Music Theory Online
The Online Journal of the Society for Music Theory
Volume 7, Number 6, December 2001
Copyright © 2001 Society for Music Theory
Mark J. Butler*
Turning the Beat Around: Reinterpretation, Metrical Dissonance, and Asymmetry in Electronic Dance Music
KEYWORDS: rhythm, meter, reinterpretation, metrical dissonance, layering, asymmetry, electronic dance music, African music, Harald Krebs, Jay Rahn, Stephen Handel, Christopher Hasty
ABSTRACT: This paper considers some of the issues involved in the analysis of rhythm and meter in electronic dance music. It begins by considering metrical dissonance and ambiguity within the context of a layered approach to musical meter; it also highlights some of the distinctive ways in which these phenomena are manifested in this repertory. The second half of the paper focuses on the use of asymmetrical patterns in electronic dance music and considers some broader questions of rhythmic and metrical theory. Throughout, the paper draws upon a number of different music-theoretical sources, including work by Harald Krebs, Jay Rahn, and Christopher Hasty, as well as scholarship in ethnomusicology and cognitive science.
Revised Submission Received 22 July 2001
 Within the past decade, music-theoretical analysis of popular music has become increasingly common. Nonetheless, until quite recently the purview of such research has been relatively limited, focusing mainly on classic and art rock from the 1960s and 70s. While these repertoires have provided fertile ground for analytical exploration, several more recent genres and styles raise interesting structural, semiotic, and aesthetic questions that also merit study.
 One of the most distinctive types of contemporary popular music is electronic dance music, a broad category that includes styles such as techno, house, drum-n-bass, and trance.(1) Electronic dance music differs from both rock and art music in a number of ways, all of which suggest avenues for productive theoretical investigation. While many genres of popular music are distinguished from each other on the basis of "surface" differences such as instrumentation or lyrical content, electronic dance music presents a distinctive overall formal structure in addition to its characteristic instrumentation. It also differs from most other types of Western music in its rejection of harmony as a primary musical parameter. Its use of pitch is typically restricted, with the majority of musical development taking place instead in the realms of rhythm, meter, texture, and timbre. Consequently, these areas should be principal concerns for the analyst wishing to pursue a close examination of musical sound in this repertoire
 Rhythm provides an ideal starting point, as it seems to be the element to which listeners and fans of EDM relate most directly. This connection is most immediately apparent on the dance floor, where audiences physically enact the rhythms of the music. Written discussions of electronic dance music also tend to highlight its rhythmic qualities--references to its "unrelenting rhythm," "pulsing dance rhythms," and "irresistible tribal rhythm" are common(2)--and fans often refer to the music simply as "beats" (as in, "let's go to the club and hear some beats"). Yet EDM poses a challenge to the analyst, for it differs in many ways from the repertoires that current music-theoretical models of rhythm and meter were developed to address. Transcription can give the impression that it is less complex rhythmically than these other bodies of music: although it typically consists of many different textural layers, each of which has a different rhythmic pattern, these patterns can appear quite simple when considered individually, and they repeat for long periods of time without changing. Nonetheless, the experience of listening to electronic dance music is neither simple nor monotonous, for it engages one's perception of rhythm and meter in a number of interesting ways. In this paper I will examine some of the issues involved in the analysis of rhythm and meter in this repertoire. In the first half of my paper, I will consider metrical dissonance and ambiguity within the context of a layered approach to musical meter. The discussion, which will be centered upon several short musical examples, will also highlight some of the distinctive ways in which these phenomena are manifested in EDM. The second half of my paper will focus on the use of asymmetrical patterns in EDM. Though this section will begin with two musical examples, it will develop into a more general discussion focusing on broader questions of rhythmic and metrical theory. Throughout the paper I will draw upon scholarly work in a number of different fields, including music theory, ethnomusicology, and cognitive science.
 Let us first consider a brief musical example, an excerpt from the song "Piku" by The Chemical Brothers. At the beginning of this example, two distinct textural layers, defined by timbre and by rhythmic patterns, are audible; they are not aligned, however (Example 1a). Accents are created by register in the synthesizer pattern and by attack-point spacing in the percussion part, thus suggesting two different streams of quarter-note pulses. This state of nonalignment continues for about thirty seconds. Although the balance between the two parts changes as the example unfolds, neither beat pattern stands out clearly as the dominant one. Thus one can easily shift attention from one pattern to another, so that the figure/ground relationship between the two patterns reverses.(3) In fact, the reader is encouraged to do so while listening to the example.
 This pattern forms what Harald Krebs would describe as a displacement dissonance, a type of metrical dissonance in which two or more layers of the same length are nonaligned.(4) The term metrical dissonance is derived from Maury Yeston's discussion of rhythmic consonance and dissonance.(5) Like Yeston, Krebs believes that meter is formed by the interaction of several different layers of motion. Usually at least three layers are present: the pulse layer, which is the fastest regular layer of motion, and two or more slower-moving interpretive layers, which group the pulse layer into larger units. In 3/4 time, for example, there is a pulse layer consisting of eighth-note motion, one interpretive layer of quarter-note motion, and a larger interpretive layer of dotted-half-note motion.(6)
 A displacement dissonance occurs when an interpretive layer sounds like it is displaced from a metrical layer; the conflicting layer is called the antimetrical layer. Since electronic sound production facilitates this sort of nonalignment, it is not surprising that it is a frequent source of metrical dissonance in EDM. In Example 1a, a sixteenth-note pulse layer is grouped by two different patterns moving in quarter notes. This excerpt differs from most of Krebs' examples, however, because it is in a certain sense prior to meter; there is no larger layer grouping the quarter notes into measures. This heightens the ambiguity of the example: it is unclear which layer is metrical and which is antimetrical.(7) If we decide to focus on the synthesizer pattern, then we hear the snare drum being displaced by three sixteenth notes, a dissonance that Krebs would describe as D4+3.(8) On the other hand, if we attend to the snare drum pattern, then we hear the synthesizer as displaced by a single sixteenth-note, or D4+1. Both of these possibilities are summarized at the bottom of Example 1a. According to Krebs, the closer a dissonance is to a state of alignment, the greater the intensity of its dissonance. In this example, then, either possibility creates a rather strong dissonance. If the layers were displaced by two sixteenth notes, however, the dissonance would be less intense; furthermore, there would be less opportunity for reinterpretation.
 Example 1b, another excerpt from "Piku," begins in the same place as the previous example but continues further into the song. I suggest listening to the example apart from the transcription at first; as before, try to experiment with different figure/ground relationships, and note the nonalignment of the layers.
 Near the end of the example, a bass drum with a regular half-note pulse enters (see Example 1b). This new layer seems to solve the problem of nonalignment. The snare drum part of Example 1a drops out, and the new synthesizer pattern seems like a union of the previously nonaligned layers, combining the two-eighths pattern on B and G-sharp with the sixteenth note/dotted-eighth rhythmic pattern previously played by the snare drum (see brackets in Example 1b).(9)
 In this example, the entrance of a new layer resolves a metrical dissonance. In other cases, such an entrance may actually create a metrical dissonance. For example, at the beginning of the song "Cups" from Underworld's 1998 album Beaucoup Fish, the synthesizer pattern clearly begins on the downbeat of a 4/4 measure, as shown in Example 2a. This hearing is soon confirmed by a cymbal pattern articulating a quarter-note pulse. After about fifteen seconds of the synthesizer pattern, however, a loud drumbeat comes in with a different pattern, whose entrance is offset from the previously established measure by a single eighth note. If one focuses on this drumbeat, it is possible to hear the beat pattern shift, so that the synthesizer and cymbal patterns now seem to occur on offbeats. As shown in Example 2b, the repetition of these patterns suggests a D8+1 displacement dissonance. At this point, though, the evidence for this hearing remains inconclusive; conservative listeners may not be willing to give up the initial metrical framework just yet.(10) The displacement becomes definitive, however, after the first eight bars of the drumbeat, when a new synthesizer pattern begins to reinforce the downbeat of the drumbeat pattern (Example 2c).
 Electronic dance music is not the only type of popular music in which displacement dissonances occur, of course. Other scholars have pointed out related phenomena in repertoires such as rock, jazz, and blues. For instance, Dave Headlam has discussed the use of metrical conflict and "metric shift" in "Rollin' and Tumblin'," a work that began as a blues song and was subsequently recorded by the rock band Cream. Headlam's analysis centers on a motive that appears in several versions of the song. In recordings by blues musicians such as Muddy Waters, the motive is syncopated, emphasizing beats 2 and 4 of each 4/4 bar. This pattern of accentuation is highlighted further in the 1966 Cream version, when a newly added drum pattern (which initially reinforced the prevailing meter) is shifted one beat over, producing a displacement dissonance. The conflict created by this dissonance is eventually resolved, however, during the B section of the song, in which both the motive (now appearing one beat earlier) and the drum pattern align perfectly with the meter.(11)
 Another example of metrical dissonance in rock is provided by John Covach, who discusses the presentation, development, and eventual resolution of a metrical dissonance across the course of Yes's song "Close to the Edge."(12) Examples of metrical dissonance are also easily found in jazz.(13) For the purposes of the present discussion, then, the significance of these metrical phenomena lies not in the mere fact of their occurrence in electronic dance music, but rather in the distinctive ways in which they are used. For instance, consider how the metrical dissonances described above are situated within the larger works of which they are a part. In the EDM examples, metrical dissonance almost always occurs as part of a process unfolding continuously in real time. In comparison, in the examples discussed by Headlam and Covach, related metrical dissonances are generally separated from each other temporally, sometimes by rather considerable distances (for instance, the time interval between the second and third appearance of the dissonance in "Close to the Edge" is approximately 9 minutes). After their initial presentations, their implications and inherent tensions are developed in various ways, as might occur with a motive in a Classical work.(14) The exploration of motivic possibilities is also a well-known feature of jazz improvisation, and Cynthia Folio's examples can be understood in these terms.(15) In electronic dance music, on the other hand, the emphasis is not so much on the long-range exploration of a motive's possibilities as it is on producing an immediate and striking effect--either to create a point of formal articulation (as in Example 1b, in which the resolution of the opening metrical dissonance coincides with the beginning of the work's first groove), or to introduce an element of surprise (as in Example 2, in which an otherwise straightforward textural buildup is complicated by the introduction of a displacement dissonance).
 Furthermore, while the types of metrical dissonance discussed by Krebs can occur in many different repertoires, they are particularly relevant to electronic dance music for a number of reasons. As discussed above (paragraph 5), the Yeston/Krebs model conceptualizes meter as the union of several layers of motion. As in the music of Robert Schumann (the composer upon whom Krebs has focused most extensively), these layers are particularly clear in electronic dance music. They often appear as discrete elements of the texture, individuated by timbre, register, and pattern repetition. For instance, in a 4/4 meter, one might find one part of the texture articulating a quarter-note pulse, another moving primarily in eighth notes, and a third emphasizing the whole-note layer, with each element being registrally and timbrally distinct. Dissonant layers usually stand out clearly as well.
 The situation just described might not seem all that unusual in and of itself. In electronic dance music, however, this layered approach to metrical construction is further emphasized by the relative equality of its textural layers. No single voice dominates the texture: in contrast to the vast majority of commercial popular music produced in America and Europe since the advent of rock-n-roll, EDM is not vocal music; in general, its only verbal elements are short, fragmentary samples.(16) Neither does it replace solo voices with instrumental ones: unlike a great deal of classical (and other) instrumental music, it does not as a general rule employ melody-and-accompaniment textures.
 Layering is also absolutely essential to the way electronic dance music is recorded and performed. In the studio, EDM is produced in a multi-channel environment, which allows all of its sound sources to be controlled independently, so that layers of motion can be combined in a variety of interesting ways. Of course, this technology is not limited to EDM; in fact, it is commonplace in contemporary popular music. In most pop traditions, however, audiences expect performers to be able to produce reasonable facsimiles of their recordings in live performance, a factor that may limit the complexity of metrical dissonance. This is not the case in EDM, though. In this repertoire, a completely different tradition of live performance exists, one in which DJs, rather than studio artists, take center stage.(17) For the most part, records are treated not as finished products but as raw material for manipulation by the DJ. While the music that one hears at a nightclub or a rave originates in the studio, records are rarely heard in their original studio-produced forms. Instead, they are manipulated and combined with other records in a real-time process that produces substantially different compositions. The DJ uses turntables to play two or more records at once and a mixing board to control the balance between the records (one record is usually more prominent). The mixing board also allows the DJ to control the high, middle, and low range of each record, which makes it possible to manipulate layers within as well as between records. This capability facilitates the real-time creation of displacement dissonances like those seen in Examples 1a/1b and 2 (which are taken from studio-produced recordings). For instance, reinterpretations (cf. Example 2b) can be created by playing a pattern such as that found in Example 2a on one record, then bringing in the bass drum of another record on the offbeats. The lower, more resonant bass drum sound "turns the beat around," but the first pattern persists, thereby producing a displacement dissonance. Conversely, given a situation like that of Example 2c, the DJ might use the high- and low-range controls to remove the bass drum and high synth pattern, thus calling a seemingly clear metrical interpretation into question.
 Thus, while a layered conception of musical meter can be applied to many different repertoires, it is particularly well suited to electronic dance music, in which a layered approach is fundamental to musical construction. As we have seen, this approach manifests itself in a variety of ways: in techniques of performance and production, in the connection between metrical and textural layers, and in the relative equality of textural layers. In fact, this last feature promotes another distinctive characteristic of this repertoire: in many cases, especially near the beginning of a track, there is no definitive accentual interpretation of a passage. Examples 1 and 2 demonstrate this phenomenon quite clearly. In Example 1a, as previously discussed, it is unclear which layer is metrical and which is antimetrical; in Example 2, a layer that is suggested as metrical eventually turns out to be antimetrical.(18) The absence of a definite interpretation in such situations encourages the listener to experiment with different interpretations. In repetitive music such as EDM, this ambiguity often plays a crucial role in the creation of musical interest. In fact, Steve Reich has noted a similar phenomenon in minimalist music. Of his own music, he says: "If I compose music that is to use repeating patterns and is also to remain interesting I must build in rhythmic ambiguity to make it possible for the ear to hear a given pattern beginning and ending in different places depending on slight differences of accent and on how one listens."(19)
 In the previous examples, ambiguity was created by the interaction of two or more patterns. As Reich's statement suggests, however, individual patterns can also be inherently ambiguous. In fact, extreme simplicity often promotes such ambiguity. For example, consider the TICK-tock-TICK-tock-TICK-tock pattern of a clock. By focusing on the "tock," one can easily hear the opposite accentual pattern: tick-TOCK-tick-TOCK-tick-TOCK.(20)
 This type of ambiguity also figures prominently in electronic dance music. As an example, let us consider another track by Underworld. This song, entitled "Moaner," begins with a drumbeat articulating a steady quarter-note pulse; between these beats, a synthesizer plays three sixteenth notes, as shown in Example 3a. It seems probable that the drumbeat, being louder and longer in duration, would play the primary role in determining the accentual pattern. The synthesizer sound also attracts attention, however, by changing timbrally and dynamically, while the sound of the drumbeat does not vary. As one focuses on the synthesizer sound, one can hear an accent on the second sixteenth note of the pattern, as shown in Example 3b. It is also possible to hear an accent on the third sixteenth, as seen in Example 3c. This last hearing begins to be especially noticeable after about fourteen seconds of the song. At this point, the timbre of the sound begins to change, as a process called a filter sweep, which gradually begins to emphasize the upper partials, is applied to it. The sound crescendos at the same time. Underworld also draws attention to the sound changes by altering the drum pattern very slightly just before the changes begin.
 In a sense, the interpretations shown in Examples 3b and 3c create displacement dissonances with the drumbeat. These dissonances differ from the one heard in Example 2c, however, since they are created by the reinterpretation of an already-sounding pattern, rather than by the entrance of a new pattern. As such, they are inherently more fragile: if one focuses too much on the drumbeat, the new interpretations of the synthesizer pattern seem to fade away.
 In addition to highlighting the ambiguity found in these examples, I have tried to emphasize the way that accentual and/or metrical interpretations evolve or emerge while one listens to them. In this respect, I am inspired by the work of Christopher Hasty and Gretchen Horlacher, who have captured this quality of musical experience very effectively in their approaches to meter. For instance, Hasty writes that "a piece of music, while it is going on, is incomplete and not fully determinate--while it is going on, it is open, indeterminate, and in the process of becoming a piece of music."(21) The processual quality that Hasty describes is particularly noticeable in electronic dance music. As in minimalist and other types of repetitive music, processes unfold very gradually in EDM. Patterns are repeated for long periods of time. For example, in "Moaner," the patterns heard in Example 3 repeat almost constantly throughout the length of the song--approximately 7-1/2 minutes. Such repetition is an asset in minimal music,(22) since it gives the listener time to rehearse different rhythmic interpretations. Although other sounds enter throughout the course of the song, masking the original, accentually indeterminate pattern, it continues to be heard beneath the surface, giving the entire song an aura of instability.
 We have now heard several examples of the rhythmic and metrical phenomena that occur in electronic dance music, including displacement dissonances and various types of ambiguity. In the next portion of this paper, I will focus on another feature--namely, the use of asymmetrical patterns. Although I will begin by looking at two musical examples, the majority of my discussion will focus on certain broader questions of rhythmic and metrical theory. I will highlight the issues raised by EDM's asymmetrical patterns and will suggest some approaches that are particularly useful for dealing with them.
 All of the examples considered thus far have contained layers of steady quarter-note or half-note motion (even though the metrical context of these layers was not always clear). However, while electronic dance music often features even rhythms very prominently, this is not always the case. For instance, in the track "Compression" by Everything But the Girl, a prominent drumbeat repeats the pattern quarter-eighth/quarter-eighth/quarter throughout. Timbral changes within the pattern suggest divisions after each of the eighth notes, and almost all of the rhythmic patterns in the track reinforce this 3+3+2 division; the only exception is the triplet pattern played by Synthesizer 1, which occurs at a low dynamic level. (See Example 4, a-e.) Thus there is very little to suggest a quarter-note pulse in this track. A 3+3+2 pattern (again divided into quarter-eighth/quarter-eighth/quarter) also underlies much of Underworld's "Pearls Girl" (though the variable fourth measure departs from this organization). (Example 5--audio only)
 Patterns that divide a measure asymmetrically occur in a number of different repertories, including folk music of the Balkans and drum ensemble music of sub-Saharan Africa. In some cases, especially in the former repertory, the number of pulses in the measure is a prime number, which means that the beat patterns comprising the meter must be irregularly spaced. In electronic dance music, however--as in much African percussion music--the measure contains a nonprime number of pulses.(23) This makes it considerably more difficult to determine the meter when asymmetrical patterns are prevalent, since the measure can also be divided evenly. Should the recurring asymmetrical patterns be considered metrical, resulting in meters such as 3+3+2/8 or 3+3+3+3+4/16, or should they be treated as syncopations against a regular background?
 In addressing this question, it is useful to see what scholars of African music have to say, as some of them have given considerable attention to the issue.(24) Thus in the next few paragraphs I will focus primarily on their arguments, though I will also relate their observations to electronic dance music. In general, Africanists' opinions on meter and asymmetrical patterns seem to fall into two different schools of thought. On the one hand, scholars such as Robert Kaufmann argue that regularly recurring asymmetrical patterns can become metrical, so that they do not seem syncopated.(25) In such cases, notes that would be accented in a 4/4 meter might sound syncopated in a meter such as 3+3+2/8. For instance (returning for the moment to EDM), in Example 4b the third note of the Synthesizer 1 part, which would fall on a half-note beat in 4/4 time, would be syncopated if the beats follow a 3+3+2 division.
 On the other hand, quite a few ethnomusicologists have rejected this sort of interpretation, arguing instead that listeners infer a background of evenly spaced beats behind groupings such as 3+3+2.(26) The view of meter as a grid--a regular background against which irregularity can occur--is also central to many music-theoretical models of meter. One of its most influential expressions occurs in A Generative Theory of Tonal Music. In this work, Lerdahl and Jackendoff posit the regular alternation of strong and weak beats as a precondition for meter. In their well-formedness rules for meter, they stipulate that "at each metrical level, strong beats are spaced either two or three beats apart" and that "each metrical level must consist of equally spaced beats."(27) And while they focus specifically on Western tonal music, another music theorist, David Temperley, has recently applied their approach to both African traditional music and Western rock music.(28)
 Scholars of African music have objected to the grid approach for a number of reasons. Some, like Kaufmann and Stone, feel that it represents a "Western" approach to meter rather than an African one.(29) Others, while not denying that the grid can be applied to African music, question whether it really suits the music. Thus the question becomes not so much "is the meter 3+3+2/8 or 4/4?", but rather "what is the best way of conceptualizing rhythm and meter in music in which asymmetrical patterns are prominent?". This question is equally applicable to EDM featuring such patterns. While it is certainly possible to accommodate the 3+3+2 rhythms of "Compression" and "Pearls Girl" within a background of evenly spaced metrical beats, such a maneuver says little about how these patterns relate to that background.(30)
 Furthermore, even if one accepts the separation between rhythm and meter that is fundamental to most grid approaches, there is much to suggest that the whole relationship between the two is different in EDM and African music. In much common-practice-era Western music, departures from the metrical structure end up reinforcing it; they pull against it just enough to call attention to it. Scholars have suggested that this phenomenon occurs with syncopation in rock as well; for instance, Temperley, arguing that rock syncopations are best understood as displacements from specific metrical beats, writes that "syncopated rhythms often seem to reinforce the metre of a song rather than conflicting with it."(31) In African percussion music, on the other hand, the rhythms of the surface tend to set up expectations of their own through persistent repetition, in spite of their irregular spacing within the measure. Thus Jay Rahn writes:
[Asymmetrical] ostinatos . . . are not mere outgrowths of the referential meters of the pieces in which they occur. In each case, they represent persistent deviations from the divisive patterns that accompany them. . . . Ethnomusicologists have observed that African musicians say these sorts of asymmetrical time lines represent an audible point of reference for the ensemble as a whole. That is, in some instances, African performers apparently find their point of rhythmic orientation within a dense texture not with respect to a pulsating pattern or a divisive, unsyncopated pattern, but rather in relation to a seemingly syncopated pattern that appears to deviate constantly from the meter of the piece.(32)
In comparison, meter seems to be more purely referential, a simple yardstick rather than the focus of compositional attention--in the words of Arthur M. Jones, "a kind of metronome which exists behind the music."(33)
 I contend that many of the same conditions apply to electronic dance music in which asymmetrical patterns are prominent. Although almost all EDM can be transcribed in 4/4 (or, less commonly, 2/4), the ways in which the music is layered (cf. paragraphs 12-14), in combination with its persistent repetition of rhythmic patterns over long spans of time, encourages the listener to attend to the periodicities of individual layers rather than focusing on how those layers deviate from a single underlying structure.(34)
 These observations suggest a shift in emphasis, a change in the way rhythm is viewed with respect to meter in electronic dance music. Rhythm begins to seem not so much like a foreground phenomenon embellishing some deep background structure, but rather as a structurally significant element in its own right. Writers studying other repertoires have suggested similar changes in focus. For instance, Dave Headlam, in a discussion of rhythm and meter in the country blues, writes:
Is it more useful to regard, for instance, Robert Johnson's songs as beginning from a regular metrical basis, as many writers do, with the surface described as "irregular"? Or is an irregular (or, better, not necessarily regular) rhythmic approach more appropriate, with any regular surface meter regarded as a compositional and performance by-product of the grouping structures?(35)
 Likewise, Jay Rahn argues with respect to African music that "one might like to determine how such syncopated patterns can be considered 'rhythms of reference' and whether there is any sense in which they can be viewed as regular. Rather than describing these patterns negatively (e.g., as deviations from a divisive organization), perhaps one can discern positive features in their structures."(36) To this end, he has explored the special properties of these rhythms. In fact, Rahn is one of a number of writers whose work suggests alternatives to grid-based views of asymmetrical patterns. In the rest of this section (paragraphs 30-36), I will explore three of these alternatives. It is not my goal in discussing this research to suggest an all-encompassing model of asymmetry (or, more generally, of rhythm and meter) in electronic dance music, for much additional research would needed before such a model could be proposed. I believe, however, that the approaches of these authors will suggest some ways in which the construction of a broader model might proceed.
 Much of Rahn's work in this area builds upon diatonic set theory, drawing parallels between pitch collections such as 7-35--a set that has long been treated as "structurally privileged"--and asymmetrical rhythm patterns.(37) For instance, following Clough and Douthett,(38) Rahn notes that these "diatonic rhythms" are maximally even: in informal terms, this means that the number of attacks within each pattern is distributed as evenly as possible among the pulses in the cycle. This property can also be seen in EDM, which contains many of the diatonic rhythms that Rahn discusses. For example, the quarter-eighth/quarter-eighth/quarter (2+1+2+1+2) pattern of Example 4 and Example 5 is a maximally even distribution of five attacks among eight pulses--in contrast, for example, to the foursquare rhythm eighth-eighth-eighth-eighth-half.(39) Furthermore, while certain foursquare patterns (e.g., four quarter notes in 4/4 time) are also maximally even, diatonic rhythms such as 2+1+2+1+2 are maximally individuated as well. In other words, each note within the pattern has a unique set of relationships with every other note. For instance, in the 2+1+2+1+2 pattern, the third note occurs three pulses after the first note, one pulse after the second note, two pulses before the fourth note, and three pulses before the last note; no other note within the pattern has the same set of relations to its surrounding notes. In addition, in contrast to maximally individuated foursquare rhythms (for example, half-quarter-eighth-eighth), notes within diatonic rhythms cannot be easily ranked in terms of metrical strength.(40)
 The properties cited by Rahn suggest possible reasons for the special presence of the asymmetrical patterns that occur in electronic dance music--the way in which they seem to stand on their own apart from any metrical grid. For instance, attacks within maximally even asymmetrical patterns are almost as regular as metrical beats. Because of the slight irregularities of these patterns, however, each attack has a unique relationship to every other attack, which is not the case in completely even rhythms. These structural features distinguish this type of organization from that of meter, even though the rhythms produced by diatonic organization can coexist with a variety of metrical structures.
 A quite different, though not unrelated, approach to the "presence" of asymmetrical patterns is suggested by some of Stephen Handel's recent work. In a 1998 article, Handel considers how metrical structure interacts with a special type of grouping known as figural organization.(41) When figural organization is in effect, tones are heard as discrete groups. The listener attends to the number of tones constituting each group, but not to the exact timing between the groups. This means that different rhythms can have the same figural organization. For instance, in a 16-pulse pattern, the rhythms X.X..X....X.X... and X.X....X..X.X... (where Xs stand for attacks, dots stand for unarticulated beats, and tones separated by only one beat are considered part of the same group) have an identical figural organization, which Handel defines as two tones, silence, one tone, silence, two tones, silence (or 2-1-2-).(42) Cognitive psychologists have claimed that meter is necessary to compensate for the imprecision of figural hearing--that listeners use it to measure the differences between otherwise similar rhythms. The three experiments that form the basis of Handel's article suggest, however, that figural organization may be just as important as meter in rhythmic perception. Handel found that listeners tend to have a hard time discriminating between rhythms with the same figural structure;(43) furthermore, the effect of meter (and of various strategies aimed at highlighting metrical organization) is particularly limited in "weakly metric" patterns (those in which the majority of attacks do not coincide with tactus-level beats). Handel's focus on the relationship between meter and "weakly metric" rhythms makes his work promising for scholars of electronic dance music, since asymmetrical patterns frequently intermingle with even rhythms in this repertory. (A classic example is 808 State's track "Cubik," in which a prominent 3+3+3+3+4 synthesizer pattern sounds against even quarter-note drumbeats.) Handel's findings, like those of Rahn, discourage us from viewing asymmetrical rhythms as embellishments of a metrical background and provide another way of accounting for their distinctive perceptual qualities.(44)
 All of the grid alternatives discussed thus far emphasize rhythmic organization over metrical structure. Nonetheless, they still preserve some sort of separation between rhythm and meter. Another possibility is to reject this division altogether. This is the approach Christopher Hasty takes in his recent book Meter as Rhythm. Instead of seeing metrical accents as a series of timepoints, Hasty characterizes meter in terms of events. He claims that meter arises when the duration of an event is replicated through a process called projection. See Example 6, a reproduction of his Example 7.2. In this diagram, capital letters A and B represent two events. At first, we do not know how long A will last. When B begins, however, the duration of A becomes definite; it now has the potential to be replicated by B. This projective potential is shown by the solid arrow Q; the dotted line Q' shows the projected duration.(45)
 Hasty also classifies events as beginnings, continuations, or anacruses. He then uses these concepts to discuss different types of meter, claiming that certain types involve more complex perceptions than others. Duple, or equal meter, is the simplest type because it involves the perception of a second event as a continuation of an initial event (see Example 7, part a, in which continuation is shown by the arrow Q). Triple meter involves a more prolonged sense of continuation, as shown in Example 7, part b; it is also more complex than duple because it denies a potential two-beat duration, as indicated by the crossed-out arrow Q in Example 8. Hasty describes this special type of denial as deferral.(46)
 In Hasty's system, meters consisting of irregularly spaced beats engender considerably more complicated perceptual processes. In duple and triple, all suggested projections are realized (although in the case of triple, the last projection is deferred); in an asymmetrical meter, however, some projections will never be realized. For example, in 5/4 meter with a 3+2 division, the three-beat projection suggested by the first part of the measure is denied when the second measure begins. See Example 9, in which the potential duration Q', indicated by the dotted line, is denied. A similar denial occurs in the second measure of a 2+3 pattern, as shown in Example 10 by the dotted line R'. Although Hasty claims that our perception of such patterns is complex, he notes that they should not be considered "unnatural or confused."(47) His approach is useful for EDM because it does not simply describe irregular patterns as syncopated, but rather provides a detailed description of the processes engendering their rhythmic complexity. In this way it can provide a convincing account of the richness that one perceives in the rhythmic surface of this music.(48)
 As the foregoing discussion has shown, the approaches of Rahn, Handel, and Hasty each provide a distinctive contribution to our understanding of the asymmetrical patterns that occur in electronic dance music. Rahn provides a structural account of the special characteristics of these rhythms, while Handel and Hasty focus more on issues of perception. Handel suggests an alternate mode of hearing that may play a role in the cognition of such patterns; Hasty, on the other hand, applies the same perceptual principle (projection) to all meters, while also showing the unique ways it plays out in irregularly spaced meters. While there are obvious differences between these approaches, they should not be considered mutually exclusive. For instance, although Rahn and Handel, unlike Hasty, preserve a separation between rhythm and meter, their respective emphases (structural properties and figural hearing) could still be situated within Hasty's method.(49) Likewise, Hasty's discussion of projection could be applied to asymmetrical patterns even if those patterns are not considered strictly metrical. In fact, I would ultimately conclude that such patterns are not generally metrical in electronic dance music (given that they usually occur in conjunction with regularly spaced patterns that can be heard as metrical more easily).(50) Nonetheless, I would argue that they should not be treated as transient foreground phenomena superimposed onto an underlying regular structure. Rather, as these three methods show us, these rhythms have a distinctive presence of their own and should be considered structurally significant in their own right.
 Our exploration has shown a variety of ways in which rhythm and meter are used to create musical interest in electronic dance music. Displacement dissonances subvert metrical stability; inherently ambiguous patterns encourage multiple interpretations; and asymmetrical patterns counteract the regularity of persistent even rhythms. The common link between all these phenomena is an emphasis on interpretive multiplicity. In other words, electronic dance music encourages us to hear it in a variety of ways. As we have seen, this multiplicity functions on many different levels. Individual patterns are often intrinsically ambiguous. Furthermore, they frequently remain so even when used in combination: when there is no definitive metrical layer, the distinction between metrical and antimetrical layers may not be apparent. Even when all the elements of a meter are in place, reinterpretations can turn the beat around, showing the listener that the metrical structure was not quite what it seemed to be. And finally, the persistent repetition of both asymmetrical and even patterns encourages multiple perspectives on rhythmic and metrical structure, thereby undermining any sense that there is a singular structure underlying the music.
 In spite of these conclusions, a number of questions remain. First, how might the instabilities and ambiguities that I have discussed be played out on a larger scale? In what ways do EDM musicians create subtlety in a work as a whole? What sorts of processes occur during the course of complete tracks, albums, and DJ sets? Second, how widespread are the phenomena considered here, and how broadly applicable are the approaches put forth to the various genres of electronic dance music?(51) Third, since EDM is first and foremost dance music, what is the relationship of dance to these rhythmic and metrical phenomena?
 Each of these questions is a potentially vast topic unto itself, and further research is needed before definitive answers can be given. Instead of trying to answer these questions at this time, I will leave them for future studies of electronic dance music to address.(52) Nonetheless, I believe that these issues, in combination with the phenomena already discussed, suggest something of the range and complexity that electronic dance music offers to listeners and scholars, both within music theory and without.
An earlier version of this paper was read at the Eleventh Biennial Symposium of Research in Music Theory, Bloomington, Indiana, February 25, 2000. I would like to thank all those who have read and commented on the paper, including the two anonymous reviewers, Gretchen Horlacher, Harald Krebs, Eric McKee, Jeffrey Magee, Felicia Miyakawa, and Tanner Menard.
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