Music Theory Online
The Online Journal of the Society for Music Theory
Volume 7, Number 1, January 2001
Copyright � 2001 Society for Music Theory
Bart�k's "Change of Time": Coming Unfixed
KEYWORDS: rhythm, meter, time, Bart�k, Hasty
ABSTRACT: I describe the metrical irregularities of Bart�k's "Change of Time" (Mikrokosmos #126) in a processive manner drawn in large part from the work of Christopher Hasty. I compare this reading with analyses based on fixed models of meter (including the work of Lerdahl and Jackendoff, Maury Yeston, Richard Cohn, and Gretchen Horlacher), concluding that a processive perspective is especially appropriate when metrical irregularity is frequent because it allows irregularity to assume a substantive role in shaping a piece's time.
Received 21 July 2000
[1.1] Bracketed above Example 1 is the opening phrase of Bart�k's "Change of Time," Mikrokosmos #126.(1) If we take the composer's barring as the perceived meter (and I will argue shortly why we might do so), we must also note that from a traditional perspective it is highly irregular, lacking periodicity both at the level of downbeat (there is no single duple or triple meter) and at the level of the tactus (there is no single quarter or eighth pulse). Nonetheless, there is strong contextual evidence for hearing downbeats where they are indicated, for these timepoints mark at bars 2, 3, and 4 a stepwise descending line B flat-A-G, each of which is preceded by ascending stepwise motion and where G is heard as a consonant arrival in the C-dominated vocabulary. Such a "radical" reading(2) places downbeats after three, four, five, or six eighths, truly a "change of time"!(3)
[1.2] While such a description (4 + 6 + 3 + 5) may accurately describe the "end result" of our perceptions, it misses how we might arrive there. If we were to consider the process of coming to this reading we might note that the passage begins in a considerably more periodic way than it ends. I borrow Christopher Hasty's terminology to describe how we might begin to count by threes.(4) When we come to understand B flat as a beginning (a perception first created by its registral status, lending to the previous two quarters an anacrustic character), we may connect "B flat-as-beginning" with the next beginning, the quarter-note, A. The characterization of A as a beginning arises from its stepwise connection with B flat, its longer duration, and its appearance after eighth notes. Because "A-as-beginning" follows "B flat-as-beginning" at a distance of three quarters, we may come to expect a third beginning three quarters hence. In this regard, G's "early" arrival is a surprise (after all, only one and a half quarters have passed since A began), and we may initially regard this pitch as a syncopation. But as the music continues, strong evidence emerges that this G (at the start of notated bar 4) may in fact have become a beginning. Its new identity comes forth as a second quarter-note G reinforces this pitch's stability, and as the reappearance of the opening material immediately thereafter confirms the start of another phrase and another counting process.(5)
[1.3] The arrows below Example 2, drawn from Hasty's usage, illustrate this sequence of events.(6) The solid arches appear gradually with each new beginning, and are ended by the next new beginning. In other words, they move through the present. The dotted arches represent the immediate future: they identify our expectations for new beginnings in the forthcoming music, based on our immediately previous experience. The dotted arches appear in full as each solid arch is ended by a new beginning. Thus, the notations at written bar 4 indicate that our decision to hear the first G as a beginning may be retrospective, graphically indicated by the "x" drawn through the dotted arch. The point is not only that we may come to consider this pitch another beginning, allowing us to label it a downbeat despite its early appearance, but also that this realization enhances G's quality: we appreciate the excitement of reaching the goal pitch on a downbeat, but well before we expected it.(7) When G comes too soon, it gives the phrase an energetic lilt.
[1.4] In this article I will describe a processive approach to metrical irregularity, differentiating it from readings where irregularity takes place within the context of fixed metrical identity. This kind of approach emphasizes the evolution of meter, and underscores our active engagement with the flow of time: our involvement extends beyond marking timepoints as strong and weak to include emerging qualitative characterizations of timespans. When meter is "irregular," we are particularly aware of time as more or less energetic or driven (or at the other end of the spectrum, as more or less relaxed or flaccid). For example, when we evaluate an event both as early and as having the potential to be accented, we allow its irregularity to assume a substantive role: we permit its unique qualities to participate in creating continuity. Our concern is not directed solely toward the maintenance of equal timespans, but rather on how events shape those spans, both as we come upon the events and as we subsequently reinterpret them. As a consequence, events designated as downbeats may not feel exactly the same: as an event that arrives early becomes accented (i.e., becomes a downbeat), its placement colors our perception of the time world it inhabits. Thus, in the opening phrase, G's early arrival, coupled with the ease with which we resume earlier counting, is a mild jolt, and sets the stage for a piece that is prone to reinterpretation and re-evaluation.
[1.5] By contrast, fixed approaches to irregularity tend to feature breaks in a pattern either as momentary disruptions whose value is local, or, if the breaks are persistent, as creators of a new (but fixed) regularity. In the first case, momentary disruptions are described as metrical dissonances that decorate an underlying metrical consonance.(8) In the second, a metrical irregularity is recast as part of a new and ongoing stream of metrical accent. This second case is commonly described as a metrical elision (in other words, the renaming of a point of time from strong to weak, or the reverse). An elision merely marks the point at which a predominant counting resumes, notably leaving aside any residue from the disruption.(9)
[1.6] Some models of metrical irregularity feature unique mixes of fixed and processive perspectives, allowing for special modes of counting in special situations. For example, Andrew Imbrie's conservative-radical distinction offers the notion that careful listeners might disagree where a metrical break occurs. A conservative listener who maintains an original meter as long as possible is drawn into a considerably longer process of re-evaluation than the radical, who lets go at the first opportunity. What happens during the re-evaluation may be very similar to what I am describing as processive counting. In another approach, William Rothstein describes unique spans of time where one might suspend regular counting, proposing that when phrases are expanded, we may momentarily let go of a predominant hypermeter in order to follow a temporary one during an expansion, and that we may return to the suspended hypermeter when the expansion is over. Suspending hypermeter permits a metrical irregularity that arises from phrase expansion to generate its own structure, while at the same time preserving intact the predominant (or fixed) earlier meter. More complex are those models that describe large-scale hemiolas.(10) These approaches compare fixed but competing interpretations over spans of music that are typically at least several bars in length. In so doing, the readings direct our attention to a span in its entirety, reinforcing the fixed identity of each reading. A processive approach, on the other hand, replaces the binary choice of strong or weak with ongoing multiple possibility, valuing the act of metrical evolution and re-evaluation over any ultimate choice of identity.
[1.7] In this article, I shall use Bart�k's "Change of Time" to demonstrate these differences, offering several readings of the opening irregularity, as well as the 5/8 music in the second part of the piece, to draw the distinctions into sharper view. I suggest that in the processive reading we understand Bart�k's title "Change of Time" to refer not only to alternating patterns of time signatures, but also to changes in the nature of the piece's time, especially as 5/8 "time" emerges more fully in the latter part of the piece. In this light, metrical "irregularity" serves as an agent, rather than a disruptor, of continuity: we enjoy Bart�k's piece because we learn how to count.
[2.1] Let us now consider what happens after we identify G as a beginning. Below bar 4 on Example 2 the dotted arch is followed by a question mark to indicate that our anticipation of a beginning three quarters hence has been considerably weakened by the interrupting G. As a beginning, it has entered so early that it significantly cuts off the emerging potential begun at bar 3, and leaves us somewhat of a loss. Or more positively, we might claim to be more open to changing circumstances provided by the return of the opening anacrustic gesture. Because this opening gesture now has a well-defined identity, its return helps us to anticipate the next beginning. The return also has larger, more important consequences. Because F and G at notated bar 5 are not a beginning (rather, they lead to a beginning), they may become attached to the previous beginning, the interrupting G at notated bar 4 (on Example 2 the solid arch continues through bar 5). In this way, F and G participate in the emergence of a new span, shown by the longer, solid arch underneath the denied, dotted arch at bar 4. The new solid arch measures a distance of nine eighths between G-as-beginning and the next beginning at notated bar 6.
[2.2] This reading has important ramifications: if we carry this new span of nine eighths forward at the start of bar 6, we are rewarded rather than surprised by the "interrupting G" the second time around (in the second phrase at bar 8). On Example 2, the dotted arch leading to bar 8 is augmented by a solid one, and in turn that solid arch gives rise to the next anticipated beginning (shown by the last dotted arch.) In other words, our experience of G's arrival may be quite different the second time than the first: if we allow ourselves to follow the new possibilities that arise out of the first phrase, we are more prepared to expect the next "interruption," or at least to be less distracted by it, for we may have come to value beginnings that occur every nine eighths.
[2.3] But what of the earlier span of three quarters? When the durational and melodic patterns of phrase one return in phrase two, can it be that triple counting has disappeared entirely? In this processive reading, the start of the second phrase (bars 6 and 7) contrasts with the piece's opening. It is not that the opening triple projection (and its subsequent denial) are absent the second time around, but that this activity is now inflected by the piece's continuation. The beginning of the second phrase at notated bar 6 is marked by moving, solid arches both six and nine eighths in length. Because the span of nine has been recently anticipated (as shown by the dotted arch above the solid one) it is therefore a stronger suggestion. Note how bar 7 contrasts with bar 3: bar 7 suggests a lengthening of the original three quarters by three eighths, a reading that differs with the metrically uncertain bar 3.
[2.4] The second phrase, then, is not merely a repetition of the first, but a development where the second phrase both draws and comments upon the metrical issues raised by the first: the phrases are joined by the emergence of two viable projective potentials (counting by six and by nine) where the second enhances and extends the first both as rhythmic and formal consequent.
[2.5] Let us now turn to three alternative readings where metrical identity is fixed. In Example 3A I have rebarred the phrase entirely in 3/4. In this "conservative" reading one continues to count periodically, thereby hearing both G's as syncopations.(11) Although the counter is rewarded for holding onto triple meter when the second phrase begins on target,(12) the opening phrase is qualitatively quite different: its goal pitch is weakened by arriving twice in metrically weak locations. I might characterize it as more unsettled than sprightly. More importantly, the second phrase proceeds exactly as the first did: its designation as a consequent phrase arises only from our recognition of the sequential pitch structure rather than from emerging metrical experience.
[2.6] Alternately, in Example 3B I have used dots to replicate the composer's barring, adapting Lerdahl and Jackendoff's usage as I myself have suggested in an earlier article.(13) In that paper, I suggested a model where breaks in periodicity may occur at the low levels of tactus and barline through the mechanism of metrical elision. A reinterpretation of a beat from weak to strong is shown by a circled dot, and the opposite action, from strong to weak, is shown by an empty circle followed by an arrow indicating the new location of the strong beat. Example 3B interprets the first G in bar 4 as being irregular at several levels of meter, all at once: it is a beat that arrives late (as we reinterpret the preceding F as part of a beat) and a downbeat that arrives early. Hypermetrical levels are unaffected by the irregularity: note that durations of nine eighths proceed periodically. In this reading, beats are fundamentally two eighths long, with the possibility of being extended by an eighth; measures are more variable, but their changing lengths are contained by the regular progression of duple hypermeter.(14) What differentiates this analysis most strongly from Example 2 is that in 3B, two fixed levels of meter are concurrently broken over bars 3 and 4. In other words, 3B assumes that levels of meter are already well established. In the processive reading, these early bars are characterized as far more unstable because "levels" are not particularly well formed. (How do we know which levels of counting have been broken?) Furthermore, in the processive reading, the second phrase is different from the first: bars 7 and 8 (the analogous location) are associated with two viable countings, one of which is more strongly projected because of immediately preceding events.
[2.7] In both Examples 3A and 3B we are still seeking a regular time that may be occasionally adjusted. The time of periodicity remains at its heart smooth, and disruptions are ultimately subsumed by a desire for equal proportion. In other words, irregularities are unable to color the fundamental nature of periodic time. "The" phrase (be it in its first or second representation) is not sprightly at its core: rather, its lilt will always be understood as a negation of periodicity.
[2.8] Example 3C provides one more reading, here focused on a hypermetrical level. Its second line of dots (labeled "6") represents the completely periodic reading of three measures given in Example 3A; its third line of dots (labeled "9") shows the hypermetrical reading given in 3B, identifying its four bars as consisting of two hypermeasures. Because eighteen is the lowest common denominator of six and nine, the two readings arrive jointly at the downbeat of bar 6, where the whole process will repeat once more. Thus, Example 3C shows a hypermetrical hemiola: eighteen eighths may be subdivided either into two groups of nine or three groups of six. The potential for dual hypermetrical interpretations originates in part from Bart�k's use of a span of eighteen eighths, for eighteen may be factored by either two (9 + 9) or three (6 + 6 + 6). The "6" reading (on the upper staff) is "fully consonant," dividing the span into one hypermeasure with three bars whose internal divisions are also periodic, whereas the "9" reading is not subdivided into measures.(15)
[2.9] Because it results from two opposing parsings, this reading is not entirely fixed. But its focus is not the moment-to-moment counting process, but rather a "high-level" metrical dissonance. This dissonance relies on one's apprehension of the entire eighteen eighths, directing one's attention to the phrase's entire span rather than its inner inflections. In other words, although it does portray spans of six and nine as interactive, Example 3C misses the experience of learning to anticipate six and then (also) nine over the first two phrases, a phenomenon that recurs when Bart�k repeats phrases one and two as three and four. That the processive counter may once again participate in the returning tug of six, this time more fully aware of its likely conversion into nine, and then anticipate the tug of six as the phrase returns yet again, is key. In fact, in the processive reading, we can come to understand the repetitions of the opening phrase as imperative for its full rhythmic-metric appreciation. We may participate in a cycle of counting that draws on a past of both sixes and nines, appreciating both the suggestion and subsequent denial of 3/4 meter and the measure of equilibrium provided by the periodicity of nine, allowing us each time to catch our breath as the anacrustic gesture returns. A processive stance proposes that this apprehension gains strength only as the music continues.
[2.10] Of course, how one chooses to count is a highly personal decision. The reading each person prefers is probably influenced by a variety of factors, including our inherent conservative or radical metrical habits, the performance we hear, and our familiarity with Bart�k's frequent use of metrically irregular folk tunes. After all, each of us values periodicity more or less highly, finding a metrical grid at times either reassuring or limiting, and it may be that the processive reading proposed here seems implausible to some. My aim, however, is not to limit our options but rather to pursue meter as a flexible activity, and to set forth the processive reading as an opportunity rather than a directive. The point is that the act of counting is not something that happens to us, but rather something in which we actively engage, and where we are continuously making choices. Let us see what opportunities arise as the piece continues.
[3.1] After the fifth statement of the original phrase (bar 21), Bart�k introduces a new durational and pitch pattern, characterized by repeating statements of quarter-quarter-eighth. The origins of the pattern are not hard to trace: the 5s replicate the final part of the original eighteen eighths. Bart�k makes the connection particularly clear by retaining the top line pitch E across bars 20 and 21; Example 4 marks this connection and brackets the first statement of the antecedent-consequent 5 + 5 melody. (The reader may return to Example 1 to see the new music in its entire context.)
[3.2] Before tackling the emergence of 5 out of the previous pattern, let us first consider the new music by itself (momentarily taking bar 21 as an admittedly artificial beginning point.) The overtaking of 5 heightens the already energetic character of this piece, a development explained best by a processive analysis. Example 4 demonstrates. First consider the repeated durational figure quarter-quarter-eighth by itself. The double-neighbor figure around E reinforces the anacrustic quality of the eighth-note D(16) and identifies the E at bar 22 as a second beginning; the reappearance of E every five eighths thereafter continuously increases the strength of E's identity as a beginning. On Example 4, the upper two rows of arches mark the new beginnings, events that contrast with the beginnings in the first part of the piece in several ways. Most obviously, the new beginnings come more quickly, giving this section an intensified forward impetus. Furthermore, they come more regularly: gone is the alternation of emerging 6s and 9s, replaced--ironically--by 5s.
[3.3] I have directed our attention to the recurring 5s, but equally engaging is the internal make-up of this duration. Any 5 has a special quality deriving from its inherent unequal subdivision. In this part of the piece, the inequality comes as a lengthened continuation, meaning that the "second beat" is consistently extended, and that it delays each successive beginning by an eighth. The little "catch" this delay imparts to the second section of music tempers its speed and regularity, matching the sprightly quality of part one.(17)
[3.4] The regularity and speed of the new beginnings also encourage us to anticipate larger beginnings. (In other words, "hypermetrical" counting is more vivid in this section of music.) On Example 4, the pair of larger arches (below the smaller ones, and also starting at bar 21) identifies the E in that bar as a large beginning, suggesting that we consider its dominance to continue beyond the repetition of E in 22 to the sequential repetition beginning at 23. The durations of 10 eighths, like those of 5, continue smoothly until nearly the end of the piece (to be considered shortly); in contrast with the earlier music, where durations of 9 eighths competed with those of 6, 5s and 10s combine to impel this music forward more directly.(18)
[3.5] Fixed readings are more problematic in the second section. From the periodic perspective, any 3/4 counting will soon disintegrate. At best, one may count a continuous quarter pulse, as shown by the reading given in Example 5A. Although this unlikely construal does exhibit parallel interpretations of each phrase, its clunky emphasis on the pitch D in each second bar misses E's decoration first by G above and then by C below. The modified periodic reading in Example 5B is more promising. Its open circles on the eighth of each durational pattern suggest the little catch we experience while waiting for the next beginning to arrive. Missing, however, are both the process of coming to count by 5s (and eventually by 10s) as well as our sense that these spans become more stable as they are repeated.
[3.6] At the risk of belaboring the reader, I wish to make one additional brief point about the second section, shown on Example 6. The piece reaches its culmination at bar 32, whose arrival is prepared not only by pitch events (i.e., the long ascending scale in the left hand) but also by rhythmic ones. The culmination is enhanced by our awareness that the phrase leading to bar 32 has gone on too long. Notice that the original 5/8 gesture (with its final anacrustic eighth) keeps reappearing from bars 27 to 31; the anticipated response that bar 22 provided for bar 21 is postponed over and over, and thereby greatly intensifies the rhythmic drive forward. As shown in Example 6, this effect is achieved by our twofold experience of the final bars: while we are becoming ever more confident about beginnings that arise every 5 eighths, the larger beginnings (those every 10 eighths) are put on hold, suspended by continuation. On Example 6, the "suspense" is shown by the "x" that appears as each possible larger arch fails to complete itself. When the consequent bar finally arrives, it brings another surprise, for the C-major triad at bar 32 is itself unique. With its unprecedented succession of 3 quarter notes, the lone 6/8 bar is not easily accommodated by any earlier experience. Rather it represents a final event in the ongoing "change of time" that this piece represents. This series of durations produces a true metrical "hiatus": it is as if counting itself must come to a halt (which of course, it shortly does!).(19)
[4.1] In some ways, a processive approach to counting is quite disarming: when an event may be characterized in more than one way, it is as if the ground is shifting under our feet. After all, a processive description rejects "the meter" of a piece, instead favoring metrical identities that may change as a piece proceeds and recognizing that some events emerge with clearer identities than others. At its heart is a perspective of time not as periodic measure, but rather as subject to constant manipulation, especially as events come into being and pass into memory. While the demands made on the counter are high, the benefits of counting in this way are at least twofold. First, processive counting is especially germane to readings that emphasize meter as a narrative device, for it suggests that one's attentions are directed to a persistently changing environment where even repetitions are "new" and forward reaching. A processive approach empowers meter to describe continuities arising from change as well as those arising from repetition. Processive readings are also especially relevant for music that is vitally "irregular," for it values such irregularities as capable of shaping the essential nature of time within the context of a given piece.
[4.2] More generally, in a processive approach meter encompasses more than measurement, for it is able to color the flow of time itself. The act of counting reaches beyond the mere identification of metrical identity; in fact, this binary act of choosing either weak or strong is superseded by our anticipation of identity, and by our ongoing experiences of fulfillment, denial, and uncertainty as a piece develops. In "Change of Time," counting at the opening is uncertain because a potential 3/4 meter is consistently interrupted, and quite possibly replaced by a larger duration of 9 eighths. As the first part of the piece unfolds, a recurring cycle of 6 and 9 becomes "regular" (in the sense that it becomes predictable), but both of these spans ultimately give way to a more strongly confirmed series of 5s. During this second part of the piece, as our counting accelerates, it also becomes more evenly directed. More colloquially, we experience the irony of counting spans more typically associated with periodicity (6 and 9) as less stable and the subsequent 5s as a resolution of that tension. Coming unfixed helps us to recognize these "changes in time."(20)
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1. The bracketed melodic and durational pattern shown in Example
1 repeats four more times, a fact that will be central as the analysis
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2. The terms radical and conservative come from Andrew Imbrie's "'Extra'
Measures and Metrical Ambiguity in Beethoven," in Alan Tyson, ed., Beethoven
Studies (New York: Norton, 1973), 45-66. Imbrie suggests that in the face of
metrical irregularity, conservative listeners prefer to maintain an established
periodic counting as long as possible, whereas radical listeners prefer
immediately to adjust to a new counting.
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3. According to Benjamin Suchoff's Guide to the "Mikrokosmos"
(New York: Da Capo Press, 1983), Bart�k remarked that the changes of time in
#126 were similar to those found in Romanian folk music (p. 109). Janos Breuer
identifies the changing rhythms in this piece as representative of the Romanian
kolinda (carol); see "Kolinda Rhythm in the Music of Bart�k," Studia
Musicologica Academiae Scientiarum Hungaricae 17 (1975): 39-58.
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4. In its emphasis on moment-to-moment decision-making, this analysis draws
upon meter as described by Christopher Hasty in Meter as Rhythm (New
York: Oxford University Press, 1997). Hasty describes meter as arising
processively, that is, by comparing one's expectations for beginnings (and
therefore accents) with how they actually occur. Any misrepresentations of his
terminology and symbols are of course my own. I also recognize that the reading
given below may not correspond with the reader's preferred one. The point is not to choose a single reading, but rather to focus on the
notion that readings emerge rather than come to us fully formed. Any
reading (not just the one given above) takes time to come about.
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5. Although Hasty does not describe processive meter as an act of counting, I
will frequently use this verb in my descriptions to draw attention to our
active, in-time engagement with meter. In other words, I do not intend
"counting" to represent an independently imposed or periodic schema.
Victor Zuckerkandl describes counting analogously (as a process); see for
example Sound and Symbol Music and the External World, trans. Willard
Trask (New York: Pantheon Books, 1956), 167-68.
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6. Directly above the first measure of Example
1 is a vertical slash followed by a diagonal one. This is Hasty's way of
indicating that the very beginning event (the first two quarter notes) becomes
an anacrusis, and therefore (eventually) does not create additional projections.
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7. Hasty describes possible "early" entries of beginning events in
his Example 7.3 (Meter as Rhythm, 87). The various cases shown there
are distinguished by how early a third event arrives. My description of G's
early arrival is most similar to Example 7.3d, where the third event comes so
early that the potential of the previous beginning comes into question. This
will be discussed below.
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8. Maury Yeston first described metrically defined strata as consonant or
dissonant with one another in The Stratification of Musical Rhythm (New
Haven: Yale University Press, 1975). The model has been considerably amplified
by Harald Krebs in "Some Extensions of the Concepts of Metrical Consonance
and Dissonance," Journal of Music Theory 31.1 (Spring 1987): 99-120,
and more recently in Fantasy Pieces: Metrical Dissonance in the Music of
Robert Schumann (New York: Oxford University Press, 1999).
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9. The metrical grid developed by Fred Lerdahl and Ray Jackendoff is perhaps
the clearest representation of fixed meter. Lerdahl and Jackendoff define meter as strictly
periodic at the levels of tactus and measure, permitting metrical
reinterpretation (their term is metrical deletion) only at hypermetrical levels
and in conjunction with grouping elision. See A Generative Theory of Tonal
Music (Cambridge, Mass.: MIT Press, 1983), especially pp. 99-104.
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10. Yeston, Stratification, first formalized hemiolas above the level
of the notated measure. Cohn has developed more sophisticated models. See
especially his "Metric and Hypermetric Dissonance in the Menuetto of
Mozart's Symphony in G Minor, K. 550," Int�gral 6 (1992):
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11. Breuer, "Kolinda Rhythm in the Music of Bart�k," discusses the
practice of rebarring difficult passages periodically as a performance aid,
citing Bart�k's suggestion to the conductor Hugo Balzer in 1930 that the
changing time signatures at rehearsals 47 to 49 in the first movement of the
First Piano Concerto might be rebarred entirely in 2/4. In his Example 10 Breuer
aligns the original notation with a 2/4 rebarring, and concludes that in
addition to being unmusical, the periodic version actually makes the orchestral
parts harder to read.
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12. This reading is similar to readings of Stravinsky's metrical irregularity
by Pieter van den Toorn. See, for example, his discussion of background
periodicity in chapter 3 of Stravinsky and "The Rite of Spring"
(Berkeley: University of California Press, 1987).
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13. See my analyses of portions of Renard and Les Noces in
"Metric Irregularity in Les Noces: The Problem of Periodicity,"
Journal of Music Theory 39.2 (1995): 285-309.
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14. In the latter part of my 1995 article, I suggest that when irregularities
are repeated, they may become contextually regular. That we may experience
repeated irregularities differently at different points of the piece is an idea
central to this paper.
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15. Full consonance, a term from Cohn's "Metric and Hypermetric
Dissonance" is the partitioning of a span where each level is exclusively
subdivided by either two or three, terminology that echoes the metrical
rules of Lerdahl and Jackendoff and highlights the typical metrical ease with
which such passages are associated. By distinguishing the "normal
situation" of full consonance, Cohn is able to describe situations outside
the norm, such as ongoing competing partitions of two and three at a single
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16. One's perception of quarter-eighth has been heavily colored already in
this piece by the repetitions of bar 3, where a double neighbor figure around G
is also anacrustic.
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17. Hasty argues that a span of 5 cannot give rise to strong projections
precisely because neither 2 nor 3 is exclusively fundamental to its
identity; he describes the quality of 5 as "limping." Instead he
focuses on the larger groupings within 5; he would label the 5 here as
"duple unequal." See his discussion in Meter as Rhythm, pp.
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18. Alternatively, some counters might come to count by 10s as early as bar
20 (i.e., beginning hypermeasures a bar earlier than shown in Example
4, in what Lerdahl and Jackendoff would call an "out-of-phase"
manner). This reading considers bar 21 as a continuation of bar 20 based on the
precedent of bars 4 and 5. An expectation for continuation after bar 20 has been
shaped by appending to bar 4's cadential arrival an anacrustic gesture (the two
stepwise, ascending quarters that begin each phrase); this joining is reinforced
in each succeeding phrase. We might hear a similar continuation take place at
bar 21, for its E and F mimic the anacrusis from earlier in the piece. The larger point remains
unaffected, however: in either reading, counting
in part two becomes smoother sailing: both 5s and 10s continue easily and
without significant interruption until near the end of the piece.
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19. Hasty, Meter as Rhythm, describes a hiatus as "a break
between the realization of projected potential and a new beginning" (p.
88); in other words, a hiatus takes place when projection is temporarily
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20. The animations in Examples 2, 4, and 6 were designed by Indiana University
music theory doctoral student Brent Yorgason. I wish to thank him for his
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