes that "the two sixteen-measure halves are hardly phrases at all; at best they are phrases manque" (1989, 10). In the face of such theoretical disparities concerning phrase lengths, it might be useful at this point to approach the issue of phrase from an altogether different angle-the perspective of the performer; in other words, what do performers tend to mean when they speak of a phrase? Perhaps many performers would hold, with Sessions, that a musical phrase is "the portion of music that must be performed, so to speak, without letting go, or, figuratively, in a single breath" (1950, 13). Rothstein, himself an active performer, draws heavily upon Sessions's viewpoint; and, in defense of a theory that frequently calls for the perception of very long phrases, he stretches that position to its furthest extreme: "Every astute listener knows that the best performances are those in which the whole composition is performed, as it were, 'in a single breath'" (1989, 13). In keeping with his looser formulation of phrase, Westergaard makes no such demands on either the performer or the listener; indeed, one of many refreshing and inspiring features of his chapter on performance is its emphasis

Schmalfeldt, Coming to Terms 107 Example 7. From ITT, 336, with formal annotations added (Haydn, Symphony No. 104, mvt. I, mm. 17-24) Reprinted by permission of Peter Westergaard. antecedent within a 16-bar period, featuring interruption (compound basic idea continuation) A II upon what performers do, and can do, rather than upon what they must do, to "communicate the structure of the music" (ITT, 409). Rothstein finds every good reason for deploring the fact that performers and editors have obfuscated the distinction between phrasing in the sense of legato performance and phrasing in the sense of "the delineation and internal shaping of phrases" (1989, 11). His comments notwithstanding, it is unlikely that performers will agree to follow his lead and banish the term phrasing from their vocabulary. Moreover, perhaps the dual meaning of the term for performers provides the very clue as to why, when performers speak of phrases, they are often referring to relatively short musical units, rather than, for example, to groups of eight four-bar hypermeasures or to complete movements. After all, sensitive performers as well as astute listeners are aware that there is a fine line between the performance of a complete movement as if in a single breath and the performance that seems breathless or fails to breathe. To avoid the latter, the stylistically attuned performer would want to give thoughtful attention

108 In Theory Only Example 8a. Durational reduction by Rothstein (1989, 8), with formal annotations added (J. Strauss, Jr., The Beautiful Blue Danube, mm. 45-60) Reprinted with permission of Schirmer Books, an imprint of Simon & Schuster Macmillan, from PHRASE RHYTHM IN TONAL MUSIC by William Rothstein. Copyright 1989 by Schirmer Books. Theme-type: sentence presentation (basic idea J. - j varied repetition) A A Example 8b. Rothstein's voice-leading reduction of J. Strauss, Jr., The Beautiful Blue Danube, mm. 45-60 (Rothstein 1989, 9) Reprinted with permission of Schirmer Books, an imprint of Simon & Schuster Macmillan, from PHRASE RHYTHM IN TONAL MUSIC by William Rothstein. Copyright 1989 by Schirmer Books. arpcggistion _I I'll 0) I (I V 1) (1 V I) 1 II1 V I to the delineation of the phrases that Westergaard proposes in the examples here reproduced from his work-whether or not these end with cadences and whether or not, from a hierarchical theoretical standpoint, these might better be understood as subphrases, segments, Zweitakter, Dreitakter, or, to use a term from Schoenberg, Grundgestalten (basic shapes, or basic ideas). Westergaard's notion of phrase has considerable pragmatic value, in that it relates directly to

Schmalfeldt, Coming to Terms 109 the notions and habits of performers, as these in turn communicate with listeners. If there are any broad observations that might profitably be drawn from this survey of conflicting views about phrase, these should be that phrase is a relative and context-dependent concept, and that disparate applications of the term have much to do with differing attitudes toward the notion of musical form in general and formal hierarchies in particular.2 It should be noted here that certain past and recent theorists influenced by Schoenberg's ideas on form in tonal music have to some extent circumvented the problem of phrase by focusing instead on such questions as what formal function does a given musical unit have within the larger formal plan, and what aspects of phrase structure point to established formal conventions within a given style? To demonstrate this kind of approach to phrase structure, I have provided larger formal contexts for several of Westergaard's phrases, as shown in examples 2c, 4b, and 6b, and I have identified the conventional theme-type within which each of these phrases plays a functional role. Those familiar with Schoenberg's Fundamentals of Musical Composition (1967, 20-22, 58-62), with the work of his disciple Erwin Ratz (1973, 21, 23ff), with Edward T. Cone's Musical Form and Musical Performance (1968, 75-76), or with certain publications of William E. Caplin (1986, 1987) will recognize from my annotations in examples 2c and 6b that both of these phrase structures are modeled upon a mid-eighteenth-century formal convention-the type of theme that Schoenberg called the sentence (Satz).3 That same theme-type serves as the basis for Strauss's theme in example 8-an observation that Rothstein might have made, since elsewhere in his study he addresses Schoenberg's concept of sentence. As shown in example 5, the theme-type that Westergaard describes as a pair of phrases, with the second phrase "completing what the first phrase has left 2Contesting with notable vehemence Rothstein's position that the Prelude to Das Rheingold has no phrases (see Rothstein 1989, 13, 265), Robert P. Morgan expresses similar observations and carries them further: "Since phrases are, if anything, groups of notes that cohere, exhibiting some sense of beginning and ending, they belong to a larger class of musical events encompassing everything from brief motives to large sections and movements. The line separating the province of'motive' from 'subphrase,' or 'subphrase' from 'phrase,' is therefore fuzzy at best; the categories involved are all relative to one another, and taken collectively they form a continuum" (1991, 79). aThe formal annotations in example 2c, as well as in examples 6b, 7, and 8a, draw upon terminology introduced by Caplin in his treatment of Schoenberg's concept of the sentence (see especially Caplin 1986).

110 In Theory Only incomplete" (ITT, 334), has also been known as the eight-bar parallel period (see Rothstein 1989, 17 or Green 1979, 65) or antecedent-consequent design; more specifically, Haydn's period (ex. 5) exemplifies Schenker's [formal] division by interruption ([1935] 1979; 36-40, 70, 132-38). Finally, the large phrase in example 7, when followed by its varied repetition, becomes an antecedent-type within a sixteen-bar period featuring interruption. To this day, the post-Schoenbergian formalist's taxonomy of theme-types is commonly mistaken for thematicism, a term chiefly associated with tracing repetitions and recurrences of melodic structures qua motives. That motivic criteria play, at best, only a subordinate role in the determination of theme-types cannot be emphasized strongly enough. Like A. B. Marx's Satz, or Schenker's Gedanke, Schoenberg's theme becomes in Fundamentals of Musical Compositionhis English term for what Caplin interprets as "not merely a melody or collection of motives... but rather... a complete musical complex that includes a soprano and bass counterpoint, a definite harmonic plan, a phrase-structural design, and cadential closure" (1987, 216). Thus, for instance, it is the immediate, varied repetition of a harmonically stable basic idea, as distinct from simply a melody or motive, that characterizes what Caplin would call the presentation phrase within the sentential themes in examples 2c and 6b. In both of these cases, the notion of a presentation rests not only upon the pattern-of-repetition structure of the phrase but also on its overall harmonic design--one that, by effecting a stable prolongation of the tonic harmony, as opposed to projecting a sequential or a cadential progression, clarifies itself as a typical beginning rather than as a middle or an ending. It can, of course, be granted that the perception of associations invited by repetitions and recurrences has always played a fundamental role in theories of musical form. If recent formal theorists have privileged any one additional dimension as pervasively form-articulative in tonal music, this has been the dimension of harmony rather than motive and rather than contrapuntal structure. But, as I have proposed elsewhere (Schmalfeldt 1991), a correspondence between the closed, nonmodulatory theme, in Schoenberg's tonal sense, and Schenker's complete middleground harmonic-contrapuntal structure, that is, a middleground replica of an Ursatz-form, arises when one acknowledges that both presume the perfect authentic cadence as the conventional formula for closure. Moreover, the theme whose cadential goal is the dominant (as in exx. 2c and 6b) or whose opening phrase ends with a half cadence (as in exx. 4b, 5, and 7) often stands in direct association with Schenker's interruption process. Put briefly, that each of the "phrases" drawn here

Schmalfeldt, Coming to Terms 111 from Westergaard the Schenkerian, when examined within larger contexts, expresses an identifiable formal function within a Schoenbergian theme-type cannot be regarded as purely coincidental. Westergaard's avoidance of\ formal-functional terminology in no way diminishes the value of his theory, nor would such an approach have been expected of him in 1975. Only recently have Schenkerians begun to reassess Schenker's outspoken condemnation of conventional Formenlehren as the hyberbolic polemic it might partially have been. But, far from altogether abandoning formal concepts, Schenker ultimately proposed a new theory of form, one that would restore the role of counterpoint as inseparable from harmony by positing the origin of form in the Ursatz itself ([1935] 1979, 130). All the more striking, then, and even for 1975, is that not once within Westergaard's text does he use the term form. One cannot help but speculate that, like the terms stylistic and style for Westergaard in 1972, form is a "dirty word"- occasionally invoked, but "rarely with serious intent" (Westergaard 1974, 71). If this is the case, then one should want to know why. In his response at the Society for Music Theory special session, Mr. Westergaard emphatically confirmed the obvious: that it was not the purpose of his work to propose how one might model a composition upon established formal or stylistic tonal conventions. Rather, and in my own words, Westergaard's metalanguage addresses how to generate a tonal composition through ever-increasing spans of time. Or, as since put to me by one of his colleagues, the topic of Westergaard's text is tonality, for which reason the domain of his theory could perhaps best be described as the "infinite set of all possible tonal pieces," rather than "the finite set of actually existent tonal pieces-the canonical repertory of the eighteenth and nineteenth centuries." IfWestergaard's species counterpoints are to be understood as veritable compositions that fulfill all "necessary and sufficient conditions" for tonality, tonal music could have evolved without thematic conventions, and form does not bear directly upon the issue of tonalness.4 Putting aside, as perhaps tangential, the question of tonality in seventeenth-century music, or the notion that the canonical (as opposed to noncanonical but documented?) repertory of the eighteenth and nineteenth centuries constitutes the finite set of actually existent tonal pieces, let us consider certain apparent contradictions arising from this interpretation of Westergaard's thought. Here is a tonal 4For these clarifications I wish to express my warmest thanks to Stephen Peles (letter to the author, 9 February 1992).

112 In Theory Only theory that is nothing if not listener-oriented; its chapter entitled "Phrases, Sections, and Movements" begins with a detailed physiological discussion of human limitations for estimating periods of time (ITT, 309) and takes this problem-the listener's difficulties in trying to grasp large-scale span structures-as the one that must be addressed "if we want to make a structure clear to our listener, either at the level of the individual phrase or for a whole movement or piece" (ITT, 312). Now, within the evolution of tonality, how is it possible that composers from one generation to the next could have avoided addressing similar problems, ones that unquestionably fall within the domain of form? How, indeed, was it historically possible for tonal works to embrace ever longer time spans within the context of ever more complex textures and tonal extensions, if the issue of form does not bear directly upon that evolutionary development? In his historical investigation of perspectives on form, Mark Evan Bonds provides ample evidence that, when sustained theoretical accounts of form began to emerge toward the end of the eighteenth century, these drew upon the metaphor of rhetoric, an art in which clarification serves the goal of persuading the listener. From this early perspective, form is understood as a process, or manner, in which a work's content is made intelligible to its audience (1991, 5, 9). As Bonds astutely notes, Schoenberg's much later stance on form echoes the late eighteenth-century rhetorical, listener-oriented viewpoint. "Form in music," according to Schoenberg, "serves to bring about comprehensibility through memorability" ([1947] 1984, 399). Despite its eschewal of the term form, Westergaard's metalanguage betrays the same concerns. The question of whether or not tonality could have evolved without thematic conventions seems to have value only if one is willing to play the game of distinguishing between, on the one hand, tonality as a concept and, on the other hand, tonal composition, an activity in which--current prejudices against conventions notwithstanding-the latter have again and again played a fascinating role. Schoenberg's reference to memorability seems relevant here: although Schoenberg and Schenker both perpetuated the nineteenth-century glorification of innovation over convention, idea over style, the genius over the average talent, Schoenberg's pedagogical interest in identifying conventional late eighteenth-century theme-types suggests that he understood an important, even innovative, feature of the late classical style: its capacity to invoke the listener's memory of tonal conventions already established, only then to thwart the listener's expectation. From this it might not be going too far to say that tonality and the dramatization of tonality evolve together. Historical contingencies of innumerable

Schmalfeldt, Coming to Terms 113 kinds most surely contributed to this spectacular development. It remains for us continually to explore the encounters with other modes of expression, the exigencies, the passions, even the accidents, that tonal music has faced as it has passed from one generation of composers to the next. Within Westergaard's theory, the composition of a phrase rests upon the very operations involved in relating notes to primary and secondary beats at the level of the measure, namely, the operations of segmentation, delay, and anticipation. It follows that for Westergaard the phrase is the next larger unit after the measure within a hierarchy of levels named "according to the time spans associated with them" (ITT, 375): the level of the beat, the measure, the phrase, the section, and the whole movement. One can only regret that even though the time-span levels of phrase, section, and movement correspond quite obviously with what formal theorists regard as hierarchical formal levels, they are not to be understood as such. Yet, the correspondence exposes what Westergaard's theory holds in common with the most compelling and influential tonal theories of our time. Like Schenker's work, and like the ardent investigations of the finest formal theorists, Westergaard's study reflects an intensive, probably lifelong, consideration of the formal as well as tonal conventions and innovations that characterize the music of the eighteenth and nineteenth centuries-the repertory that, perhaps more than any earlier body of Western-European music, clarifies its time spans through well-articulated formal units. Finally, Westergaard's work sheds new light on the most fundamental of tonal conventions, the composing-out of an Ursatz-structure.

114 In Theory Only References Bonds, Mark Evan. 1991. Wordless Rhetoric: Musical Form and the Metaphor of the Oration. Cambridge, Mass.: Harvard Univ. Press. Caplin, William E. 1986. Funktionale Komponenten im achttaktigen Satz. Musiktheorie 1/3: 239-60. _ 1987. The "Expanded Cadential Progression": A Category for the Analysis of Classical Form. Journal of Musicological Research 7: 215-57. Cone, Edward T. 1968. Musical Form and Musical Performance. New York: Norton. Green, Douglass M. 1979. Form in Tonal Music. 2nd ed. New York: Holt, Rinehart and Winston. Koch, Heinrich Christoph. [1787] 1969. Versuch einer Anleitung zur Composition. Vol. 2. Hildesheim: Georg Olms. Morgan, Robert P. 1991. Review of Phrase Rhythm in Tonal Music by William Rothstein. 19th-Century Music 15/1: 75-80. Ratz, Erwin. 1973. Einfuhrung in die musikalische Formenlehre. 3d ed., rev. and enl. Vienna: Universal Edition. Riepel, Joseph. 1755. Anfangsgrande zur musikalischen Setzkunst. Vol. 2, De Grundregeln zur Tonordnung insgemein. Regensburg: Johann Leopold Montag. Rothstein, William. 1989. Phrase Rhythm in Tonal Music. New York: Schirmer. Schachter, Carl. 1980. Rhythm and Linear Analysis: Durational Reduction. In Music Forum 5, ed. F. Salzer and C. Schachter, 197-232. New York: Columbia Univ. Press. Schenker, Heinrich. 1926. Mozart: Sinfonie G-Moll. In Das Meisterwerk in der Musik. Vol. 2. Munich: Drei Masken Verlag. Reprint Hildesheim: Georg Olms, 1994.

Schmalfeldt, Coming to Terms 115 _ [1935] 1979. Free Composition. Trans. and ed. by E. Oster. New York: Longman. Schoenberg, Arnold. 1967. Fundamentals of Musical CompositionEd. G. Strang and L. Stein. New York: St. Martin's.._ [1947] 1984. Brahms the Progressive. In Style and Idea, ed. L. Stein, 398-441. Berkeley: Univ. of California Press. Schmalfeldt, Janet. 1991. Towards a Reconciliation of Schenkerian Concepts with Traditional and Recent Theories of Form. Music Analysis 10/3: 233-87.. 1992. Cadential Processes: The Evaded Cadence and the "One More Time" Technique. Journal of Musicological Research 12/1-2: 1-52. Sessions, Roger. 1950. The Musical Experience of Composer, Performer, Listener. Princeton: Princeton Univ. Press. Westergaard, Peter. 1974. On the Notion of Style. In Report of the Eleventh Congress of the International Musicological Society, Copenhagen, 1972. Vol. 1. Ed. H. Glahn, S. Sorensen, and P. Ryom. Copenhagen: Edition Wilhelm Hansen.. 1975. An Introduction to Tonal Theory. New York: Norton.

forum Analysis-What Is it Good For? Just as the following essay derives its title from a well-known rock-'n'-roll anthem, so the title of this forum itself borrows lyrics from a hit song by Edwin Starr (1970). The question embodied by the forum's title has been raised repeatedly at least as long as structural analysis has been practiced, and answers have varied widely. While analysis has had its detractors (e.g., Kerman 1980) among scholars of the repertoire that spawned it (Western art music), resistance has proven even stronger in popular-music studies. A provocative essay by John Covach launches this forum. It thoughtfully considers the issues and comes out in favor of structural analysis of popular music. Whether you agree or not, we hope you find it stimulating. We welcome responses, which may lead to a continuation of this forum. To appropriate yet another popular-music title: "Who's Next?" References Kerman, Joseph. 1980. How We Got Into Analysis, and How to Get Out. Critical Inquiry 7/2: 311-31. Reprinted in Write All These Down: Essays on Music, 12-34. Berkeley: Univ. of California Press, 1994. Starr, Edwin. 1970. War. Gordy Records 7101.

We Won't Get Fooled Again: Rock Music and Musical Analysis1 John Covach I In 1971 the British rock band The Who released their seventh album, entitled Who's Next. The final track on the LP, the eight-and-a-half minute "Won't Get Fooled Again," is for the most part a hard-driving rock number.2 The arrangement of this song is perhaps most noteworthy for its use of a repeated-note figure played by the organ, occurring especially in the introduction and in two instrumental interludes, and for Roger Daltry's two excruciating screams, the second of which, occurring immediately before the final verse, must surely be considered among the most famous screams in all of rock music. The lyrics of "Won't Get Fooled Again" represent Pete Townshend at his cynical best. Writing in the first person, Townshend portrays a feeling that political revolutions change very little for those not in political power; for the average person--or in this case, the 1 An earlier version of this paper was presented at the annual meeting of Music Theory Midwest in Madison, Wisconsin, 16 May 1993. I would like to thank Susan Cook, Marianne Kielian-Gilbert, Allen Forte, Walter Everett, and David Schwarz for reading that earlier version and offering many helpful comments. The opinions that follow are, of course, my own. 2"Won't Get Fooled Again," words and music by Pete Townshend, from the LP Who's Next, Decca DL 79182, 1971.

120 In Theory Only average restless youth-nothing seems to be significantly changed the day after the revolution. Following the second organ interlude, which terminates with a Keith Moon drum solo, Daltry lets out his famous scream. It is a scream of recognition and horror: our singer discovers that the "new boss" is no different from the "old boss."3 As its title indicates, this essay will explore issues in the analysis of rock music. Within the academic community of musical scholarship, musical analysis is usually considered to fall within the domain of the discipline of music theory. Certainly musicologists and popular-music scholars incorporate musical analysis into their work to some degree, but it is music theorists who have developed and routinely employ a number of sophisticated techniques and systems for analysis. This essay, then, will focus in large part on how music theorists might approach the analysis of rock music within their own disciplinary contexts. It will, however, also be concerned with the ways in which music theory and analysis can make a contribution within the disciplinary context of popular-music studies generally. Indeed, the question of how music theorists might approach the analysis of popular music-and even whether they should consider popular music at all-is one that affects both the disciplines of music theory and of popular-music studies. These two disciplines, however, have tended to ignore one another: theorists have been occupied almost entirely with the analysis of music within the European art-music tradition, and popular-music scholarship has tended to focus its attention more on cultural, social, and economic contexts and less on the musical texts themselves. I will suggest in this essay that there is an interdisciplinary middleground that shows a potential for enriching both disciplines. I will focus my remarks in this essay especially on rock music, since that is the area within popular music that my own research addresses. I will argue that Townshend's parable-like lyrics sound a warning that must be heeded as we consider, first, the role popular music might play in the ways in which theorists will think about music and music theory in the future; and second, the role that musical analysis should play in the study of popular music in a broad sense. In the first case, theorists might well ask themselves why they should be concerned with popular music at all. Having adapted 3it should be pointed out that, according to this interpretation, Daltry should have employed the word "nol" for this scream. Instead, he employs the word "yeah!" The latter is, however, the standard word/syllable employed in rock screams; "yeah!" should thus not be understood literally in this case-and this is obvious from the context-as suggesting that illusory political change is somehow positive.

Covach, We Won't Get Fooled Again 121 Townshend's song title for my essay, I am obliged to explain how I feel theorists have been in some sense "fooled"; after all, if one has not been fooled initially, how could one be fooled again? I will therefore address this issue first. Second, I will consider two positions that address the analysis of popular music that have been forwarded outside of the discipline of music theory; the first position comes from the field of musicology, and the second arises from popular-music studies. Both of these positions are critical of the notion that traditional analysis can offer much to popular-music studies and even assert that such analytical perspectives can distort an interpretation of the music in fundamental ways. I will argue that both of these positions have problems, and I will identify and discuss these. Finally, I will suggest a number of reasons why I feel some theorists may want to consider investigating rock music, why the study of rock music can make a positive contribution within the music-theoretical discourse, and how the analysis of popular music can make a significant contribution to the field of popular-music studies. UI It is probably safe to say that music theory as a professional discourse is currently in a period of critical self-reassessment; at music theory conferences one often hears such Kuhnian terms as paradigm shift and post-paradigm period (Kuhn 1970) bandied about by colleagues engaged in informal discussion. Much of this discussion can be organized around two intimately related questions: (1) How should theorists study music? and (2) What music should they study? In terms of analysis, for example, techniques and methods influenced by literary theory have made significant inroads in the discourse.4 In terms of the musical works that theorists analyze and theorize about, criticism that the canon of "great works" is too narrow and must be expanded to include a wider range of styles and cultures has perhaps 4To observe the influence that literary criticism has had on music theory in the last decade-and many articles could be cited-one need only consider such widely read articles appearing in the journal of the Society for Music Theory as Littlefleld and Neumeyer (1992) and McCreless (199 1). Straus (1990) was presented the society's annual book award in 199 1. See also the 1992 Society for Music Theory Keynote

122 In Theory Only caused some theorists to explore the analysis of nonwestern and popular musics.,5 In order to examine the question of analytical paradigms in music theory, I would like to turn first to a brief consideration of the work of the Viennese theorist Heinrich Schenker. I do so in part because his theories form one of the dominant paradigms within the discipline of music theory, and if one can propose that theorists have been fooled to some extent by their own theories, then one should expect that this situation may be found in the work of Schenker and his various students and followers. I also choose to consider Schenker because the relationship of his theory to the musical literature it addresses is clear cut. This is important because I will focus below on whether or not Schenker's theories-and analytical theories generally-can be separated from a literature with which they are intimately bound. As one surveys Schenker's written work, and especially his theoretical writings after 1904, one notes that Schenker is principally concerned with the music of a relatively limited group of composers, all of whom are German, Austrian, or strongly identified with the German musical tradition.6 Schenker's notion of the superiority of the German musical genius is in fact central to his musical world view.7 Schenker's well-known position, in a nutshell, is that a certain group of German composers, living over a period of roughly two-hundred years, raised music to the status of the masterwork; music before Bach is viewed as evolving toward the masterwork, music after Brahms (as well as the music of Wagner and his followers) is degenerate. Many critics of Schenker's view would probably label it "ethnocentric" and "elitist."' Conservative supporters of Schenker's position might suggest that Schenker was right; the music he discusses is superior to other music. More moderate supporters of Schenker's 5 Lewis Rowell's work on the music of India, for example, culminates in his 1992 book devoted to music in early India. 6Joseph Kerman's harsh criticism of Schenker on this point is well known. He writes that "in his tacit acceptance of received opinion as to the canon of music's masterpieces, Schenker exemplifies more clearly than any of its other practitioners one aspect of the discipline of analysis" (1980, 317). In Schenker's defense., it might be added that while he may have focused his attention on the music of a restricted number of composers, within that particular body of musical literature Schenker knew and studied a great number and tremendous variety of works. 7Fra careful consideration of the role of the genius in Schenker's thought, see Cook (1989). 8See, for example, Cook (1987, 57-9) for discussion of Schenker's eltit

Covach, We Won't Get Fooled Again 123 theories might claim that whether or not Schenker was right about the literature he explored, we must study his writing as if he were right, suspending judgment for the sake of a hermeneutic understanding. But I am not concerned here with engaging the question of Schenker's musical values in any absolute sense. I would merely suggest that the success of Schenker's theories-and by Schenker's theories I mean not only the Ursatz-dominated late writings, but also the early- and middle-period works--depends in large part on the constraints Schenker placed on the body of musical works that he considered. Schenker's writings are as powerful as they are because Schenker was able to draw out generalizing principles from a body of musical works that he knew were related to one another before he ever began. Schenker started with a repertoire of German masterworks and, with the famous exception of Wagner's works, studied these masterworks-or other works in the same tradition-throughout his entire career.10 His theory is not intended to prove that these pieces truly are masterworks-for after all, Schenker felt it was his responsibility not to test masterworks, but to learn from them '-his theory instead tells us with only a few exceptions how these masterworks are related to one another.12 9In the United States, theorists tend to view Schenker's theoretical work as culminating in Free Composition and the Meisterwerk essays. Thus, earlier writings tend to be viewed as teleologically oriented toward the later writings, and Oswald Jonas's annotations in the English translation of Schenker's Harmony are just the most obvious instance of this teleological approach. When one therefore speaks of "Schenker's theory," one almost always means Schenker's late theory, and so the Schenkerian paradigm with which American theorists work is essentially one founded on Schenker's late work. For a discussion of the issues surrounding the reception of Schenker's work in the United States, see Rothstein (1986). For a critique of what Allan Keiler calls the "teleological straightjacket" in Schenkerian writing, see Keiler (1989). See also McCreless (1997). 10In the Harmonielehre, Schenker describes a passage from Wagner's Tristan und Isolde as a "masterpiece of poetry and articulation" ( [1906] 1954, 112). Cook (1989) discusses Schenker's later, more negative assessment of Wagner. 1"Throughout his writings Schenker maintains a high level of respect for the music he considers to be of masterwork caliber. Schenker assumes that it is he who must rise to the greatness of the masterwork, not that the masterwork must be vindicated by analytical scrutiny. 12 1 do not mean by making this claim to also assert that Schenker himself would necessarily have seen his theory in this way. Instead, I posit that a Schenkerian graph "situates" the particular musical work within the much larger group of works that constitute Schenker's masterwork literature. Graphing a piece, then, tells us less about the piece in isolation from other works (i.e. in an absolute sense) than it does about how that piece is similar or differs from other works within the specified literature. According to this interpretation of Schenker's theory, meaning in a graph is

124 In Theory Only Now if Schenker's theories can really be seen as generated from a specific literature, then one might wonder how effective Schenker's theories can be when applied to literatures other than the one upon which he focussed."1 While Schenker certainly believed, as he states in Free Composition, that his "concepts present, for the first time, a genuine theory of tonal language" (Schenker [1935] 1979, 9), one is tempted to add "in pieces that are important to me." Of course from Schenker's point-of-view, that there are no great tonal pieces outside the tradition with which he is concerned is a priori, so the whole question is-for him at least-meaningless. One might, however, accuse Schenker of making too broad a claim for his theory; perhaps there are pieces that are tonal but operate according to principles that are in some significant way different from those principles that Schenker describes. Considering the relationship of Schenker's theories to the specific repertoire that it describes and generalizes, two approaches have tended to dominate Schenkerian thought: theorists stick with the analysis of pieces within the repertoire circumscribed by Schenker himself; or theorists attempt to modify Schenker's late theory in order to apply it to the analysis of music outside of that repertoire. The pioneering work of Felix Salzer (1952) is a clear instance of this second practice, and in recent years Lori Burns's work on modal middlegrounds in Bach (1993) and Matthew Brown's analyses of Wagner (1989), Debussy (1993), and Jimi Hendrix (1997) have helped revitalize this approach. But if there is a real danger of a Schenkerian being fooled in some sense, it lies in the alluring analytical power of Schenker's theory when it is applied to the repertoire for which it was designed. Schenker's late theory provides the theorist with a powerful analytical apparatus for approaching the music of Mozart, Beethoven, and Brahms. Is it any wonder that some theorists are only too happy to remain within the world of the great German masterworks and rarely stray into other repertoires? All of this discussion of Schenkerian theory has ultimately been in the service of making a very simple point: when a theorist has a strong paradigm from which to work-and this is not restricted to the Schenkerian paradigm-repertoire decisions can sometimes be made relational and contextual. See Covach (1994). 13That Schenker's theory arises from the pieces themselves, and not from preconceived theoretical notions that are subsequently applied to pieces, has a number of strikdng parallels with Johann Wolfgang von Goethe's scientific method, a method that falls into sharp relief in Goethe's critique of Isaac Newton's scientific methodology. See Sepper (1988).

Covach, We Won't Get Fooled Again 125 on the basis of what pieces are likely to work best within that paradigm. The paradigm under consideration could just as easily be pitch-class-set theory or twelve-tone theory; in either of these cases it is perfectly possible to choose repertoire in terms of the theoretical paradigm itself. In pointing this out I do not also mean to object to such a practice; but a common image (or caricature) of music theorists held outside the discipline-and held especially, as I will argue below, by popular-music scholars-is that the only music that theorists value is music that they can get to fit into their established analytical models. Ultimately, this opinion goes, theorists ignore any music that does not fit into one of their pre-established conceptual molds.'4 While this characterization of the discipline is certainly exaggerated, it is not entirely without foundation. To return to the issues outlined a moment ago, theorists may at times determine what music they study by how they plan to study it. If we return for a moment to Pete Townshend's lyrics (and work the metaphor of political revolution for paradigm shift a little harder), the established theoretical paradigms can be thought of as a kind of "old boss"-an old boss that may have the effect of overdetermnining the repertories to which a theorist is drawn. But as theorists endeavor to expand their work to include the analysis of new reportories-in this case popular music-the question that might follow naturally is: What threatens to assume the role of the "new boss"?, and How can music theorists avoid being fooled again? As the traditionally trained theorist turns his analytical attention to popular music, are there traps lying in wait that need to be identified and avoided? In order to pursue these questions, I will now turn to two arguments that have been made outside the field of music theory with regard to -the relationship between analysis and popular music. Mf In her 1989 article, "Terminal Prestige: The Case of Avant-Garde Music Composition," musicologist Susan McClary offers what proves 14 Popular-music scholar John Shepherd states the case as follows: " historical musicology's tendency to neglect popular music as a legitimate object of enquiry is in turn symptomatic of a number of fundamental and related premises within the discipline, namely, that all music conforms to one set of technical criteria... that all music can be judged in terms of these criteria, and that in terms of these criteria tserious-'art music emerges as inherently more valuable than popular music" (1982, 147-48). Shepherd considers theorists to be specialists within historical musicology, and it is clear from the remarks that immediately precede this quotation that this

126 In Theory Only essentially to be a response to Milton Babbitt's 1958 essay, "The Composer as Specialist" (an essay frequently referred to by the title given it by the editors of High Fidelity, "Who Cares If You Listen?"). McClary argues that avant-garde composers-she quotes Arnold Schoenberg, Roger Sessions, Pierre Boulez, and Babbitt--essentially relish the difficulty of their music. The average concert-goer finds this music impossible to understand, and this is something McClary contends provides this cerebral music with a kind of vindication in the eyes of its avant-garde composers. The problem, as McClary sees it, is that these composers insist that their music be understood strictly in terms of its structure; any attempt to understand this music in more "human terms" (whatever that is supposed to mean) is discouraged by the very composers themselves, despite the fact that such a perspective might actually provide a kind of aesthetic entree for at least part of this alienated audience. The prestige of this music depends on its difficulty, and since relatively few are listening, avantgarde composers suffer from a condition McClary diagnoses as "terminal prestige." I will not consider here the many problems in McClary's portrayal of Schoenberg, Sessions, Boulez, and Babbitt. Instead, I want to focus on the way McClary employs popular music rhetorically in the course of her argument. There is much with which the advocate of popular music can agree in McClary's article. She points out that music scholars in the academy, or at least those in music theory, have tended to ignore almost all forms of popular music; while avant-garde composers are predicting the end of music, vital music is breaking out all around them. McClary claims that various popular musics have played a crucial role in musical life in the twentieth century and calls for the serious and careful study of popular music. So far, then, we know what music she thinks we need to study, but the big question is how we should study it; and it is the answer to this second question that I find troubling in McClary's argument. McClary examines an Earth, Wind, and Fire song, "System of Survival."'5 She is mostly interested the way the tune addresses social and political issues, and in her perception that it was conceived, in contrast to the music of Babbitt, by musicians who do care if you listen. McClary is careful to stress the fact that "System of Survival" is not a simple song; it is carefully produced and recorded. But she adds, "the kind of intelligence that shines through this song is of quite a different order: it is an intelligence that accepts the experiences of the 15Skylark, "System of Survival," on Earth, Wind, and Fire, Touch the World, produced by Maurice White for Kalimba Productions, Columbia CK 40596, 1987.

Covach, We Won't Get Fooled Again 127 body-dance, sexuality, feelings of depression and elation-as integral parts of human knowledge that accrue value as they are shared and confi~rmed publicly" (1989, 80). McCla-ry is referring mostly to the lyrics of the song, though she also makes some very brief remarks about the rhythm tracks, the singing, the harmonic structure, and various other aspects of the tune. What I find troubling about McClary's reading of this Earth, Wind, and Fire song is this: she seems to have accepted uncritically the notion that popular music is uncomplicated in the traditional sense, or if it is complicated structurally, or engages our attention along structural lines, then this is not how the song was meant to be heard anyway. In fact, McClary seems to be saying to those difficultymongering avant-garde critics of popular music, "OK, you're right, this stuff isn't very interesting structurally, so here are some ways in which it is interesting." Based on her description, one might think that a large part of the Earth, Wind, and Fire album on which this track appears is given over to the type of hip political statements that she praises so warmly in her article. But in fact there are only three other numbers that contain lyrics addressing social issues.16Frtems part, the rest of the album is, like much art music, music about itself or music about other music. There are a number of extremely interesting structural moments, some clever references to other tunes and styles, and a lot of masterful playing, singing, production work, and song-writing. As McClary states, this music is popular, and that popularity is certainly the result of a number of factors, including such things as the marketing of the product, radio and MTV airplay, etc. But surely one factor that accounts for the effect of this music is the way the pitches and rhythms go-the structure of the music itself."1 Along with McClary, I believe we need to devote more scholarly attention to popular music, and I agree with most of the historical reasons she gives for doing So.18 But in McClary's argument a "new boss" emerges that threatens to replace the old one; for her, the most valuable interpretation of a piece is the one that is most informed by its social and cultural context. Like the Schenkerian paradigm 16The lyrics to "Evil Roy," "Money Tight," and "Touch the World" suggest that the individual needs to rise above material circumstances. In the case of "Touch the World," the lyrics by Rev. Oliver Wells suggest turning to Jesus. These songs, then, are not nearly as cynical as "System of Survival" and cannot be thought of as political in the sense of suggesting the need for change in one's external circumstances. 17Consider, for example, the instrumental cut "New Horizons," composed, produced, and arranged by Bill Meyers. lf.1i oiini frhrage i clr^n Wle 19)

128 128 In Theory Only discussed a moment ago, a strong sociological paradigm can also attract the scholar to a certain repertoire; in this case, McClary is drawn to consider this particular Earth, Wind, and Fire tune because it can be shown to do some of the things that she presumably wants music to do. But if there was a chance of being fooled before, it is equally possible in the present instance. McClary's choice of a musical example in this case is just as motivated by ideology as are the choices of any theorist she might care to cite, and I am not sure that she would disagree on this point."1 By carefully considering her examination of "System of Survival," the reader unquestionably comes to a greater understanding of McClary's intellectual position, but I am not sure one learns very much about the music of Earth, Wind, and Fire, or about popular music generally. McClary clearly directs the reader's attention to popular music because it raises issues that are routinely ignored in the professional discourse of academic musicology and music theory. In McClary's argument, popular music constitutes a kind of disciplinary disruption to the standard picture that scholars routinely paint of music in the twentieth century. But while the example she chooses is effective in such a role, it is certainly not the case that all popular music--or even all Earth, Wind, and Fire music-would have such a marked disruptive effect. Casting popular music in the role of the significant disruption, and in so doing emphasizing the ways in which it is different from art music (a difference that is asserted rather than argued), proves ultimately to be just as distorting as any attempt to emphasize the similarities between the two broad styles could ever be.20 IV While McClary devotes only a few sentences to an examination of the music-technical aspects of "System of Survival," she does use technical terms that are usually employed in the analysis of European art music. She writes, for example, of "pungent dissonances that refuse to resolve," of the "continual resistance to harmonic closure," and of the "absence of the secure harmonic foundation that usually grounds such music" (1989, 78). These technical descriptions are, of course, used to support her sociologically informed reading of the song. But 19 For her critique of the field of music theory, see MoClary (1985). 20The tendency of popular-music scholars to assume that popular music and art music are fundamentally dissimilar is taken up in greater length in Covach (1997b). See also the discussion of this issue in Moore (1993), pp. 11-15 esp.

Covach, We Won't Get Fooled Again 129 there are a number of scholars in the discipline of popular-music studies who might advise McClary to be cautious in her use of such traditional analytical terms. For popular-music scholars such as Richard Middleton, John Shepherd, or Peter Wicke, descriptive terms derived from the study of western art music are ideologically loaded; by even employing such technical terms and the conceptual prejudices that they are thought to imply, one risks interpreting the music according to analytical criteria that are foreign to the music itself. In other words, since most of our music-analytical paradigms have been developed to examine music in the European tradition, they are inherently unsuited for the analysis of popular music. Here, surprisingly, our conservative Schenkerian and radical cultural theorist take a similar position with regard to the what and how of analysis. Both maintain that an analytical system and the musical repertoire it describes should be perfectly matched: the analytical system should be developed out of the specific repertoire under consideration. The Schenkerian may avoid analyzing rock music because the music may not produce satisfying results according to the paradigm; the sociologically oriented popular-music scholar will likely be glad to see the conservative analyst avoid popular music altogether. For example, in proposing that rock music must not be analyzed according to what he terms "the Beethoven tradition," Peter Wicke writes, "Thus, in order to take rock seriously as music, we need to investigate the conception of music which underlies it rather than apply aesthetic criteria and musical models that are completely alien to its cultural origins" (Wicke 1990, 2).21 Along similar lines, John Shepherd writes, "While it is true that historical musicology has developed a formidable range of analytic techniques and terms for coming to grips with the internal parameters of 'music,' such techniques and terms have a very limited application. It is not possible, for example, to agree with Wilfrid Mellers [1973, 15-16] that there are such things as objective 'musical facts,' necessarily susceptible to explanation through a terminology 'which has been evolved by professional musicians over some centuries" (1982, 146). Richard Middleton, in a careful and instructive assessment of the applicability of traditional modes of analysis to popular music, writes, "On the other hand, terms are commonly ideologically loaded. 'Dissonance' and 'resolution' immediately suggest certain harmonic procedures, and a string of associated technical and emotional 21It seems to me that the categorial assertion that the European tradition is completely alien to the origins of rock music is wrong-headed and worth far more careful consideration than it has been given in popular-music studies.

130 In Theory Only associations. 'Motive' immediately suggests Beethovenian symphonic development technique" (1990, 104). I will return below to Middleton's position in regard to the application of traditional analytical techniques to popular music. For now, however, I would like to focus on the problem that these popular-music scholars have posed for music theorists interested in the analysis of rock music. According to Middleton, Shepherd, and Wicke, the application of analytical paradigms developed in the study of art music to popular music (and rock) is likely to produce distorted interpretations. A common target for this kind of criticism is the work of Wilfrid Mellers. Mellers's books on the music of the Beatles and Bob Dylan are often accused of presenting the music of these artists as if it were art music. Middleton, for example, takes Mellers to task for privileging "the areas of tonality, melodic contour and, especially, harmony" (Middleton 1990, 113). Middleton is especially concerned with Mellers's Beatles analyses, where "almost any analysis can be taken as an example of the way harmonic progressions are automatically seen as the most interesting, the most interpretively important, aspects of the music" (113). Rock critic and sociologist Simon Frith finds Mellers's lack bf attention to the social dimension of this music troubling. Frith states that "Wilfrid Mellers's scholarly books on the Beatles and Bob Dylan, for example, describe in technical terms their subjects' transcendent qualities; but they read like fan mail and, in their lack of selfconscious hipness, point to the contradiction at the heart of this aesthetic approach" (1987, 136). I find at least two serious problems with this critique of traditional analytical approaches. First, none of these authors demonstrate a close familiarity with music theory and analysis as it has been practiced in the discipline recently; one often wonders about whom they might be writing.22 Middleton does devote a considerable amount of discussion to possible applications of Schenkerian analytical techniques to popular music; still though, he writes not as a professional theorist, but rather as someone who has explored theoretical approaches to popular music.23 Writers in popular-music scholarship sometimes set up the theorist or musicologist as a straw 221n light of this, one might even wonder whether these critics are reacting principally to the discourse of music theory at all; it sometimes seems as if these writers are reacting against the way theory and analysis were taught when they were students. 2-3The chapter on analysis in Middleton (1990) is, along with Moore (1993), Brackett (1995), and Tagg (1982), undoubtedly the best music-analytical work to come out of the field of popular-music studies.

Covach, We Won't Get Fooled Again 131 man, as a caraciture that serves as a foil to their own ideas. It is as if these writers were against the idea of theorists examining popular music as a matter ofprinciple. The second problem I find with this approach has to do with the theme I have been following throughout this paper; that is, by insisting that we cannot view popular music through our current set of analytical lenses, these scholars risk allowing a set of a priori assumptions to dictate methodology. It seems clear that the time to judge the fruitfulness of an approach is after a significant amount of sophisticated work has been done, not before it has been done. If such work is to be done, it seems obvious that music theorists and analysts possess the technical skills to do it. V To summarize my argument up to this point, then: it is a mistake to ignore the analysis of rock music because it does not fit current musictheoretical and analytical paradigms in obvious ways; it would also be unwise to take up the study of rock music simply because it works nicely according to a more sociologically oriented paradigm and to insist in so doing that rock music cannot work in more traditional ways too. Finally, rejecting the applicability of current analytical methods to rock music is premature and ultimately unnecessarily limiting. In considering why theorists might be interested in the analysis of popular music in general, and rock music in particular, it is helpful to return to the questions, What music should we study? and How should we study it? Is it, for example, possible to adapt current analytical approaches to the task of analyzing rock? Recent analytical work has suggested that it is. Brown (1997), Everett (1985, 1986, 1992, 1995, 1997), Kaminsky (1992), and Burns (1997) have used modified Schenkerian approaches in work examining the music of Jimi Hendrix, the Beatles, Paul Simon, and k. d. lang.24 Far from demonstrating that this music is somehow just like art music, these analyses have suggested that while rock music can at times hold certain structural characteristics in common with Schenker's masterwork literature, it also has certain musical characteristics that are all its own. Rock music raises issues in tonal theory that simply do not come up in the consideration of the masterworks, and that this music is different in these often tacit ways is partly what we mean 24See also Allen Forte's (1993) application of Schenkerian analysis to the songs of Cole Porter and the American popular ballad generally (1995).

132 In Theory Only when we say that rock constitutes a different style of music. To the extent that theorists are interested in developing their theories of tonal music in ways that cross repertory boundaries, the analysis of rock music can make a significant contribution. As mentioned above, Schenker's theory is one that arises from a particula~r repertoire. There are other theoretical approaches, however, that do not arise from some specific body of works. Style theory, for example, especially as articulated by Leonard Meyer (1989), considers how styles can evolve in general ways. Meyer's notion of style change, for example, generalizes across a number of historically and geographically situated styles. Rock music provides a ready testing ground for many of Meyer's notions of how styles change; because rock music is disseminated almost immediately after it is produced, the timneframe within which style change can and does occur is drastically shortened. Consider, for example, the amazing development of rock music from the British invasion of the 1964-66 period, to the psychedelia of the 1967-69 period, to the explosion of widely divergent rock styles in the early 1970s. One only needs to compare an early Beatles album with, say, King Crimson's In the Court of the Crimson King of some five years later to underscore the speed and magnitude of this stylistic transformation. This rapid style change is sometimes thought to be due to a superficial demand on the part of the rock consumer for constant variety. But one might also posit that styles changed so quickly because innovation was absorbed and adopted almost instantaneously by the musicians involved. In any case, I am convinced that the general music-technical mechanisms of style change in the rock music of the 1960s and 1970s are ultimately not much different from those operative in other historical periods. As mentioned at the beginning of this essay, music theory has begun to incorporate techniques and methods drawn from other disciplines. Rock music can serve as a focal repertory for testing the effectiveness of some of these ideas in musical analysis, especially because approaches borrowed from other disciplines are not as repertoire-dependent as those developed inside of the discipline. Thus, as I have attempted to show in recent a-rticles (Covach 1990, 1995, and 1997a), notions of stylistic competency and intertextuality can be very useful in unpacking the effect of certain kinds of rock music. Intertextuality and stylistic competency are especially applicable, for instance, to the new-wave groups of the latter third of the

Covach, We Won't Get Fooled Again 133 1970s-groups whose music depends upon the listener's ability to identify references to earlier styles in rock music.25 In addition to enriching our perspective on current analytical paradigms, the study of rock music also suggests that there are particular analytical issues that arise in the study of rock that may not arise as obviously in other, more traditionally studied repertoires. Consider tone color and instrumentation, for example. A large part of the aesthetic effect of much rock music depends upon certain precise timbres: what would the Moody Blues be without their Mellotron, Jimi Hendrix without his Stratocaster, or the Byrds without Roger McGuinn's Rickenbacker electric twelve-string guitar? These sounds can become referential in precise ways and this referentiality in the work of later groups can take on a tremendous significance. An analytical apparatus that accounts for these and other kinds of timbral relationships in rock music could in turn be applied to other repertoires, even those in the art-music tradition. Thus, while rock music benefits from the application of established analytical approaches, it also, potentially at least, has something to give in return; it can perhaps address our attention to aspects of familiar repertoires that have been less carefully examined within the discipline. In short, my position is not that theorists and analysts should consider rock music simply because it is there, although that may be a good enough reason for musicologists to consider it.26 Theorists should pay more attention to rock music because it is interesting, and it is interesting because as a repertory it challenges disciplinary assumptions about what music is, how it can work, and how we experience it. I do not think one should use rock or popular music as a kind of club with which to beat the avant-garde and the structuralism or formalism that that music may be seen to represent. I also do not think that one must necessarily adopt a sociological orientation in the study of rock music.27 Certainly the socially 25Moore (1993) and Brackett (1995) also employ style-based approachs that depend on notions of competency in the analysis of rock music. 26I take up the relationship between rock music and musicology, as well as the relationship between musicology and popular-music studies, in Covach (1997b). 27In making this point I do not mean to suggest that I am opposed to approaching popular music in a way that is essentially sociologically oriented. McClary's discussion of Madonna's music (1991), for example, employs analytical techniques usually associated with the analysis of art music in the service of a sociologically oriented approach. Walser (1992, 1993) also uses technical analysis effectively in studies that ultimately focus on social and cultural issues. I am merely arguing that an investigation of popular music need not be motivated exclusively or

134 In Theory Only determined elements in rock music must be considered and this will enrich analysis, but whether or not the analytical argument principally addresses sociological concerns is ultimately a question of interpretive emphasis and, ultimately, an issue of intellectual freedom. While I have argued that music theory would benefit greatly through a closer engagement with popular music, popular-music studies would also be tremendously enriched by the kind of careful and close musical analysis that theorists could bring to the field. As John Shepherd has pointed out, popular-music scholars often treat the music itself as a kind of "inscrutible black box"; they are keenly aware that music works its effect on some particular audience, but are at the same time almost totally unaware of how these effects are achieved in music-technical terms (1991, 206). In his important 1981 book Sound Effects, Simon Frith characterizes the situation as follows: "Most rock musicians lack formal training, and so do all rock commentators. They lack the vocabulary and techniques of musical analysis, and even the descriptive words that critics and fans do use-harmony, melody, riff, beat-are only loosely understood and applied. I share this ignorance" (1981, 15). Later in the book, Frith argues that the primary focus in the critical evaluation of rock music should always be a social one; he does, however, admit that musictechnical analyses of rock music that address what he calls the "aesthetic question"-"how does music achieve its effects"--could be included as a secondary concern (54-55).28 In fact, most popular-music scholars would not object to the suggestion that some kind of analysis of the actual musical text needs to be done;29 the real questions thus become: What kind of analysis should this be? and Can analysis be a primary focus? Popular-music scholar Allan Moore has argued that musical analysis must be considered a key component in the study of popular music. Reacting against the position taken by Frith in the passages cited above, Moore writes, "The problem is that a commentary that does not have a sound theoretical underpinning is liable to be of uncertain quality at best" (1993, 16). In a later passage, Moore argues primarily by sociological concerns. 281n his keynote address at the recent conference of the International Association for the Study of Popular Music (1993) at the University of the Pacific in Stockton, California, Frith seemed to stake out an even more conciliatory position with regard to close music-technical analysis, suggesting, for instance, that the work of Leonard Meyer might be useful in such endeavors. In his most recent book (1996), Frith draws on the music-theoretical work of Nicholas Cook. 29See, for instance, Middleton 1990, 115-26; and Moore 1993, 15-17.

Covach., We Won't Get Fooled Again 135 for technical analysis as a primary focus. "it seems clear to me that our concern with the music includes, but does not begin from, the ways in which it is used: in other words, the aesthetic question is primary.... Our concern has to begin from the sounds, because until we cognize the sounds, until we have created an internal representation on the basis of their assimilation, we have no musical entity to care about, or to which to give value'" (17). Thus, following Frith, there does seem to be a place-albeit secondary-for technical analysis in popular-music studlies; and following Moore, there is a very important role-a ftmdamental role, even-that musical analysis can play. In terms of the analytical techniques employed in such close textual readings, it seems clear that aftempting to force popular music into models created for the analysis of European art music is bound to produce distortions; at the same time, however, asserting that an entirely new approach to musical analysis needs to be devised especially for popular music seems extreme.30 One problem with the project of developing entirely new modes of analysis is that it presumes that popular and art music are entirely different from one another. This certainly need not be the case, however, as work by Peter Van der Merwe (1989) has suggested. In considering the question of using analytical approaches developed in the study of European art music to study popular music, the matter comes down, in large part, to disciplinary assumptions and prejudices. Popular-music scholars are quick to note what they take to be the silent assumptions of theorists and analysts, and clearly the analyst must admit that there are significant differences between the music of, say, Elvis Presley and Richard Wagner. Residing outside the discipline of music theory, these scholars are able to detect interpretive biases that often go undetected within it. But the crucial point in sounding out our silent prejudices is to avoid replacing one set of assumptions with another, equally insidious 30See Middleton 1990 (172-246), however, for a broad survey of analytical techniques for popular music. Despite the wide range of approaches Middlleton explores, there is little in these pages that could not be represented equally well by relatively mainstream analytical techniques. E~arlier in his book, Middlleton suggests that once the musical field is "freed from the distorting grip" of what he considers to be an ideology that is inscribed on musicology as a discipline, then "the ground is cleared for a useful musicology to emerge" (122). If the disciplinary assumptions associated with mainstream analytical techniques are thus uncovered, it remains unclear why one would adopt new techniques that merely produce similar analytical results in a different way-unless the difference is important, and in that case one might question how effective and complete the initial uncovering of assumptions had

136 136 In Theory Only set. Thus we must be cautious of too quickly rejecting an entire approach to musical analysis, with all the sophisticated techniques that theorists have developed for accounting for the musical text, on the assumption that because such techniques were developed to study art music they could never produce anything but a distorted reading of popular music. Indeed, if we as theorists or as popular-music scholars do not want to be fooled again in regard to musical analysis, we must resist the temptations that disciplinary paradigms can create, or at least be keenly aware of the ways in which these pressures can operate.3' As musical scholarship pays increasing attention to popular music, we need to be sure that we avoid falling into traps that silently reside within our own disciplines: we must avoid creating a "new boss, just like the old boss." 3 'Though I have been considering these assumptions as disciplinary ones-which in itself is a kind of distancing technique of which one must constantly be aware-it should not be overlooked that the assumptions to which I refer are also likely to have some basis in each scholar's own personal experience and background. Thus, for a popular-music scholar the assumption may not simply be that the discipline of music theory has nothing interesting to say about rock music, but also that music theorists like the ones I've known couldn't possibly have anything interesting to say. This, of course, applies equally to the biases of music theorists and analysts. The argument then is not just about bringing disciplines together-something that seems comfortably abstract-but also about bringing people together.

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review The Analysis and Cognition of Melodic Complexity: The ImplicationRealization Model by Eugene Narmour Stephen W. Smoliar Introduction Narmour's second volume, subtitled The Implication-Realization Model, begins by reacquainting the reader with the basic theory presented in his previous book, The Analysis and Cognition of Basic Melodic Stnuctures (Narmour 1990). Since I reviewed this earlier book (Smoliar 1991), I would like to take a similar approach here. My original review began with an attempt to summarize the basic thesis of the book, after which I undertook to challenge its approach on three interrelated issues. First, I asked whether isolating melody was a valid approach. Second, I turned to Narmour's presentation of his goal ("to discover and explain melodic syntax on the lowest level") and asked whether or not melody actually has a syntax. Finally, for the sake of argument, I granted the possibility of a melodic syntax and asked whether or not notes could serve as valid lexical primitives for that syntax. I then argued that confronting these questions would require an alternative to Narmour's approach; I introduced Gerald Edelman's research into perceptual categorization (Edelman 1987) as one such alternative.1 1I have discussed this alternative in greater depth in Smoliar (1992).

144 In Theory Only Finally, I provided a general "balance sheet" of the book's liabilities and assets. I shall structure this review in the same manner, trying to take account of how Narmour's position has changed, how it has developed, and how it has remained the same. What It's All About Narmour's first volume, The Analysis and Cognition of Basic Melodic Structures, begins with the objective of discovering and explaining the nature of "melodic syntax." More specifically, Narmour undertook to describe the syntactic nature of note-to-note relations. Thus, the operative metaphor is one of linguistics; if we view this metaphor as a mapping from linguistic concepts to those of music theory, then Narmour's objective was to trace the mapping of the linguistic concept of parsing. Narmour's second volume, The Analysis and Cognition of Melodic Complexity, takes a slightly different tack in sailing the sea of metaphors: This book and its predecessor explore a surprising idea: that a cognitive "genetic code" enables both naive and experienced listeners to comprehend the entire world of melody.... This code governs (1) complete prospective realizations of melodic implication, which cognitively generate three simple archetypal structures; (2) partial realizations of these, which produce five archetypal derivatives; (3) retrospective realizations and denials of implication, which double the number of archetypal and derivative structures; and (4) the contextual sharing of intervals between all these various types, which geometrically increases both the number and the complexity of the available structures. (ix) The shift from linguistics to biology is not as radical as it may first appear. Narmour's concern has been and remains matters of structure-identifying a suitable foundation of primitive elements and a set of operators through which those primitives may be combined, where both primitives and operators must be expressive enough to accommodate the wide gamut of melody. Following the Saussurian tradition (Nattiez 1990), what we choose to call primitives and operators need not be anything more than a matter of convention unless we deliberately wish our names to invoke very specific associations. Narmour's invocation of biological associations is not new to music theory. In the second volume of his Das Meisterwerk in der Musik yearbook, for instance, Heinrich Schenker (1926) drew upon the adjective "organic" for the titles of two of his articles concerned, respectively, with sonata form and fugue. Narmour, on the other hand,

Smoliar, Review of Narmour 145 does not exercise this adjective; nor does he pursue any specific associations with genes as primitives or with operators of molecular biology such as mutation and crossover. Such analogies tend to be extremely far-fetched with regard to melody, although not entirely out of the question (Iverson and Hartley 1990). Rather, Narmour appears to have shifted his metaphors to biology in order to distance himself from syntax as a model for describing structure, perhaps because associations with natural language can also be highly questionable (Smoliar 1991). In so doing, Narmour has not explicitly addressed any dissatisfaction with a linguistic stance but has simply chosen to minimize his invocation of linguistic concepts, essentially to the point of eliminating them entirely from this second volume. The primitives that provide the foundation for Narmour's structures are the intervals between successive pitches in a melodic line. More specifically, those primitives serve to represent the variety of relationships that may exist between two successive intervals, where the first of these is called the implicative interval and its successor is called the realized interval. The objective of The Analysis and Cognition of Basic Melodic Structures was to introduce the reader to the full gamut of these primitives, illustrating each with musical examples. Unfortunately, as I have already observed (Smoliar 1991), Narmour's initial volume is not an easy book to read. The basic exposition gets buried beneath excursions into psychology and philosophy, not to mention considerable rhetoric about the state of the art of music theory. As a result it is entirely possible that a reader will emerge from this book with some knowledge of the symbols Narmour has chosen but with little intuition for how they actually serve as primitives for his system. It is equally unfortunate that the introductory chapter to The Analysis and Cognition of Melodic Complexity is no more helpful in this regard, but the dedicated student should not lose heart. Psychologist Carol Krumhansl has recently done an excellent job of identifying both Narmour's primitives and the criteria that distinguish them (Krumhansl 1995), and it will be useful to invoke her summary in this review. Krumhansl's summary depends on establishing a necessary set of primitives for describing the primitives, so to speak. It begins with the recognition that each primitive describes a relationship between an implicative interval and its succeeding realized interval. That relationship is "measured" with respect to two parameters: the direction of the realized interval relative to the implicative interval, and the size of the realized interval relative to the implicative interval. Directional relationships are simply classified as the same or different, while relations of size are classified into similar, smaller, and larger.

146 In Theory Only The criteria for what constitutes the same direction are laid out in a "principle of registral direction"; and a "principle of intervallic difference" similarly accounts for what constitutes similar size. Finally, implicative intervals are classified as either small or large: those less than a tritone are small, and those greater than a tritone are large, leaving this interval as an unclassified hedge that may be adjusted to suit specific interpretations. Once all these "meta-primitives" are understood, Narmour's primitives may be defined as in table 1, reproduced as it was presented by Krumhansl (1995). This table does not include Narmour's primitives for duplication (D) and intervallic duplication (ID), which represent exact repetition. These are clearly subsumed by Krumhansl's summary; and it is unclear whether or not they really need to be separated into their own categories, as they were originally presented by Narmour (1990). However, all the labels for the structural primitives follow Narmour's notation. If The Analysis and Cognition of Basic Melodic Structures is all about these primitives, then The Analysis and Cognition of Melodic Complexity is basically about operators that combine them. Fortunately, this exposition is not quite as complicated because, at the end of the day, there is really only one way in which they may be combined-through concatenation. Thus, much of the book discusses what such concatenation entails, again providing illustrative musical examples. However, concatenation is not simply a matter of stringing out primitive symbols from beginning to end in a melody. If we return to the linguistic metaphor momentarily, melodies are articulated by "punctuation marks" that indicate varying degrees of closure between consecutive notes. These points of closure must be identified and honored within the structuring process. Thus, specific notes serve to begin and end structural elements delimited by closure. Those notes then serve as bases for score reduction and are elevated to a higher hierarchical level, where they may be analyzed by the same structural principles. All this is potentially very neat. Furthermore, Krumhansl (1995) has been attempting to justify Narmour's approach through experiments in the perception of melody. The claim that a particular approach to music theory may be grounded in evidence from experimental psychology is seldom made, although Irene Deliege (1987) has undertaken a similar experimental approach towards the "Generative Theory of Tonal Music" of Fred Lerdahl and Ray Jackendoff (1983). From the point of view of the study of music theory, Narmour has benefited from the clarifications of his work teased out by Krumhansl's need to design effective experiments; furthermore, the dropping of the metaphorical baggage of linguistics will probably prove

Smoliar, Review of Narmour 147 advantageous, since there was little to be gained from those metaphors in the first place. Nevertheless, there are other challenges which I have raised previously (Smoliar 1991) that I believe still need to be recognized. At the risk of being accused of raking over dying embers, I would like to return to some of these issues. Table 1. Narmour's structural primitives Structural Primitive Direction Size For small implicative intervals Process, P same similar Intervallic Process, IP different similar Registral Process, VP same larger Retrospective Reversal, (R) different smaller Retrospective Intervallic Reversal, (IR) same smaller Retrospective Registral Reversal, (VR) different larger For large implicative intervals Reversal, R different smaller Intervallic Reversal, IR same smaller Registral Reversal, VR different larger Retrospective Process, (P) same similar Retrospective Intervallic Process, (IP) different similar Retrospective Registral Process, (VP) same larger Outstanding Questions Can Melody Be Isolated? Before I challenged Narmour's appeal to linguistic metaphors, I first raised the question of how realistic it is to isolate melody as an object of study. I observed that while "divide and conquer" is a necessary strategy for just about any large and unmanageable problem, eventual success in conquering has a lot to do with judicious dividing. Narmour certainly does not try to hide the fact that there is more to music than melody. However, his one attempt to address this question explicitly reveals a "vision of division" which, while reasonably intuitive, may not be that all judicious: Obviously, individual parameters-of which I count at least ten (melody, harmony, duration, meter, dynamic, register, tessitura, texture, tempo, and timbre)-are each capable of independently producing groupings whose closural nodes may or may not coincide with the grouping anchor points established by other parameters. (328) From a scientific point of view, the principal problem with this division is that it entails a somewhat arbitrary mix of the quantitative

148 In Theory Only and the qualitative. Thus, we know from the work of Deliege (1987) and others how one may approach duration and meter systematically. Considerable work has also gone into the development of quantitative representations of visual texture (Tuceryan and Jain 1993), but the extent to which any of that work would benefit our understanding of musical texture remains uncertain. When we then progress to a parameter such as tessitura, which is so tightly coupled to the physiology of vocalization in ways that remain far from being modeled, the tools for quantitative analysis seem even more remote. However, even if we ignore Narmour's particular enumeration of parameters and concentrate on his attention to melody, there is a more general ontological question that he appears to wish to avoid: What, after all, is a melody, and how do we know one when we hear it? Is Syrinx a melody simply by virtue of the fact that it is played by an instrument which can only sound one note at a time? Can we apply the same line of reasoning to the gigue movement of Johann Sebastian Bach's D minor partita for solo violin, as Narmour does in his example 12.25 (287), in spite of any intuitions we may have to interpret many of the broad skips as delineating multiple voices of counterpoint realized by this single instrument? Then, when we are actually confronted with notes that sound simultaneously, how do we know where melodies reside within those simultaneities? In examining Etude III from Robert Schumann's Opus 13 (example 4.21 b, p. 108), Narmour has decided that the right-hand passage of the first section, consisting of arpeggiated thirty-second notes in generally broad skips, should be analyzed as a melody, whereas most pianists would interpret it as accompanying embellishment to the melodic line in the left hand. Narmour never defends his decision to call it a melody, leaving the reader wondering whose responsibility it should be to provide a good reason why we should not call it a melody! What may be necessary for an understanding or definition of melody is less of an appeal to the traditions of music theory and music history and more of an attempt to approach music in terms of the sounds it produces and the behaviors that lead to those sounds. If we wish to approach perception from an acoustic point of view, we may think about the sound of music either in terms of how it progresses in time or in terms of how it sounds at a particular moment in time.2 Thus, whether we are listening to a solo instrument, a homophonic choir, or the rich texture of counterpoint, one of our concerns is with the flow of time; we address that concern by trying to identify specific 2If we appeal to the printed score page as metaphor, we might then call these the horizontal and vertical dimensions, respectively, of an auditory experience.

Smoliar, Review of Narmour 149 features that relate the sounds we are hearing in the present to those we have heard in the past and those we anticipate in the future. Similarly, we may concentrate our attention on the immediate present through data available to us by Fourier analysis. If we wish to reconcile this approach of thinking both "horizontally" and "vertically" with our intuitions about melody, then it makes sense to concentrate on features such as pitch or intervals. So, within the context of this approach, Narrnour has probably set off in the most sensible direction; but how sensible is the context itself if we have the courage not to sweep the ontology of melody under the rug? We can certainly approach Syrinx through an acoustic-feature analysis that will provide us with information on pitches and intervals, but can we do the same with the operas of Verdi (or even the solo arias in those operas)?3 If we really want to go "back to the basics" of "the sounds themselves," what are the features we should be identifying? This remains an open question for which this review cannot provide any easy answers. In may be worth considering, however, that physics may be a better source of metaphor than either linguistics or biology. Thus, there are mathematical definitions of energy that may be computed from waveform and spectral representations of musical performances. Are any of those definitions relevant for the examination of music? Can they help us ask questions concerned with how energy changes in the course of a performance, where it goes through peaks and valleys, or whether it is articulated in ways that are consistent with our intuitions of perceptual closure? By concentrating on melody, Narmour has set an agenda, consistent with most of the rest of music theory, based on the assumption that the answers to these questions may be found in music scores. However, there remains the possibility that the scores are not the right place to look and that the answers have more to do with the sounds themselves. Perceptual Categorization Revisited. My personal polemic attitude is that music theory can only advance to the extent that we prevent excessive attention to scores from impeding our progress, and the best way we can wean ourselves away from those scores is by trying to develop a better understanding of how we form perceptual categories. 3lIronically, Narmour never has anything to say about Syrinx (despite its great popularity among music theorists who have chosen to concentrate on melodic analysis) in either The Analysis and Cognition of Basic Melodic Structures or The Analysis and Cognition of Melodic Complexity, while Verdi receives considerable attention in both books and is practically a focal point in Chapter 14 of the latter

150 150 In Theory Only The necessity of perceptual categorization for any pursuit of music theory was the basic thrust of an earlier article (Smoliar 1992), and I have continued to pursue the thread since then. While this is too complicated a question to admit of simple answers, elsewhere I have tried to set a clearer agenda for how we may cultivate our understanding (Smoliar 1995); I would like to elaborate briefly on that agenda in this review. The problem of perceptual categorization may be approached by dividing it (hopefully in such a way as to conquer it) into three subproblems: identifying objects, discriminating objects, and associating objects. Identifying objects entails the problem of establishing which stimuli correspond to objects in the first place. For purposes of just about any systematic analysis, the waveform of an audio signal (such as a musical performance) may be suitably digitized and represented as a sequence of numbers, but that sequence is usually extremely opaque. Nothing in those numbers themselves provides any clues as to what "audio objects" may be or how we might be able to find them. Perception depends heavily on our ability to represent those numbers in terms of quantitative features, which are nothing more than other numerical structures that tend to share descriptive properties with the phenomena we perceive. Being able to identify what those features are is a problem unto itself, and not an easy one at that. Indeed, the primary thesis behind Edelman 1987' is that the problem is best solved through selection from a vast population of options, rather than through the reasoned construction we tend to associate with physics and higher mathematics. If this is the case, then our only option may be to attempt to model listeners, bearing in mind that such models are likely to be extremely demanding on computational resources. The categorization part of the story enters when it becomes necessary to discriminate one object from another. Once we have features that allow us to identify objects-to tell the difference between figures and ground, for example-we need to know when we are dealing with a single object or with a collection of different objects. In 4 Much of this work is based on biological evidence concerned with how feature detectors develop and function in the neuronal structures associated with vision. It goes without saying that the path from ear to brain is decidedly different from that from eye to brain (Gibson 1983). We are probably safe in generalizing that auditory perception relies as much on feature detectors as visual perception does; and, if those feature detectors are developed by a selective process for vision, it seems reasonable to assume that a similar process would determine audition. However, it is very unlikely that we shall learn very much about the nature of the auditory feature detectors themselves by trying to generalize on what we know about visual feature detectors.

Smoliar, Review of Narmour 151 the auditory domain this area has been explored most extensively by Albert Bregman (1990), who has concentrated on features related to spectral analysis. However, Bregman has probably turned up more problems than solutions, which is an indication of just how complicated this particular subproblem is. It is also important to recognize that there are two kinds of discrimination that come into play (and need to be discriminated). One concerns being able to distinguish different categories, such as the sound of a piano and the sound of a harpsichord. However, any category is a category because it is an abstraction of some population of instances; and it is often just as important to discriminate instances within a category. As an example, we need to know how it is that we can distinguish a recorded performance of a Chopin ballade by Vladimir Horowitz from one by Artur Rubinstein. When we deal with both similarities and differences, we then move into the subproblem of associating objects. Categorizations are very rarely cut and dried. Given any two objects, they will have both similarities and differences; and we need to appreciate the nature of both. An excellent (not to mention humorous) example of the subtleties of association was cited in the 1947 essay "Brahms the Progressive" by Arnold Schoenberg: A musician or a music lover might intend to display his own great understanding, good judgement of music, and acquaintance with "some" of Brahms' music. Hence he dared say he had observed that Brahms' First Piano Sonata was very similar to Beethoven's Hammerklavier Sonata. No wonder that Brahms, in his straightforward manner, spoke out: "Das bemerktja schon jeder Esel." ("Every jackass notices that!") ([1947] 1975, 398) We all appreciate this joke, but we have yet to understand the mechanism behind that process of reminding that makes the joke funny. Past research in artificial intelligence has attempted to model such reminding on the basis of the mechanisms of propositional logic (Kolodner 1984); in the realm of human behavior, however, it is very easy to go beyond the bounds within which propositional logic can be effective (Rosenfield 1988). Categorization may provide the key to memory, but we still need to tease out how the key actually unlocks the door. The Value of the Book Liabilities. In my earlier review (Smoliar 1991), I discussed the contentious nature of Narmour's writing style. The tone of The Analysis and Cognition of Melodic Complexity is a little more affable,

152 In Theory Only but Narmour has still not overcome his general preferences for overinflated language, both in the choice of specific words and in the higher-level structures of sentences and paragraphs. If one believes in plain speaking, one is likely to be put off by this book unless one is forearmed with a healthy sense of humor. Indeed, a lighter touch could probably go a long way in making all this material far more accessible; it is most unfortunate that Narmour seems yet to appreciate this fact. Another problem with over-inflated language is that it is often inaccurate. Narmour has insisted on invoking words, such as "network" and "time tag," that already carry significant semantic baggage in other intellectual communities. Those of us familiar with that baggage can only view his own choice of these words as misplaced scientific pretensions. Again, the only word that comes to mind is "unfortunate," particularly given the recognition Narmour is now receiving from the psychology community. Assets. At the end of the day, however, the work remains interesting, and that is what matters most. If melody is not necessarily the right place to look in our quest for a useful set of primitives and operators that may be applied to analytic descriptions of music experiences, then we may still gain from being better acquainted with Narmour's methodology. He is not afraid to take on hard problems; and that courage alone makes his work interesting, even when his manner of delivery is at its most trying. Furthermore, the reader who gets beyond the style of that delivery will quickly discover how much of Narmour's actual technique is admirable and a model for future investigators. Just being able to command as many musical examples as he has done in his two books in impressive in itself.5 It should also remind us that, no matter how carried away we may get with the intellectual intricacies of our questions, if the answers cannot be traced back to the music, they are unlikely to be the answers we seek. 50f course, when there are such a large number of musical examples, minor flaws are inevitable. Thus, the aforementioned Schumann Opus 13 actually has two disjoint listings in the Index of Musical Examples: the example of Etude III is listed as "Symphonic Etudes, III," while another example, based on the theme, is listed as "Etudes in the Form of Variations, op. 13." (There is also the problem that the original index was incomplete, but an erratum sheet has been furnished to provide the missing material.)

Smoliar, Review of Narmour 153 References Bregman, Albert S. 1990. Auditory Scene Analysis: The Perceptual Organization of Sound. Cambridge, Mass.: MIT Press. Deliege, Irene. 1987. Grouping Conditions in Listening to Music: An Approach to Lerdahl & Jackendoffs Grouping Preference Rules. Music Perception 4/4: 325-59. Edelman, Gerald M. 1987. Neural Darwinism: The Theory of Neuronal Group Selection. New York: Basic Books. Gibson, James J. 1983. The Senses Considerd as Perceptual Systems. Westport, Conn.: Greenwood Press. Iverson, Eric and Roger Hartley. 1990. Metabolizing Music. In Proceedings of the International Computer Music Conference. Ed. S. Arnold and G. Hair. Glasgow. Kolodner, Janet L. 1984. Retrieval and Organizational Strategies in Conceptual Memory: A Computer Model Hillsdale, N. J.: Lawrence Erlbaum Associates. Krumhansl, Carol L. 1995. Music Psychology and Music Theory: Problems and Prospects. Music Theory Spectrum 17/1: 53-80. Lerdahl, Fred and Ray Jackendoff. 1983. A Generative Theory of Tonal Music. Cambridge, Mass.: MIT Press. Narmour, Eugene. 1990. The Analysis and Cognition of Basic Melodic Structures: The Implication-Realization ModeL Chicago: Univ. of Chicago Press. ____ 1992. The Analysis and Cognition of Melodic Complexity: The Implication-Realization Model. Chicago: Univ. of Chicago Press. Nattiez, Jean-Jacques. 1990. Music and Discourse: Toward a Semiology of Music. Trans. C. Abbate. Princeton: Princeton Univ. Press. Rosenfield, Israel. 1988. The Invention of Memory: A New View of the Brain. New York: Basic Books.

154 In Theory Only Schenker, Heinrich. 1926. Das Meisterwerk in der Musik, Band II. Munich: Drei Masken Verlag. Reprint Hildesheim: Georg Olms Verlag, 1974. Schoenberg, Arnold. [1947] 1975. Brahms the Progressive. In Style and Idea: Selected Writings of Arnold Schoenberg. Ed. L. Stein, 398-427. New York: St. Martin's Press. Smoliar, Stephen W. 1991. Review: The Analysis and Cognition of Basic Melodic Structures: The Implication-Realization Model by Eugene Narmour. In Theory Only 12/1-2: 43-56.. 1992. Elements of a Neuronal Model of Listening to Music. In Theory Only 12/3-4: 29-46.. 1995. Parsing, Structure, Memory, and Affect. Journal of New Music Research 24/1: 21-33. Tuceryan, Mihran and Anil K. Jain. 1993. Texture Analysis. In Handbook of Pattern Recognition & Computer Vision. Ed. C. H. Chen, L. F. Pau, and P. S. P. Wang. Singapore: World Scientific.

Response to Buchler William Thomson A flush of excitement naturally follows disclosure that one has created a latter-day Greek tragedy-no small accomplishment. But my own flush faded rapidly as I advanced further in Michael Buchler's (1994) review of Schoenberg's Error (hereafter S'sE), which bears that message. The fade crept in with the dawning realization that Mr. Buchler understood little more than the surface of the book's contents, thereby blunting the force of his flattering judgment. Indeed, it would be hard to conceive of more plentiful and deep veerings from verity, more specious interpretations in a single document. I'm reminded of the nice lady who begged Albert Einstein's explanation of his general theory: although she understood most of the words, she understood none of the sentences. But there is a significance to Mr. Buchler's ambitious essay that goes beyond the merits or demerits of S'sE. His difficulty in coping with the book is in part symptomatic of a malaise that has inhibited musical discourse in the United States for some six decades. Carefully nurtured in the periods-perspective of our rich Germanic tradition, a praxis freighted in the minutiae of stylistic description, some novitiates are left unreceptive to systemic generalizations, antagonistic toward attempts to cross (or sometimes blur) border lines in describing the nature of and causes for fundamental musical matters. It is a glorious tradition, but it sometimes makes us reluctant even to engage in discussions of theoretical concepts rather than in nominalistic descriptions of particular artifacts. Careers are made, turfs relentlessly

156 In Theory Only defended, by attending to the every blip of an age, a century, a period, a cathedral, a composer. Acts interpretable as antithetical to such store-housings can be regarded only as seditious, as unhealthy for the greater (the entrenched) cause. For instance, Mr. Buchler repeatedly imputes aesthetic judgments of particular composers and eras to S'sE that are not there, missing the more embracing nature of judgments that are. For these broader reasons, it is incumbent on me, the least disinterested but best-equipped spectator, to respond in detail to his review, not only as a gesture of self-defense but also in the name of a bigger and impersonal cause as well. These motivations and justifications in mind, and in view of the inevitable length imposed by such a line-item task, my response is organized as a succession of paired parts, a kind of recitativo-aria form, if you will. FICTION: "The author places Arnold Schoenberg squarely on the side of Richard Wagner and Dionysius.... However, one could also say that Schoenberg later leaned in precisely the opposite direction, citing the composer's essay 'Brahms the Progressive' (Buchler, 59). FACT: Curiously, the second sentence above discloses what other readers may think I overplayed in S'sE, in observing that "Schoenberg's perspective on the world... [was] marked by oscillating tension between poles of being" (22), deepening this judgment by the suggestion that he inherited distinct genetic traits of opposition from his parents. In conceptually nailing down this equivocating psyche, I continue by saying that Even his frequent equivocations concerning Wagner the avant-gardiste and/or Brahms the revolutionary-classicist suggest the depth of this inner conflict.... Which was his real idol of the immediate musical past? Was it Wagner, the revolutionary dramatist-composer, who showed that tonal indecisiveness performed admirably as an ingredient in the musical mix? Or was it Brahms, the symphonist who continued where Beethoven left off, perpetuator of the classicism begun with Haydn and Mozart? (S'sE, 22) For good reasons, these questions are left unanswered in S'sE, although I later return to this issue of ideological conflict when I define the principal error in Schoenberg's perspective, again alluding to the conservative/radical split: "He opted for Wagnerian chromaticism, but then he relentlessly adapted it to creations of Brahmsian autonomy, as well as to works with programmatic ties or elaborate texts" (S'sE, 175). FICTION: "While Thomson provides extended critical discussions of Schoenberg's aesthetic as reflected in his music... it becomes

Response to Buchler 157 clear, as one moves farther into this book that Thomson's goal is to use Schoenberg's writings on historical and theoretical issues as a springboard for a broad criticism of Schoenberg's music" (Buchler, 60). FACT: I beg Mr. Buchler to cite a page in S'sE on which I share my personal judgment of the aesthetic import of a Schoenberg composition. I observe that "Our vantage point enables us to evaluate the... results of his musical decisions" (178), but I decline the opportunity, keeping to the decision outlined in the book's foreword to talk about theories of music, not Schoenberg's music. Schoenberg may very well have composed divine music; what I argue is that his justifications for doing it the way he did were ill-begotten. The comment in S'sE regarding "our vantage point" leads into a discussion of (1) the failure of such works as his solo concerti (potential crowd-pleasers by the nature of the species) "to establish a performance base with soloists, with orchestras, or with concert-going patrons";1(2) the continuing occasional performances of older works such as Verkldrte Nacht and Gurrelieder (whose very size is a deterrent to performances); (3) the general neglect of his atonal and dodecaphonic works; (4) the anomalous state of works he composed with a "secret" program in mind, as if hard listening over the years would unlock them, much as it would emancipate dissonance; and (5) remarks made by (a) sociologist John Mueller regarding posterity's apparent decision to build a box of honorable neglect around Schoenberg's music, (b) composer G. Rochberg's recognition of the compositional cul-de-sac mapped out by Schoenberg, and (c) composer John Adams's conclusion that Schoenberg's music will remain permanently difficult to approach (S'sE, 178-80). Lastly, my own later observation is that A larger part of [his] music seems destined to survive mainly in a confining academic purgatory thereby permanently illustrating an ever-widening gap that separates his image as "great composer" from the reality of neglect, in public performance, that prevails for all but a handful of his early works and transcriptions he made of others' music..... The Rejection-Acceptance-Assimilation- Endearment cycle for truly great music that shaped his aesthetic perspective appears to have been another case of unfulfilled prophecy. (S'sE, 179-80) These are statements of posterity's conclusions about Schoenberg's music. If I say that "the Edsel was a failure as a marketing venture by 1I delay judgment that a groundswell is represented by Alfred Brendel's recent testimonial ( 1995).

158 In Theory Only Ford," I am not making a personal value judgment; I'm making an empirically verifiable statistical observation. If Schoenberg's music is one day revived, like J. S. Bach's in the nineteenth century, to become a string of veritable household hits, my position remains: however he accomplished it, the apologia he provided for having done it the way he did was myth. FICTION: "Thomson refrains from challenging a more obvious weakness in Schoenberg's (and Schenker's) concept: that the use of the major scale as tonally defining breaks down when any chromaticism is inserted" (Buchler, 61). FACT: I refrain from nothing. It never occurred to me because I'm not aware that the insertion of "any chromaticism" into a tonally defined arena does any such thing, either to Schoenberg's and Schenker's or to my theory. They both, through their respective explanations of secondary and tonicizing harmonic actions within diatonic contexts, deal with such matters convincingly. Where did Mr. Buchler get such a novel idea? FICTION: Thomson pulls the old, out-of-context trick, omitting the parts that really tell us what Schoenberg meant. Then, after quoting a Schoenberg excerpt (S'sE, 58) regarding the remarkable phenomenon of dual keys projected in the "church-modes," Mr. Buchler jabs with "Thomson has distorted Schoenberg's message by means of omission" (61), delivering the knockout punch with Schoenberg's continuation: "This contradiction was first resolved when the two principal modes used today were evolved out of the church-modes into a predominant position. Up to that time music can scarcely be regarded as tonal, in the present sense of the word. On the contrary we must concede that the church-modes do not at all conform to the law of tonality" (Schoenberg 1975, 276). FACT: I search in vain for my distortion of Schoenberg's message. His words could not be more clear about a very simple matter. They describe an imaginable perceptual phenomenon, one he said occurs when listening to music based on the "church-modes." He is saying that there are two structural hierarchies copresent: the key of the underlying tonal series and the key in which the piece really exists. The second quotation Mr. Buchler shoves into the ring, regarding the resolution of "this contradiction," is irrelevant to the perceptual reality Schoenberg reports, regardless of what may have happened in a subsequent evolution of pitch paradigms (major and minor scales). Buchler's special pleading is just that. It merely blows a justifying fog into an area already confused ontologically. For Schoenberg, the potential contradiction (the copresence) was resolved by the invocation of the biscalar system in later music, not in what Schoenberg calls

Response to Buchler 159 music of the church-modes, which in his judgment had prevailed "up to that time." FICTION: "Much of [S'sE] is spent contending that this last point [that human perception has changed over the centuries] does not hold true when examined through the spectacles of Gestalt psychology" (Buchler, 62). FACT: Gestalt psychology would make a poor courtroom for such a trial, so I would hope not to have made the simple epistemological blunder my reviewer alleges. Anthropological and genetic evidence are relevant; I bring those to bear indirectly when I say that "even if Lamarckian hypotheses about acquired characteristics proved to be feasible in such areas of human experience, their working out-in the absence of an instantaneous mutation-in the timetable of this world's evolutionary crawl would have required at least several millennia" (S'sE, 74). Gestalt perceptual theory can offer confirming evidences to anthropological/genetic conclusions, but to my knowledge it has always been couched in language meant to describe a universal authenticity, not trivial (because statistically so) variations that have been environmentally imposed. I do discuss this issue of human perceptual "progress" in S'sE (74). I judge there that Schoenberg's reliance (perhaps unconscious) on the trinity of Hegelian dialectic, its synthesis stage providing a sense of metaphysical improvement, betrays his entrapment in a clumsily distorted version of Darwinian evolutionary change. There is no mention of Gestalt psychology in these contexts. Finally, this notion of Homo sapiens' improved perceptual prowess (by no means unique to Schoenberg) occupies at most two pages in S'sE. The word and/or concept Gestalt (and Gestaltist) is present in ten locations designated in the index (214). None of these bears an argument of the kind alleged to dominate the book. FICTION: Mr. Buchler rejects the claim that Schoenberg believed that the harmonic series was a "progressive ladder of values" (Buchler, 63). He counters with a reference (63, n. 4) to Pat Carpenter's "detailed critique of this claim" in her review of the book (1993). This feint to another quarter seems meant to convey, without saying so, that Ms. Carpenter's word ends the matter. FACT: No one, including Prof. Carpenter, can remove nor explain away the simple statements made by Schoenberg regarding the remoteness of intervals judged dissonant as they lie within the harmonic series, and his conviction that this remoteness is progressively alleviated-transformed into the comprehensible-with time and familiarity. Note: "In my Harmonielehre I presented the theory that

160 160 In Theory Only dissonant tones appear later among the overtones, for which reason the ear is less intimately acquainted with them.. -. Closer acquaintance with the more remote consonances-the dissonances, that is-gradually eliminated the difficulty of comprehension"~ (Schoenberg 1975, 216-17).2 And seven years later: "Today's ear has become as tolerant to these dissonances [higher in the series] as musicians were to Mozart's dissonances. It is in fact correct to contend that the emancipation of the dissonance is at present accomplished and that twelve-tone music in the near future will no longer be rejected because of 'discords'" (Schoenberg 1975, 246). And much earlier in his life, speaking of the equality of consonance and dissonance: "Consonances are the sounds closer to the fundamental [of the harmonic series], dissonances those farther away; that their comprehensibility is graduated accordingly, since the nearer ones are easier to comprehend than those farther off' (Schoenberg 1975, 260-6 1).3 "But... the chromatic scale flows from the same source as the major: from the elements which are the constituents of every tone [the harmonic series]. The difference is only that the one imitates the natural sound up to the sixth overtone, while the other reaches about twice as far, to the thirteenth overtone; in other words, the chromatic scale brings the more distant overtones within the possibility of relationship" (Schoenberg 1975, 272). Schoenberg seems oblivious in some of these passages that his remarks allude to two kinds of events: the kinds of chords that inhibit the metabolism of real-world pieces, and isolated simultaneities (a unison, a tritone, a m2, etc.) I welcome an explanation (from Buchler or Carpenter) of why Schoenberg is not describing in al of these instances a ladder concept (the metaphor is mine, obviously). If these are not explanations of the harmonic series as a collection of pitch rungs, humanity valianily climbing them through the ages to reach the (more complex, less comprehensible) heaven of pitch emancipation, then I do not understand what Schoenberg was saying nor the meaning of the word ladder. FICTION:- "Thomson dismisses all nature-based systems as 'lame explanations for the manifestations of consonance and dissonance in music.'... His principal argument against the overtone series is that it is not perceptible" (Buchier, 63). 2 Written in 1941. 3Written in 1926. It is hard to know just what he means by comprehension here. I find a minor ninth quite as easy to comprehend (qua m9) as I do a perfect fifth (qua P5).

Response to Buchler 161 FACT: Assuming that I understand what Mr. Buchler means by "nature-based," this is one of his more fanciful fictions. That I do not accept such specious geometrical excuses as Kepler's, Max Meyer's powers-of-two, or Schoenberg's simpler/more-complex comprehension, along with reams of similar effluvia that have clogged the drains of music theory, is no basis for charging me so irresponsibly. If Mr. Buchler had bothered to read any number of attempts I have made to clarify the pitch-hierarchical conditions of musical perception (a not unreasonable expectation of a book reviewer), he would have found his remarks inadmissable. As for the overtone series, it has been the daily furniture of my auditory world since I started playing a horn at the age of five, eminently hearable, both as a set of successive fundamentals for a blown pipe and as a composite wave form of any vibrating medium.4 A cursory review of S'sE turns up the following: "It seems certain that the harmonic series plays a primal role in our conversion of the auditory signal into musical meaning" (122); "The immediate perceptual archetype for pitch available to us is the 'perspective' that squares with the shape of the harmonic series, the shape known to us from birth" (123); and "It is the whole that serves as a shaping matrix for our tonal perceptions, what we carry with us to our every musical encounter" (124-25). I find strange Mr. Buchler's supercilious comment (63, n. 5) that perhaps debate regarding temperament of intervals should have ended with Walter Odington. Odington had no way of empirically supporting any claim he might have made about major thirds produced by singers. This suggests to me that Mr. Odington cannot be trusted in such remarks, imposing though they may at first seem as passing quotations in book reviews. FICTION: Critiquing my claim that the necessity of alternating tension and release in musical experience is the reason for absolute standards of consonance and dissonance, Mr. Buchler claims this stance is a "not-so-thinly veiled criticism of atonality" (64). FACT: The stance is far from a criticism of atonality; conditions of tension and release are eminently available via other musical properties. Indeed, I must confess a soft aesthetic spot in my heart for stretches of atonality located in diverse collections of musical sources. But what I say in S'sE is that the wholesale eradication of this harmonic potential, which resides in shifting levels of sonance, is an aesthetic cop-out (not to mention loss), one supported by Schoenberg 4Mr. Buchler can review my discussion of such matters in my entry "Sound, musical," in the 1974 edition of the Encyclopedia Brittanica.

162 In Theory Only for purely ideological reasons (not unlike the fool who amputates his finger to prove he can survive without it). If every sonority within a stretch of music (whether of five seconds or fifty minutes) possesses identical (or very nearly so) sonance levels, then it necessarily follows that "Harmony ceases to have power as an agent of contrast.... Harmony [qua chords] is neutered" (S'sE, 71), and such a condition, as Adorno sagely observed, "enchains music by liberating it" (1973, 75).' But this hardly proves that harmonic neutrality is inherently bad. It just risks utter boredom. FICTION: Mr. Buchler accuses me of ignorance or confusion (or both) in holding to what he condemns as a "clean distinction between consonance and dissonance" (64). My flawed mental set surfaces in ignoring that some sounds of Mozart's "Dissonance Quartet" were "striking for their time" (65). He chides my penchant for historical correctness, informing us that "if we hear it within its historical frame as a piece by Mozart, then it truly is new and will always be considered a maverick composition" (65). FACT: The ground beneath Mr. Buchler is rumbling. (1) I do recognize a clear distinction between consonance and dissonance; on last check, all of my friends do. Indeed, we would feel distinctly unmusical without that faculty. But this is not to say that we find octaves especially dull nor minor seconds immoral. We do agree that the FCC was wise to choose the Ab/Bb sonority it employs for its auditory warning systems. Why? That conjunction is just dissonant enough to grab our attention, yet not quite so "desperate" (as a minor ninth or minor second might be) as to become grating before the sixty seconds is up. (2) I truthfully do not know what I would actually do to "hear [Mozart's Quartet] in its historical frame," so I find this comment unhelpful. (I've never heard the work without being aware that Mozart composed it, for whatever that's worth.) I would welcome Mr. Buchler's instruction. (3) If the "Dissonance Quartet" is a maverick, then I have several dozen other "mavericks" for Mr. Buchler's herd. He might begin with m. 18 of the Eb Symphony K. 543, mvt. I, which, with my highly developed sense of historical correctness, I know to have been influenced by Shostakovich. 5I give as an example of this sterile potential of redundancy the harmonic stasis of music of the gagaku tradition (S'sE, 76, n. 12). Lots of music capitalizes on the effect, from folksong settings (unadorned triads) to passages in Win. Schuman (like the "Chorale" of the Third Symphony, which contains a long section of unvaried [0,2,4,7] chords).

Response to Buchler 163 (4) Mr. Buchler misses in this discussion both my point and Schoenberg's prescient support for it. These piquant resonances project their engaging qualities because of their contexts, sounding as strikingly unusual today as they did when he played them over for Constanza's approval. Circumstances, as I reaffirm, do indeed affect our tonal expectations, but those circumstances are musical, not social, political, religious, nor economic. FICTION: The issue of tonality "that plays a constant role in this book has little or nothing to do with Schoenberg" (Buchler, 65). FACT: Half of this is real news to me. For the other half, it is true that tonality is a major issue of S'sE, for good reason: it had an enormous lot to do with Schoenberg and with his legacy for the history of ideas. The quintessential ingredient in the Schoenbergian saga-what motivated him to emancipate dissonance and liberate the chromatic pitch collection-was his hypothesis that tonality was just a sometime thing in music's history, that it was his evolutionary destiny to disclose its expendability. FICTION: "The governing power of the dominant-tonic relationship does not have much of a role in Thomson's tonality" (Buchler, 65). FACT: The fifth relation (which subsumes the more exclusive condition of "dominant/tonic") enacts a major (but not necessary) empowering role in "Thomson's tonality." For pre-S'sE testimonials to this fact, Mr. Buchler might examine the archetypal "tonality frames" obsessively pressed into service in my published textbooks (Christ et al. 1963, ch. 2; Thomson 1965; 1971; Thomson and DeLone 1967), or for a post- S'sE exposition my article in Music Perception (1993). Direct proof of the enormity of his error is available on pages 98-99 of S'sE. There hell find a hard push on the idea of the fifth as a powerful (and historically prominent) condition of the pitch focus of tonality. FICTION: "[Thomson] broadly considers tonality to be a goal-oriented motion of a piece toward one pitch-class, the tonic" (Buchler, 66). FACT: Mr. Buchler nudges my position, making it sound more like Schoenberg's (with a touch of Schenker) by injecting issues of goal orientation, "motion... toward," and the bonding of a whole piece with a singular tonic. None of the three is a necessary ingredient of generic tonality, unless one's definition is irretrievably anchored in a style-based Rameau/Riemann variant of harmonic functionalism (which was the Schoenberg and Schenker base). FICTION: "Thomson claims that Schoenberg did not know (or did not convey) how the role of tonic arises in a context as the datum of listening" (Buchler, 66).

164 In Theory Only FACT: The claim is true; but the implication of my position's inaccuracy is untenable. My opinion is the same whenever I discuss Schoenberg's explanation of tonality qua chord-succession: at best he was consistent, but when he hints at causes, the tonic turns out to be a prefacto posited element. In S'sE I turn to a Krenek quotation because it reveals better than any sentence I know the inherent tautology of the generic position. It is that "A tone becomes the tonic only when the central triad is built over it" (S'sE, 79). FICTION: "In what seems to be a largely semantic distinction, Thomson separates modality from tonality by using the terms melodic and harmonic tonality" (Buchler 66); and later, "After all his time insisting that 'harmonic tonality' did not supersede 'melodic tonality,' Thomson now waxes on about the grave loss of 'tonality' (in the conventional sense of the word)" (76). FACT: I do not know a singularity that with confidence I can call modality. This renders fragile any distinction I might make between an "it" and several other musical conditions. But generalizing, as one must about such matters, both the conceptual literature of modality and the music it attempted to explicate was concerned with melodic matters. Well-intentioned struggles to apply modal precepts to harmonic music (i.e., textures of simultaneities), Renaissance through George Russell, run from clumsy and indecisive to downright misleading (as reliable Renaissance theorists openly admitted). And, on the other hand, conventional eighteenth-to-twentieth-century presumptions of the nature of tonality (Blaze, Rameau, Schenker, Dahlhaus), have been that it arises from a menage c trois, Mama I flanked by Her empowering dominants IV and V. In S'sE I posit the notion that the term harmonic has been unjustly (and misleadingly) confined in the parlance of the past few centuries to chords cohabiting in fields of tonic-dominant bliss, that the term in its Greek etymology can as properly (and for discourse profitably) apply to successive pitches (81-82). Harmony has a more inclusive meaning than just tones sounding at the same time as a singularity. For these reasons alone, I should wish to avoid any suggestion of bickering, like the melody vs. harmony prattle of the Guerre-du-Buffons variety. The two simply are not mutually exclusive. FICTION: "Thomson seems to believe that modality was just another form of tonality" (Buchler, 66, n. 10). FACT: The accuracy of this statement turns on some immodest historical and ontological qualifications. As I mentioned above, there is no collective body of precepts one can call upon as modality; it is a fickle target in the history of our art, varying substantively from its

Response to Buchler 165 Carolingian initiation through rumored dabblings with the dorian mode by Miles Davis. The "Modality as Pre-Tonal Confusion" section of S'sE (56-58) defines Schoenberg's perspective, not mine. Even here we must sharply distinguish the music commonly (and sometimes loosely) known as modal and conceptual expositions of same. From what I know of both the corpus of west-European music from c. 350 B.C. to A.D. 1550, certain clearly delineated pitch-structural characteristics recur enough for me to conclude that they are not coincidental-similarly within conceptual explications of modal theory of the same historical slot. I am struck-not in all cases, but in most-by recurring axioms of pitch-kinetics thought to be cogent to musical cohesion. These artifactual/conceptual correspondences subsume such primal ingredients as fifth/fourth-divided octaves as framing paradigms (and thus harmonic values) of the melodic trajectory and recognition of one note [pitch class] as hierarchical nexus. The oversimplistic claim my reviewer makes is precontradicted by such statements as "Nothing in the bales of chant theory harvested by reporters through the Renaissance puts before us an unassailable correspondence between the finalis of former times and the tonic of today" (S'sE, 110). In this respect, and with these qualifications, I regard the musics usually designated as modal as carriers of what I call tonality. They are merely a sidebar manifestation of the larger generic program. FICTION: (1) "Thomson uses Schoenberg's overt disregard for historical correctness [!] as his raison d'etre in constructing a more unified description of tonality in historical perspective" (Buchler, 67). (2) "He criticizes Schoenberg for his... inability to specify a date when the break between modality and tonality occurred" (Buchler, 67). (3) "Current research has shown that Odo of Cluny was not the author of the Diologus" (Buchler, 67, n. 11). FACT: (1) Schoenberg was hardly a singular motivation; much broader considerations have fostered my struggles to inject reason into the way we talk about pitch structuring in all music.6 Schoenberg made a best-of-all-possible points of departure for such a pilgrimage: it was a celebrated factor in his professional life. To be accurate, it is not "Schoenberg's overt disregard for historical correctness" that falls under my scalpel. It is his indifferent ignorance of the music and of its 6A glance at my Ph.D. dissertation (Thomson, 1952), which preceded S'sE by four decades, will dispel any notion that Schoenberg was the priming pump.

166 In Theory Only theoretical conceptualizations, both of which he feigned to draw upon as a base for pontifications about music's past and future. (2) Schoenberg deserves no blame (nor do I wag a critical finger at him) for supplying no date, and for good reason: nobody could pinpoint such a date, mainly because there was none. (3) Finally, regarding Mr. Buchler's truly remarkable page 67, whether Odo of Cluny or Odette of Chagny wrote the Diologus is wholly irrelevant to the argument of pages 105-106 of S'sE. The point is humbly simple: somebody back there, pre-renaissance, was saying some rather prescient things about pitch organization in that old nontonal music. FICTION: "Thomson gets into some logical dilemmas... in trying to decide how one can talk about the perception of early music" (Buchler, 68). FACT: Since Mr. Buchler does not tell us where these logical dilemmas reside, I forge blindly ahead to say that my text contradicts his judgment. No words are wasted in S'sE in deciding any such thing. I point out the obvious: we cannot hope to replicate "the mentation processes of remote ancestors" (109). We furthermore cannot hope to verify some of the shady hypotheses served up over the past century regarding whether those remote ancestors responded (or did not) to pitch patterning (or to anything!) as we do. Finally, these unanswered questions leave us with "only one prime bit of evidence pertaining to ancient musical perception: the record of actual music, however securely or insecurely it may be preserved for us" (109). If confusion or illogic or indecisiveness reign in these conclusions, they are unique to Mr. Buchler. FICTION: "[Thomson's] categorization of other cultures as 'primitive' and undeveloped is inaccurate and makes Thomson seem ethnocentric and socially biased" (Buchler, 69). FACT: I plead hopelessly guilty to the ethnocentric part. My friends share the same affliction. But I presume that this flourish of name-calling is motivated by Mr. Buchler's wish to forget my passion for "historical correctness," now to favor the more fashionable political kind. Many musical cultures on this planet are with precision (and correctness) called primitive. The washboard player, accompanying his own plaintive rendition of a cotton-field holler, is a primitive musician compared to, let us say, the late John Birks Gillespie. For similar reasons, the Lapp tune quoted in S'sE (102) is primitive music compared to, let us say, David Raksin's Laura. I understand precisely and broadly the meaning of primitive, as it can be applied nonpejoratively to artifacts of music and to social groups. I learned its

Response to Buchler 167 proper use from the late George Herzog while a student at Indiana. He had learned it in the ethnocentric kitchen of von Hornbostel and Stumpf in Berlin. But there is a more troubling element here than dirty names. By the time Mr. Buchler got to the second hundred pages of S'sE, he should have kept in mind that I posited much earlier a tight, nonstylistic definition of tonality (as rooted in pitch focus), which I employ throughout the book, in opposition to the chord-based tonality that has dominated west-European musicology. This opposition acknowledged, Mr. Buchler might then have remembered (especially from my pages 98-101) that when the notable figures of early- and midcentury ethnomusicology speak of tonality, they have in mind what they learned in their genteel German academic tradition, the style-based chord progressions of the Rameau/Riemann fork. They had no other point of reference. Concluding, as Mr. Buchler does, that "these quotations reveal a great deal about one of the fundamental misconceptions of this book: Thomson rework[ing] non-Western music to fit his own tonal norm" (69), is an outrageous misconception of its own. For one to recognize common verifiable elements among a large array of artifacts and to formulate an encompassing hypothesis for those found elements is not to "rework" the substance of the artifacts themselves. In this, Mr. Buchler (and perhaps his mentors as well) appears to be ignorant of the essential and primal act of scientific inquiry, in this case of behaving like a music theorist. FICTION: -"Thomson uses his own criteria for tonality as a basis for judging musical value" (Buchler, 70). FACT: My claim regarding the wholesale abandonment of hierarchic pitch-structuring-the pitch focus of tonality-is that "what music loses by this replacement [through pitch serialization] is far greater than [Schoenberg] bargained for" (S'sE, 176-77). I'm not so foolish that I invoke tonality's presence or absence as an exclusive criterion of musical value. I make that clear when I observe that the other musical properties that are potentials "for achieving local coherence and formal integrity make music without tonality possible" (S'sE, 177). But I add the crucial observation that those other properties are shared by other art forms, while tone, for which pitch is one of the most powerful organizing components, is music's unique property. For music, pitch is "nothing less than the primary musical property, the paradigmatic quality of the medium" (177). What does this have to do with "judging musical value?" Does Mr. Buchler imagine me plugging my ears during tonally fluid sections of a Schubert development, its unfixed pitch condition compromising its

168 In Theory Only musical value? Must I be repulsed by the tonal abandon of Tristan's opening, preferring the E b stasis of Das Rheingold? Come now. The point either missed or ignored by Mr. Buchler is that Schoenberg's (and his followers) imposition of atonality's historical necessity was pure bunkum, in itself a false basis for valuing of any kind. FICTION: "To use Thomson's logic, if one cannot hear the row, the structure itself must be invalid, hence the music must be poor"; and further, "[Thomson] deals only with Schoenberg's serial works" in criticizing his music (Buchler, 70). FACT: Heaven forbid that any listener (excepting the most fervid serialist illuminati) "hear the row" in anything they might wish to experience as music.7 Where does such a notion surface in S'sE? Nowhere! Mr. Buchler draws it from his unique interpretative style. The only focused references in S'sE to a single Schoenberg composition occurs in a discussion of the composer's alleged derivation within the Kammersymphonie, Op. 9, of the second theme from the first (32-34). But Op. 9 is not serial. FICTION: Thomson stacks the deck of cognitive psychology, misrepresenting the conclusions reached by Diana Deutsch (1980) aid Carol Krumhansl et al. (1987) in their respective empirical studies of the experiential togetherness projected by twelve-note rows (Buchler, 70-73). Those writers' conclusions are distorted to suit a devious scenario. For instance, Thomson calls conclusions "tenuous" that Krumhansl et al. describe as positive. "The excerpts quoted by Thomson seem to provide great evidence against the row's perceptibility" (Buchler, 70). The real truth, as Mr. Buchler interprets it, has been skewed in the wrong direction. Considerations for ensuring reliable empirical test results of such things demand, Mr. Buchler tells us, recognition that the "composers who claim that the row is a unifying principle likely are more skilled listeners" (71) than the subjects who were tested. Furthermore, (a) the subjects of these studies (Deutsch, Krumhansl et al.) were tested in "an artificial (i.e., non-musical) environment," some distracters using "unrhythmicized rows in a non-musical texture;" (b) "Krumhansl et al. even used Shepard tones to mask contour," etc., etc.; and-as if prefacto apologies are not already getting a bit deep-(c) the "results of an experiment [using a full musical texture] like this would naturally depend in large part on which piece was chosen and how well the performers brought out the row" (Buchler, 71, italics mine). 7In this I heartily agree with Schoenberg.

Response to Buchler 169 FACT: This is a big one. In sorting out the rich barrenness of Mr. Buchler's position, one needs recall the two elements at stake: the goals of such controlled perceptual studies and the claims Schoenberg made in his frequent apologias for the serialization of pitch. Both entail that the serialized collection take on a Gesamtqualitdt (not, to my knowledge, Schoenberg's term) that is preserved in the permutations undergone in the compositional process. This wholeness projected by the series, and its cohesiveness in the face of radical transformations, must persist independent of any other properties, rhythmic, textural, contoural; they are pcs, after all! Fudging via other properties, say by holding this pitch longer, articulating it more decisively, sounding it sul ponticello, or such, would all qualify as Use of Excessive Force. The row must do it by itself, even though it cannot even exist without the copresence of those other messy entanglements. I do not, as stated, "dismiss... an entire milieu on cognitive grounds" (Buchler, 71), glibly or otherwise. (1) All I dismiss in S'sE is the validity of Schoenberg's conviction (repeated by others after him) that the dodecaphonic row provides a glue for the pitch continuum, a perceived property that replaces what is forfeited in junking the harmonic hierarchy. I furthermore question that empirical grounds yet exist to confirm Schoenberg's self-serving claim. (2) Prof. Deutsch concludes, "Sequences [of pitches] whose tonal structure could be parsimoniously encoded in hierarchical fashion [which excludes atonal] were recalled with a high level of accuracy. Sequences that could not.-.. produced substantially more errors in recall" (Deutsch 1980, 381; quoted in S'sE, 85). (3) Since I explicitly recognize that the Krumhansl et al. study "holds a ray of hope for dodecaphonic perceptual solvency" (S'sE, 85), by no means have I concluded that all answers are in. (4) In view of the celebrated pc status of row elements and the alleged collective cohesion of same, the use in such empirical tests of Shepard tones and "unrhythmicized"8 serialized soundings strikes me as a stroke of genius. Furthermore, I shudder at the prospect of a study incorporating the provisions Mr. Buchler would invoke to load the perceptual dice, in order to assure that rows are shown to possess all powers claimed: stacked subject population (those "more attuned to modem music"), a congenial musical environment, the employment of rhythmicized rows encapsulated in approved "musical textures," careful selection of compositions utilized (Webern in, Berg out?), and even performers skilled in "[bringing] out the row." Mr. Buchler's 81 think we know what is meant by this, but let us note that tones just do not happen without being rhythmicized.

170 In Theory Only formidable recipe for empirical verity suggests that he has not yet been taught that hallowed old dictum of empiricism, Know Thy Variables. (5) What I declined to add in S'sE (since it is a supporting rather than primary issue) is that even Krumhansl's guarded optimism for an eventual positive conclusion is unfounded, for reasons III enunciate here presently. In that she calls her study "exploratory," its conclusions hardly merit recognition as emblematic evidence for the defense. Its exploratory status is further underlined when it is noted that there were inexplicably "large individual differences... found throughout the experiments" (Krumhansl et al. 1987, 72). Circumspection forces me to doubt universal validity for such results because one group of subjects, the "experts," seemed excessively better able to intercept messages of cohesion. In her later book, The Cognitive Foundations of Musical Pitch, Krumhansl describes these bicameral differences as "diametrically opposite one another" (1990, 243). Thus far, my "tenuous" (S'sE, 85) strikes me as a reputable characterization of Prof. Krumhansl's little ray of hope. But a more serious problem plagues the Deutsch and Krumhansl studies. It will plague the conclusions of any study so-designed that purports to determine whether a twelve-pc row can project a cohesiveness, a unity, a semantic something, that is ipso facto importable via serial processings. Unity, cohesion, meaning: they're big words. They do not at all equate with the recognition of isolated parts. Or put another way, they are misnomers in thoughtful parlance unless they can pertain to wholes, unless they can refer to a Gesamtqualitcit that is more than an enumeration of parts. For example, suppose I wish to determine auditors' abilities to find meaning-to recognize semantic cohesiveness-in a verbal phrase. Just to highlight the expert vs. novice divergences of the Krumhansl et al. study (1987), let's say that my subject population consists of 50% University of Rochester graduate students, the remainder drawn randomly from secondary school dropouts of Monroe County. Under proper clinical circumstances, my subjects are fed the following phrase, reasonably paced and with clear enunciation: "Pas de lieu Rh6ne que nous." Some subjects report that the phrase "sounds French." Others (perhaps the academically unwashed) merely give a twist of the head: no comment beyond "sounds like words." (Any Gallic subjects present may report a suspicion of fractured French.) Now the real test. Subjects are fed the following permutation of the original (presumably "meaningful, cohesive, unified") phrase: "Nous que Rhdne lieu de pas." Several subjects-perhaps those originally reporting "French"-might observe that some sort of reversal is

Response to Buchler 171 present: the last sound was like the first. Voil3d-a retrograde. But not so fast with the conclusions. All we actually can infer from this exercise is that some subjects, perhaps those with language awareness, detected the identity of terminal members, with the further observation (perceived fact or maybe just deep inference) that one phrase is the obverse of the other. Doing this makes one an expert. But a nagging question dogs our tracks: Is this recognition of bits and pieces an exhibition of experiential cohesion, of unity, of meaning? If it is (and my position is that it is not), it is paltry, by nature fragmentary, and begging of deep inferential faith. It hardly qualifies as a Gesamtqualitdt. In fact, the cohesiveness, the unity, the semantic gist-whatever we might call it-that inheres as a semantic potential in this intriguing string of phonemes (only superficially French) has not been touched by my subjects. They have not come close to the encoded message; they have merely noticed relational resemblances between a few of its elements. The forest remains invisible. In one sense, there is no forest. But in fact a potential message does occupy the string of phonemes, and it is aurally detectable.9 So it goes with those experts whom Krumhansl and Buchler praise as superior interceptors of dodecaphonic hierarchies. They have learned to recognize isolated parts as they go by: a major third here, a semitone there (or their inverses). Why should one marvel that on the whole (but with some haunting divergences) they score higher in recognizing Ts and Is and Rs and RIs?'0 One need only detect in T- and I-related forms that the first pc pair of one realization bears the same interval-class relation as the first pair of the other, and one's better-than-chance scoring probabilities have increased by 33%. Who among Mr. Buchler's readers would not likely recognize that an initial interval-class succession of 4-1-6 in one row form is reflected in the terminating ic succession of 6-1-4 in another (RI-related), thereby raising a positive score potential by 50% beyond chance?" 9I refrain from divulging the "sentence's" semantic content, because figuring it out vocally/aurally is more fun than knowing the answer. Failing such, the key can be found in William James (1981, 726). 10The transposed forms in the Krumhansl study (which do not involve R or I permutations) would be a snap for anybody who has passed sophomore aural skills. 1I'here are even shorter shortcuts available to seasoned listeners, methods for increasing probabilities of informed decisions, still without witnessing cohesion. In this light, it is important to note that the second probe-tone experiments of the Krumhansl et al. study used Schoenberg excerpts that incorporate only T permutations. Any listener who can recognize pcs 0-4-6 (Eb-G-A; Wind Quintet) and pcs 0-11-7 (G b -F-C 0; Fourth Quartet) can make a high score in these recognition tests. The excerpts are shown in Krumhansl 1990 (249-50).

172 In Theory Only As Buchler's quotation from the Krumhansl et al. study makes clear, "The results for only some of the listeners [the expert group] were those that might be predicted from twelve-tone theory" (73). She makes a telling revelation in her later book: something more devious than perceptions of cohesion was operative in the expert subjects' deductive processings. She does this in the summary of her study of local key implications on subjects' responses to probe tones, a condition that might, or might not, violate the serialist dictum of no pc repetition within the series. One ploy is revealed when she observes that some subjects apparently reversed the orderings of the ratings from those appropriate to tonal contexts, thereby giving higher ratings to tones that work against these suggestions of a local key area (1990, 269-70). In other words, they were using a negative principle from one domain (tonal) to come up with a positive conclusion in another (dodecaphonic). This is cohesion?-... hierarchy? With the hindsight of five years, my characterization "tenuous" in S'sE was charitable. FICTION: "Assuming, perhaps, that his readers are satisfied that the structural elements of atonal music cannot be perceived" (Buchler, 73). FACT: Again, Mr. Buchler attempts to paint his author as a naive optimist. He furthermore confuses, once again, whole with part. Structural elements of atonal music are not a part of this discussion; only one contributing property to atonality is: the pitch-continuum cast as a numerically preordered series. Also, Mr. Buchler misses the point. I refer emphatically to "the continuing debate over the perceptual fertility of twelve-tone constructs" (N.B. Mr. Buchler's quotation of my words, 73). FICTION: "Thomson also fails to suggest a method for taking rhythm into analytic consideration," and "the construction of a theory of harmonic rhythm... would almost certainly lead to a deconstruction of Thomson's definition of a universal tonality" (Buchler, 74, n. 14). FACT: Mr. Buchler should carefully review pages 119-21 of S'sE. He will find there a homely algorithm of pitch empowerment as by-product of temporal play (rhythm).-12 Who needs a "theory of harmonic rhythm" in explaining tonality? That was a handy pedagogical tool of the Piston/McHose era of statistical indexing. It is a questionable cause of tonality, even in the music for which the concept has some explanatory relevance. The only deconstruction 12He will find a more thorough discussion of the same kinds of facilitating in Thomson 1993 (407-411i).

Response to Buchler 173 uncovered is Mr. Buchler's motivation for using the chic word "deconstruction." FICTION: "The final three chapters [of S'sE] attempt to discredit Schoenberg's music by claiming that it was not adequately foreshadowed by the path of history" (Buchler, 74). FACT: Only the middle of the last three chapters deals even generally with Schoenberg's music. Further, if I had in fact written directly about such matters, I would have emphasized, on the contrary, that most of Schoenberg's music suffers its indebtedness to one of the crampingly narrow paths of history that immediately preceded it. I have in mind the turgid Sturm und Drang, the sonic hand-wringing, the unflagging high emotion that motivated him and his sounds-what critic Martin Bernheimer of the Los Angeles Times has referred to (with regard to Op. 9) as crumbling romanticism. FICTION: "Reger is classified as merely an academic composer who was not very musical" (Buchler, 75). FACT: Recognizing Reger's "astounding compositional technique," as I do (S'sE, 169), hardly squares with an appraisal of musical dysfunctionalism. In observing what seems to be a judgment widely held by academics and other music professionals, I do recognize that Reger sometimes clogs his arteries with artifice. (Sometimes he does not.) FICTION: "While he defends the music of the late-romantic composers as tonal, Thomson's preference clearly lies in more traditional circles" (Buchler, 75). FACT: What could be more traditional than late-romantic composers? But again, my reviewer is speculating out of context, imputing to me what he cannot find in S'sE. I make amply clear (160-63) that Wagner was a veritable master in accomplishing musically what he set upon, that in his dramas he "applies every resource of orchestral nuance and vocal tone to create utmost reality and the most intense expression of feeling, which, after all, was the paramount goal of Romanticism" (163). I make no value judgment of any kind re the music of Richard Strauss; I merely comment about the programmatic motivations for the atonal section in Also Sprach. As for Liszt, his problem was not chromaticism per se. (Listen to Les Preludes.) Scriabin--one of the musical incomparables of his time-composed some music that, like Reger's, "exhibits ultrachromatic mixtures of pitch that warrant our well-established maxim: in the hands of a few of the most talented composers, music has truly reached [by the early 1900s] a dramatic peak of development in harmony" (S'sE, 169-70). These do not strike me as hate mail for the late romantics.

174 In Theory Only I catalog the works of Borodin through Verdi to confirm something that has nothing to do with my personal musical affections: plenty of "not inconsiderable music" (S'sE, 170) was composed between 1880 and 1910 that makes a mockery of the tale that a veritable tsunami of chromaticism left the nineteenth century awash, threatening to drown art music (which is where lifesaver Schoenberg dived in). How can one mistake an observation of historical fact for an expression of personal aesthetic? FICTION: "Only in the final section of the book does Thomson reveal his not-so-hidden agenda" (Buchler, 76). FACT: Revealing "not-so-hidden" things is in itself worth a long discussion. But I'll pass that up to note that the agenda of S'sE is trumpeted from page x, wherein the final three prefatory paragraphs state that (1) "only Schoenberg's theories of music... occupy... this study," (2) Schoenberg is the focal point because of his enormous influence on theoretical thought in the twentieth century, and (3) Ill deal with the primal questions Schoenberg's early conclusions and predictions posed for us as thinking people. What's left in my agenda to be revealed in the final section? FICTION: Footnote 16 (Buchler, 76) charges as unfair my paraphrase of Lerdahl and Jackendoff (1983), wherein I rephrase their conclusion as "no amount of injections [from other musical parameters] are adequate replacements for the relatedness that comes from hierarchical structure" (S'sE, 177). FACT: Fred Lerdahl kindly and methodically read the S'sE manuscript twice before publication. He found nothing amiss in my representation of his book's ideas. Perhaps Mr. Buchler has understood A Generative Theory of Tonal Music no better than he has understood S'sE. I must add that I can only agree 100% with Mr. Buchler's conclusion that Lerdahl and Jackendoff "do not deny that serialism provides at least a locally audible structure" (Buchler, 76, n. 16). FICTION: Mr. Buchler condemns my rhetoric of pages 185-88. It "only contributes to the fanciful misconception that serial technique is, by its very nature, complex.... Thomson is attempting to scare people away from this technique by merely listing some associated big and technical-sounding words, without offering any explanation of what [they] mean. (This is particularly disappointing from an author whose specialty is pedagogy.) Of all terms to complain about, Thomson singles out one [pitch-class] which he himself uses in his first chapter.... No one, to my knowledge has ever claimed that pitch-class is the only parameter of music" (Buchler, 77).

Response to Buchler 175 FACT: My goal on pages 185-88 is eminently clear: to display the rhetorical flamboyance (and occasional silliness) of the "budding grove" of serialist verbiage. My comments have nothing to do with the actual processes of composing serial music, whether simple or complex, only with how they are described. Mr. Buchler is correct: a child can compose twelve-tone music. I have taught a few how. They do not do it very well, of course. Like adults, some do it better than others. People needed no help from me, when S'sE was published in 1991, to be scared away from serial procedures as the creative wave of the future. So far as I can tell, that condition persists today, and my pedagogical "specialty" notwithstanding, S'sE was not conceived as a how-to primer of serial techniques. That should be obvious.'" Pitch class "as a referential datum" (S'sE, 180) is a lousy term; it refers to nothing one can hear-simple as that. As Mr. Buchler fails to observe, I happily endorse the term as useful in the universal rhetoric of musical relationships when I note that "it is hard to imagine life without the handiness of the term pitch class, which so readily hurdles several linguistic barriers" (S'sE, 184). Like Mr. Buchler, I know no person so confused as to think "that pitch-class is the only parameter of music" (Buchler, 77). In fact, I did not know that pitch class could even qualify as a parameter of music, as Mr. Buchler suggests. But again, my reviewer imputes ideas to me: I would be among the last to make such a foolish statement. The whole of pages 185-88 is devoted to unmasking (1) the vacuity of many terms of serialism used to describe musical events and (2) the fact that most of those explications speak to us as if pitch (not pitch class) were the only musical parameter. This is the point of my quotations (S'sE, 185-86) from two representatively sterile descriptions (John Rogers's "Some Properties of Non-Duplicating Rotational Arrays" [1968] and Milton Babbitt's misguided tour of his own Relata I [1970]). The crowning irony of the latter discussion is provided by Prof. Babbitt's assurance that his "analysis" is framed for the listener, things the listener probably would not learn through that defective old process called audition. Finally, as I point out, these verbal/numerical assaults of serialism describe with gripping precision "a note-generating process" (S'sE, 186). They fail miserably (as does Babbitt) to describe what is audible. Thus it is by omission that they suggest pitch to be the sole malleable property of music. 13For an extended discussion of such, Mr. Buchler might look into my Introduction to Music as Structure (Thomson 1971, 184-99), where I use a George Rochberg Bagatelle and Schoenberg's Op. 33a for dissections. This book was written for advanced placement students in high school.

176 In Theory Only FICTION: "Thomson blames Schoenberg for everything he considers wrong with music today" (Buchler, 78). FACT: This is culpably irresponsible reporting. The wrongs of music credited to Schoenberg happen to be the only wrongs discussed in a book that purports to concern itself exclusively with Schoenberg's statements regarding musical structure. A bumper crop of musical rubbish was produced during the '60s and 70s (a bit more even in the '80s) for which Schoenberg shared no guilt. Most of these were the nonsensical-although sometimes fun, in a social sense-post-Dada diddlings of avant-garde mavens, residues of the sensuous sixties. These musically desolate goings-on may one day be traced to reactions to the extrarationalism of serialism. But nothing in S'sE loads Schoenberg with the blame. It is our good fortune that in the midst of-indeed, in spite of-occasional lapses from the simple realities of perceptual mores, a wealth of individual, engaging, and moving music was (and is) turned out by composers who have felt no need to be guided by insensitive ideologies. I cannot end this extended response without noting that my reviewer failed even to mention, must less provide meaningful discussion of, the many hard-core theoretical matters dealt with intensely in S'sE. That he missed those vital messages, while skimming the parts he did not understand (thus denying their synopses for the readers of In Theory Only) is regrettable. He is right, of course, in his assessment of Schoenberg's place in history; my book does indeed help to confirm that he was "one of the most influential composers in music history" (Buchler, 79). I have never argued otherwise. For Mr. Buchler to remark that the book enables its author to ride Schoenberg's famous coattails into posterity seems only mean-spirited. It is reminiscent of one wag's observation that Churchill would not have achieved fame without Hitler, neglecting to recall what had happened to Neville Chamberlain. We can enter hosts of "most influential" names into our log of history without forfeiting the right to check out their credentials, without for a moment feeling forced to admire what got them there.

Response to Buchler 177 References Adorno, Theodor W. 1973. Philosophy of Modem Music. Trans. A. Mitchell and W. Blomster. New York: Seabury. Babbitt, Milton. 1970. On Relata L Perspectives of New Music 9/1: 1-22. Brendel, Alfred. 1995. On Playing Schoenberg's Piano Concerto. New York Review of Books 42: 27-29. Buchler, Michael. 1994. Review of Schoenberg's Error by William Thomson. In Theory Only 12/7-8: 59-80. Carpenter, Patricia. 1993. Review of Schoenberg's Error by William Thomson. Music Theory Spectrum 15/2: 286-99. Christ, William et al. 1963. Materials and Structure of Music, vol. 1. Englewood Cliffs, N.J.: Prentice-Hall. Deutsch, Diana. 1980. The Processing of Structured and Unstructured Tonal Sequences. Perception and Psychophysics 28/5: 381-89. James, William. 1981. The Principles of Psychology, vol. 2. Cambridge, Mass.: Harvard Univ. Press. Krumhansl, Carol L. 1990. The Cognitive Foundations of Musical Pitch. Oxford: Oxford Univ. Press. Krumhansl, Carol L., Gregory J. Sandell, and Desmond C. Sargeant. 1987. The Perception of Tone Hierarchies and Mirror Forms in Twelve-Tone Serial Music. Music Perception 5/1: 31-78. Lerdahl, Fred, and Ray Jackendoff. 1983. A Generative Theory of Tonal Music. Cambridge, Mass.: MIT Press. Rogers, John. 1968. Some Properties of Non-Duplicating Rotational Arrays. Perspectives of New Music 7 /1: 80-102. Schoenberg, Arnold. [1911] 1978. Theory of Harmony. Trans. R. Carter. Berkeley: Univ. of California Press.

178 In Theory Only S1975. Style and Idea. Ed. L. Stein. Trans. L. Black. Berkeley: Univ. of California Press. Thomson, William. 1952. A Clarification of the Tonality Concept. Ph.D. diss., Indiana Univ.. 1965. Introduction to Music Reading. Belmont, Calif.: Wadsworth.. 1971. Introduction to Music as Structure. Reading, Mass.: Addison-Wesley.. 1991. Schoenberg's Error. Studies in the Criticism and Theory of Music. Philadelphia: Univ. of Pennsylvania Press.. 1993. The Harmonic Root: A Fragile Marriage of Concept and Percept. Music Perception 10/4: 385-416. Thomson, William and Peter DeLone. 1967. Introduction to Ear Training. Belmont, Calif.: Wadsworth.

Rejoinder to Thomson Michael Buchler In his lengthy response to my review, Professor Thomson asserts that I misunderstood and therefore misrepresented his book. In his response, however, I believe that he has dramatically misunderstood or misread some of my criticisms. I stand by my review; however, for the sake of brevity (and for reasons outlined at the end of this rejoinder), I will address only a few of Thomson's comments rather than respond to all of them. I was most surprised to learn from Thomson's response that I deny that Schoenberg believed the harmonic series to be a "progressive ladder of values" (159). While I did mention in a footnote that Patricia Carpenter (1993) offers a critique of this claim in her review of Thomson's book, I did nothing more than report Thomson's rejection of this idea. I did not disagree with him that Schoenberg made this assertion; rather, I questioned Thomson's particular argument against the overtone series as a progenitor of consonances. Actually, I agree with Thomson that the overtone series offers rather poor (though interesting) justification for consonances and dissonances, but I question his proof of why it is a poor explanation. Thus, when I stated that "Thomson dismisses all nature-based systems as 'lame explanations for the manifestations of consonance and dissonance in music"' (Buchler, 63), it was not a complaint but rather a simple

180 In Theory Only account of what I read in his book. Indeed, though Thomson may have read it as such, not everything in my review was meant as criticism. In his response, Thomson again discusses the imperceptibility of atonal structures. In my review, I did not criticize the studies that Thomson cites,' but rather the misleading way in which he presents them and, moreover, the questionable connections drawn between their results (as he reads them) and the enjoyment of atonal-particularly serial-music (Buchler, 70-73). However, any further debate concerning Thomson's interpretation of the experiments would be tangential to my rejoinder, perhaps overshadowing my own views. As I stated in my review, Thomson is laboring "under the dubious assumption that one must perceive the row in order to enjoy serial music. [I shall not even address the odd implication that a cognitively imperceptible structural element renders a piece unenjoyable.] If the row facilitates composition, does it really matter whether the element itself is heard? Furthermore, one could easily initiate a study to see how many people perceive the structural elements of, for example, a Brahms symphony in a way that they could accurately describe" (Buchler, 72). I would certainly never suggest (as Thomson implies I do [168-69]) that, as a general principle, the row should be brought out by performers. Sometimes it is thematic; sometimes it is not. In Thomson's rebuttal of my criticism that he "fails to suggest a method for taking rhythm into analytic consideration" (172), he points me toward pages 119-21 of Schoenberg's Error (hereafter S'sE). In fact, it was precisely those pages, along with his treatment of the cadential six-four (135-37),2 that led me to criticize his treatment of rhythm. Since Thomson believes that pages 119-21 remedy the situation, a close examination of their contents should precede my renewal of this criticism. On those pages, Thomson demonstrates the workings of pitch focus (the central facet of his "generic tonality") using the melodic line from the first twelve measures of Mozart's Piano Sonata in F Major, K. 332. 1In fact, given that Thomson criticizes atonal music and particularly atonal music theory for ignoring all musical parameters beyond pitch class (Schoenberg's's Error, 189-90), it is rather surprising that he would embrace perceptual studies that abstract tone rows from the compositions that utilize them, without regard for how composers employ the structures in their music. 20n pages 135-37, Thomson attempts to demonstrate how a cadential six-four could be replaced by a root-position tonic triad without altering the effect of the music.

Rejoinder to Thomson 181 Example 1: Mozart, K. 332, mm. 1-12, from S'sE (120; phrase marks are Thomson's) Reprinted by permission of the University of Pennsylvania Press. A a_________Q 92 1 F: F I He begins by dividing the melody into three awkward segments. Unfortunately, he does not actually provide specific measure numbers for the divisions but instead describes them as follows: Observe first that this melody can be divided (or perhaps we should say "divides itself") as three segments, P1, P2, P3. These segments are projected most primitively through rhythm and contour. Phrase P3 overlaps with P2, its first note serving as P2's last. All three segments begin with the same pitch, F. The two phrases P2 and P3 also end with it. (S',sE, 119) If I understand Thomson correctly, he is segmenting the passage as follows: P1 = nmn. 1-4, P2 = mmn. 5-7, and P3 = mm. 7-12. (This is the only possibility wherein P2 ends and P3 begins with an F.) But the melodic line should not be the sole factor in phrase division. The interpretation of this passage as a three-phrase structure (instead of an expanded two-phrase structure, which I prefer) with a division at m. 7 overlooks the bass's imnitation in mm. 7-8 of the melody from mm. 5-6. Indeed, despite Thomson's assertion that this passage "divides itself" in the manner described above, he felt it necessary to include the following footnote in his discussion of the latter two phrase boundaries: This interpretation becomes less reasonable when the melody is heard in full texture, with its imitative line in the bass of measures 5-7. This "disruption" also urges a three-phrase structure, but each phrase consisting of four-measure segments. (S'sE, 128, n. 1 1)3 Thomson claims that this passage is in F major not because of its harmonic support, but because of its melodic line. Since he wants to 3The reader will note that Thomson's slurring of the passage suggests three four-measure phrases rather than the asymmetrical segmentation that he describes

182 182 In Theory Only move away from the idea that the triad is the primary building block of tonality, he cites four harmony-neutral reasons that this passage "is dominated by the note F": a. as pitch boundaries for the tonal pattern (high/low, first/last); b. as time boundaries for interior patterns (phrases); c. as a pitch of agogic and metric accent; d. as 25 percent of total consumed time in the melody's twelve measures. (S-'sE; 120) Regarding his first claim, the high/low boundaries are actually E' and G', not F' and F5- Thomson rationalizes this false claim by saying that "the G in measure three and the E just before the final F are fleeting, and they fall on unaccented beats, so they are not comparable to highs and lows that occur in more providential locations" (S'sE, 120). In fact, while the G' in m. 3 may appear "fleeting" and unaccented, it is not merely neighboring (at least locally) to F'. Rather, that F' is an accented passing tone to E'. His second claim-that F is a note that occurs at phrase boundaries-relies upon his awkward phrase divisions. In the last of the harnony-neutral reasons, Thomson seems to be promoting a new statistical analysis. Apart from the problems such an approach would raise in much romantic and lateclassical music (especially in development sections), Thomson's use of this method is ironic, considering his response to me that harmonic rhythm as an explanation of tonality was a "handy pedagogical tool of the Piston/ McHose era of statistical indexing" (172). Having examined the contents of pages 119-2 1, it is now time to return to the initial point. Setting aside my criticisms of Thomson's analytic technique, what Thomson provides here is a bit of analysis and some suggestions for how a particular pitch-class can acquire salience. I fail to see how Thomson's "homely algorithm of pitch empowerment as a by-product of temporal play (rhythm)" (172) in any way amounts to a method for taking rhythm into analytic consideration. I stand firmly behind this criticism. In addressing the second of my alleged "fictions," Thomson begs me to cite a page in the book where he shares his "personal judgment of the aesthetic import of a Schoenberg composition" (157). Here I will plead only partially gumilty. While he may not actually state, "I, William Thomson, dislike Schoenberg's Op. 25," he expresses this sentiment by echoing what he considers to be the consensus of the people (whoever they might be). For instance, on the first page of his preface, he states:

Rejoinder to Thomson 183 Even today, most serious music lovers do not find Schoenberg's suite of little pieces [Op. 25] especially delightful. In fact, the composition's principal claim to fame was not its substance, but Schoenberg's use of his "method of composing with twelve notes." (S'sE, ix) Later in the book, he claims that Schoenberg's removal of the distinction between consonance and dissonance caused "the intrinsic loss to music of one of its most effective structural ingredients, one of its most potent conveyors of feeling" (S'sE, 72). Throughout both the book and the response to my review, Thomson cleverly avoids making aesthetic claims in his own name. Instead, he cites others who, I expect, share his point of view.4 If, after all, Thomson does not share Rochberg's view that Schoenberg mapped out a "compositional cul-de-sac," why are no opposing views presented? Interestingly, what Thomson does here is precisely what he accuses me of doing when I cited Patricia Carpenter's review of SsE in a footnote (see my earlier discussion of this issue). Thomson writes, "This feint to another quarter seems meant to convey, without saying so, that Ms. Carpenter's word ends the matter" (159). By the same token, Thomson seems to imply that the very partisan words of John Mueller, George Rochberg, and John Adams will end this matter (157). Is it any wonder that Thomson was so quick to believe that I would employ such a tactic? I have chosen to respond to this particular point (although it is not especially substantive) not because I feel that I somehow must defend Schoenberg's music, but because Thomson's portrayal of "facts" in this case is symptomatic of his treatment of "facts" throughout both his response and his book. Thomson seems to regard himself as the sole guardian of what he calls "facts" (this will be evident to anyone who even skims his response). Is it a fact that atonality causes the intrinsic loss of harmony as a powerful structural ingredient? Is it a fact that most serious music lovers do not find Schoenberg's suite especially delightful? Or are these simply reflective of Thomson's musical experiences and personal opinions (both of which are certainly valuable, though not when represented in this manner)? If what Thomson claims to be facts are indeed "empirically verifiable statistical observations" (158), then let's see the data. Short of convincing data (and I don't know how such data would even be collected), Thomson's 4Sometimes the people he cites are not even named, but are simply referred to as his friends. In the course of his response, we learn that all his friends share his unbending separation of consonance and dissonance (162) and are similarly "ethnocentric" (166).

184 184 In Theory Only facts sound suspiciously like the disgruntled student's protest that "nobody in the class really understood the concept just tested." We can disagree on concepts or interpretation, but let's not disguise our beliefs as somehow uniquely factual or fictitious. Such writing does not foster academic dialog. I found it particularly ironic, therefore, that Thomson has diagnosed me as being afflicted with some mysterious "malaise that has inhibited musical discourse in the United States for some six decades" (155).' Professor Thomson dismisses me and "perhaps [my] mentors as well" as ignorant of "the essential and primal act of scientific inquiry, in this case of behaving like a music theorist" (167). (He says this in response to my mention of his rather unfortunate and needless categorization of some non-Western cultures as "primitive"-a categorization he proudly restates in his response.) Aside from the rudeness of the remark (particularly the implication that my writing does not represent my own thoughts), I fail to see how my mention of this (or, for that matter, my footnoted correction of his citation of Odo, which Thomson found to be needlessly picky) makes me behave any less like a music theorist. Does Thomson believe that we theorists are exempt from remaining current on research trends? Citing findings from older research is fine, but one need not retain dated and potentially offensive language in one's own prose. If my desire not to offend brands me with the recently unfashionable label "politically correct," so be it; I far prefer it to the alternatives. Finally, I would like to say that I am quite sorry that Professor Thomson believes I in any way resorted to calling him "dirty names" or have written a "mean-spirited" review. While I did not find much to like in S'sE, I have no personal vendetta against Professor Th *omson, and I certainly did not believe that anything I wrote was mean. I regret that he was moved to respond with such an ill-tempered, and frequently personal, attack on me. I have largely resisted the urge to respond to those points that liken me to a fool or that criticize my scholarship or my musicianship. I hope that those who read Thomson's response will do so with a copy of my review and, if possible, a copy of his book in hand and will draw their own conclusions. 5Personally, I cannot think of a time when musical discourse in the United States was more healthy. While this might not be such a weighty statement coming from a "novitiate" (as Thomson has labeled me), I sincerely doubt whether the discipline called "music theory" has ever had more facets and outlets than it enjoys today.

Rejoinder to Thomson 185 References Buchier, Michael. 1994. Review of Schoenberg's Error by William Thomson. In Theory Only 12/7-8: 59-80. Carpenter, Patricia. 1993. Review of Schoenberg'Js Error by William Thomson. Music Theory Spectrum 15/2: 286-99. Thomson, William. 1991. Schoenberg'~s Error. Studies in the Criticism and Theory of Music. Philadelphia: Univ. of Pennsylvania Press.

con trib utors Justin London is associate professor of music at Carleton College in Northfield, MN. His primary research interest involves musical meter and rhythm and their perception. He is also interested in musical aesthetics and the philosophy of langulage, as well as the history and analysis of Delta blues. Tuina Koivisto received her Ph.D. in music theory from the University of Michigan. Stephen Peles is a composer and theorist who currently teaches at the University of Alabama School of Music. Janet Schmalfeldt is professor of music at Tufts University and currently serves as President of the Society for Music Theory. John Covach is associate professor of music at the University of North Carolina at Chapel Hill and co-editor of this journal. He is the author of many articles on twelve-tone music, rock, and the aesthetics and philosophy of music. Stephen W. Smoliar is a member of the Society for Music Theory. His main areas of research interest are in knowledge representation, perceptual categorization, and cognitive models. He is currently interested in the role of hypermedia in communication.

Music Theory Online The Online Journal of the Society for Music Theory Current Editors and Editorial Board Members: Lee Rothfarb, David Headlam, Justin London, Brian Alegant, Ann McNamee Robert Judd, Thomas Mathiesen, Bo Alphonce, Jonathan Bernard, Benito Rivera, John Clough, John Rothgeb, Nicholas Cook, Arvid Vollsnes, Allen Forte, Robert Wason, Marianne Kielian-Gilbert, Gary Wittlich, and Stephen Hinton. RECENT AND FORTHCOMING ARTICLES INCLUDE John R. Covach, "Schoenberg's Turn to an "Other" World" Scott Burnham, "Theorists and "The Music ItselfP Joseph Dubiel, "On Getting Deconstructed" Matthew Brown, "Adrift on Neurath's Boat" Kofi Agawu, "Analyzing Music Under the New Musicological Regime" Eytan Agmon, "The Bridges that Never Were: Schenker, Counterpoint, and Tonal Theory" Diana Deutsch, "Music Read and Music Heard: Some Surprising Differences" Jonathan Walker, "The Framing of Schenker: Another Look at Kerman's "How We Got Into Analysis..." In addition to these target articles, each issue of MTO contains commentaries, conference and employment announcements, communications, reviews, recent dissertations, all presented in a timely fashion (and best of all, it's free!). SUBSCRIBING TO MTO MTO is available via the MTO server, FTP, Gopher, and through the World-Wide Web. To subscribe to MTO, send an e-mail message to: [e-mail] which contains the single line: subscribe mto-list <yourfirstname yourlastname> (As usual with list servers, please leave the subject line blank). Subscribers will receive information on MTO formats, retrieval procedures, as well as submission guidelines. For those of you with World-Wide Web connections, check out the MTO homepage at: http://boethius.music.ucsb.edu/mto/mtohome.html

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