NORTHWESTERN UNIVERSITY
Cognitive Frameworks for the Production of Musical Rhythm
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
for the degree
DOCTOR OF PHILOSOPHY
Field of Music
By Ives Chor
EVANSTON, ILLINOIS
June 2010
2
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3 Abstract This dissertation introduces the theoretical concept of rhythmic frameworks and provides empirical evidence of their role in shaping musical behavior. Many popular musics of the African Diaspora, such as rhythm and blues, funk, and Afro-Cuban rumba, feature highly repetitive, syncopated rhythms. The dissertation outlines a theory according to which these figures interact with musical meter to form rhythmic frameworks, which provide context for composition, performance, and dance. Additionally, the dissertation offers empirical evidence that these frameworks influence cognitive processes involved in the production of musical rhythm. This work has implications for the fields of music theory and cognitive science. For music theory: rhythmic frameworks serve as a key element of rhythmic organization, and understanding their function is essential to the analysis of rhythm in styles based on repetition and variation. For cognitive science: rhythmic frameworks serve as cognitive constraints on production, and empirical studies of musical performance can provide new models of cognitive representations of musical time. Three projects examine the role that rhythmic frameworks play in the production of musical rhythm. The first project consists of the development of a music theoretical method for analyzing rhythm in relation to relevant rhythmic frameworks, and the application of this method to several examples of African-American and Afro-Cuban popular music. The method identifies patterns of recurrence in performed rhythms, considering individual instrumental parts as well as multiple parts in an ensemble. The second project is a statistical analysis of rhythm in relation to Afro-Cuban clave in a corpus of MIDI performance data. The results of the analysis support the hypothesis that composers and improvising musicians vary rhythmic structure and microtiming to
4 communicate elements of meter and rhythmic frameworks to listeners. The third project, drawing on empirical methods from cognitive science, is a set of laboratory studies designed to illuminate the roles that meter and clave play in constraining microtiming aspects of rhythm production. The studies, which measure performances by professional musicians, support the hypothesis that timing in performance is shaped by musicians‘ internal representations of rhythmic frameworks and metric structure.
5 Acknowledgments I would like to express my gratitude for the generosity of the following organizations at Northwestern University that supported the creation of this work: the Henry and Leigh Bienen School of Music, the Graduate School, the Cognitive Science Program, and the Northwestern Institute on Complex Systems. I would also like to acknowledge several people who supported me during my graduate studies and helped to shape my thinking: Ji Chul Kim, Kyung Myun Lee, Caroline Davis, Jung Nyo Kim, Dana Strait, Alexandra Parbery-Clark, Ben Duane, Ben Anderson, Matt Gilmore, and Karen Chan, my friends and colleagues in the doctoral program in Music Theory and Cognition; Lisa Margulis and Candace Brower, for introducing me to new ways of thinking, talking, and writing about music; Bruno Repp, Anne Danielsen, and Fernando Benadon, for their thoughtful feedback on my research and writing; Ed Large, Petri Toiviainen, Marc Leman, Rolf Inge Godøy, Mari Riess Jones, David Temperley, Mark Butler, Matt Butterfield, Peter Martens, and David Huron, for a number of valuable conversations that have left a lasting impact on me; Bob Gjerdingen, Matt Goldrick, and Justin London, for teaching me what research and scholarship are all about; Ric Ashley, whose depth of knowledge, breadth of interests, and length of leash were invaluable to me throughout the entirety of this project; and my wife Aimee, whose love and admiration make it all worthwhile.
6
For my son, Simon
7 Table of Contents Abstract ......................................................................................................................................3 Acknowledgments .......................................................................................................................5 Table of Contents ........................................................................................................................7 List of Figures ........................................................................................................................... 11 Chapter 1: Introduction ............................................................................................................. 13 Where Does the Time Go?: Methods of Temporal Organization in Music .............................. 13 Hierarchical Structure and Musical Meter .............................................................................. 13 Subdivisions of the Beat ........................................................................................................ 14 Other Models of Microtiming Deviation................................................................................. 16 Time-Line Patterns ................................................................................................................ 17 Popular Music of the African Diaspora .................................................................................. 19 Research Overview ................................................................................................................ 22 Limitations and Assumptions ................................................................................................. 23 Contributions to Knowledge .................................................................................................. 24 Chapter 2: Musical Theoretical Analyses ................................................................................... 25 Introduction and Theoretical Context..................................................................................... 25 Repetition of short phrases................................................................................................. 25 Interplay between instruments ............................................................................................ 30 Movement, dance, and the body ......................................................................................... 30 Description of Analytical Method .......................................................................................... 31 Grid notation ..................................................................................................................... 32
8 Summaries of note onset distribution.................................................................................. 34 Applied Analyses ................................................................................................................... 35 “Bernadette” ...................................................................................................................... 36 “Tumbling Dice” ................................................................................................................ 43 “Everybody’s Everything”.................................................................................................. 47 Discussion ............................................................................................................................. 53 Chapter 3: Quantitative Analysis of Clave-Based MIDI Data ..................................................... 57 Clave ..................................................................................................................................... 57 Clave as surface rhythm ..................................................................................................... 58 Clave as rhythmic framework ............................................................................................. 59 Rhythmic Frameworks as Behavioral Constraints: Previous Studies ....................................... 60 How do rhythmic frameworks affect rhythmic structure?.................................................... 60 How do rhythmic frameworks affect patterns of microtiming? ............................................ 61 Methodology ......................................................................................................................... 63 Analysis: Rhythmic Structure ................................................................................................. 64 Syncopation ....................................................................................................................... 66 Beats 2, 2.5, and 3 ............................................................................................................. 68 Beats 4.5 and 1 .................................................................................................................. 70 Analysis: Microtiming ............................................................................................................ 70 Average asynchrony........................................................................................................... 71 Variance of asynchrony ...................................................................................................... 73 Discussion ............................................................................................................................. 74
9 Chapter 4: Behavioral Studies of Microtiming............................................................................ 77 Introduction........................................................................................................................... 77 Beat induction and preferred tempo ................................................................................... 77 Global tempo and the magnitude of microtiming deviations ................................................ 78 Research questions ............................................................................................................ 78 Experiment 1: Meter .............................................................................................................. 79 Method .............................................................................................................................. 79 Results............................................................................................................................... 84 Discussion ......................................................................................................................... 90 Experiment 2: Clave .............................................................................................................. 92 Method .............................................................................................................................. 92 Results............................................................................................................................... 95 Discussion ......................................................................................................................... 99 Figural Aspects of Stimulus Patterns .................................................................................... 100 General Discussion .............................................................................................................. 103 Microtiming as an index of temporal representation.......................................................... 103 Mechanisms for beat subdivision ...................................................................................... 104 Alternative rhythmic frameworks ..................................................................................... 105 Chapter 5: General Discussion and Conclusions ....................................................................... 107 Concepts ............................................................................................................................. 107 Methods .............................................................................................................................. 107 Findings ............................................................................................................................... 108
10 Future Directions ................................................................................................................. 109 References ............................................................................................................................... 112 Appendix A: “Tumbling Dice” Grid Notation .......................................................................... 127 Appendix B: “Everybody’s Everything” Grid Notation ............................................................ 141 Appendix C: Songs Included in Chapter 3 Data Set ................................................................. 164
11 List of Figures Figure 1. TUBS notation for “I’ve Been Working on the Railroad”. .......................................... 32 Figure 2. Example of grid notation, “Twinkle, Twinkle, Little Star”. ......................................... 33 Figure 3. Rhythmic figure shown in both standard and grid notation. ......................................... 34 Figure 4. “Bernadette” chorus, grid notation. ............................................................................ 36 Figure 5. “Bernadette” chorus, distribution of note onsets. ........................................................ 38 Figure 6. “Bernadette” chorus, basic rhythm pattern at 50% threshold. ...................................... 39 Figure 7. “Bernadette” verse, grid notation. .............................................................................. 40 Figure 8. “Bernadette” verse, distribution of note onsets in odd- and even-numbered measures. 41 Figure 9. “Bernadette” verse, distribution of note onsets in all measures. ................................... 42 Figure 10. “Bernadette” verse, basic rhythm pattern at 70% and 50% threshold. ....................... 43 Figure 11. “Tumbling Dice”, grid notation summary. ................................................................. 44 Figure 12. “Tumbling Dice”, distribution of note onsets by instrument. ...................................... 45 Figure 13. “Tumbling Dice”, distribution of note onsets, overall. ............................................... 46 Figure 14. “Everybody’s Everything”, grid notation summary. .................................................. 48 Figure 15. “Everybody’s Everything”, basic rhythm figure, bass. ............................................... 49 Figure 16. “Everybody’s Everything”, distribution of note onsets, bass. ..................................... 50 Figure 17. “Everybody’s Everything”, distribution of note onsets, timbales. ............................... 51 Figure 18. “Everybody’s Everything”, distribution of note onsets, cowbell. ............................... 51 Figure 19. “Everybody’s Everything”, distribution of note onsets, drum set. .............................. 52 Figure 20. (a) 3-2 son clave; (b) 3-2 rumba clave....................................................................... 58 Figure 21. (a) 2-3 son clave; (b) 2-3 rumba clave....................................................................... 58
12 Figure 22. Distribution of note onsets by metric position. .......................................................... 66 Figure 23. Piano montuno rhythm in 3-2 clave. ......................................................................... 67 Figure 24. Syncopation ratio in the two sides of clave. .............................................................. 68 Figure 25. Distribution of notes between the sides of clave. ....................................................... 69 Figure 26. 3-2 son clave with corresponding beats..................................................................... 70 Figure 27. Average asynchrony by metric position. .................................................................... 71 Figure 28. Average asynchrony by metric position. .................................................................... 72 Figure 29. Variance of asynchrony by metric position. ............................................................... 73 Figure 30. Stimulus rhythm patterns 1 through 8. ...................................................................... 80 Figure 31. Stimulus rhythm patterns 9 through 16. .................................................................... 81 Figure 32. Mean asynchrony across participants by metric position in Experiment 1. ................. 85 Figure 33. Standard deviation of asynchrony by metric position in Experiment 1. ...................... 87 Figure 34. Skewness of asynchrony in Experiment 1.................................................................. 89 Figure 35. Skewness of asynchrony by metric position in Experiment 1. .................................... 90 Figure 36. Son clave, 2/4 notation. ............................................................................................ 92 Figure 37. Son clave, 2/4 notation in single-measure 4/4 context. .............................................. 93 Figure 38. Stimulus rhythm patterns 1 through 8. ...................................................................... 93 Figure 39. Stimulus rhythm patterns 9 through 16. .................................................................... 94 Figure 40. Mean asynchrony across participants by metric position in Experiment 2. ................. 96 Figure 41. Standard deviation of asynchrony by metric position in Experiment 2. ...................... 97
13 Chapter 1: Introduction Where Does the Time Go?: Methods of Temporal Organization in Music Music is made up of sounds occurring in time. With some exceptions, musical composition and performance is generally regulated by one or more methods for organizing the continuous flow of time into discrete segments. These methods yield temporal structures that serve not only as references for the timing of sound events, but also as potential foundations for cognitive representations of time. Hierarchical Structure and Musical Meter One such method is meter, common in musical styles throughout the world. Listeners tend to group beats into larger temporal units of two or three beats, often described as relatively strong or weak; these units can be further grouped into yet larger units, including measures. The recursive nature of grouping results in a hierarchy of temporal layers that is widely thought to serve as the basis for the structure of musical time (Cooper & Meyer, 1960; Lerdahl & Jackendoff, 1983; London, 2001; Yeston, 1976). Evidence suggests that listeners, as well as experimental participants tapping in synchrony with rhythmic stimuli, possess and rely on cognitive mechanisms for monitoring multiple hierarchical levels simultaneously (Chen, Ding, & Kelso, 2001; Drake, Penel, & Bigand, 2000; Jones & Boltz, 1989; Large, Fink, & Kelso, 2002; Large & Jones, 1999; Large & Kolen, 1994; Large & Palmer, 2002; Repp, 2008). Meter is often not heard explicitly, but may be implied in the music. Thus, the temporal organization of musical materials may be influenced by the composers‘ desire to communicate hierarchical metric structure to listeners. In a corpus of Western art music, the frequency of occurrence of note onsets in a metric position was found to be closely correlated with the
14 theorized strength of that position (Palmer & Krumhansl, 1990). In the course of performance, musicians may also communicate information about metric structure. Several studies investigated the performance of rhythm patterns rotated to different metric positions (Clarke, 1985b; London, 2004, pp. 150-152; Repp, 2005a; Repp & Saltzman, 2001; Semjen & Vos, 2002; Sloboda, 1983, 1985). These studies, designed to isolate the effect of position within the metric hierarchy on microtiming, identified a number of ways in which musicians may vary performance timing in order to communicate elements of hierarchical structure: lengthening inter-onset intervals (IOIs) on strong metric positions and shortening them on weak ones; and marking measure and halfmeasure boundaries with expressive timing variations. Other studies employing less comprehensive methods also noted expressive timing variations at the measure and half-measure level (Bengtsson & Gabrielsson, 1980, 1983; Gabrielsson, 1988; Gabrielsson, Bengtsson, & Gabrielsson, 1983). Musical structure at hierarchical levels higher than the single measure is well-known to play a role in tempo rubato and other variations in microtiming. In many studies, expressive lengthening or decreases in local tempo were observed at phrase or section boundaries (for example, Clarke, 1987b; Palmer, 1989, 1996; Repp, 1990, 1992b, 2000b; Shaffer, 1994; Timmers, Ashley, Desain, & Heijink, 2000; Todd, 1985, 1989, 1995). Subdivisions of the Beat A phenomenon often observed in musical performance is a pattern of uneven durations or IOIs subdividing a beat. Such a pattern is perhaps most well-known in jazz; a number of studies investigated ―swing‖ eighth-note IOIs (Busse, 2002; Collier & Collier, 1996, 2002; Ellis, 1991; Friberg & Sundström, 2002; Pressing, 1987; Prögler, 1995). The phenomenon has been examined
15 in non-jazz contexts as well (Gabrielsson, 1974). Honing (2002) proposed a representational system for performed rhythms: a continuous n-dimensional plot of durations for sequences of n notes. Subsets of this space can be useful for analysis. As an example given in the article, the subset of three-note sequences that share the same aggregate duration make up a planar triangle in the three-dimensional rhythm space. This triangle can then be used to study timing, rhythmic structure, and expressive character, as in Desain and Honing‘s (2003) subsequent experiments, which showed that expressively-timed rhythms are typically still perceived categorically (though their studies focused on beat-level rather than subdivision-level rhythms). The hierarchical structure of musical time described above does not necessarily imply recursion; different cognitive strategies may be employed by performers and listeners at different levels of hierarchy (Serafine, Glassman, & Overbeeke, 1989). Indeed, Cooper and Meyer (1960) took care in defining the ―primary rhythmic level‖ – that of pulses – to be qualitatively different from superior and inferior levels (pp. 2-3). How do performers pace motor programs for musical output on these different levels? Clarke (1985a) proposed that two different mechanisms are used for the timing of beats and subdivisions: while beats are timed with some sort of internal clock, subdivisions are carried out too rapidly for that. Instead, patterns of subdivisions are overlearned in rehearsal and carried out according to ―relatively fixed strategies for accomplishing particular rhythmic figures‖ (p. 314). Shaffer, Clarke, and Todd (1985) analyzed piano performance data that supported Clarke‘s (1985a) dual timing mechanism proposal. If beats represent temporal markers in performers‘ cognitive representations of time, it would seem to follow that timing would be more consistent on beats than between them. However, at certain tempi, the opposite appears to be the case: antiphase tapping – halfway
16 between beats – tends to be less variable than in-phase tapping, at least at rates with IOIs greater than 500 ms, where the act of temporal subdivision may lend an advantage. This advantage is negated or even reversed at faster tempi (Repp, 2005a, 2005b; Semjen, Schulze, & Vorberg, 1992). Keller and Repp (2005) found that antiphase tapping may be aided by phase-resetting mechanisms elicited by a metric context created by a regular pattern of metric accents. Other Models of Microtiming Deviation Much of the existing research on microtiming has focused on deviations from regularity in timing across formal sections of pieces, especially within a single instrument. These deviations are usually described as expressive variation or tempo change on a local or global level, and are observed and measured as discrepancies between the simple time integer ratios of music notation and the IOIs between musical notes as they are actually performed by human musicians. This method of inquiry is most appropriate for the study of music in which the pulse varies widely. Repp (1997) was careful to draw a distinction between ―highly expressive, nearly ametrical music‖ like that of Schumann or Debussy and ―less expressive music that exhibits a more rigid metrical structure, such as a dance rhythm‖ and considered it ―likely that expressive timing and metrical subdivision are, to some extent, mutually exclusive, since the latter depends on the integral relationships that the former destroys‖ (p. 533). Clynes (1987) theorized a construct called the pulse matrix, ―a metarhythm which provides a microstructure for any arbitrary combination of rhythmic elements a composer might choose‖ (p. 205). The pulse, as he defined it, quantitatively summarizes the relative duration of each recurrent metric position in a piece. More controversially, Clynes claimed that composers each have their own characteristic pulse that remains stable from piece to piece and ―seems to
17 apply to both fast and slow pieces‖ (p. 211); Beethoven‘s pulse, for example, is 106-89-96-111; the numbers indicate relative inter-beat intervals, with a norm of 100 corresponding to nominal duration. Repp (1990) refuted the latter claim with counterevidence from recordings of 19 performances of a Beethoven minuet. However, synthesized performances generated using composers‘ pulses were most often preferred by expert listeners over performances with other timing profiles (Clynes, 1995; Repp, 1989). Studies have shown that musicians play with at least a trace of expressive timing variation even when trying to playing mechanically or with a metronome. Common sources of timing variations in such scenarios are rhythmic grouping and phrase-final lengthening (Drake & Palmer, 1993; Palmer, 1989; Penel & Drake, 1998, 2004; Repp, 1999a). Repp (1999c) observed that when called upon to perform mechanically (either with or without a metronome), pianists still performed sixteenth notes expressively, demonstrating ―that it is possible to regulate timing at a higher metrical level without much effect on timing at a lower metrical level. It is likely that the timing variations at the sixteenth-note level represent obligatory effects of rhythmic grouping that are perceived as subjectively regular‖ (p. 201). Time-Line Patterns In much of the music of sub-Saharan Africa, temporal organization is not necessarily governed by meter. Instead, it uses time-line patterns, which differ from meter in several important ways. First, they are explicitly heard in the music, usually performed by a percussive instrument. Second, the notes of time-line patterns are spaced unevenly, unlike the nominally isochronous beats of a measure. Third, they do not a priori possess a structural levels of hierarchy
18 below the notes of the pattern; that is, the notes are not organized into strong and weak (Arom, 1991). Kubik (1994) describes common properties of time-line patterns, which he identifies as a key principle of timing in African music and dance: These constitute a specific category of struck motional patterns, characterized by an asymmetric inner structure, such as 5 + 7 or 7 + 9. They are single-note patterns struck on a musical instrument of penetrating sound quality, such as a bell, a high-pitched drum, the rim of a drum, the wooden body of a drum, a bottle, calabash or percussion beam, concussion sticks (such as the Cuban claves) or a high-pitched key of a xylophone. They are a regulative element in many kinds of African music… [A] time-line pattern represents the structural core of a musical piece, something like a condensed and extremely concentrated representation of the motional possibilities open to the participants (musicians and dancers). Singers, drummers and dancers in the group find their bearings by listening to the strokes of the time-line pattern, which is repeated at a steady tempo throughout the performance. (p. 45) Kubik‘s claim that the time-line pattern serves as the core of a piece is another element that distinguishes it from meter. Contrasted with time-line patterns, the time signature of a piece of metric music says relatively little about the music‘s character; London (2004, pp. 113-115) further discusses relationships between the two. Agawu (2003, 2008) considers time-line patterns not as analogous to meter, but as members of the general category of dance topoi, rhythmic figures that originated as accompaniment for specific dances.
19 The phenomenon of time-line patterns can be seen, in altered form, in musical styles derived from or influenced by African music. Several such styles are associated with Latin America but most closely with Cuba, including son, rumba, bolero, merengue, salsa, and Latin jazz; due to their connections to African music, they are often (as in this dissertation) referred to as varieties of Afro-Cuban music. These styles exhibit characteristics of European as well as African music. In particular, they are rhythmically governed by a time-line pattern but are based on musical meter as well. The resulting hybrid framework, known as clave, is discussed in depth in Chapters 3 and 4 (Mauleón, 1993; Morales, 2003; Spiro, 2006; Washburne, 1997, 1998). Popular Music of the African Diaspora In addition to Afro-Cuban music, many forms of music of the African Diaspora, especially styles of popular music intended for dance, feature a nearly isochronous pulse and a timekeeping rhythm section (Iyer, 2002; Manuel, 1988; A. F. Moore; 2001; R. Moore & Sayre, 2006; Mowitt, 2002; van der Merwe, 1989; Vincent, 1996). Often, individual instrumental parts consist not of continuous streams of notes but of much sparser rhythmic patterns – including, but not limited to, time-line patterns – lying in juxtaposition with other instrumental parts. Performance of music with this sort of temporal organization places demands on performers different from musical forms that lack these features. Not only must the performers communicate elements of musical structure to listeners, but they must also maintain a high degree of synchrony with the ongoing pulse and the rhythm section (the members of the ensemble playing rhythmic accompaniment instruments such as drums, percussion, bass, guitar, or keyboards). Thus, timing relationships internal to a musician‘s performance are not the only index of his or her conception of time and
20 musical structure; timing relationships between the performance and the ongoing pulse, metrical grid, and rhythm section are also relevant. Keil and Feld (1994) introduced the widely used term participatory discrepancies – deviations in musical performance from either mechanical regularity or the notated score – and argued for their centrality in the efficacy of music, shunting even musical syntax to the periphery. While the term can be used to describe phenomena in any dimension, including pitch and timbre, it is evoked most commonly in studies of rhythm and timing. Iyer (2002) built directly on this work, focusing specifically on the connection between musical time spans, expressive microtiming (participatory discrepancies in the temporal dimension), and the body. Ultimately, Iyer hypothesized the role of the body in mediating music perception and cognition, and described in detail several recorded musical examples and how their microrhythmic deviations engender various feelings. Butterfield (2006) restored syntax to a position of importance alongside participatory discrepancies, hypothesizing that the two aspects of rhythm interact and work together to drive listeners‘ experiences of groove patterns. He also argued against Iyer‘s (2002) hypothesis that the phenomenon of backbeat delay results from physiological constraints. Using examples from jazz and rock, Butterfield concluded: Some grooves seem to call for backbeat delay, and others backbeat push… Backbeat timing can then be varied for strategic purposes. Pushing the backbeats to the top of the beat can add energy at crucial moments in a performance without rushing the tempo; conversely, laying back on them can dissolve some of that energy as the need arises. (para. 54)
21 The basis of Danielsen‘s (2006) analyses of music by James Brown and Parliament was the premise that rhythm is the interaction between sound events and an unheard reference structure. Strong beats, instead of deriving simply from an alternation with weak beats, are ―more a center of gravity or concentration of energy than a fixed point in a metrical framework‖ (p. 79). In this view, notes played slightly earlier (by as much as an eighth note or by as little as milliseconds) than a metric position can both gesture towards it and stretch the beat, making it dynamic. Danielsen also addressed repeated patterns that do not align neatly with metric hierarchy: ―If a regular pattern of syncopation appears in the music for a long time, sooner or later it ceases to be heard as deviation from the main pulse: it ends up forming an independent and equally relevant layer of pulses‖ (p. 62). Benadon and Gioia (2009) examined the boogie groove as performed by blues musician John Lee Hooker. While the approximately 2:1 subdivision of the beat was similar to that of jazz, Hooker‘s boogie was more regular, presumably aimed at moving listeners to dance. The authors analyzed note durations and IOIs, as well as timing relative to Hooker‘s foot taps. It was unclear to them which part of the groove served as the temporal reference: ―the syncopation generated by the omnipresent upbeats is so prominently featured that one is led to wonder whether they, and not the beats, carry the groove‘s timekeeping function‖ (p. 25). This idea of upbeat primacy echoed Friberg and Sundström‘s (2002) observation that jazz soloists performed with positive temporal asynchrony (that is, later than the rhythm section) when playing on the beat but in synchrony with the rhythm section on swung eighth-note upbeats. Alén (1995) and Gerischer (2006) took parallel approaches to their respective studies of Cuban tumba francesa and Brazilian samba. Each described microtiming relationships in terms of a sequence of IOIs, similar in a way to Clynes‘s pulse matrix – Alén using milliseconds and
22 Gerischer using ratios centered around the ideal value of 100, from the Nominal Units of Time (NUTs) system outlined in Jairazbhoy (1983). While Alén never related his data to meter, referring instead to the ordinal position of notes, Gerischer did, and she reported several conclusions. First, sixteenth-note positions preceding beats were lengthened. Second, ―doubletime offbeats and simple offbeats are consistently stressed by playing them a little earlier than an equidistant division of cycles and beats would suggest‖ (p. 114). A good portion of her analysis focused less on deviation from metronomicity and more on temporal asynchrony between percussion instruments. Using the metaphor of kinetic energy, she proposed that the swing (suingue) of samba resulted in part from its systematic variations in asynchrony, citing as evidence certain beats with relatively low and high asynchrony. Research Overview The central thesis of this dissertation is that rhythmic frameworks, resulting from a combination of musical meter and time-line patterns or other recurrent rhythmic figures, serve as a temporal reference and a basis for rhythm production for composers and performers of popular musics of the African Diaspora. Original analyses of musical transcriptions and performance data provide empirical support for the claim that rhythmic frameworks, and not only meter, are reflected in structural and microtiming aspects of rhythm production. Three investigations of musical composition and performance examine the role of rhythmic frameworks in shaping the production of musical rhythm. Chapter 2 introduces a music theoretical method for ascertaining patterns of repetition and variation from musical transcriptions, and applies the method to several examples of popular music. Chapter 3 outlines an analysis of rhythmic structure and microtiming in relation to clave in a corpus of MIDI performance data.
23 Chapter 4 discusses the results of two behavioral experiments designed to identify patterns of microtiming in musical performance in the contexts of meter and clave. Limitations and Assumptions The proposed research has several limitations and assumptions. The first is inherent to cognitive psychology: the assumption that quantified outcomes of behaviors provide evidence related to internal mental processes. The second is heuristic in nature: to simplify the research problem space, the investigation of musical parameters is limited to the temporal domain. Even though other factors such as dynamics and melodic contour may be relevant to the function of rhythmic frameworks, findings related only to the temporal dimension of music are still presumed to be valid and useful, independent of these other factors. The conceit of the analytical methods proposed and applied here is that composition and performance across multiple musical measures and phrases can be thought of as participation in multiple trials of an experimental task. In approaching the music this way, the methods do not make use of some potentially useful information, including the serial order of the repeated phrases. Further, the basis used as an index of timing precision is the comparison of performed note onset timing to ideal, metronomic timing. Bengtsson (1987) and Repp (2000b), as well as other researchers, have cautioned against drawing inappropriate inferences from this practice, arguing that metronomic timing is an arbitrary and artificial norm. The music theoretical analyses, by necessity, examine only a limited number of examples, and these are taken to be representative of not only their styles but, to a lesser extent, African Diasporic popular music as a whole. The microtiming studies (in the behavioral experiments and the MIDI corpus analysis) are limited to one participant playing at a time with sequenced
24 accompaniment. Additionally, it is assumed that in the behavioral studies, the aural presentation of a rhythm pattern, in conjunction with the sound of either a metronome or a clave pattern, induces a sense of meter or clave in participants. Finally, the experimental setup for the behavioral studies, involving musicians tapping on a pad and listening to sampled sounds, is not entirely ecologically valid as a proxy for musical performance. Contributions to Knowledge This work makes contributions to the fields of both music theory and music cognition. For music theory, understanding the function of rhythmic frameworks is essential to the analysis of rhythm in styles based on repetition and variation, particularly those styles that utilize a rhythm section for accompaniment. For music cognition, these empirical studies of performance demonstrate how rhythmic frameworks serve to inform and constrain musical behavior, and provide insight into cognitive representations of musical time.
25 Chapter 2: Musical Theoretical Analyses Introduction and Theoretical Context One of the primary concerns of music theory is the explication of musical works. The regulative and descriptive traditions of music theory include a host of techniques for the analysis of musical structure (Christensen, 2002). However, given the variety of musical styles, as well as the multifaceted nature of any form of music, theorists must limit their scope, strategically illustrating only specific styles and certain structural dimensions. Thus, analytical methods devised for particular musical styles often focus on the specific features that are unique or especially germane to those styles. This chapter introduces a new method intended for the analysis of popular musics of the African Diaspora. This analytical method focuses on the patterns of note onsets in repetitive rhythmic phrases performed by rhythm section instruments. It is demonstrated through its application to bass and percussion parts in three different recorded and transcribed examples of popular music from the 1960s and 1970s. The analytical method was not developed in a vacuum; it has a number of intellectual antecedents. It is informed by prior theoretical work and empirical research on African and African Diasporic musics, Western art music, musical performance and perception, and dance. Since it is, as noted above, necessarily limited in scope, it should be thought of not as a replacement for other theories and methods, but as an additional perspective on the organization of musical materials. Repetition of short phrases. A number of scholars examine the repeated rhythmic phrases, ranging from two beats to two measures in length, that provide a rhythmic foundation for a piece. Repetition of short phrases is effectively distinct from repetition of larger-scale formal
26 sections – in Middleton‘s (1983, 2006) taxonomy, the difference between musematic and discursive repetition. By parceling musical time into consistent but irregular subdivisions, such phrases create ―a firmly structured temporal matrix, typically called a ‗feel‘ or ‗groove,‘‖ within which other musical events are located (Pressing, 2002, p. 287). Traut (2005) posits that the basis of hooks in many popular music recordings is a syncopated accent pattern one or two measures in length. As evidence, he lists over 150 examples of hit songs from the 1980s, classified by accent pattern, for example, <332> or <3445>. These patterns can be instantiated in either the vocal melody or an instrumental riff, and they function as hooks because they are catchy, high-impact, and memorable. Butler (2006) presents a theory of rhythm in electronic dance music, which exhibits some important characteristics common to the genres considered here. First, it can be organized into short, repeated rhythmic cells; second, it is intended for dancing. Butler identifies three distinct categories of rhythm patterns: even, syncopated, and diatonic. Even patterns reinforce the duple nature of 4/4 measures by dividing them evenly into quarter-note, eighth-note, and sixteenth-note partitions. Syncopated patterns include accents on metrically weak beats but still lie on top of an even, hierarchical metrical structure. Such patterns ―are defined by a dynamic tension between our perception of a note‘s position and our sense of where it should be. This interplay creates a kind of gravitational pull toward the beat, a sort of negative emphasis on the position from which the note is displaced‖ (p. 87). Diatonic patterns – so named for their maximal evenness, similar to the diatonic scale – involve an uneven core rhythm like 3+3+2 or 3+3+3+3+4. By orienting his musical analyses of certain pieces around their diatonic patterns rather than strict meter, Butler
27 suggests the existence of rhythmic frameworks in which notes are not heard relative to meter, but to an intermediate layer. The account of syncopation in rock in Temperley (1999) finds no reason to reduce the status of meter as the basis of rhythm perception. In his view, syncopation (of vocal melodies, in this article) can be thought of as the result of the displacement of one or more notes from a deep surface representation, a ―de-syncopated‖ version that is congruous with metric hierarchy. One of the excerpts analyzed is the first line of ―Let It Be‖, which Temperley normalizes to what is essentially a series of eighth notes, accented on each beat. Working with the same excerpt, Iyer (2008) takes serious issue with Temperley‘s interpretation: ―It is evident that this fragment of ‗Let It Be‘ contains more than a trace of this clave rhythm – the two are nearly identical… Although this rhythmic correspondence doesn‘t exactly prove anything, neither does a forced alignment with an antiquated, uniform conception of 4/4 meter that one would use to analyze a Haydn symphony‖ (p. 274). Indeed, Temperley‘s simplification of the syncopated melody removes much of what is interesting from the song‘s rhythm. Instead of trying to map the notes (via desyncopation) directly to strong metric positions, it seems more sensible to find an intermediate rhythmic layer to which the notes might relate. As will be discussed in Chapter 2, a rhythmic framework can serve in this function. Danielsen (2006) takes the stance that meter itself is malleable. In her view, meter does not imply an alternation of strong and weak beats; instead, ―how the beats are weighted varies from genre to genre‖ (p. 45). The basis of her analyses of music by James Brown and Parliament is the premise that rhythm is the interaction between sound events and an unheard reference structure. Strong beats are ―more a center of gravity or concentration of energy than a fixed point
28 in a metrical framework‖ (p. 79). She also addresses repeated patterns that do not align neatly with metric hierarchy, similar to Butler‘s diatonic patterns: ―If a regular pattern of syncopation appears in the music for a long time, sooner or later it ceases to be heard as deviation from the main pulse: it ends up forming an independent and equally relevant layer of pulses‖ (p. 62). While Danielsen is correct in positing the existence of an intermediate rhythmic layer, it need not take the form of a sequence of isochronous pulses; instead, it may be more complex, as in son clave. Further, this dissertation argues that regular deviations from the main pulse can be attributed to an operant rhythmic framework, rather than to an altered sense of meter. In any case, the principle of repetition and variation is central to African-American aesthetics: ―Repetition and revision are fundamental to black artistic forms, from painting and sculpture to music and language use‖ (Gates, 1988, p. xxiv). While no claim is made here about their cultural source or transmission, these properties of African Diasporic music share some commonalities with the traditional musics of Africa. The short, repeated rhythmic phrases provide context for variation and improvisation in both traditions. Chernoff (1979) observes that in African drumming ensembles, ―without the other rhythms, the improvisations of a great drummer would be meaningless. In short, a drummer uses repetition to reveal the depth of the musical structure‖ (p. 112). Arom (1991) notes: ―All [Central African] musical pieces are characterised by cyclic structure that generates numerous improvised variations: repetition and variation is one of the most fundamental principles of all Central African musics, as indeed of many other musics in Black Africa‖ (p. 17). Repetition leads to periodic structure, with a sort of hierarchy of phrase length. However, in his analyses, meter does not exist in these genres, at least as it appears in
29 Western music: ―No use whatsoever is made of the notion of matrices of regular contrasts of strong and weak beats. African music is thus based, not on measures in the sense of classical musical teaching, but on pulsation, i.e., on a sequence of isochronous temporal units which can be materialised as a beat‖ (p. 180). Arom argues that beats do not exhibit relative strength or weakness; his rejection of an alternating pattern of strong and weak beats in African music leads him to deny its use of syncopation as traditionally defined: ―there is no suitable term to describe the extension of [an] off beat note onto the next beat, when the latter cannot be called ‗strong‘, although this is quite frequently found in the field of African music‖ (p. 207). Instead, he prefers the term contrametric (Kolinski, 1973) to refer to rhythmic organization in which figures and accents conflict with the ongoing pulsation. His term regular contrametricity seems to apply to many rhythmic figures found in many examples of African Diasporic music: ―Contrametricity is said to be regular when the position of the marked element with respect to the pulsation is always the same‖ (p. 242). This useful concept will be discussed again later in this chapter. Rahn (1996) introduces a set of analytical concepts for African and Diasporic rhythms. Intended as an alternative to methods developed for the study of Western art music, this account centers around fundamental African aesthetic principles of complementation, circularity, and calland-response. From this perspective, a rhythmic pattern like 3+3+2 has different status: ―Rather than depicting syncopated rhythms merely as deviations from a four-square metrical hierarchy, I try to show how they can be portrayed as highly integrated wholes in their own right‖ (p. 71). This approach is similar to Iyer‘s (2008), noted above, as well as that of this dissertation. Further, these patterns can replace meter as the framework of temporal reference for performance: ―in some instances, African performers apparently find their point of rhythmic orientation within a
30 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 (Rahn, 1987, p. 25). Interplay between instruments. The analytical picture becomes more complex when multiple rhythmic patterns, performed by multiple rhythm section instruments, are layered on top of one another. These patterns can be juxtaposed with one another in a number of ways, including call-and-response, heterophony, and hocketing, such that listeners experience perceptual rivalry or multiplicity (Chernoff, 1979; Pressing, 2002). Cross-rhythms and counter-rhythms, discussed later in this chapter, are two other types of rhythmic pattern combinations (Danielsen, 2006; Kubik, 1994). Rhythmic interactions between instruments are often likened to conversation, particularly in jazz (Berliner, 1994; Hodson, 2007; Monson, 1996). When the multiple instruments involved exhibit different timbres, they can embody the heterogeneous sound ideal of AfricanAmerican music; further, the temporal juxtaposition of different sounds can lead to the perception of emergent patterns (Greenwald, 2002; Wilson, 1992). Movement, dance, and the body. An important characteristic of many African traditional musics and African Diasporic popular musics is their intended role in dance. As such, the length of salient musical time spans is often related to the periodicities of various body movements, and the body can be said to be implicated in the perception of the music. Zbikowski (2004) points out that ―our understanding of rhythm is shaped not only by our tendency to locate rhythmic periodicities within a somewhat limited range of temporal intervals, but also by our regard of rhythmic patterns as targets onto which we can map real or imagined bodily motion‖ (pp. 296-297). Rhythmic categories correlated with the periodicities of body movements include: phrase (breathing, body
31 sway); pulse (heartbeat, walking); and subdivisions of the beat (hand and finger motions, speech) (Iyer, 2002, p. 393). Musical rhythms are linked to motor periodicities not only for listeners, but also for musicians. It is considered important for African musicians to move correctly in order to produce the desired sounds; such motion is often centered on multiple body parts (with corresponding periods) simultaneously (Kubik, 1994; Tracey, 1994). Tracey states the operant principle: ―The various parts of a dancer‘s or an instrumentalist‘s body should each move, as much as possible, at a regular speed, with their own natural periodicity or swing, like a pendulum. And it is an unstated assumption that each part should move at different speeds, or at the same speed but encompassing a different amount of space‖ (p. 277). Taken further, ―musical patterns may be represented cognitively by the performer as patterns of movement rather than as patterns of sound‖ (Baily, 1985, p. 237). Another way to consider rhythmic repetition is in terms of the opportunities it creates for physiological entrainment (Clayton, Sager, and Will, 2005; Todd, O‘Boyle, & Lee, 1999). In addition to the motor level, entrainment to music is believed to occur on the neural level (Large & Kolen, 1994; Todd, Lee, & O‘Boyle, 2002). Entrainment is also posited as a mechanism for focusing the attention of listeners toward important future events (Jones & Boltz, 1989; Large & Jones, 1999; Large & Palmer, 2002; London, 2004). London (2004) connects the recent work on attentional energy to earlier theories likening meter to the motion of waves (Zuckerkandl, 1956). Description of Analytical Method The method described here is designed to reveal core rhythmic patterns that serve as the basis for repetition and variation. It utilizes grid notation to depict note onsets in a way that
32 makes repeated patterns easy to identify visually. The distribution of note onsets is then calculated manually or with commonly available software. Grid notation. The visual method of representing rhythm borrows from the Time Unit Box System (TUBS), in which a horizontal row of boxes is used to represent equal lengths of musical time (Koetting, 1970). These lengths of time correspond to the shortest relevant categorical duration in a given piece; this duration has several terms, including minimal operational value, fastest pulse, elementary pulse, and density referent (Arom, 1991; Koetting, 1970; Kubik, 1969; Nketia, 1974). Notational symbols are entered in boxes corresponding to note onsets. These can be perfunctory (a dot or ―X‖) in analyses of a single instrumental part, or can provide a shorthand to instrument names in an multi-part analysis (see Figure 1). Figure 1. TUBS notation for ―I‘ve Been Working on the Railroad‖ (Koetting, 1970, p. 127). X
X X
X X
X X
X
While note onsets are represented in this method, note durations are not, consistent with the TUBS method. In African music, ―what matters is the position of the note – to be more precise, of the beginning of the note – and not, as in European music, its length… The important thing is the crisp and precise articulation of the note: and if it then rapidly dies away this does not matter, for it has served its purpose in the rhythmic scheme of things‖ (van der Merwe, 1989, p. 39). It is hypothesized in this dissertation that the same property holds in some, if not all, African Diasporic music. One of the goals of the development of the TUBS method was to allow Western musicologists to transcribe African rhythm patterns without imposing their own interpretive
33 biases. Thus, it includes no mechanism for combining fastest pulses into larger groups that might represent beats or measures, units which, unlike in Western conceptual models of temporal organization, do not necessarily play a role in African conceptual models or at least are not necessarily shared among musicians in an ensemble (Kubik, 1985). This is a key difference between African music and the African Diasporic music examined here; the latter possesses regular beat and metric structure, which are encapsulated by the grid notation introduced here. Fastest pulses are grouped together to form one beat length; these groups are separated either by thick vertical lines or by the shading of alternating beats. Each horizontal line of notation represents one measure (see Figure 2). However, the one-measure length can be altered to reflect shorter or longer periodicities, say, of one-half or two measures. Figure 2. Example of grid notation, ―Twinkle, Twinkle, Little Star‖. 1 X
X
2 X
X
3 X
X
4 X
This notational system has two main advantages over standard notation. It displays the metric position of note onsets clearly, even when they are located in positions not directly on the beat, a common occurrence in African Diasporic musics. In standard notation, a note falling threefourths of a beat after the downbeat might be depicted with a dotted-eighth rest followed by a sixteenth note, here it is shown by an X in the appropriate box (see Figure 3).
34 Figure 3. Rhythmic figure shown in both standard and grid notation.
1
2
3
4
X
The other advantage of this system is that it makes it simple to compare measures and detect not only repeated patterns but variations. Since box widths represent equal durations, note onsets in any given metric position are aligned vertically in a column with the same metric position in other measures. Repeated patterns are apparent in the regular occurrence of note onsets in a given set of metric positions, while variations appear as less regular occurrences. While this is useful when visually examining the notational grid, the advantage is even greater when summarizing the distribution of note onsets quantitatively, as described in the next section. Summaries of note onset distribution. The second component of the analytical method is the tabulation of note onsets in each metric position across measures. As a technique for ascertaining rhythmic structure, it echoes the previous study by Palmer and Krumhansl (1990), who made similar calculations on a set of Western tonal compositions. While the aim of that study was to determine the contributory role of meter to the distribution of note onsets, here it is hypothesized that the repetitive rhythm section parts in Diasporic popular music serve to reinforce
35 the operant rhythmic framework, at least as much as they do the hierarchical organization of meter (Lerdahl & Jackendoff, 1983).1 The total number of note onsets in each metric position is entered at the bottom of each column of the grid. The resulting set of totals can then be used a variety of ways. It can be displayed in a graph, creating a visual representation of note onset distribution. It can be compared to summaries of other musical data sets: other instrumental parts or formal sections of the same piece, other pieces, or other styles. It can also be subjected to more rigorous quantitative analysis, using statistical and computational methods (see Chapters 3 and 4). In the analytical examples that follow, whether a metric slot is filled is typically not an allor-nothing proposition. Instead, metric slots tend to contain note onsets in some measures and not in others. This can in some cases be seen to reflect variations on a core repeated pattern, variations that can be the product of compositional or improvisational processes (Ashley, 2009; Pressing, 1988). Such variations on a measure-to-measure level are a distinguishing feature of many examples of human-performed popular music, in contrast with the strict repetition found in electronic dance music (Butler, 2006). Applied Analyses The three musical works analyzed in this section were written and recorded between 1967 and 1972 in the United States or England. All of them can be said to be derived from or influenced by the musics of African Diasporic cultures. Most importantly, they exhibit
1
In any musical style, rhythmic strategies for composition and performance may be shaped by statistical learning
(Huron, 2006, pp. 190-191).
36 characteristics of those musics that are relevant to this investigation: repetition and periodicity in the parts played by rhythm section instruments; and compositional or improvisational variation of core repeated patterns. They were selected for inclusion in this study based on their status in the popular music canon, the acknowledged excellence of the performing musicians, and the availability of high-quality transcriptions of the instrumental parts. “Bernadette”. In 1967, the vocal group the Four Tops recorded their single ―Bernadette‖ (Holland, Dozier, & Holland, 1967). As with many other hits from Motown Records, the label‘s regular stable of creative talent contributed to the making of ―Bernadette‖, including the songwriting team of Edward Holland, Jr., Lamont Dozier, and Brian Holland, and the Motown house band known as the Funk Brothers. At the center of the Funk Brothers was the brilliant electric bassist James Jamerson. After a two-measure introduction, the song utilizes an ABABACBA form: A sections are choruses; B sections are verses; and the C section is the eight-measure bridge. The following analysis focuses on the song‘s bass line on the choruses and verses, as performed by Jamerson and as transcribed by Jamerson biographer Allan Slutsky, also known as Dr. Licks (Slutsky, 1989). Regular patterns are apparent in the grid (see Figure 4). The downbeat is played in every measure, while the three subsequent sixteenth-note positions rarely are. With a few exceptions, beat 3 and the sixteenth note immediately preceding it are always played. Other similar patterns can be seen at a glance. Figure 4. ―Bernadette‖ chorus, grid notation.
Chorus 1
1 X X
2 X
X X
3 X X
X X
X X
4 X X
X X
37
Chorus 2
Chorus 3
Chorus 4
X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 35
X X X
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X X
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X X X X
X X X
X X X X X
X X X X X X
X X
X X X X X X X
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24
X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 35
X X X X X
X X X X
X X X X X X X X X X X X X X X X X
X X X X X X
X X
X X
X X
X X X X X X X X
X X X
X X X X
X X
X X X
X X X X X X X X X 31
X X X X
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8
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28
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X X X X X X X X X X X X X
X
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X
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X
6
X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 34
X
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X
X
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5
The next analytical step is the construction of a graph of the sums located in the bottom row of the grid. This graph presents an additional visualization of the note onset distribution. The X axis of the graph shows metric positions, with the major beats labeled. The Y axis can be scaled to show the total number of onset per metric position, but here it displays the number of onsets
38 per metric position as a proportion of the number of measures in the grid. This allows for a consistent scale when comparing multiple graphs. Figure 5. ―Bernadette‖ chorus, distribution of note onsets. 100% 90%
Note onsets as % of measures, chorus
80% 70% 60% 50% 40% 30% 20% 10% 0% 1
2
3
4
In the graph (see Figure 5), the observations noted earlier are once again apparent. In addition, the typical pattern of beat 4 is revealed to be two eighth notes, with the other two sixteenth-note positions played a small proportion of the time. What is obvious is that many of the metric positions contain note onsets in something between zero and 100% of the measures. Is there, then, a basic rhythmic pattern that serves as the core for repetition and variation in the choruses of the song? Rather than dictating a hard-and-fast threshold for how high the proportion must be in order for a metric position to be perceived as part of the basic pattern, it can be left as an analytical parameter. If desired, one can specify a threshold for perceptual salience. In this case, eight of the metric positions exceed a threshold set at 50% of the measures, and they can be expressed as a whole in standard musical notation (see Figure 6). It should be emphasized that the basic rhythm pattern at a given threshold is not necessarily to be considered the end result of
39 analytical distillation. Instead, it is just one product of the method; the grid and its row of sums is also important, particularly for additional quantitative analyses of the kind discussed in Chapter 3. Figure 6. ―Bernadette‖ chorus, basic rhythm pattern at 50% threshold.
In similar fashion to the song choruses, we can construct a grid displaying the note onsets in the verses of ―Bernadette‖. In this case, though, we use a period length of two measures, warranted by an examination of the score, which reveals that Jamerson tends to play different rhythmic patterns in odd and even measures (see Figure 7).
Figure 7. ―Bernadette‖ verse, grid notation. Verse 1
Verse 2
Verse 3
1 X X X X X X X X X X X X X X X 15
2
0
X X X X X X X X X X X X X X X 15
X
X X X
X X X
X X X X X X X X X X X
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3 X X X X X X X X X X X X X X 6
8
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X X
4 X X X X X
X
X X X X X
X X X X
X X X X X X X X X 14
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X X X X X X X X 13
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3
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40
41 In both the odd and the even measures, the beat 1 span (that is, from beat 1 up to, but excluding, beat 2) is usually occupied by two eighth notes. Beat 2 contains a rest followed by a series of sixteenth notes; in the even measures, this series tends to increase in density within each verse, while the odd measures do not exhibit a consistent pattern. (The grid is the only product of this analytical method that allows such observations to be made about the temporal order of measures, since temporal order is lost in the process of tabulation.) In odd measures, beat 3 is typically played and beat 4 is not, while the opposite is true in even measures. The summary graph shows that odd- and even-numbered measures are nearly identical through the first half, but diverge in the second half (see Figure 8). An explanation of this difference in musical terms is that Jamerson concludes each two-measure phrase with a distinct two-beat rhythmic figure. Figure 8. ―Bernadette‖ verse, distribution of note onsets in odd- and even-numbered measures. 100% 90%
Note onsets as % of measures, verse
80% 70% 60% 50% 40% 30% 20% 10% 0% 1
2
3 Odd measures
4 Even measures
To place the verse on an equal footing with the chorus for the sake of comparison, we can generate an equivalent graph using a one-measure period (see Figure 9).
42 Figure 9. ―Bernadette‖ verse, distribution of note onsets in all measures. 100% 90%
Note onsets as % of measures, verse
80% 70% 60% 50% 40% 30% 20% 10% 0% 1
2
3
4
The differences between Figure 5 and Figure 9 reveal at least one key contrast. The and of beat 1 (beat 1.5) is played seldom in the chorus, but almost always in the verse. Given its presence near the beginning of the measure, this contrast can be interpreted as a way that Jamerson signals, perhaps to fellow band members as well as to listeners, the current location in the form of the song. The graph in Figure 9 also reveals that in several metric positions in the verse, note onsets occur at various frequencies in the moderate range between 20% and 80%. Because of the fairly continuous spread of frequencies, a rhythm pattern determined by a single threshold will not tell the whole story of repetition and variation within the verses of the song. Instead, the selection of two threshold levels demonstrates that some metric positions are more central to rhythmic framework of the verses than others (see Figure 10). The metric positions that appear only at the lower threshold – the second and third sixteenth notes of beat 2, in this case – might be thought of as occasional locations for composed or improvised variation, contributing to the sense of groove
43 in the instrumental part but not central to it. Of course, for other pieces, different numbers of thresholds and at different levels can be selected as appropriate. Figure 10. ―Bernadette‖ verse, basic rhythm pattern at 70% and 50% threshold.
“Tumbling Dice”. The British rock group the Rolling Stones released their classic double album Exile on Main St. in 1972. The LP contains the song ―Tumbling Dice‖ (Jagger & Richards, 1972). As with all songs by the group, ―Tumbling Dice‖ is driven by the elegantly understated drumming of Charlie Watts. It features simple variations, often involving one or two added notes, on the basic rock pattern: bass drum on beats 1 and 3, snare drum on beats 2 and 4 (the latter two of which are also referred to together as the backbeat). The high-hat cymbal provides a nearly constant eighth-note background throughout the song; since its contribution to rhythmic structure and note distribution is fairly uninteresting, it is not included in the analysis that follows. The analysis encompasses the use of the bass drum, snare drum, and tom-toms, as transcribed in Leonard (1999). Unlike the analysis of the bass line of ―Bernadette‖, the ―Tumbling Dice‖ drum set analysis examines multiple instruments, so for clarity the analytical notation employs multiple shorthand symbols: S for snare drum; K for kick or bass drum; and T1, T2, and T3 for the three tom-tom drums, pitched high to low. Each instrument is also given its own horizontal row in the grid, so
44 that a measure of notation is expressed in five horizontal rows. These rows are presented in descending order of pitch, as in the percussion staff notation in the original transcription – tom 1, tom 2, snare, tom 3, kick (see Appendix A). The count of note onsets for each instrument in each metric position are shown below (see Figure 11). Figure 11. ―Tumbling Dice‖, grid notation summary. 1 Tom 1 0 Snare 13 Tom 2 0 Tom 3 8 Kick 89 Toms total 8
0 0 0 0 0 0
0 1 0 1 3 1
0 0 0 0 0 0
2 0 72 7 8 0 15
0 0 0 0 0 0
2 2 5 1 7 8
3 0 1 12 3 0 5 0 5 0 89 0 11
0 8 0 0 0 0
6 6 6 1 3 13
0 8 0 0 0 0
4 3 1 1 69 21 28 7 0 5 7 0 1 1 0 32 17 1 7
0 7 0 0 0 0
The analytical goal in this example is not, as in ―Bernadette‖, to compare formal sections, but instead, to elucidate Watts‘ approach to the drum set throughout the entire song. The graph below summarizes note onsets by metric position, with each instrument plotted as a separate data series; the three tom-tom drums, played relatively infrequently, are grouped together as a single instrument (see Figure 12).
45 Figure 12. ―Tumbling Dice‖, distribution of note onsets by instrument. 110
Note onsets by instrument
100 90 80 70 60 50 40 30 20 10 0 1
2
3 Kick
Snare
4 Toms
The four highest peaks, on the four main beats, illustrate the basic rock pattern played on the kick and snare drums. The next highest frequency of occurrence for both instruments is beat 4.5, an anacrusis to the downbeat that Watts plays often as part of a drum fill. However, this visual representation does not demonstrate much else about his improvisations or use of variations. The tom-toms as a group essentially provide occasional color distributed in eighth-note positions throughout the measure. When all of the instruments of the drum set are combined into one data series, though, Watts‘ strategy emerges (see Figure 13). Again, the highest peaks are on the four main beats, and beat 4.5 is prominent. In between the main beats, though, on the segments of three metric positions x.25, x.5, and x.75, a pattern is apparent: note onset frequency in these segments increases with metric position. In other words, the drumming variations become increasingly dense as the measure progresses.
46 Figure 13. ―Tumbling Dice‖, distribution of note onsets, overall. 110 100
Note onsets, overall
90 80 70 60 50 40 30 20 10 0 1
2
3
4
One likely reason for this strategy is that Watts leaves space in the song for the vocal line, which generally adheres to the early portions of measures. Another possibility is that increasing density throughout the phrase is a part of his signature sound, present in other songs in the Rolling Stones oeuvre. As noted by Mowitt (2002), Watts uses this strategy in the repeated twomeasure phrases that make up the verses of ―Get Off of My Cloud‖: What is remarkable about the pattern is the way it activates a micro-narrative, by which I mean that through the multiplication of snare strokes (one on the 2, at times two on the 4, then one on the 2, and six on the 4), it is as though the drums are building, almost accelerating. But then, through the fill, the drums reach a high point, or flurry of beats, that immediately falls off, only to build again… The percussive narrative is thus one of failed striving repeated over and over until the recording apparatus itself intercedes to end things through the fade‖ (p. 62).
47 The existence in both songs of short, repeated patterns of increasing density suggests that the improvisational approach Watts takes in ―Tumbling Dice‖ is related to the compositional approach he takes in the basic rhythm pattern of ―Get Off of My Cloud‖. While the former takes shape in a distributed way over the 98 measures of the song, and the latter is repeated explicitly in each two-measure phrase in the verses, they develop in similar fashion. One observation to be made of the overall graph (Figure 13) is that note onsets are most frequent on beats x, followed by eighth-note subdivisions (x.5) and then sixteenth-note subdivisions (x.25 and x.75). Unlike the other two analytical examples in this chapter, then, the drum set part in ―Tumbling Dice‖ exhibits a note onset distribution that follows theoretical models of metric hierarchy. In this sense, it is consistent with the examples of Western tonal music studied by Palmer and Krumhansl (1990). In any case, what is made clear by the relative utility of the overall graph versus the individual instrument graph (Figure 12) is that Watts‘ conception of the drum set is, at least in the interstices between the beats of the basic rock pattern, a unified whole, rather than a set of distinct voices. Instead of confining instruments repeatedly to the same metric positions, he deploys them flexibly in service of his compositional goals as an accompanist. “Everybody’s Everything”. After seemingly coming out of nowhere to shock the music world with an incendiary performance at the Woodstock Festival in 1969, Santana introduced mainstream rock audiences to Latin rhythms and percussion. Led by Mexican guitarist Carlos Santana, the band‘s music was not a strict fusion of rock and Afro-Cuban music; it integrated elements from other African Diasporic genres as well, including funk, R&B, jazz, blues, and Brazilian music (McCarthy, 2004). By 1971, it had refined its raw sound and reached number one
48 on the Billboard album chart with Santana III, which included the hit single ―Everybody‘s Everything‖. The instrumentation on the album included voice, guitar, electric bass, organ, drums, and a variety of Latin percussion (Santana, Moss, & Brown, 1971). On ―Everybody‘s Everything‖, the guitar and organ generally play sustained notes or chords; even when they play in more rhythmically active fashion, they do so without a sharp attack. The main rhythmic drive comes from the bass, drums, and three-part percussion section: timbales, shekere, and cowbell (Leonard, 1998). The following analysis examines the parts played by these instruments, with the exception of the shekere, an instrument whose beads produce a diffuse attack. Additionally, in a small number of instances, the rhythm patterns performed in the song include triplet or sextuplet figures. For the sake of simplicity, note onsets outside the confines of the sixteenth-note grid are omitted from the analysis. As in the ―Tumbling Dice‖ analysis, each instrument is given its own horizontal row in the grid; the drum set instruments are separated into their own group by a line of demarcation (see Appendix B). The count of note onsets for each instrument in each metric position are shown below (see Figure 14). Figure 14. ―Everybody‘s Everything‖, grid notation summary.
Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick
1 109 102 95 65 9 79 29
3 3 1 2 1 0 0
21 88 6 11 0 5 26
83 99 1 89 7 82 48
2 3 24 15 95 2 97 83 100 3 94 11 97 9 89 0 2 0 85 0 18 1 80 16 35 27 11 2 4 5 5 1 9 3 85 10 53 43 14 6 97 3 24 18
31 95 4 36 3 60 51
94 91 4 40 3 54 43
4 23 91 83 35 6 49 26
15 104 11 97 0 2 13 35 4 4 25 68 0 96
5 3 1 4 2 0 0
49 The electric bass part is dominated by a repeated one-measure rhythmic figure (see Figure 15). However, the standard notation of the figure fails to communicate the similarity between the first two beats of the figure and the last two. In particular, the dotted-eighth/sixteenth figure on beat 1 looks markedly different from the sixteenth/eighth/sixteenth figure on beat 3. Figure 15. ―Everybody‘s Everything‖, basic rhythm figure, bass.
By contrast, both the grid notation and the ensuing graphical representation make it clearer that the only difference is the addition of one note on beat 3.25. In fact, not only is the basic repeated figure identical in both halves of the measure (save for the added note), but the pattern of variations is nearly identical as well. This is apparent when the note onset distributions of the two halves are graphed against each other (see Figure 16). Save for beat 1.25/3.25, the shapes of the distributions track very closely with each other, suggesting that bassist David Brown takes similar approaches to varying the rhythm pattern in both halves of the measure.
50 Figure 16. ―Everybody‘s Everything‖, distribution of note onsets, bass. 110 100
Note onsets, bass
90 80 70 60 50 40 30 20 10 0 1, 3
2, 4 First half
Second half
The off-beat notes of the repeated bass pattern, consistently in conflict with the pulse, exhibit regular contrametricity (Arom, 1991). For example, the notes on beats 1.75 and 3.75 of each measure have a regular position relative to the ongoing pulse. It should also be noted that the notes on beats 1, 1.75, and 2.5 (as well as beats 3, 3.75, and 4.5) are spaced evenly by threefourths of a beat length. Danielsen (2006) describes such patterns in terms of counter-rhythm, a layer of isochronous pulses in competition with the main pulse. In her analyses, such counterrhythms typically last only briefly (three or four notes), stopping for a short time before resuming with the next repetition of the phrase. While this is an intriguing reading, it is unclear whether musicians conceptualize rhythmic figures in these terms, given the short length of the counterrhythms as well as the need to track a second layer of pulses. The timbale part is dominated by a repeated one-measure rhythmic figure, just like the bass part. In the graph of note onset distribution, one can see that variations on the basic figure are seldom introduced (see Figure 17).
51 Figure 17. ―Everybody‘s Everything‖, distribution of note onsets, timbales. 110 100
Note onsets, timbales
90 80 70 60 50 40 30 20 10 0 1
2
3
4
Similarly, the graph of note onset distribution in Figure 18 shows that the cowbell performance hews closely to a regular quarter-note pattern. Figure 18. ―Everybody‘s Everything‖, distribution of note onsets, cowbell. 110 100
Note onsets, cowbell
90 80 70 60 50 40 30 20 10 0 1
2
3
4
The drum set on ―Everybody‘s Everything‖ is played by percussionist Jose ―Chepito‖ Areas, normally the timbale player in Santana, rather than regular group drummer Michael Shrieve
52 (McCarthy, 2004, p. 91). His minimal use of the snare drum (see Figure 14) and idiosyncratic use of bass drum and tom-toms hint at his unusual approach to rock drumming (see Figure 19). Figure 19. ―Everybody‘s Everything‖, distribution of note onsets, drum set. 110
Note onsets by instrument
100 90 80 70 60 50 40 30 20 10 0 1
2
3 Kick
Tom 1
4 Tom 2
Two observations about this data are of interest. First, the bass drum and two tom-toms are used in nearly identical manner. In fact, as seen throughout the Figure 14 grid, Areas uses these three drums interchangeably, possibly to vary pitch or timbre, rather than to serve distinct functional ends. Second, there is little emphasis on beats 1 and 3, especially with the bass drum. Instead, distributional weight is shifted to beats 2.5 and 4.5 and, to a lesser extent, beat 1.75 (another example of regular contrametricity). Given Areas‘s background in Afro-Cuban percussion, this is not surprising, since these beats are part of the Afro-Cuban clave. The relation between performed rhythm patterns and clave will be discussed in considerably more depth in Chapters 3 and 4. To consider the ensemble as a whole, one can compare the note onset distributions of the various instruments. There are some metric positions of near-unanimous agreement, particularly
53 beats 1, 1.75, 2.5, and 4.5; these can be thought of as a surface accent pattern. Also, while the drum set distribution pattern is clear in the first half of the measure but less so in the second half, the other instruments make little distinction between the two halves. This difference, in which the drum set evokes a clave-like polarity, and the other instruments forge ahead in a more straitlaced common time, is indicative of Santana‘s characteristic Latin rock fusion. Discussion The method outlined in this chapter is optimized for the analysis of music that features a significant amount of rhythmic repetition. By highlighting repeated or quasi-repeated phrases, it shows how the phrases serve as the structural core of rhythmic organization in a piece. Through the visual representation of the distribution of note onsets, it also identifies the contribution of composed or improvised variations to the rhythmic shape of the phrase, whether in single or multiple instrumental parts. Two of three analytical examples exhibited note onset distributions that differed greatly from the hierarchical distributions found by Palmer and Krumhansl (1990) in Western tonal music. This might suggest that the compositional or performative intent of the musicians was to communicate not a hierarchical sense of meter as hypothesized by Lerdahl and Jackendoff (1983), but an unevenly weighted system of energy as described by Danielsen (2006). Unlike Danielsen, I see no pressing need to reject the prevailing definition of meter as a system of alternating strong and weak beats. Instead, I argue for the existence of a rhythmic framework, akin to Butler‘s (2006) concept, that lies intermediate to meter and the rhythmic surface. It is this rhythmic framework, and not meter, that can impart an irregular or asymmetric structure to a piece and
54 provide the most salient ongoing temporal context for performers, listeners, and dancers, and is perhaps most closely related to the phenomenological concept of groove. The analytical method is additionally motivated by the idea that the perception of African Diasporic musics is deeply rooted in body movement and its various periodicities. Much of this music is, after all, intended for dancing, and it cannot be fully understood without the notion that the body is involved in processing it. It has been argued that the rhythmic repetition inherent in many Diasporic genres is, to some degree, experienced corporeally, and draws its power from its ability to appeal to motor periodicities in listeners (Clayton, et al., 2005; Iyer, 2002; Zbikowski, 2004). It may be linked to motor periodicities in performers, as well (Clayton, 2007; Kubik, 1994; Tracey, 1994). Considering the music from this viewpoint, each recurrent metric position can be thought of as a unique phase angle in a physiological period, and every successive time a sound event occurs in that position is further evidence, providing positive reinforcement for whatever motor action is engaged. Perhaps this partially explains why moving and dancing to repetitive music is so rewarding. Such rewards may also accrue to musicians experienced in performing for appreciative dancing audiences. In most musical and musicological contexts, metric positions are referred to by their serial position within the measure: beat 1, beat 2, and so forth. While this nomenclature is no doubt accurate, it fails to communicate the idea that within certain styles or pieces, there is often a qualitative difference between metric positions. Some beats have special names used by practitioners of a style: rock musicians call beats 2 and 4 the backbeat; Afro-Cuban musicians call beat 2.5 bombo; and of course, beat 1 is widely referred to as the downbeat. These names derive
55 in large part from the fact that within these styles, the beats are typically played by specific instruments. The backbeat is usually played by the snare drum, and bombo is played by the bass drum, also called bombo. To draw an analogy, consider the hours of the day. Like metric positions, they too are labeled with numbers, but everybody recognizes that they are qualitatively different, so there are special names: dawn, noon, afternoon tea, bedtime. Schedule planners are not laid out in completely serial fashion, in a line with the hours of one day continuing on into the next. They are instead laid out as a grid, to make it easy to see the correspondence between, say, 11 AM of multiple days. Similarly, a method of analyzing musical rhythm, at least for music of a highly cyclical nature, should make it easy to see the correspondence between notes that share the same metrical position. Certain metric positions in a recurring pattern can be conceptualized and felt in their own qualitatively unique way. There is a cognitive correspondence between sonic events occurring in the same metric position but in different measures or phrases, a sense that they are experientially the same. (However, this sameness does not necessarily hold across musical pieces or styles.) The concept of an experiential quality of a metric position is related to that of time-order qualia in serial music theory (Boretz, 1970). However, the experience is not ineffable, since it is linked to corporeal sensation, so it may or may not fall under strict definitions of qualia (Raffman, 1993). The use of the less controversial term quality sidesteps philosophical arguments about the nature of musical qualia. The analytical examples in this chapter demonstrate the validity of the method of summarizing note onset distributions by metric position across repeated phrases. They also
56 provide evidence that the rhythmic surface of certain pieces or styles may be based not solely on meter but on an intermediate rhythmic framework. Both of these findings warrant the use of statistical and computational tools to extend the analytical method to larger sets of musical data. These extensions are discussed in Chapters 3 and 4.
57 Chapter 3: Quantitative Analysis of Clave-Based MIDI Data2 Theoretical and empirical studies of rhythm have identified the important role that meter plays in temporal aspects of musical behavior. Most of these studies, limited in scope to Western classical music, have assumed a hierarchical structure of musical time in which strong events alternate with weak events on various time scales. However, many other styles of music, particularly those of the African Diaspora, feature highly repetitive rhythms that can interact with meter to create unique rhythmic frameworks, which provide context for composition, performance, and dance. Understanding the function of such rhythmic frameworks is essential to understanding the organization of rhythm in these styles. One such style is Afro-Cuban music, in which rhythms are based on clave. This chapter utilizes empirical investigations to examine the ways in which microtiming and rhythmic structure in Afro-Cuban musical performance are informed and constrained by the rhythmic framework of clave. These investigations support the claim that timing accuracy and the distribution of note onsets are modulated by the rhythmic framework resulting from the interaction between meter and clave. Clave Clave is the foundation of rhythm in Afro-Cuban music. It serves as ―a kind of asymmetrical metronome, purely structural, whose only function is to provide an organizational spine for the rhythm‖ (Sublette, 2004, p. 95). While clave can be understood as a single musical phenomenon, it is manifested in two primary ways: as a rhythm pattern played on percussion
2
Much of the material contained in this chapter appeared previously in Chor (in press).
58 instruments, and as a framework for rhythmic organization. 3 For the sake of clarity, these will be referred to, respectively, as the surface rhythm and the rhythmic framework of clave. Clave as surface rhythm. The surface rhythm of clave is a repeating pattern consisting of five notes distributed across two measures. The two variations of clave, known as (a) son clave and (b) rumba clave, are shown in Figure 20, written in 4/4 time. The only difference between the variations is the displacement of the third note by one-half beat. Figure 20. (a) 3-2 son clave; (b) 3-2 rumba clave.
In both variations, one measure contains three notes, while the other contains two. These are commonly referred to as the three side and the two side (Spiro, 2006). The patterns in Figure 20, in which the three side comes before the two side, are known as 3-2 clave. In some songs, clave begins with the two side rather than the three side. These versions of clave are known as 2-3 clave and are shown in Figure 21. Figure 21. (a) 2-3 son clave; (b) 2-3 rumba clave.
Whether 3-2 or 2-3, son or rumba, all variations of clave display a two measure pattern in which each measure is diametrically opposed. The two measures are not at odds, but rather, they are balanced opposites like positive and negative,
3
The term can also be used to refer to the wooden sticks on which the rhythm pattern is typically played.
59 expansive and contractive, or the poles of a magnet. As the pattern is repeated, an alternation from one polarity to the other takes place creating pulse and rhythmic drive. (Amira & Cornelius, 1992, p. 23) As described below, this alternating pattern applies not only to the surface rhythm of clave, but to all rhythms performed in clave-based music. Clave as rhythmic framework. In addition to its instantiation as a surface rhythm, clave serves as a framework for rhythmic organization, informing and constraining rhythms and how they are performed, such that ―instrument patterns, melodic phrases and even improvisation revolve around it‖ (Mauleón, 1993, p. 48). While it is considered to play a central role in the music—a typical example from the pedagogical literature describes it as ―the most important organizing principle within which all the instrumental patterns must fit‖—exactly how these patterns should fit within clave is not well understood (Garibaldi, Diaz, & Spiro, 1999, p. 66). Instead, teachers and fellow musicians typically suggest certain informal guidelines or rules of thumb: rhythms should reinforce, rather than conflict with, the surface rhythm of clave; and rhythms on the three side should contain more upbeats and be more syncopated than rhythms on the two side (Spiro, 2006; Washburne, 1998). Additionally, clave-based music exhibits an unusual characteristic related to meter. Metric stress and phrase initiation—two phenomena usually attributed to the downbeat—tend to occur not on beat 1, but on the preceding beat 4 or 4.5 (García, 2008; Manuel, 1985; Spiro, 2006). Pressing (2002) observed that this effective displacement of the downbeat also occurs in other styles, including jazz; he dubbed it the fake 1 technique and suggested that its use has ―a psychologically disorienting effect on listeners‖ (p. 301). As discussed below, this removal of
60 emphasis from beat 1 and virtual shift of the bar lines, summarized in the aphorism ―four is the Latin one,‖ may partially account for the sense of lightness and forward motion in Afro-Cuban music. Rhythmic Frameworks as Behavioral Constraints: Previous Studies In investigating the role of rhythmic frameworks in modulating musical behavior, previous studies, following a two-pronged approach to rhythm outlined by Clarke (1985), have generally focused on either structure or expression. Clarke defines these, respectively, as ―a relatively fixed canonical representation equivalent to the notations in a score and a more flexible and indeterminate representation that is evident in expressive performance‖ (p. 211). These are referred to here as rhythmic structure and microtiming. How do rhythmic frameworks affect rhythmic structure? Researchers have typically relied on corpus analyses to provide evidence of the role of rhythmic frameworks in informing and constraining musical composition and improvisation. Palmer and Krumhansl (1990), for example, hypothesized that composition is influenced by mental representations of metric hierarchy of the kind postulated in Lerdahl and Jackendoff (1983), and supported that hypothesis by calculating the frequency of occurrence of note onsets by metric position in piano literature from the Western canon. Their study reported that in compositions in 2/4 time, for example, note onsets occurred most often on beat 1 and progressively less often on beat 2, eighth-note subdivisions, and sixteenth-note subdivisions. However, in its focus on Western classical music, the study by Palmer and Krumhansl did not encompass musical styles of Africa and the African Diaspora, in which rhythmic organization may be determined not only by meter, but by other rhythmic frameworks. In such styles, a
61 rhythmic framework can often be derived from a topos, defined as ―a short, distinct, and often memorable rhythmic figure of modest duration (about a metric length or a single cycle), usually played by the bell or high-pitched instrument in the ensemble, and [which] serves as a point of temporal reference. It is held as an ostinato throughout the dance-composition‖ (Agawu, 2003, p. 73). As mentioned earlier, Rahn (1996) argued that in such styles, syncopated rhythms—in particular, 3+3+2, as in the three side of son clave—should be analyzed not ―merely as deviations from a four-square metrical hierarchy‖ but ―as highly integrated wholes in their own right‖ (p. 71). Taking this approach tacitly, Washburne (1998) transcribed a relatively small number of excerpts from notable performances of recorded Afro-Cuban music, which he intentionally selected as exemplars of adherence or non-adherence to the principles of clave. Washburne‘s study analyzed the excerpts in terms of their correspondence with these principles and described in detail the ways in which specific passages reinforced or conflicted with clave. Though anecdotal in its selection of examples, it remains one of the only empirical investigations of rhythmic structure in clave-based music, and more research is needed. How do rhythmic frameworks affect patterns of microtiming? Several published studies have investigated microtiming patterns in the context of various rhythmic frameworks. Controlling for other factors, Sloboda (1985) found that pianists systematically use expressive lengthening to mark measure and half-measure boundaries, concluding that ―skilled music performance is intimately mediated by mental representations of metrical structure‖ (p. 291). Similarly, Iyer (2002) argued that ―deviations from strict metronomicity both convey information about musical structure and provide a window onto internal cognitive representations of music,‖
62 citing as evidence the phenomenon of backbeat delay, in which drummers strike the snare drum slightly later than the metronomically precise locations of beats 2 and 4 (p. 397). Additionally, temporal structures of a scale larger than one measure can affect microtiming. Combining music theoretical analysis with microtiming measurements of recorded jazz performances, Ashley (2002) found evidence of cadential anchoring, ―the tendency of the soloist to align with the accompaniment at important cadential positions‖ and determined that ―[t]he primary musical function of this technique in these recordings … would seem to be that of clarifying the hierarchic phrase structure of the composition‖ (p. 320). Certain style-specific rhythmic frameworks can serve to modulate microtiming in a systematic manner as well, or, as Butterfield (2006) put it, ―different groove patterns offer different potentials for expressive timing‖ (para. 4). Bengtsson and Gabrielsson (1983) discussed the well-known non-isochronicity of the beats of the Vienna waltz, in which ―the second beat ‗starts too early‘ if compared with a mechanical performance‖ (p. 42). Gerischer (2006) analyzed microtiming phenomena in samba drumming, and Alén (1995) presented data on microtiming patterns in Cuban tumba francesa, a style not based on clave but on the related cinquillo pattern (Floyd, 1999; Sublette, 2004). In these and other examples, non-isochronicity may remain fairly consistent across repetitions within a piece or style; London (2004) observed that ―what may appear as a local form of variance among successive IOIs [inter-onset intervals]—what are known as expressive deviations or expressive variations—can emerge as part of a larger pattern of temporal regularity‖ (p. 142). Rasch‘s (1988) analysis of chamber ensemble performance presented a model for describing timing and synchronization in musical styles in which the beat reference is not explicitly
63 voiced by one instrument, but is instead derived as a function of onset times of the multiple instruments in the ensemble. Such a model is most applicable to styles that make regular use of tempo rubato. However, as Iyer (2002) argues, most groove-based musical styles of the African Diaspora feature ―a steady, virtually isochronous pulse‖ (p. 397). As such, measurements of microtiming in these styles can often be made using an explicit beat reference. In analyses of recordings, a single instrument such as a drum can provide the beat reference, and in behavioral studies such as Prögler‘s (1995) jazz microtiming experiments, a metronome can do so. Methodology The data in this study were taken from the instructional software package The Latin Pianist, published by PG Music.4 The manual for the software specifies: ―All selections are complete live performances artistically played by top studio pianists on an 88-note weighted MIDI piano keyboard and recorded ‗Live-to-MIDI‘ in real time. These performances are never quantized or step-recorded‖ (PG Music Inc., 1998). The pianist and composer for The Latin Pianist is Rebeca Mauleón, a San Francisco-based performer and educator as well as the author of Salsa Guidebook: For Piano and Ensemble and 101 Montunos. The package contains 50 songs, each consisting of the live piano tracks accompanied by quantized, sequenced rhythm tracks (R. Mauleón, personal communication). Since the rhythm tracks are sequenced, the tempo of each song is invariant. Songs for this study were selected from the original set of 50 based on several criteria, the first of which was style. The instructional software lists a short description of each song, including
4
A similar package by the same publisher was used by Busse (2002) to derive models of jazz piano performance.
64 its style, and only songs from Afro-Cuban styles were considered for the study. Thus, songs in son, guaracha, mambo, and son-montuno styles were included, while those in Brazilian (for example, samba or bossa nova) or hybrid Caribbean styles (for example, calypso-son) were excluded. The second criterion was the explicit presence of the surface rhythm of son clave. To ensure the integrity of the study and its focus on the effects of clave, each song in the study had to have the clave pattern performed throughout by a percussion instrument. In addition, to keep comparisons consistent, songs using the rumba clave were excluded.5 The third criterion was meter. Only songs in common or cut time were considered; this excluded songs in 6/8 time. Seven songs, ranging between 140 and 190 beats per minute, fulfilled these criteria and were included in the study.6 The MIDI files for the seven songs were then processed with MIDI Toolbox, a set of analytical functions for the statistical software package MATLAB (Eerola & Toiviainen, 2004). The resulting data set included instrument, pitch, onset timing, and velocity information for all 30,864 notes from the seven songs. At that point, the percussion parts performing the surface clave pattern were removed from the data set, with the goal of demonstrating the influence of clave as a rhythmic framework in the ensuing analyses. Removing these percussion parts resulted in a slightly smaller data set containing 28,828 notes. Analysis: Rhythmic Structure
5
Studies of rhythmic organization in rumba clave, and comparisons between son and rumba clave, are areas for
further investigation. 6
Titles are listed in Appendix C.
65 The rhythmic structure portion of the analysis examined all of the MIDI data in the sevensong set with the exception of the percussion parts performing the surface clave pattern. The remaining data included piano, melody instrument, guitar, bass, and other percussion parts. 7 Onset timing in the MIDI data, expressed in terms of beats, was interpreted relative to 3-2 clave in common time. To do this, note onsets from songs in 2-3 clave were shifted by four beats, while those from songs in 3-2 clave were left unchanged. Then, given that a two-measure clave phrase spans eight beats, the beat value of each note onset was reduced modulo eight (yielding the remainder after dividing the beat value by eight) to show its metric position relative to clave. To get past the counting problem—MIDI data begins with beat zero, while measures begin with beat 1—one was added to each MIDI note onset. The resulting value was then quantized, or rounded, to the nearest eighth note, the standard density referent 8 in Afro-Cuban styles, with the microtiming remainder recorded as the temporal asynchrony. For example, a MIDI note onset with a beat value of 37.5187 from a song in 2-3 clave was shifted by four beats to 41.5187. Then it was reduced modulo eight to 1.5187. Adding one yielded 2.5187. Its quantized value, then, was 2.5, with a temporal asynchrony of .0187. It was interpreted as a note falling on beat 2.5 of the three side of clave, with a positive asynchrony of 1.87 per cent of the beat IOI. The distribution of the data set among the 16 eighth-note metric positions (presented as a histogram in Figure 22) was the focus of the rhythmic structure portion of the analysis. The
7
For the study, each tone of a chord was treated as an individual note.
8
As defined by Nketia (1974).
66 position with the highest note count was beat 2.5 of the three side (beat 2.5 in the histogram), with 2,946 notes (10.22 per cent of the data set). The position with the lowest note count was beat 4 of the two side (beat 8 in the histogram), with 1,117 notes (3.87 per cent of the data set). The data showed large deviations from the average of 1801.75 notes per metric position, as well as from the model of metric hierarchy utilized by Palmer and Krumhansl. An analysis of these differences follows, focusing on three elements: syncopation; beats 2, 2.5, and 3; and beats 4.5 and 1. Figure 22. Distribution of note onsets by metric position.
Syncopation. One rule of thumb governing Afro-Cuban stylistic performance states that the three side of clave ―contains more upbeats and syncopated material‖ than the two side, which contains more notes on the beat (Spiro, 2006, p. 14). As an example, the rhythm of a basic piano montuno (accompaniment figure) in 3-2 clave is shown in Figure 23 (Spiro, 2006, p. 16).
67 Figure 23. Piano montuno rhythm in 3-2 clave.
To test whether the rule of thumb is reflected in practice, this study uses the ratio of notes falling on an upbeat to notes falling on the beat as an index of syncopation. Other definitions and quantitative measures of syncopation exist (Longuet-Higgins & Lee, 1984), but it is not clear that the intention of the rule of thumb extends beyond the use of upbeats and beats. Analysis of the data shows that the rule is followed in practice, at least by the single musician included in the study; the ratio is higher on the three side (8,753 upbeats to 5,823 notes on the beat, or 1.503:1) than on the two side (7,733:6,519 or 1.186:1) to a significant degree, X2(1, N = 28,828) = 98.73, p < .001. Interestingly, the contrast between the two sides in the degree of syncopation is even stronger if we compare one-measure spans beginning on beat 4 instead of beat 1 (see Figure 24). This may be appropriate, given the aphorism that ―four is the Latin one‖: harmonic, and other, changes in Afro-Cuban music often occur on beat 4 rather than beat 1 (García, 2008; Manuel, 1985). With this redrawing of the borders, the ratios calculated above change to 9,145:4,979 or 1.837:1 for the three side, and 7,341:7,363 or 0.997:1 for the two side.
68 Figure 24. Syncopation ratio in the two sides of clave.
This difference in syncopation is seen not only in the aggregate comparison of the three side to the two side, but also on the level of individual beats. As can be seen in the left half of Figure 22, on the three side, notes falling directly on any beat are less common than notes falling on the following eighth-note subdivision (for example, beat 1 versus 1.5). As seen in the right half of the figure, the same pattern does not apply universally on the two side. Beats 2, 2.5, and 3. One metric span notable for the high degree of contrast between its instantiations on the two sides of clave is the one comprising beats 2, 2.5, and 3 (see Figure 25). The two side exhibits little variability in frequency of occurrence of notes in these positions (1,904:1,949:1,948), but on the three side a clear pattern is evident: notes falling on beat 2.5 are especially common (2,946), and notes falling on beats 2 or 3 are especially uncommon (1,311 and 1,165).
69 Figure 25. Distribution of notes between the sides of clave.
What explanation can be offered for this contrast? The overall difference in syncopation, as described above, is likely to be accountable for some of the difference. However, another factor may be partially responsible: the surface rhythm of clave. The two sides of clave are perfect opposites during this metric span. On the three side, the pattern reads rest–note–rest, while on the two side, it reads note–rest–note (see Figure 26). This suggests that clave is a framework of elicitation and inhibition; it encourages the playing of notes in certain metric positions while discouraging it in others. In this metric span, the playing of notes is encouraged when clave notes occur (beat 2.5 on the three side, beats 2 and 3 on the two side), and inhibited when clave rests occur (beats 2 and 3 on the three side, beat 2.5 on the two side). It is plausible that this practice helps to communicate the sense of clave in musical performance.
70 Figure 26. 3-2 son clave with corresponding beats.
The other two metric positions containing notes of the surface clave pattern (and thus potentially affected by this same phenomenon) are beats 1 and 4. Beat 1 will be discussed in the section below. Beat 4 shows a note count that is consistent with the beat 2/2.5/3 span: it is performed more often on the three side (1,961 times), when it coincides with the surface rhythm of clave, than on the two side (1,117), when it does not. Beats 4.5 and 1. The data also exhibit a low frequency of occurrence of notes on beat 1 (on both sides of clave), contrasted with a high frequency of occurrence of notes falling on the preceding beat 4.5: 2,936 times for beat 1 and 4,880 for beat 4.5, a ratio of 0.602:1. A likely explanation is the common practice of anticipating downbeats by one eighth note (Spiro, 2006). When this is seen in conjunction with the ―four is the Latin one‖ aphorism, one might posit a larger phenomenon related to the downbeat: in clave-based music, the displacement of beat 1 to beats 4 and 4.5 results in a shift of emphasis away from beat 1, so that it is simultaneously anticipated, destabilized, and omitted, giving the music a sense of lightness and forward motion. In funk music, by contrast, syncopations and rhythmic displacements often direct attention and weight toward beat 1, resulting in a feeling of heaviness and remaining in place (Danielsen, 2006). Analysis: Microtiming
71 The microtiming portion of the analysis focused on the human-performed piano data. The piano portion of the data contained 10,865 notes, with onset information for each note. The MIDI data from the seven songs were processed using the same method as the structural analysis. Average asynchrony. The relative asynchrony values were summarized statistically, yielding an average and variance for each metric position. The average asynchrony was positive for each metric position, ranging from 0.2 to 4.1 per cent of beat IOI (see Figure 27). Thus, while some individual note onsets occurred early relative to quantized location, note onsets on average occurred late relative to quantized location for each metric position, with varying degrees of lateness. Figure 27. Average asynchrony by metric position.
While the average asynchrony varied from position to position, it was fairly consistent between the sides of clave, as shown in Figure 28. The consistency between the three side and the
72 two side of clave suggests that the pattern of variance is not driven by the differences between the sides, but rather that it is a function of position within the measure. On each side, the position of highest positive asynchrony is beat 3, and the next-highest value is on beat 1. The position of lowest positive asynchrony is beat 4. Figure 28. Average asynchrony by metric position.
On each side of clave, the peak average asynchrony is on beat 3. Given that clave-based music is often notated in cut time and is considered to possess ―a half-note pulse on beats 1 and 3‖ (Mauleón, 1993, p. 47), the peak on beat 3 echoes Iyer‘s (2002) observation of backbeat delay in postwar African-American popular music. The troughs in average asynchrony are somewhat inconsistent, but appear to be centered around beats 2.5 and 4. Perhaps not coincidentally, these beats represent precisely the tumbao pattern played by the bass in Afro-Cuban music (Manuel, 1985). This suggests two plausible but unconfirmed explanations: that musicians tend to play in
73 close synchrony with the notes of the bass tumbao, and that they make a special effort to give the music a sense of forward motion by playing ―on top‖ of those beats. Variance of asynchrony. The variance of asynchrony was calculated for each metric position, in similar fashion as the average asynchrony. As seen in Figure 29, these values ranged from 0.19 to 0.58 per cent of beat IOI. However, there were two peak values worthy of mention: 0.58 per cent for beat 3 of the three side, and 0.47 per cent for beat 4 of the two side. Outside of these peak values, no variance exceeded 0.37 per cent. Figure 29. Variance of asynchrony by metric position.
To explain the source of these peak values, we refer to the surface rhythm of clave, as seen in Figure 26. The beats in question not only fail to coincide with the notes of clave, but are, in a sense, on the wrong side of clave. That is, beat 3 on the three side is not a clave note, but the same beat on the two side is. Similarly, beat 4 of the two side is not a clave note, but the same
74 beat on the three side is. According to these data, it seems, then, that clave serves to modulate variance in microtiming. In certain metric positions, the variance of asynchrony is comparatively lower on the clave side in which that metric position coincides with the surface clave pattern. However, this relationship is clear in only two of the five metric positions corresponding to clave; it is also present but to a lesser degree on beats 1 and 2.5. The fact that a strong pattern is not seen on beats 1, 2, and 2.5 might suggest that clave modulates microtiming variance in such a way that only beats 3 and 4 are affected. Discussion This study analyzed musical performance data to investigate how microtiming and rhythmic structure in Afro-Cuban musical performance are affected by the rhythmic framework of clave. The results indicate that clave informs and constrains production in several systematic ways, and they provide empirical support for several informal rules of thumb governing performance in Afro-Cuban musical styles. One way in which it seems that clave functions as a production constraint is as a framework of elicitation and inhibition. An example is the metric segment spanning beats 2, 2.5, and 3. On the three side of clave, the surface rhythm in this segment reads rest–note–rest, while on the two side, it reads note–rest–note. In parallel with this contrast, the data set exhibited both a high frequency of occurrence of notes played in coincidence with the notes of clave and a low frequency of occurrence of notes played in coincidence with the rests. Thus, it can be argued that the stylistically expert musician composes, performs, and improvises in a manner that helps to communicate and reinforce clave.
75 An analogous study found that the frequency of occurrence of musical events in a corpus of Western tonal music followed a hierarchical distribution that reinforced a sense of meter (Palmer & Krumhansl, 1990). The fact that the clave-based data did not follow the same distribution suggests that highly repetitive grooves or topoi, common in musical styles of the African Diaspora, can interact with meter to create different rhythmic frameworks. This comports with the argument that in such styles, syncopated rhythms may be best understood not ―merely as deviations from a four-square metrical hierarchy‖ but ―as highly integrated wholes in their own right‖ (Rahn, 1996, p. 71). Syncopation is another example of how clave serves to constrain production. The study found that the ratio of notes falling on an upbeat to notes falling on the beat was significantly higher on the three side than on the two side, consistent with the principle that figures on the three side should contain more upbeats and be more syncopated than those on the two side (Spiro, 2006; Washburne, 1998). The difference in degree of syncopation was found to be even greater when the sides of clave were considered to begin on beat 4 instead of beat 1, an appropriate shift given the common practice of emphasizing beat 4 using bass notes and harmonic changes (García, 2008; Manuel, 1985). Further evidence of a shift of emphasis away from beat 1 was found in the infrequent occurrence of notes in that position compared with beat 4.5. The study also found regular patterns of microtiming in the data: piano notes were generally played later than the quantized percussion notes with which they nominally coincided, but the magnitude of these temporal asynchronies varied by metric position. The average asynchrony appeared to be a function of meter, with little difference between the sides of clave. Piano notes were on average played especially late on beat 3. Given the half-note pulse of clave-
76 based music, this is akin to the phenomenon of backbeat delay—also a common practice in African Diasporic rhythm—in which the snare drum is played slightly late on beats 2 and 4 (Iyer, 2002). The variance of asynchrony was calculated for each metric position and exhibited important differences between the two sides of clave. For beats 3 and 4, the variance was notably lower when the metric position coincided with the surface rhythm of clave. Thus, microtiming variance seems to be modulated by clave; timing in clave-based musical performance is more consistent in certain metric positions than in others. It should be mentioned that the present study was based on performances by only one musician, accompanied by sequenced tracks. The results reported in this chapter warranted more thorough, controlled behavioral experiments involving multiple participants; these experiments are discussed in Chapter 4.
77 Chapter 4: Behavioral Studies of Microtiming Introduction Timing in musical performance is imperfect. Performed rhythms tend to deviate from metronomic precision by small amounts of time on the order of milliseconds. This chapter investigates patterns of microtiming in a set of behavioral experiments designed to demonstrate the influence of cognitive processes, such as the representation of musical time and the organization of temporally ordered auditory events into a hierarchical structure like musical meter. By isolating the effect of metric context while keeping other musical aspects constant, the experiments provide evidence of the role of meter and clave in informing and constraining rhythm production. The experiments described in this chapter utilized only one tempo for the study of musical performance. The findings reported are thus bound by certain limitations. A summary of relevant literature on the relationship between tempo and timing precision is given below. Beat induction and preferred tempo. The primary phenomenological unit of musical time is the beat, also known as the pulse or tactus (Lerdahl & Jackendoff, 1983). Listeners are often able to hear within a complex musical texture a series of nearly isochronous auditory events and infer a regular series of beats. Auditory events occurring at very fast rates, though, are typically grouped by listeners into sets of two, three, or four events, such that the inferred beats fall within a moderate tempo range. Conversely, events occurring at very slow rates are typically subdivided such that the inferred beats fall within that same range. The range within which beat perception is most likely to occur is 85 to 120 beats per minute, with a peak at 100 beats per minute. These tempi correspond roughly to inter-onset intervals (IOIs) between beats of 500 to
78 700 milliseconds (ms), with a peak at 600 ms (London, 2002, 2004; Parncutt, 1994; van Noorden & Moelants, 1999). Empirical evidence of the existence of this preferred beat range has led some researchers to suggest that the human body and/or neural architecture is implicated in beat induction and rhythm perception (Clayton, Sager, & Will, 2005; Todd, Lee, & O‘Boyle, 2002; Todd, O‘Boyle, & Lee, 1999). Global tempo and the magnitude of microtiming deviations. Several studies examined the relationship between expressive timing and global tempo and found that the magnitude of deviations from metronomic precision did not scale proportionally with tempo (Desain & Honing, 1993, 1994; Povel, 1977; Repp, 1995b). These contradicted a study in which the timing patterns in performances of Schumann‘s ―Träumerei‖ at three different tempi were close to relationally invariant, that is, expressive timing varied nearly linearly with tempo (Repp, 1994). The nonproportional relationship between timing deviations and tempo was also found to hold in the perceptual domain as well; studies found that listeners were able to identify tempo-transformed versions of recorded musical excerpts as sounding unnatural or of lower quality (Honing, 2007; Repp, 1995b). Research questions. The study described below examined the performance of musical rhythms in metric contexts featuring an external timekeeper maintaining an invariant pulse. The dependent variable in the study was the temporal asynchrony between note onsets, as performed by human participants, and the ongoing pulse. Several research questions informed the investigations:
79 Microtiming as an index of temporal representation. What can we learn about cognitive representations of metric structure from patterns of microtiming in the context of an audible timekeeper? Is the variability of microtiming correlated with the hierarchical strength of beats? Mechanisms for beat subdivision. How do musicians perform notes on subdivisions of the beat in the absence of auditory cues providing explicit temporal markers? Does the antiphase advantage hold for irregular (non-isochronous) rhythms or only for the regular offbeat rhythms previously studied? Alternative rhythmic frameworks. Can a rhythmic framework with a non-isochronous surface rhythm alter a musician‘s representation of time? In such a context, does meter still serve to shape performance, or is its function supplanted by the rhythmic framework? Experiment 1: Meter Method. This experiment was aimed at obtaining a timing profile produced by a set of participants, aggregated by sixteenth-note position in 4/4 meter. Participants. Eight musicians (one female) from the Chicago area were recruited by email. The participants identified themselves as having normal hearing and professional experience playing Afro-Cuban music, five as percussionists and three as pianists. All participants gave informed consent according to the procedures approved by the Institutional Review Board of Northwestern University. Stimuli. 16 rhythm patterns served as the stimuli, each a single 4/4 measure in duration and consisting of four notes falling on sixteenth-note subdivisions of the measure. The notes were realized as open high conga tones (MIDI pitch 63) with MIDI velocity 90. The conga tones
80 featured a rise time of 11 ms, followed by a 124 ms decay. The patterns were presented at 100 beats per minute; consecutive beats were thus separated by a 600 ms IOI. Patterns 1 through 8 were all based upon the same basic figure, a pair of notes with a 300 ms IOI followed 1050 ms later by another pair of notes with the same IOI. In musical terms, this figure can be represented as a pair of eighth notes followed one and three-quarters beats later by another pair of eighth notes. Patterns 1 through 8, presented in standard as well as grid notation, were generated from this figure by iteratively rotating it by 300 ms, or one eighth note (see Figure 30). (All notes were presented with equal duration; the standard notation in Figure 30 and Figure 31 represents them otherwise for ease of reading.) Figure 30. Stimulus rhythm patterns 1 through 8. Note durations are provided here for ease of reading but are not indicative of their actual length, which were all equal. 1
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81 In similar fashion, patterns 9 through 16 were based upon a second basic figure, a pair of notes with a 450 ms IOI followed 1200 ms later by another pair of notes with the same IOI. In musical terms, this figure can be represented as a dotted-eighth and sixteenth note pair followed two beats later by an identical pair. Patterns 9 through 16 were generated from this figure by iteratively rotating it by 150 ms, or one sixteenth note (see Figure 31). Figure 31. Stimulus rhythm patterns 9 through 16. Note durations are provided here for ease of reading but are not indicative of their actual length, which were all equal. 1
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The patterns were designed this way in order to isolate the effect of metric position on timing while keeping IOI and other musical aspects, including rhythmic grouping, constant. In addition, each set of eight patterns was designed such that in performing the entire set, a participant would produce an equal number of note onsets in each of the 16 metric positions.
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82 Equipment. The stimuli were produced by a program running in Max 5.0.5 on a Power Mac G4 computer. The MIDI messages sent by the program were converted to audio by the MIDI capabilities included in Mac OS X 10.4.11. The audio signal was sent by analog cable from the computer‘s headphone output jack to a Mackie HR824 studio monitor speaker for monaural playback. Participants performed the rhythm patterns on a single electronic drum pad from an Alesis DM5 Pro Kit. Performance data were sent as MIDI messages from the Alesis DM5 drum module to an M-Audio Audiophile USB MIDI interface and then to the USB port of the Power Mac G4 computer. These data were played back through the same system as the stimuli so that the participants heard their own performances through the speaker in real time. The entire system was tested for any latencies in the signal path. Procedure. Participants were seated on a chair or drum throne and allowed to position it in front of the drum kit so that they could comfortably play the drum pad and view the computer monitor placed behind it. They then went through an acclimation period to get used to the appreciable amount of force necessary to trigger the pad. They were instructed to use only one hand, and most participants chose to use an open or cupped palm stroke to strike the pad. During the acclimation period as well as the experiment, whenever the pad was triggered, the sound played through the speaker was the open high conga tone at MIDI velocity 90, no matter how hard the pad was struck. Participants were instructed to use their foot to trigger the bass drum pedal to signal that they were done with the acclimation period and ready to begin the experiment. The experiment consisted of 16 trials, one for each of the stimulus rhythm patterns. The patterns were presented in random order using the urn (unchosen random number) function in
83 Max, which takes as its seed the amount of time elapsed since system startup. Each trial was divided into three segments: familiarization, practice, and test. At all times, the computer monitor displayed the current trial number and segment. Throughout each segment, participants heard a simulated metronome playing quarter notes at 100 beats per minute. Downbeats in common time were realized as high woodblock tones (MIDI pitch 76) while the other beats were realized as low woodblock tones (MIDI pitch 77), with all notes at MIDI velocity 127. The woodblock tones featured a rise time of 11 ms, followed by a 103 ms decay. After the metronome played for two measures, it continued while the stimulus rhythm pattern for the trial repeated four times using the open high conga tone. In the familiarization segment, after participants heard the rhythm pattern repeat four times, the metronome stopped and participants were instructed to strike the bass drum with the foot pedal to continue. The practice segment began the same way, but during the fourth repetition of the rhythm pattern, a one-measure count-in was displayed on the computer monitor, with beat numbers appearing in succession: 1… 2… 3… Play! After the fourth repetition, the rhythm pattern stopped playing, and participants began performing it on the pad as the metronome continued. In the practice segment, participants repeated the pattern four times. After this, the metronome stopped and participants were instructed to strike the bass drum with the foot pedal to continue. The test segment was identical to the practice segment, except that participants performed the pattern 32 times instead of four. In addition, the Max program, using the mtr (multi-track sequencer) function, recorded the amount of time elapsed (in milliseconds) between the beginning of the segment and the first note played, and the IOIs between all subsequent notes. Once again,
84 after sufficient time had passed for 32 repetitions of the pattern to be played, participants were instructed to strike the bass drum with the foot pedal to continue to the next trial. This process was repeated for all 16 trials. The entire session took approximately 45 minutes. Results. The eight sessions yielded a total of 16,297 notes played. For each note, the timing information recorded by the Max program was used to calculate the time elapsed (in milliseconds) since the beginning of the measure. Each note was then matched to the nearest of the four target notes of the stimulus pattern for the trial, with the timing difference between the performed note and the nominal target recorded as positive or negative asynchrony. Since the focus of the study was on within-category variability, notes with an absolute value of asynchrony greater than or equal to 150 ms (one sixteenth note) and thus exhibiting evidence of categorical error were excluded from the data set. To further reduce the possibility of categorical errors in production, entire trials were excluded if at least 10% of their notes had an absolute value of asynchrony greater than or equal to 75 ms. After these exclusions, a total of 15,154 notes remained in the data set, each of which was within 150 ms of an accurate production of a note from its trial‘s stimulus pattern. The signed asynchrony and nominal metric position of each note were appended to the data set. The overall mean asynchrony of the data set was 15.098 ms. For each of the 16 metric positions, the mean asynchrony for each participant was calculated; the participant means were then averaged to give a mean asynchrony across participants. Mean asynchrony across participants varied widely among metric positions from -0.866 ms to 33.218 ms (see Figure 32). The one-way repeated-measures ANOVA revealed a main effect of metric position on mean asynchrony across participants, F(15, 105) = 12.951, p < 0.001.
85 Figure 32. Mean asynchrony across participants by metric position in Experiment 1. Standard error bars denote standard error of participant means.
Visual inspection of Figure 3 suggested a regular pattern of beat-to-beat correspondence. The four positions of highest mean asynchrony across participants were, in sequential order: 1.25, 2.25, 3.25, and 4.25 (x.25) – all one sixteenth note after the beat. Similarly, the four positions of lowest mean asynchrony across participants were, in sequential order: 1.5, 2.5, 3.5, and 4.5 (x.5) – all one eighth note after the beat. Pairwise comparisons between the four positions x.25 and the four positions x.5 were made, using two-sided t-tests with a Bonferroni adjustment. All 16 pairwise differences were significant at the p < 0.01 level (with t values ranging from 5.334 to 6.482), consistent with the claim that the mean of any x.25 beat was significantly different from any x.5 beat. Similar Bonferroni-adjusted two-sided t-tests found no significant pairwise differences at the p < 0.05 level among any of the four metric positions x, among any of the four metric
86 positions x.25, among any of the four metric positions x.5, or among any of the four metric positions x.75. (These 24 comparisons and the 16 comparisons above were all made with a Bonferroni adjustment for 40 comparisons.) The high degree of beat-to-beat correspondence warranted an analysis of mean asynchrony across participants by beat subdivision; that is, a comparison among metric positions x, x.25, x.5, and x.75. The data were collapsed into these four categories and the mean asynchronies across participants were, respectively, 11.951, 32.729, 1.190, and 13.330 ms. The one-way repeated-measures ANOVA revealed a main effect of beat subdivision on mean asynchrony across participants, F(3, 21) = 17.285, p < 0.001. Bonferroniadjusted two-sided t-tests were used to make pairwise comparisons between beat subdivisions and found significant differences (p < 0.05, with t values ranging from 3.585 to 6.137) only between x.25 and each of the three other beat subdivisions x, x.5, and x.75. The overall standard deviation (SD) of the data set was 15.098 ms. SD varied among metric positions from 18.044 ms to 32.677 ms (see Figure 33). Levene‘s test for equality of variance indicated a main of effect of metric position on SD, F(15, 112) = 29.106, p < 0.001.
87 Figure 33. Standard deviation of asynchrony by metric position in Experiment 1. Standard error bars denote standard error of the standard deviation (Sheskin, 2004, pp. 167-168).
In a pattern of regularity analogous to that seen in the mean values, the four positions of highest SD were, in sequential order: 1.75, 2.75, 3.75, and 4.75 (x.75) – all three sixteenth notes after the beat. Closer examination revealed that the two highest positions were 2.75 and 4.75 followed by 1.75 and 3.75, and the two lowest positions were 1.5 and 3.5. Taken along with a visual inspection of Figure 33, this suggested that the periodicity of the SD of asynchrony function might best be modeled with a period length of two beats. The SD pattern was found to be quite consistent when collapsed across beat subdivision and broken out by participant. Seven of the eight participants produced their lowest SD on beat x.5, and five of the eight produced their highest SD on beat x.75. For further evidence of periodicity in the SD function, all 28 pairwise comparisons among the eight positions x.5 and x.75 were made, using Levene‘s test for equality of variance with a
88 Bonferroni adjustment. These comparisons revealed that the SD value of any x.5 beat was significantly different from any x.75 beat (p < 0.001, with F(7, 56) values ranging from 19.133 to 163.441). The 12 pairwise comparisons within each beat subdivision group (for example, 1.5 vs. 2.5; 2.75 vs. 4.75) lent further support to the claim of periodicity on the two-beat level, as evidenced by similar SD values found in pairs of metric positions located two beats apart. Four such comparisons were not significant at the p < 0.05 level (with F values ranging from 0.003 to 3.963): 1.5 – 3.5; 2.5 – 4.5; 1.75 – 3.75; 2.75 – 4.75. Only two other comparisons were not significant at the p < 0.05 level: 1.75 – 2.75 (F = 6.001) and 3.75 – 4.75 (F = 4.946). It is possible that participants sometimes played triplet-like figures rather than sixteenthnote figures. In such cases, participants attempting to play notes on beat x.25 or x.75 would have played them closer to x.33 or x.67. However, the distribution of asynchrony within any metric position failed to reveal a pattern of skewness or bimodality that would support this type of categorical performance error. The data exhibited other patterns in skewness, though. The overall skewness of the data set was -0.265, using the measure of skewness adopted by MINITAB (Joanes & Gill, 1998). This negative skew translated to a concentration of asynchrony values on the right (positive) side of the distribution and a longer tail to the left (negative), as seen in the full data set and, more easily, in beat 2.75, the position of largest negative skew (see Figure 34).
89 Figure 34. Skewness of asynchrony in Experiment 1.
Skewness varied widely among metric positions from -0.909 to 0.175 (see Figure 35). Of particular interest were the low values at metric positions 2.5, 2.75, 4.5, and 4.75, especially in light of the fact that they immediately followed the two highest values at 2.25 and 4.25. Coming at the end of each two-beat segment, this pattern suggested a two-beat periodicity to skewness, as seen above in mean and SD of asynchrony.
90 Figure 35. Skewness of asynchrony by metric position in Experiment 1.
Discussion. The data exhibited periodicities on multiple levels, lending credence to the method of using temporal asynchrony by metric position as an index of cognitive representations of temporal hierarchical structure. The mean and SD of asynchrony were highly periodic at the beat level but, in the case of SD, to an even higher degree at the two-beat level. Skewness also exhibited periodicity at the two-beat level. For all three indices, the two-beat level accounted for most of the periodicity; the measure level did not appear to contribute much more. The regularities in the data are consistent with the claim that at least two distinct levels of temporal structure, the one- and two-beat levels, are relevant in cognitive and/or motor strategies for rhythm performance. Representations of musical time, however, are not independent of tempo. An important caveat about this study is that participants performed at only one tempo, 100 beats per minute
91 (600 ms IOI), around which pulse salience is greatest (Parncutt, 1994). The finding that the twobeat level (1200 ms period) was relevant, while the measure level (2400 ms period) was not, should not be abstracted from the absolute lengths of time involved. Mean asynchrony on subdivisions of the beat was highly consistent from beat to beat, even though the stimulus rhythms performed were relatively sparse compared to the ones found in Satie‘s Gnossienne No. 5. The latter were used as material for the studies that yielded the theory of dual timing mechanisms, one for timing beats and one for executing motor procedures to perform subdivisions (Clarke, 1985a; Shaffer et al., 1985). If such a scheme is also at work in the performance of sparse rhythms, and motor strategy is indeed involved even in the timing of eighth and sixteenth notes lying amidst rests, it would seem to follow that the performance of syncopated beat subdivisions relies on literally feeling the unplayed note that coincides with the beat. In addition, the experiment obtained results consistent with previous determinations of antiphase advantage in timing. On beats x.5, notably low values were found in both the mean and SD of asynchrony. Again, the experiment utilized solely a beat IOI of 600 ms, the same tempo at which Semjen et al. (1992) observed antiphase advantage. As was found in that study, it is expected that the antiphase advantage would disappear or be reversed at faster tempi. In many forms of music, notes on beats x.75 are typically followed by notes on the subsequent beats x; Huron (2006) described such relationships between neighboring elements in terms of event binding. (It should be noted, however, that Afro-Cuban styles and such African Diasporic genres as funk often violate this pattern.) In this interpretation, participants could be seen as expecting to play two consecutive notes but then experiencing a sensorimotor failure of
92 event binding. It is possible, then, that the high SD values found on beats x.75 are related to their relatively unusual juxtaposition with rests located on the subsequent beats x. Experiment 2: Clave Method. This experiment was aimed at obtaining a timing profile produced by a set of participants, aggregated by sixteenth-note position not only in 4/4 meter but also in the context of son clave. The experiment actually took place three weeks prior to Experiment 1 but is presented here as Experiment 2 for conceptual clarity. The 4/4 method of notating clave used in Chapter 3 is not the only one found in regular practice. There is also a 2/4 notation in which note durations and IOIs are notated half as long while maintaining the same relative proportions (see Figure 36). This 2/4 version is often used for slower tempi and historically predates the use of the 4/4 method of notating clave (Mauleón, 1993, p. 50-52). Figure 36. Son clave, 2/4 notation.
The method of notation utilized in this chapter takes the note durations from the 2/4 version and places them in the context of a single measure of common time (see Figure 37). The use of this method is appropriate here for several reasons beyond the relatively moderate tempo of 100 beats per minute. First, it matches the one-measure length and sixteenth-note density referent of the stimuli rhythms. Second, it allows direct comparisons between the clave period (and subsequent timing data) and Experiment 1‘s single 4/4 measure.
93 Figure 37. Son clave, 2/4 notation in single-measure 4/4 context.
Participants. The participants were the same eight musicians who participated in Experiment 1. Because the participants in all likelihood listened to and played music during the three-week break between the two sets of sessions, it is presumed that there were no effects of order on the results of the two experiments. Stimuli. The stimuli were the same as in Experiment 1. Their alignment with the clave pattern is shown in Figure 38 and Figure 39. Figure 38. Stimulus rhythm patterns 1 through 8. Metric positions of clave notes are marked with a ―C‖. Note durations are provided here for ease of reading but are not indicative of their actual length, which were all equal. C
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94 Figure 39. Stimulus rhythm patterns 9 through 16. Metric positions of clave notes are marked with a ―C‖. Note durations are provided here for ease of reading but are not indicative of their actual length, which were all equal. C
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Equipment. The equipment was the same as in Experiment 1. Procedure. The procedure was the same as in Experiment 1 except for the audible timekeeper. Instead of a simulated metronome playing quarter notes, participants heard the clave pattern at 100 beats per minute throughout each segment of the 16 trials. The notes of the clave pattern were realized as high woodblock tones (MIDI pitch 76) at MIDI velocity 120. After the clave played for two measures, it continued while the stimulus rhythm pattern for the trial repeated four times using the open high conga tone. The remainder of the experiment was the same as in Experiment 1. The entire session took approximately 45 minutes.
95 Results. The eight sessions yielded a total of 16,166 notes played. The timing information recorded for each note by the Max program was processed in the same way as in Experiment 1. After applying the same criteria for excluding data on the basis of categorical errors, a total of 13,336 notes remained in the data set, each of which was within 150 ms of an accurate production of a note from its trial‘s stimulus pattern. The signed asynchrony and nominal metric position of each note was appended to the data set. The overall mean asynchrony of the data set was 15.188 ms, almost identical to that seen in Experiment 1 (15.098 ms). Mean asynchrony across participants by metric position as in Experiment 1 and varied among metric positions from -2.892 ms to 24.467 ms (see Figure 40). After several individual trials were excluded based on the criteria described earlier, one participant had no remaining notes produced in one metric position, beat 4.25. The one-way repeatedmeasures ANOVA revealed a main effect of metric position on mean asynchrony across participants, F(15, 104) = 4.794, p < 0.001.
96 Figure 40. Mean asynchrony across participants by metric position in Experiment 2. Standard error bars denote standard error of participant means. Metric positions of clave notes are marked with a “C”.
As in Experiment 1, visual inspection of Figure 40 suggested a regular pattern of beat-tobeat correspondence. The four positions of highest asynchrony were, in sequential order: 1.25, 2.25, 3.25, and 4.25 (x.25) – all one sixteenth note after the beat. Similarly, the four positions of lowest asynchrony were, in sequential order: 1.5, 2.5, 3.5, and 4.5 (x.5) – all one eighth note after the beat. Both of these patterns matched those found in Experiment 1. The high degree of beat-to-beat correspondence warranted an analysis of mean asynchrony across participants by beat subdivision; that is, a comparison among metric positions x, x.25, x.5, and x.75. The data were collapsed into these four categories and the mean asynchronies were, respectively, 12.043, 25.449, 3.489, and 18.474 ms. The one-way repeatedmeasures ANOVA revealed a main effect of beat subdivision on mean asynchrony across
97 participants, F(3, 21) = 8.418, p < 0.001. In addition, an upward drift in mean asynchrony beginning on beat 1.5 was apparent when comparing subdivision-over-subdivision data, e.g., 1.75 vs. 2.75 vs. 3.75. For each subdivision, this function increased nearly monotonically. The overall standard deviation (SD) of the data set was 25.239 ms. SD varied among metric positions from 19.300 ms to 29.211 ms (see Figure 41). Levene‘s test for equality of variance indicated a main of effect of metric position on SD, F(15, 111) = 15.729, p < 0.001. Figure 41. Standard deviation of asynchrony by metric position in Experiment 2. Standard error bars denote standard error of the standard deviation. Metric positions of clave notes are marked with a “C”.
The SD values exhibited a markedly different pattern from that seen in Experiment 1. The points of lowest SD through the first two full beats (the three side, in this version of clave notation) are beats 1, 1.75, and 2.5, coinciding perfectly with the first three notes of clave. However, the final two full beats (the two side) do not continue in coincidence with the surface
98 rhythm of clave. If anything, they nearly parallel the three side; with the exception of beat 4.25, the three points of lowest SD are beats 3, 3.75, and 4.5. Of particular interest is the metric span encompassing beats 2.25, 2.5, and 2.75. This span includes the lowest and the two highest SD values of the entire measure; in Figure 41 this appears as a V-shape. This pattern was also seen when the data were broken out by participant. Seven of the eight participants produced a similar V-shape in this span, with a lower SD on beat 2.5 than on beats 2.25 and 2.75. As in Experiment 1, all 28 pairwise comparisons among the eight positions x.5 and x.75 were made, using Levene‘s test for equality of variance with a Bonferroni adjustment. Unlike in Experiment 1, though, the SD value of several x.5 beats were not significantly different at the p < 0.05 level from x.75 beats, as in the following pairs: 1.5 – 1.75; 1.5 – 3.75; 1.5 – 4.75; 3.5 – 2.75; 3.5 – 3.75; 3.5 – 4.75. Each of these pairs included beat 1.5 or 3.5, which had notably low SD values in the metronome context but not in the clave context. The fact that these beats each occurred one sixteenth note before a clave note may have accounted for the relatively high timing variability in the clave context. More generally, the aperiodic shape of the clave SD pattern would appear to support the idea that the clave context disrupted the pattern of timing consistency seen in the metronome context. A one-way repeated-measures ANOVA was conducted to test the interaction between metric position and experimental context (metronome versus clave) on asynchrony. There was a significant main effect of metric position, F(3, 21) = 12.999, p < 0.001, echoing the separate findings for each experiment. There was no significant main effect of experimental context on asynchrony at the p < 0.05 level, F(1, 7) = 0.0205, thus the metronome condition was not found
99 to have more or less variation overall than the clave condition. Importantly, though, there was a significant interaction between metric position and experimental context on asynchrony, F(3, 21) = 6.876, p < 0.01, suggesting that the choice of context modulated the effect of metric position on asynchrony. Discussion. Mean asynchronies in the clave data shared several commonalities with the meter data in Experiment 1. The overall mean asynchrony was almost exactly the same in the two experiments, and in both, mean asynchrony was highly periodic at the beat level, with a regular pattern of variability among subdivisions (see Figure 32 and Figure 40). Also, the data in both experiments exhibited timing patterns consistent with antiphase advantage, with the four lowest points of mean asynchrony lying on beats x.5. The most notable differences between the mean asynchronies found in the two experiments were found on beats x.25 and x.75. The mean asynchrony across participants on beat x.25 was significantly lower in the clave condition than the meter condition, one-sided paired t(7) = 2.77, p < 0.05, but higher on beat x.75, though not at a significant level, one-sided paired t(7) = -1.43, p = 0.10. It is possible that the sixteenth-note resolution established by the audible clave pattern provided participants with an explicit timekeeping reference for beat subdivisions that was not present in Experiment 1. This reference would likely discourage a tendency, noted above in the Results section of Experiment 1, of participants to round sixteenth-note figures into tripletlike figures. Such a tendency might have been partially responsible for the higher mean asynchronies on beat x.25 and lower mean asynchronies on beat x.75 in Experiment 1, but as discussed earlier, this was not supported by measures of skewness or bimodal distributions in asynchrony.
100 These differences notwithstanding, the periodic peak/trough pattern seen on the beat level of Experiment 2, as well as the overall similarities with the results of Experiment 1, suggest that meter still functioned as a basis for the pacing of rhythm production, despite the use of clave instead of a metronome as the audible timekeeper. The upward drift found in mean asynchrony, followed by a restoration of timing accuracy shortly after the sounding clave note on beat 1, was consistent with Keller and Repp‘s (2005) observation of timing stabilization due to phaseresetting mechanisms. Unlike the means, the pattern of SD values took on a different character from the one seen in Experiment 1. Here, clave seemed to serve an important role in modulating timing variability, which was attenuated on beats 1, 1.75, 2.5, 3, 3.75, and 4.5, all of which are highly salient in various ways to performers in the context of clave: beats 1 and 3 represent the half-measure downbeats of the two sides of clave; beats 1.75 and 3.75 are named bombo for the bass drum that commonly plays there; and beats 2.5 and 4.5 are the points of rhythmic origination known as ponche (Mauleón, 1993). In addition, together the six beats represent two instances of the tresillo rhythm characteristic of the danzón and rumba guaguancó genres (Manuel, 1985). Figural Aspects of Stimulus Patterns The rhythm patterns used as the stimuli in this study were each made up of four notes. In each case, the four notes can be grouped into two sets of note pairs, each representing a different figure type: patterns 1 through 8 as two pairs of eighth notes, and patterns 9 through 16 as two pairs of dotted-eighth-note/sixteenth-note figures. Further, notes can occupy one of two positions within a pair; each note pair can be thought of as consisting of a first note and an end note. The
101 behavioral data collected in the study were analyzed for any evidence that these figural aspects of the rhythm patterns affected the participants‘ production data. A one-way repeated-measures ANOVA was conducted to test the interaction between metric position and figure type on asynchrony. In each experiment, there was a significant main effect of metric position, F(15, 99) = 13.128, p < 0.001 for Experiment 1, and F(15, 93) = 6.566, p < 0.001 for Experiment 2, echoing the findings reported earlier. There was no significant main effect of figure type on asynchrony at the p < 0.05 level, F(1, 6) = 0.692 for Experiment 1, and F(1, 4) = 0.694 for Experiment 2; thus, patterns 1 through 8 were not found to have more or less variation overall than patterns 9 through 16. However, there was a significant interaction between metric position and figure type on asynchrony, F(15, 99) = 3.528, p < 0.001 for Experiment 1, and F(15, 90) = 2.682, p < 0.01 for Experiment 2, suggesting that the figure type modulated the effect of metric position on asynchrony. Based on a visual inspection of the mean asynchrony curves in the combined data from the two experiments, the primary difference between the two figure types was that mean asynchronies on beats x.75 were considerably lower for dotted-eighthnote/sixteenth-note figure types than for eighth-note figure types; in fact, they were lower (for dotted-eighth-note/sixteenth-note figure types) on all four beats x.75 than on the subsequent beats x. Bonferroni-adjusted Levene‘s tests were conducted to compare variance at each metric position across figure type. In Experiment 1, the SD values were found to be significantly different between figure types at the p < 0.05 level for three metric positions: beats 1, 2.75, and 4.5. In Experiment 2, the SD values were found to be significantly different between figure types at the p < 0.05 level for four metric positions: beats 1.75, 2, 3, and 3.5. No discernible pattern was found among these metric positions.
102 Similar statistical tests were performed to investigate the role of position within a note pair on asynchrony, with a hypothesis that notes in end position (and thus possessing a final note accent) would exhibit less asynchrony and possibly less variance. A one-way repeated-measures ANOVA was conducted to test the interaction between metric position and pair position on asynchrony. In each experiment, there was a significant main effect of metric position, F(15, 105) = 12.383, p < 0.001 for Experiment 1, and F(15, 94) = 5.187, p < 0.001 for Experiment 2, echoing the findings reported earlier. There was no significant main effect of pair position on asynchrony at the p < 0.05 level, F(1, 7) = 0.272 for Experiment 1, and F(1, 5) = 3.347 for Experiment 2. Thus, first-note stimuli were not found to have more or less variation overall than end-note stimuli. A significant interaction between metric position and pair position on asynchrony was found in Experiment 1, F(15, 105) = 4.085, p < 0.001, but not at the p < 0.05 level for Experiment 2, F(15, 93) = 1.281. The reason for this discrepancy is unclear, but a threeway repeated-measures ANOVA (metric position by pair position by experimental context) confirmed that the difference between experimental contexts was reliable, F(15, 93) = 2.449, p < 0.01. Based on a visual inspection of the mean asynchrony curves in Experiment 1, the primary difference between the two pair positions was that for notes in first position only, beats x.25 exhibited markedly higher mean asynchronies than all other beats. Bonferroni-adjusted Levene‘s tests were conducted to compare variance at each metric position across pair position. In Experiment 1, the SD values were found to be significantly different between pair positions at the p < 0.05 level for nine metric positions: beats 2, 2.5, 2.75, 3.5, 3.75, 4, 4.25, 4.5, and 4.75. In Experiment 2, the SD values were found to be significantly different between pair positions at the p < 0.05 level for three metric positions: beats 1, 4.25, and 4.75. Again, no discernible pattern
103 was found among these metric positions. Implications of these figural aspects of rhythm for future research will be discussed in Chapter 5. General Discussion Many styles of music, including some popular musics of the African Diaspora, utilize a model of performance timing that features an isochronous beat and one or more timekeeping instruments. These styles are often performed within the context of a rhythmic framework, such as son clave, that coexists with meter and plays an important role in rhythm perception and production. This chapter described an analytical method designed for the study of performances of music with this conception of time. Microtiming as an index of temporal representation. The data collected in the study exhibited periodic patterns of microtiming on single- and double-beat levels. This finding is consistent with two related ideas. First is the theory that musical time is represented internally by a multi-tiered hierarchical structure (Cooper & Meyer, 1960; Lerdahl & Jackendoff, 1983; London, 2001; Yeston, 1976). Second is the empirical observation that the production and perception of deviations from metronomic precision in music provide evidence of this internal representation (Bengtsson & Gabrielsson, 1980, 1983; Chen et al., 2001; Clarke, 1987b; Drake & Palmer, 1993; Drake et al., 2000; Gabrielsson, 1988; Gabrielsson et al., 1983; Jones & Boltz, 1989; Large et al., 2002; Large & Jones, 1999; Large & Kolen, 1994; Large & Palmer, 2002; Palmer, 1989, 1996; Penel & Drake, 1998, 2004; Repp, 1990, 1992b, 1992c, 1995a, 1999a, 1999b, 1999c, 2000b, 2005b, 2008; Shaffer & Todd, 1994; Sloboda, 1983, 1985; Timmers et al., 2000; Todd, 1985, 1989, 1995).
104 It is important to note that the finding of single- and double-beat periodicities applies only to the tempo used in the experiment (100 beats per minute) and is not necessarily generalizable to other tempi, since many rhythmic tasks do not scale proportionally with tempo (Desain & Honing, 1993, 1994; Honing, 2007; London, 2002, 2004; Parncutt, 1994; Povel, 1977; Repp, 1995b; van Noorden & Moelants, 1999). It is likely that if a similar experiment were to be conducted at considerably faster tempi, periodicities would be detected at higher levels, such as the measure. Conversely, at slower tempi, periodicities would likely be detected at lower levels, such as the eighth note. Mechanisms for beat subdivision. Microtiming for notes occurring on subdivisions of the beat evinced a pattern distinctly different from that of the beat level. This finding was consistent with a model of rhythm production that proposed separate mechanisms for the timing of note onsets on the beat level and the subdivision level, with the latter relying on motor procedures (Clarke, 1985a; Shaffer et al., 1985). That model, however, was derived from performances of solo piano music with a fairly dense, consistent rhythmic texture. The experimental task in the current study, designed to simulate the performance of a single instrument in an ensemble, utilized rhythm patterns that were sparser and more syncopated. Seen in light of the dual timing mechanism model, the observed consistency of timing of subdivisions across metric rotations suggests that musicians experienced in performing syncopated rhythms employ a motor strategy that accounts for not only the notes of a rhythm but also the rests. Thus, they time the syncopated onset of a note occurring, for example, one sixteenth note after the downbeat (beat 1.25) by actually feeling, on a neural or muscular level, the unplayed beat.
105 In addition, antiphase advantage in timing was found in the form of lower mean asynchronies on the eighth-note slots between beats than in any other metric position. Given that the rhythm patterns were fairly complex and non-isochronous, this finding can be seen to represent a more generalized form of antiphase advantage than that reported in previous studies, which were restricted to isochronous patterns (Keller & Repp, 2005; Semjen et al., 1992). Alternative rhythmic frameworks. In Experiment 2, participants performed the same rhythm patterns as in Experiment 1, but in the context of the audible clave instead of the metronome. However, the pattern of mean asynchronies was similar and still highly periodic on the beat level despite this difference, consistent with the idea that meter was still an important element of the participants‘ representation of time. Unlike the pattern of mean asynchronies, the SD values obtained in Experiment 2 reflected the influence of clave. SD was found to be attenuated on certain beats that were salient to the participants, who all had extensive experience performing clave-based music. It thus appears that the rhythmic framework of clave serves to constrain timing variability in key metric positions. These findings echo those of Chapter 3: mean asynchrony is largely a function of meter, while SD of asynchrony is a function of clave. Perhaps we can therefore close the loop begun by Butterfield (2006). He claimed in his analysis of Herbie Hancock‘s ―Chameleon‖ that the perception of microtiming deviations helped listeners to reach a certain interpretation of a syntactical rhythm pattern, based on Hasty‘s (1997) model of metric projection. However, in Experiment 2 we found empirical support for the notion that a rhythmic framework can have a systematic effect on microtiming in performance, independent of the rhythm pattern performed within the context of that framework. Putting it all
106 together: the performer utilizes variations in microtiming to communicate information about the underlying rhythmic framework to the listener, who in turn uses that information to interpret the framework and the syntactical patterns performed within it. Whether or not it is performed in the context of a Diasporic rhythmic framework, whether the music features free-flowing rubato or a steady pulse, the phenomenon of microtiming is, as so many have said, essentially about the communication of musical structure.
107 Chapter 5: General Discussion and Conclusions Concepts The central concept of this dissertation is the rhythmic framework. As discussed throughout, a rhythmic framework is a cognitive construct resulting from a combination of meter and repeated or quasi-repeated rhythmic figures. It serves as a structural core of rhythmic organization in a piece and provides a temporal context for musical composition and performance. Rhythmic frameworks are common in musical styles that rely on rhythmic repetition, especially using instruments that produce sharp, percussive attacks. Some of these styles fall under the category of popular musics of the African Diaspora. These are of particular interest, in that they in many cases exhibit characteristics not only of African musics, including repetition, variation, time-line patterns, asymmetry, and close associations with movement and dance, but also of Western musics, including metric organization and hierarchical temporal structure. Because the repetition of patterns in a given piece tends to occur with fixed position relative to the metric context, the rhythmic organization of the piece can be viewed in terms of a set of more or less closely-related phrases of equal length. Each phrase can then be considered to include a set of note onsets. This set of note onsets, and its consistency and variability from phrase to phrase, provide a compact summary of the rhythmic identity of the piece. Methods This dissertation introduces methods for processing and analyzing musical performance data from notated scores, MIDI, and behavioral experiments. Two indices are used to quantify temporal aspects of musical performance. Each of the methods utilizes one or both of these indices, tabulated by metric position across repetitions. The first index, the distribution of note
108 onsets (discussed in Chapters 2 and 3), pertains to the metric categorization of note onsets and is related to the rhythmic structure of a piece. The second index, the pattern of temporal deviations from metronomic timing (discussed in Chapters 3 and 4), operates within metric categories and is measured in millisecond-level variations in performance timing. The analytical and behavioral studies are intended to show that both meter and rhythmic frameworks serve to inform and constrain musical performance. They are additionally intended to provide evidence that meter and rhythmic frameworks play a role in cognitive representations of musical time for performers. One rhythmic framework examined in depth is son clave, ubiquitous in Afro-Cuban music. Findings The findings of the studies are consistent with the claim that musicians produce rhythm – in composition and performance – in a manner that helps to communicate information about operant rhythmic frameworks. More specifically, note onset distributions in African Diasporic styles exhibit asymmetries that signal a key contrast with Western art musics, which have been shown to reflect the symmetrical structures of musical meter. Additionally, in the microtiming studies reported here, the standard deviation of asynchrony between note onsets and metronomic timing is seen to be a function of the rhythmic framework, and not only of meter. However, the presence of rhythmic frameworks does nothing to make meter irrelevant. The findings of the studies are also consistent with existing models of meter and music cognition in which time is represented, internally to musicians, with a hierarchical structure. In the microtiming studies, the mean asynchrony between note onsets and metronomic timing appears to be a function of meter, rather than the rhythmic framework.
109 The behavioral studies in Chapter 4 also yield findings related to mechanisms and strategies for the timing of subdivisions of the beat. These findings are consistent with previous claims that musicians execute motor programs to carry out the timing of beat subdivisions. They also provide evidence of an off-eighth-note advantage, in which notes are performed most accurately when they fall on the eighth-note metric positions halfway between beats. This can be considered to be a generalized form of antiphase advantage, one that exists not only when notes are performed halfway between the sounds of an isochronous pacing stimulus, but also when they are produced as part of more complex rhythm patterns in a metric context. Future Directions As with all human endeavors, this dissertation is necessarily limited in scope. There are several directions in which one might extend it, pursuing the questions raised herein along new avenues of inquiry. For example, it would be quite simple to use the analytical and/or behavioral methods to study the ways in which different tempi, time signatures, and rhythmic frameworks shape musical performance. It would be helpful as well to study the function of rumba clave and 6/8 claves, and to examine how clave direction (3-2 vs. 2-3) and its interaction with formal structure affect rhythm and timing. Another logical extension would be to examine other musical pieces and styles. Our understanding of many supposedly well-known styles often runs no deeper than their surface rhythm patterns. This tends to lead observers to conflate frameworks that are superficially similar, such as son clave and the Bo Diddley beat. More thorough accounts – of various forms of reggae, tango, shuffles, and specific Afro-Cuban and Brazilian styles, to start – would clarify these
110 frameworks by describing their affordances for composed or improvised variations, as well as their points of relative timing freedom and stability. The interaction of instruments in an ensemble should be studied in terms of rhythmic frameworks. How do microtiming patterns in different instruments relate to each other? Can an entire ensemble be modeled within a single framework as proposed here, or does a multiinstrument microtiming framework require a more complex model? How are structural and microtiming aspects of rhythmic performance negotiated within an ensemble? It is written in several instances here that rhythmic frameworks inform and constrain performance. Interviews would contribute to our understanding of what this means experientially. Musicians might describe the sensation as a guide or heuristic, or perhaps to being in the groove, but they also might report that they feel inhibited from playing notes in metric positions inappropriate to the framework. The performances studied in this dissertation were made by stylistically expert musicians. Behavioral experiments could be conducted with a wider range of musicians in order to identify the role of expertise in timing accuracy and variability. Data from the experiments (or from audio recordings) could also be used to create models of timing in individuals‘ performance styles. Such models could then be used as the basis for generating synthesized performances, either for artistic purposes or for the subjective judgment of the quality of the models. Statistical tests indicated that figural aspects of the stimulus patterns used in Chapter 4 may have affected the production data collected from the participants. In order to eliminate this issue from future studies, one could use patterns consisting of only a single note; however, this would of course make it more time-consuming to collect data. As an alternative, one might
111 control for figure type and sequence position by considering them as factors in a more complex study. The methods of processing and analyzing performance data ignore some potentially useful information, in particular the serial order of all of the note onsets and phrases in a given piece or experimental trial. Statistical techniques other than those employed here, including autocorrelation and functional data analysis, could make use of this information. In addition, the information could be used to fit the performance data to oscillator models, thus placing the work into closer conversation with existing and ongoing research on neural and physiological entrainment. This dissertation is focused on only the production half of the musical equation, leaving unasked and unanswered questions about the perception of music in the context of rhythmic frameworks. Certainly, standard methodologies from cognitive psychology and electrophysiology could be utilized to test, for example, how well a rhythm pattern fits with a given framework. However, the tight linkage between rhythmic frameworks and dance, especially in African Diasporic popular music, suggests that methods from embodied music cognition might be most fruitful: video and sensor technologies, motion capture, and the analysis of movement and gesture in response to music. Such methods might shed light on the connections between musical durations and motor periodicities, as well as the cues in music that are salient to listeners and dancers. If it turns out that rhythmic structure is the key to bodily engagement with music, perhaps it can truly be said that listeners are dancing to architecture.
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127 Appendix A: “Tumbling Dice” Grid Notation
1 Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2
2
3
S
4 S
K
K S
K S
K S
K S
K
K S
K
S
K
K
S
K
K
K S
K S
K
K S
K S
K S
128 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick
K
K
K
S
K
S
S
S
S
S
S
S
K S
K
K
K
S
S
K
K S
S
K
K S
K
S
S
S
S
S
K S
K
S
K S
K
S
K S
K
K
129 Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare
S
K
S
K
K
S
K
S
S
S
K S
K
S
S
S
K S
K
S
K S
K
S
K 1 S
K
S
K S
K
S
K S
K S
S
K S
130 Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3
K
K S
S
K
K S
S
K
S
S
K S
S
K
K S
S
K
K
K S
S
K
S
K S
S
S
S
131 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1
K
K
K
S
S
K
K
K
S
S
K
K
K
S
S
K
K
K
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K
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K S
K
K S
S
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K S
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S
132 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2
S
K
S
K 1 S
S
K
1 S
K S
K
S
K
K
S
K
S
K
K
S
K
S
K
K 1
S
K
K
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S
K S
K
K 1 S
S
S
S
133 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick
K
K
K 1 S
S
S
K
K 1 S
S
S
S
K
K
K
S
S
S
S
K S
K S
K
S
K S
K
K
134 Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare
1 S
K
S
K
K
S
S
K 1
K 1
S
K
S
S
K S
K
S
S
S
S
S
K S
K
S
K S
K S
S
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K S 2
K
2
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S
135 Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3
K
K
K
S
K
S
K
K
S
K
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K S
S
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2 K
K S
K
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K S
136 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1
S
K S
S
K
K
K
S
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K S
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137 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2
S
K
K
K
S
K
K
S
K
K
2 3 K
3
3
3 K
3 K
2
3
3
2 3
2
K S
3 K
2 3
S
2 3 K
2 3
2
2 3
2
2 3 K
2
2 3
2
2
2
2
2
2
2
3 K
138 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick
3 K
3 K
3 K
3
3 K
2 3
2
2 3
2
3
2 3 K
2
3 K
2 3
2
2 3
3
1 S 3 K
2 3
S
S
S
S
S
3 K
S
S
K
S
S
K 1 S
S
K
S
1 S
S
S
S
S
S
S
K S
S
K
S
K S
S
S
S 2
K
S
K
S
139 Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare Tom 2 Tom 3 Kick Tom 1 Snare
S
S
K
K S
S
S
S
S
S
S
S
S
S
2 K
K S
S
K
K S
S
K
S
K S
K
S
K
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K 1 S
K
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S
S
S
K S
K
S
K S
S
140 Tom 2 Tom 3 Kick K Tom 1 Snare Tom 2 Tom 3 Kick K Tom 1 Snare Tom 2 Tom 3 Kick K Tom 1 0 Snare 13 Tom 2 0 Tom 3 8 Kick 89 Toms total 8
K S
S
K
0 0 0 0 0 0
0 1 0 1 3 1
0 0 0 0 0 0
S
S
0 72 7 8 0 15
K 0 1 12 3 0 5 0 5 0 89 0 11
0 0 0 0 0 0
2 2 5 1 7 8
S
0 8 0 0 0 0
K 6 6 6 1 3 13
S
S
S
S
S
0 8 0 0 0 0
3 1 1 69 21 28 7 0 5 7 0 1 1 0 32 17 1 7
0 7 0 0 0 0
141 Appendix B: “Everybody’s Everything” Grid Notation 1 Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2
T C 1
B T C 1
B T C 1
B T C 1
B T C 1
2 T
B T
1
1
T
B T
1
1
K B T
T
1
1
K B T
T
1
1
K B T
T
1
1
T C 1
T C 1
B T C 1
B T C 1
B T C 1
B T 1
1
K B T 1
K
1
K B T 1
1
1
K B T 1
B T C 1
K
K B T 1
3 B T C 1
1
B T C 1
B T C 1
B T C 1
K B
K B
B T
B T
1
1
K B T
B T
1
1
K B T
B T
1
1
K B T
B T
1
1
K B T
T
1
1
4 B T C 1
K B T C 1
B T C 1
K B T C 1
K B T C 1
B T
2 K B T
2 K B T
2 K B T
2 K B T
2
142 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass
B T C 1
B T C 1
K B T
B T
1
1
K B T
T
1
1
B T C 1
B T C 1
K B T
B
1
1
K B T 1
1
B T C 1
B T C 1
K B
K B
K B T
B T
1
1
K B T
T
1
1
B T C 1
K B T 1 2 K B T
B T C 1
2 B T C 1
K B T
B T
1
1
B T C 1
K B T C
T
B T
T C
K B T
B
1
1
K B T
B T C 1
B T C
K B
K B
K B T
B T
1
1
2 B
2
T
B T
T C
2 K B T
T
1
1
2
2 K B
B T 1
2 B T C
2 B
2 K B
2 K B T
T C 1
1
2 K T
2 K B
B T
K
1 2 K B T C 1
K B T C 1
2 K
B T
T C 1
K B T 1
1
2 B
K B
143 Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell
T C
2 K B T C 1
T
T
2 B T C 1
T
2 B T C 1
T
2 B T C 1
T
2 B T C
T
T
T C
T
1
1
2
2
B T
T C
B T
1
1
2
2 K B T
B T
T C
1
1
2
2 K B T
B T
T C
1
1
2
2 K B T
B T
T C
1
1
2
2 K B T
B T
T C
T C
2 B T C
T
2 K B
T
K T
2 K
B T C
B T
T
B T C
2 B T C
2 B T C
2 K B
B T
2 K T
2 K B
T C 1
2 K T
2 K B
T C 1
2 K
1 2
T 1
2
1 2
T C 1
B T
T C 1
K B T 1
K B T 1
K B T 1
2 B T
T C 1
K B T 1
2 K T
B T
T C
K B T
144 Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare
1
1
1
2
2
B T C 1
B T
2 K B T
T
2
T C
1
1
1 2
B T C
T
T
1
1
1
1
2
2 K
2
2
B T C
1
2
2
B
B
1
1
K
2
2
K
B T
B T
T
2
2
2
2
1
1
1
1
1
1
2
B T C 1 2
B T
2
B
B
B
2 K
2 K
2 K
T
T
T
T
1 2 K B T C
T
B T 1
2 K T C
B T 1
B T C
B T
B T
T C
B T 1
145 Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick
2 K B T C 1
T
2 K B T
T C
2 K B T
1
1
2
2
B T C 1
B T
2 K B T
T
T C
1
1
2
2
B T C 1
B T
2 K B T
T
T C
1
1
2
2
B T C 1
B T
2 K B T
T
T C
1
1
2
2
B T C 1
B T
2 K B T
2
T
T C
1
1
2
2 K
2 K B T C
B T
2 B T
2 K B T
T C
1 2 B T C
2 K
B T
2 B T
2 K B T
T C
1 2 K B T C
B T
B T
2 K B T
T C
1 2 K B T C
B T
2 B T
2 K B T
T C
1 2
2
B T C
B
2
2
T
B T
2
T C
2 B T 1
2 K
2
2 K
1
146 Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales
B T C 1
T
B T
B T
T C
1
1
2
2
B T C 1
B T
2 K B T
T
T C
1
1
2
2
B T C 1
B T
2 K B T
T
2
T C
1
1
2
2 K
B T C
B T
B T
B T
T C
1 2 K B T C
B T
2 B T
2 K B T
T C
1 2 K
2
B T C
T
T
2 K B T
B T C
1 2
2
2
2
2
2
B T C 1 2
2
2
B
B
B
B
B
K
K
K
K
K
1
B
B T
B T
T
T
T
T
T
1
147 Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1
1 S 2 K B T C
T
K B T
K T C
B T
1
1
2
2
B T C 1
B T
2 K B T
2 K B T C
T
T
T C
1
1
2
2 K B T
B T
T C
1
1
2
2
B T C 1
B T
2 K B T
T
T C
1
1
2
2
B T C
B T
2 K B T
T
1
T C
1
B T C
B T
B T 1
2
2
B T C
B
2
B T
T C 1
2 K T
B T
1
1
T C 1
2 K B T 1
2 K
B T C
1
B T
B T 1
2
2
B T C
B
2
2 K B
K B T
T C 1
2 K T
B T
2 K B T
T C
1
B T C 1
1
2 K
2
T
B T
1
1
2
T C 1
2
2 K B T
148 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2
2 B T C 1
2
T
2 B T C 1
T
2 B T C 1
T
2 B T C 1 2
T
B T
T C
2 K B T
1
1
2
2 K B T
B T
T C
1
1
2 K B T
2 K B T
T C
1
1
2 K B T
2 K B T
T C
1
1
2 K
2 K
B T C 1
B T C 1
B T C
T
K B T
1
1
B
1
B T
K B T
2 K B T
T C 1
2 K B T
T C
1 2 B T C 1
2
2
T
2 K B T
1
1
B
2 K
K
T C 1
K B T 1 2 K B T
K B
T C
2
T
T
2
2
B T C
2 K
1 S 2
2
2
149 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass
B
B
B
B
B
K
K
K
K
K
1
B T
B T
T
T
T
T
T
1 S 2 K B T C 1 2 K B T C
2 K B T C 1 2 B
T
T
T
K B T
B T C
B T
1
1
2
2 K B T
B T
T C
1
1
2
2 K B T
B T
T C
1
1
2
2 K B
B
K B T C
K T
B T
B T
T C
1 2 B T C
2
2 K
B T
2 B T
2 K
K B T
T C
1 2 B T C
2
2 K
B T
2 B T
2 K B T
T C
1 2 B
2 B
2 K
2 B
2 K B
K B
150 Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell
T C
T
2
T
T C
T
1
1
2
2 K B T
B T C 1
B T 1
1
2 K B T C 1
2 K B T
2 K B T
1
1
2
2 K B T
2 K B T
B T C 1
T
2 B T C 1
T
2 B T C
T
T C
T C
T C
1
1
2
2 K B T
B T
T C
1
1
2
2 K B T
B T
T C
T C 1
K B T C
1
B T C
B T C 1
B T C
T
1
1
1
1
B T
B T
B T
1
1
B T
K B T
1
1
B T
T C
T
C 1
B T C 1
T
B T 1
1
1
B T
K B T
1
1
T
K B T
1
1
B
B
1
B T
B T
1
1
B T
B T
C 1
T C 1
T C 1
T C
K B T 1
K B T 1
K B T
151 Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare
1
1
1
2
2 K B T
2 K B T
B T C
T
2 B T C 1
T
2 B T C
T
T C
1
1
2 K B T
2 K B T
T C
1
1
2 K B T
2 K B T
T C
1
1
2 K B T C 1
2 K B T
2 K B T
2
2
B T C 1
B T
T C
2 B T C 1
B T C
T
1
T C
1
2 K B T
B T
1
1
1
K B
K T
2
2
B T C 1
B
1
2
B T
2 K B T
2 B T
1
1
K B T C 1
1 2 K B T
2 K B
B T
T C 1
2 K B T 1
K B T
T C
2 K
2 K B T
T C 1
1
K
T
B T
2
2
B T
B T
1
T C
T
2
2
K B T
1
2 B T C 1
2
B
1
T C
2 K B T
152 Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick
2
2
B T C 1
B T
T
2 K B T
T C
2
B T C
B T
T
2 K B T
T C
1
1
2
2
B T C
B T
2 K B T
K B T C
T
2 K
T C
2 K
2
2 K
2
2
T
B T
2
2
B T
B T
2 K
2
T C
T
T
2 K
1
B T C 1
1
2
2 B
2 K B T
T C
1
2 B T C
2
B
2
T C
2
2 K B T
2 K B T C
2 K B T
2 K
2 K
2 K
1
1
1
B
2
B
B
B
B
B
K
K
K
K
K
1
153 Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales
B T
B T
T
1
K B T C
T
2 B T C 1
T
2 B T C 1
T
2 B T C
T
2 B T
T
K B T
T C
K B T 1
2 K B T
2 K B T
1
1
2 K B T
2 K B T
T C
1
1
2 K B T
2 K B T
T C
1
1
2 K B T
2 K B T
T
T
T
2
2 K
1
1
T C
T
B T C
B
2
2
B T C
B
2
2
B T C 1
B
T
2
T
1
2
B T
2 K B T
T
2 K B T
1
1
T C
2
B T C 1
T 1
1
B
1
2
B T
B T
2 K B T
T C
2
2 K B T
2
2 K B T
T C
2 K B T
2 K B T
T C
2 K B T
B T
T
154 Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1
C
C
2 B T C
T
2 B T C
T
2 B T C
T
2 B T C 1
1
1
2 K B T
2 K B T
T C
1
1
2 K B T
2 K B T
T C
1
1
2 K B T
2 K B T
T C
C 1
K B T C
1
1
B T C 1
T
K B T
1
1
B
1
B T C 1
T
K B T
1
1
B
1
B T C
B T
1
1
2 K B T
2
2 K
T C
2
2 K B T
2
2 K B T
T C
2 K B T C
B T C
2 K B T
T C
2 K B T
2
T C
2
1
B T
1 2
T
2 K B T
1 S
2 K
C
2 K
S
S
S
S
S
S
S
155 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2
S
S
S 2
B
B
B
B
B
K
K
K
K
K
2 B T
B T
T
T
T
T
1 S 2 K B T C
T
2 B T C 1
C
T C
2 B T
2
T C 1 2
K B T
K T C
B T
1
1
2 K B T C 1
2 K B T
2 K B T
2
T C
1
T C
K B T C 1 2
B T C
B
2
2
B T
B
2
2
B T
B
2
2
T
2
T
B T
2 K B T
T C
2
T
B T
2 K B T
1 2
T
2 K B T
2
T
2 K B T
2
2
2
2
156 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass
B T C
T
K B T
K B T
T C
B T C 1
1 2 B T C
B T
2 B T C 1
B
2 B T C 1 2 B
B T C
2 K B T
1
1
2 K B T
2 K B T
C
B T C
T
T C
1
2
2 B T C
2
K B T
B T
2
2
B T
T
B T
2
2
2
B T C
B
2
2
2 K B T
2 K B T
2
2 B
T C
2
T
B T
2
2
2 K B T C
2 K B T
1
2 B T C
2 K B T
B
K B T
T
2 K B T
2
T C
2 K B T
1
1
2 K B T
2 K B T
T C
1
1
2 K B
2 K B
T
T
B T C
2
2 K B T
2 K
B T 1
C 1
T C
2 B T C
2 B
B
K B
2 K B T
2 K B T
2 K B
157 Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell
T
T
2 B T C
T C
2 B T C 1
T
T
2 B T C
T
1
T C 1
2 K B T
2 K B T
2
T C
1
1
2 K B T
2 K B T
T
T
T
T
T
1
1
1
T
2
B T C
B
2
2
B T
B
2
2
B T C
B
2
2
B T C 1
B
B T
2
K B T
2 K B
T
B T C
2
T
2 K B T
2 K B T
1
2 B T C 1
T C
C
2 B T C
T
T
2 K B T
T C
2 K B T
1
1
2 K B T
2 K B T
T C
1
1
2 K B T
2 K B T
T C
2
B T C
2
T
1
2
2 K B T
T
2 K B T
1
1
B T
K B T
2
T C
2
T C 1
T C
2 K B T
2 K B T
2 K B T
158 Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare
2 B T C 1
T
2 B T C 1
T
2 B T C
B T
2 B T C
B T
2 B T C
B T
1
1
2 K B T
2 K B T
T C
1
1
2 K B T
2 K B T
T C
1
1
2 K B T
2 K B T
B T C
1
1
2 K B T
2 K B T
B T C
1
1
2 K B T
2 K B T
1
B T C
1
1 2 B T C 1
1
2 B T
K B T
1
1
B T C
B
K B T
2
2
B T C 1
T C
1
B T C
2 B T C
1
2
2 B T
2 K B T
1
1
B T
K B T
2 B T
2 K B T
2 K B T
T C 1
1
K B T
T C
2 B T C 1
1
B T C
2
2 B T
C
K B T
K B T
2 K B T C
159 Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick
2 B T C
B T
2 B T C
B T
2 B T C
B T
2 B T C
T
2 B T C 1 2
T
2 K B T
B T C
2 K B T
1
1
2 K B T
2 K B T
B T C
1
1
2 K B T
2 K B T
B T C
1
1
2 K B T
2 K B T
T C
1
1
2 K B T
2 K B T
T C
1
1
2 K
2 K
2
2
B T C
2
B T
2 K B T
2
2 B T
2 K B T
B T
2 K B T
T
T
2 K B T
1
1
1
T
K B T
C
2
B T C
2
2
2 B T
2 K B T
2
C
2
B T C
2 B T C 1
B T C
B
2
2
B T C
B
2
2
T
2 K
2 B
K B T C
2
T
2
2 K B T
2 K
2
T C
2
2 K B T
2 K
160 Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales
B T C
T C
2 K B T C 1
B T
B T
T C
B
1
1
T C 1
2 K
2 K
2 K
T C 1 2 K
T
B T
B T
1
1
2 K
2 K
1 2
2
B
B
B
B
B
K
K
K
K
K
1
B
B T
B T
T
T
T
T
1 2 K B T
K T
T
2
K B T
B T
T
B T
1
B T
K B
B T
K B T
B T
K B T
B T
S 2 K T
2 K B T
B T
1
S 2
B T
K B
K B
S 2 K B T
161 Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1
1
1
1
1
1
S 2 K
K B T
K B T
T
2 K B T
2
1
K B
B
1
1
K B
B
K B
2 K B T
B T
S 2 K B
K T
2 K B
T
2
2 K B
T
K B T
1
K B T
K B T
T
1 S
1 S
K B T C
2 K
2 K B
T
T
S 2 K B T
2
2 K B
1 S 2 K B T
T 1 S
S
S K B T C
K T
T
B
2 K B
1
2
S 2
B
K B
1
1
S
K B
S 2
K B T
K B T C
K B T
2 K B T C
2 K B T
B T C
K B T C
B
B
S 2 K B T C
S 2
K B T C 1
T
B T
2 K B T
1 S
2 K B T
2
K B T C
K B T
B
1
1
B T C 1
2 K B T C
2 K B T
B
162 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2
S 2 K B T C 1 S K B T C 1 S
K T
T 1 S
K B T
T 1 S
K B
K
S 2 K B T C 1 S K B T C 1 S K B T
T
2 K B T
S 2 K
2 K B T C S 2
2 K B T C
T
S 2 K B
2
2 K
B T
S 2 K B
K B T
B T
S 2 K B T
B T C
K B T C
2
B
2
2 K B T
S 2
S 2 K B
K B T C
T C
S 2 B
2 K B T
K B
B
C 1 S
S K B T
K
K B
S
K B T
T 1 S
S K B T
1 S
1 S K B T
1 S K B
S 2 K B
1 S K
S 2 K B T
S 2
S 2 K
B T
B T
2
2
1 S
K B
1 S
T
1 S 2 K B
K
K B
B
T 1 S
K B
1 S 2
2
2
163 Kick Bass Timbales Cowbell Tom 1 Snare Tom 2 Kick
K 109 102 95 65 9 79 29
3 3 1 2 1 0 0
K 21 88 6 11 0 5 26
83 99 1 89 7 82 48
K K K 24 15 95 2 97 83 100 3 94 11 97 9 89 0 2 0 85 0 18 1 80 16 35 27 11 2 4 5 5 1 9 3 85 10 53 43 14 6 97 3 24 18
K 31 95 4 36 3 60 51
94 91 4 40 3 54 43
K 23 91 83 35 6 49 26
15 11 0 13 4 25 0
K 104 97 2 35 4 68 96
5 3 1 4 2 0 0
164 Appendix C: Songs Included in Chapter 3 Data Set Songs selected for the study in Chapter 3 (all composed and performed by Rebeca Mauleón): Café Olé; La Jefa; Montuneando; Nueva York Mambo; Peruchineando; Sabroso; Sonando.
