Mensural Determinacy

The concept of mensural determinacy is borrowed from Hasty's (1997) Meter as Rhythm. In Hasty's theory, mensural determinacy emerges from a durational event, which, upon completion, becomes imbued with projective potential, that is, specific temporal expectations related to the duration of future events. 4 It is important to note that for Hasty, a measure (as opposed to a "bar-measure") results from the perception of a determinate duration and its projective potential. In other words, meter does not require the realization, but only the setting-up of an expectation of temporal regularity. Although it might be tempting to make a categorical distinction between mensural determinacy and indeterminacy, in practice it is more fruitful to consider the relative salience of an underlying periodicity based on a predetermined set of variables. Beyond musical parameters, individual preferences for tactus also need to be considered. Thus, in assessing mensural determinacy, one may be concerned with the likelihood that a particular rhythmic sequence will give rise to beat percepts whose periods are closely related to one another across listeners. While questions of mensural determinacy could certainly be explored in the context of nonisochronous beats, for practical and stylistic reasons, the present study is limited to isochronous beats.

The proposed conceptual framework is particularly attractive for the exploration of beat perception in Western art music of the twentieth century. For one, despite the explicit use of time signatures, the temporal structure of many of these works is much more irregular than that exemplified by the musical practice that gave rise to this notational convention. Such irregularity often means that while it might be possible to identify one periodicity, it is much less likely to find two or more nested periodicities that are sustained for a significant amount of time, which is a requirement for one to address questions of metric hierarchy. Furthermore, when considering beat induction across listeners, a pulse finding paradigm that relies on spontaneous tapping and measures accuracy across participants for a single beat period is problematic. Even if a given temporal structure is highly regular, individual preferences for tempo are likely to give rise to several different tapping periods related by simple pulse ratios (e.g., Drake, Penel, & Bigand, 2000; Jones et al., 1995; Martens, 2011; Parncutt, 1994). For irregular rhythmic sequences, this issue is magnified as a higher degree of syncopation also means that there are fewer structural constraints on induced periodicity. In this context, a more flexible measure might prove useful. Because of its emphasis on the projection of an underlying periodicity that may or may not be confirmed by a subsequent auditory event, mensural determinacy is well suited to investigate questions of temporal expectation without having to deal with assumptions pertaining to metric hierarchy. In this context, performance accuracy measures can be used to compare the relative influence of a given beat period or phase on participants' tapping, regardless of tapping profile.

Another advantage offered by the concept of mensural determinacy is its implicit appeal to some form of communicative process. The large majority of experimental studies dealing with beat induction focus almost exclusively on the temporal features of stimuli, and the percepts they give rise to, without considering compositional strategies and aesthetic goals. In other words, the exploration of musical communication is mostly one-sided. An important exception to this practice is exemplified by performance studies, particularly microtiming analyses that address questions of both random and systematic deviations from a notated rhythmic structure from the dual perspective of expressed and perceived structure (although in this case, expressed structure is determined based on performance rather than compositional strategies). While mensural determinacy can be assessed from the perspective of the listener (with or without the explicit consideration of the role of the performer), it can as easily address questions of compositional intent. Emphasizing the process of communication that is at play between composers, performers, and listeners, and recognizing the importance of a shared musical language (understood as a style-specific set of tools for signification) encourages us to address not only questions of compositional and performance strategies, but also of familiarity, competency, and expertise.

Rhythmic Complexity

Defining and measuring complexity in the temporal domain is not an easy task, especially if one considers rhythm perception as well as production. 5 Rhythmic complexity is commonly operationalized following structural criteria that rest on some form of isochronicity. 6 One widespread approach to measure complexity is based on the specific IOI ratios exhibited by a given rhythmic sequence, with the identity ratio (1:1) judged to be the simplest, followed by simple integer ratios (2:1, 3:1, and 4:1), and non-integer ratios such as 3:2 or 4:3; an alternative formulation of this principle involves a contrast between small-sum and large-sum ratios (e.g., Carson, 2007). The preference for smaller integer ratios has been linked to cognitive limits involving working memory (Drake & Bertrand, 2003), and it would seem that the total duration of a rhythmic pattern also plays a role. For example, a rhythmic sequence characterized by a 6:5 duration ratio and a referent pulse unit of 250 msec (a total duration of 2.75 sec) might be conceived as "less complex" than a sequence characterized by a 3:2 duration ratio and a referent pulse unit of 900 msec (4.5 sec total duration), based on the higher demand placed on working memory by the latter. Finally, if the stimuli involve human performance, microtiming variations need to be taken into account (e.g., by applying some form of quantization), and in some cases, larger-sum and smaller-sum ratios can be assumed to be perceptually equivalent. For example, given a very short referent pulse unit, 32:21 might be judged to be equivalent to, rather than more complex than, 3:2.

When the hypothesized temporal structure involves two pulse levels related by simple integer ratios, as is the case with metric hierarchies based on the duple or triple grouping of isochronous beats, the resulting framework provides a convenient grid by which to measure relative complexity, an approach we might refer to as the "goodness-of-fit" model. Rhythmic sequences in which more event onsets (or intensity accents) coincide with the implied accentual pattern of the underlying structure are judged to be simpler (or less syncopated), and those with a lower number of coinciding onsets, more complex (e.g., Jones & Yee, 1997; Povel & Essens, 1985). Rhythmic sequences that can accommodate more than one framework are described as being "ambiguous," and the temporal structure that is judged to be the simplest or most "economical" (Povel, 1984) will be presented as the "preferred" structure. Furthermore, sequences that exhibit highly irregular accentual patterns are often deemed "a-metric," without attempting to measure their relative complexity. An alternative approach might be to count the number of operations that need to be applied in order to transform a given complex (irregular) sequence into a simpler (normative) one (e.g., Tanguiane, 1993); this approach has also been proposed to measure rhythmic similarity/dissimilarity (Toussaint, 2013, pp. 246-257). Similarly, Lempel and Ziv (1976) measure the complexity of a rhythmic sequence by counting the number of steps required to generate the sequence. Taking perceptual experience into consideration, studies that have tested the predictive reliability of different complexity measures generally agree that models based on an isochronous framework (i.e., "clock" models) are more consistent with participants' complexity judgments (Schmulevich and Povel, 2000; Vinke, 2010).

Still, defining rhythmic complexity mostly on structural criteria is problematic, even if the criteria are revealed to be consistent with findings from behavioral experiments. Experimental research has shown that rhythms that are more familiar to some listeners will appear less complex to these listeners than rhythms with which they are unfamiliar (Snyder, Hannon et al., 2006). For example, metrical structures that involve nonisochronous beats (e.g., Balkan music) are not necessarily detrimental to task performance for enculturated listeners, as attested by recent developmental and cross-cultural studies on meter perception (Hannon, Soley, & Ullal, 2012; Hannon & Trehub, 2005a; Hannon & Trehub, 2005b; Hannon, Vanden Bosch der Nederlanden, & Tichko, 2012). Based on these findings, we may speculate on the influence of culture-specific expertise on the cognitive processing of temporal structures (Iversen, Patel, & Ohgushi, 2008). For example, in a study that combined structural equation modeling and covariance analysis, Magill and Pressing (1997) were able to demonstrate that a cognitive clock model based on nonisochronous pulses provided a good fit for most of the polyrhythmic patterns performed by an expert master drummer from West Africa. More generally, musicians have been found to exhibit enhanced skills in rhythmic tasks, such as lower thresholds for detecting timing deviations (Rammsayer & Altenmüller, 2006; Jones et al., 1995), access to a larger number of pulse levels (Drake, Jones, & Baruch, 2000; Drake, Penel, & Bigand, 2000), and greater facility in finding and subdividing an implied beat (Jones & Yee, 1997; Palmer & Krumhansl, 1990). Furthermore, although they are more sensitive to timing deviations, musicians are also more resilient to timing irregularities in pulse attribution tasks (Madison & Merker, 2002). Finally, when asked to judge the complexity of rhythmic sequences, musically trained listeners tended to judge these rhythms as less complex than musically untrained listeners (Vinke, 2010). Thus, it would appear that despite the observed correlation between different models of complexity measures and perceptual findings, structural complexity cannot be easily equated with perceptual complexity, and that a specific rhythm might be experienced as being more or less complex depending on the cultural background and musical experience of a given listener.

These issues come to the fore when one considers the rhythmic sequences typically found in Western art music of the twentieth century, the repertoire selected for this study. Composers of that period are generally recognized for their predilection for breaking with tradition and experimenting with novel musical structures. In the rhythmic domain, composers have employed a wide range of techniques, such as asymmetrical and mixed meters (e.g., Béla Bartók and Igor Stravinsky), proto- and total-serialism (e.g., Olivier Messiaen and Pierre Boulez), polytempo (e.g., Conlon Nancarrow and György Ligeti), and stratification (e.g., Charles Ives and Elliott Carter). In particular, complex rhythmic structures that result from textural as well as temporal stratification, i.e., the layering of contrasting rhythmic strands—each of which may be associated with a different speed, characteristic duration pattern, and/or rhythmic process (e.g., accelerating and decelerating strands against one another)—present a special challenge to our understanding of how rhythm and meter relate to one another in perception and cognition. In this repertoire, one is often faced with a musical surface that exhibits very irregular rhythmic sequences that are nonetheless built upon a regular metric framework characterized by simple ratios. In some cases, the situation is made even more complicated by the presence of two such frameworks, or polymeter, and thus more than one set of nested periodicities, each of which might be only weakly represented in the musical surface (Poudrier & Repp, 2013). While the underlying framework might not be readily apparent to most listeners, style-specific expertise, which assumes a certain degree of familiarity with the composer's aesthetic strategies, might enhance a listener's ability to detect an implied beat.

For the purpose of the present study, a complex rhythmic sequence is defined as a series of IOIs that cannot be readily mapped on a metric grid that results from the interaction of two pulses related to one another by simple integer ratios. Following the proposed conceptual framework, the relative degree of complexity of the experimental stimuli is not measured a priori, but is considered a product of style-specific expertise that is expressed through mensural determinacy, as measured by listeners' spontaneous synchronization with an implied beat period and phase.

Previous Studies

Previous studies most relevant to the present research include research on auditory streaming and multipart rhythm, and pulse-finding studies whose stimuli were derived from actual music. Although there are an increasing number of empirical studies that use actual music as source materials, few of these studies feature rhythmic sequences comparable to those that result from the use of temporal stratification as a compositional technique. Nonetheless, studies that measured the influence of musical expertise on task performance are of special relevance.

In considering questions of beat perception in polyphonic textures, it is useful to think of auditory streaming as involving two distinct types of streaming, i.e., textural streaming (the extraction of a specific auditory sequence) and mensural streaming (the extraction of a specific underlying period or SOP). The seminal work on auditory streaming is Bregman's (1990) Auditory Scene Analysis. Although Bregman's findings are based on very simple auditory stimuli, his proposition that schema-based streaming might be important in the context of musical stimuli is suggestive. Schemas are generally associated with learned behavior. Musicians are especially skilled at using various forms of selective attention, for example tracking changes in an ensemble while performing their own part, and we might speculate that schema-based streaming of both types (textural and mensural) might play a role in pulse finding in actual music.

While most experimental research using stimuli derived from actual music is likely to involve SOP, there is only a small number of studies that have focused explicitly on the perception of SOP. One notable example is Carson (2007), who investigated SOP in non-metric contexts. Using newly composed materials with nonisochronous sequences that exhibited "dissonant" (large-sum) first-order ratios (e.g., 11:2:10:16), but "consonant" (small-sum) second-order ratios (e.g., 1:2) in a range of tempi, Carson found that participants were able to detect underlying periodicities (referred to as "pulsedness") under certain loudness and pitch streaming conditions, with loudness and pitch distance resulting in higher "pulse confidence" ratings. Research on rhythm perception using polyrhythms also offers some pointers on the influence of musical parameters. For example, in a study with polyrhythms combining two or three 2-, 3-, 4-, 5-, and 7-element pulse trains, Handel and his colleagues found that participants' choice of temporal structure varied based on the relative amplitude (loudness), frequency (pitch), and duration of the rhythmic events (Handel, 1984; Handel & Lawson, 1983; Handel & Oshinsky, 1981). At the slowest rate, participants tended to integrate events from superposed pulse trains ("cross-rhythms"), while faster rates yielded more isochronous tapping, with a preference for pulse trains of lower cardinality at the fastest rate. And while Handel and colleagues did not find an influence of musical experience, they did observe a reliable effect of individual preference on tapping profile ("cross-rhythms" vs. "meter rhythms"). Unfortunately, the relative influence of each pulse train on participants' tapping profiles was not measured.

Another strand of research has focused on attentional issues related to competing rhythmic patterns (Jones et al., 1995; Keller & Burnham, 2005). Keller and Burnham's (2005) study is of special interest because it was not limited to isochronous pulse streams but developed an approach whereby several rhythmic sequences with varying degrees of alignment or "metric compatibility" are combined. In the first experiment, participants had on average 11.5 years of experience playing an instrument, and performed in an ensemble regularly. As expected, when asked to memorize embedded rhythmic patterns (recognition memory paradigm), participants performed better when concurrent rhythmic materials provided a good metrical match. The second experiment tested only professional percussionists with extensive performance experience in symphony orchestras and ensembles specializing in contemporary music, and extended the procedure to include reproduction (dual-task paradigm). While the findings were consistent with those of the first experiment, the deficit in performance between matched and mis-matched meters was less pronounced, with nonmetrical stimuli still yielding the poorest performance. Thus, while these experiments provide more evidence that hierarchical structure facilitates the perception and production of simultaneous nonisochronous sequences, they also point to differences in performance based on the specific type of musical experience of its participants.

The three pulse-finding studies used as models for the experimental design of the present study (Drake, Penel, & Bigand, 2000; Snyder & Krumhansl, 2001; Toiviainen & Snyder, 2003) featured contrasting musical materials ranging from J. S. Bach to ragtime and Bartók. However refined in construction these pieces are, none of them even comes close to the type of rhythmic complexity encountered in the source materials used in the present study. Furthermore, although Drake and colleagues (2000) and Snyder and Krumhansl (2001) investigated the influence of musical training, musical training was defined in the most general terms. Participants' experience with the styles exemplified by the source materials was measured, but its influence on participants' performance was not assessed. Nonetheless, the findings point to some of the benefits of musical expertise as well as musical factors that may enhance the determinacy of a specific pulse. Drake and colleagues (2000) compared mechanical (with and without intensity accents) and expressive (performance by a concert pianist) versions of six musical excerpts of different styles and tempi (tempo range = 90–150 beats per minute or bpm). Overall, they found that both musical expertise and microtiming had an influence on participants' performance. In particular, not only did musicians (who had on average 7 years of musical training or instrumental playing) tend to attune to a slower pulse than nonmusicians (who had no formal musical training or experience playing an instrument), but their tapping period also corresponded more often with the theoretically correct meter (as determined by the authors based on an analysis of the notated score), suggesting that musicians had access to a more complete and stylistically valid mental representation of the meter. Also of note is the finding that the addition of intensity accents on downbeats induced nonmusicians to tap at a slower rate, in a manner similar to musicians. Finally, although synchronization was generally poorer with the expressive version, the presence of expressive timing reduced the range of tapping rates, which suggests that microtiming patterns can function as specific cues for beat induction.

Tapping studies that seek to investigate the influence of different musical parameters and use actual music as source materials are faced with the challenge of balancing experimental control and ecological validity. In two experiments with piano ragtime music, Snyder and Krumhansl (2001) presented their participants with eight ragtime excerpts (tempo range = 86–102 bpm) in four different versions with texture (full and right-hand only) and pitch information (varied-pitch and monotonic) as variables. While there was no significant difference based on pitch information, synchronization performance was poorer in the right-hand only version, a finding that was attributed to its higher degree of syncopation. Post-hoc autocorrelation analysis of the temporal structure of the two parts revealed that there were no significant peaks for event onsets in metrically strong positions in the right hand, while the left hand exhibited peaks at two, four, and eight sixteenth notes. In the first experiment, the researchers investigated the influence of musical experience (musically experienced participants' mean years playing music was 13 as compared with 3 for musically inexperienced participants). Contrary to the findings of Drake and colleagues (2000), there was no significant effect of musical experience on participants' performance. It is worth noting that the average reported familiarity with ragtime music (on a 7-point scale) ranged only from 2.2 to 3.9, and that it may be that the lack of style-specific expertise weakened the potential influence of musical experience on the task.

Toiviainen and Snyder (2003) investigated the musical features that support pulse finding by varying the amount of information (left-hand, right-hand, and whole) and metrical orientation (starting and ending on the first vs. third eighth-note beat of a 2/2 meter) in eight excerpts of eight bars each, taken from a mechanical performance of J. S. Bach's organ duetto for solo performer, BWV 805 (tempo = 75 bpm). Because of the difficulty of the task (suggested by the results of a pilot study), all participants were musically trained, with at least 8 years of experience playing an instrument or singing (M = 12.1), and a reported average familiarity of 4.5 and 1.7 on a 7-point scale with the music of J. S. Bach and the source materials, respectively. Findings pointed to a strong bias for tapping to the phase corresponding to the starting metrical position, as well as to a facilitation effect when the two parts were presented together. A post-hoc principal component analysis of four quantifiable musical variables (onsets at the eighth-note pulse level, melodic direction changes, syncopation at the half-note pulse level, and notes longer than an eighth note) revealed that a high degree of syncopation resulted in a less stable tapping performance.

As the findings from these few studies suggest, musical experience can give rise to significant differences in the performance of both production and perceptual rhythmic tasks. The use of actual music as source materials, while enhancing ecological validity, may also require participants to be musically experienced, thus limiting the researchers' ability to test the effect of musical experience on participants' performance. As the present study will show, such limitations can be avoided by using more specific measures of expertise.

Structural Features of Source Materials

The source materials chosen for this experiment consisted of the molto espressivo section of Elliott Carter's 90+ for piano (1994), a work that is representative of textural and temporal stratification in twentieth-century Western art music. In particular, this excerpt provided the opportunity to investigate beat induction in two contrasting rhythmic sequences that are nonetheless based on a common referential periodicity, with the respective underlying quarter-note beats in anti-phase with one another (interleaved beats). The upper layer is characterized by various instances of FOP resulting from the irregular subdivision of an underlying duration pattern, while the lower layer is characterized by an irregular rhythmic sequence with a few instances of FOP, but explicit SOP.

Fig. 1. Interonset interval (IOI) series for the left-hand (LH) and right-hand (RH) layers of Elliott Carter's 90+ for piano, m. 68, fourth eighth note, to m. 88, fifth sixteenth note. The two event sequences exhibit contrasting rhythmic profiles, with gradual tempo changes in the RH and an alternation of periodic and non-periodic events in the LH, with only eight first-order periodicities that correspond to the quarter-note beat (quarter = 72 bpm or 833 msec).

The excerpt extends from m. 68, fourth eighth note, to m. 88, fifth sixteenth note, inclusive; a reproduction of the entire excerpt is included as Appendix A and B. Figure 1 presents each of the two layers as an IOI series. In the original score, the passage is notated with a 4/4 time signature and a prescribed tempo of quarter note equal to 72 bpm; its mechanical performance lasts about one minute. The two component layers correspond roughly to the right and left hands of the pianist (designated RH and LH in Figure 1), although the two parts cross in register a few times. The right-hand layer consists of a mostly stepwise but sweeping melody built from subdivisions of an underlying 1:2 duration pattern; in notational value, these subdivisions range from septuplet sixteenth to triplet quarter notes. This temporal design suggests at least two nested pulse levels in a 3:1 ratio, that is, a quarter note at 72 bpm and a dotted-half note at 24 bpm, corresponding to chronometric durations of 0.833 and 2.5 sec, respectively. The left-hand layer consists of an accompaniment with regularly accented lower tones whose IOIs correspond to a SOP of 18 bpm (3.333 sec), the rate and duration associated with the whole note. These slow pulsations alternate with short staccato notes in a higher register, most of which are notated as triplet eighth and sixteenth notes, resulting in a highly irregular rhythmic sequence. The compound layer suggests measures of four quarter notes at 72 bpm, and this structure is explicitly presented only once toward the beginning of the passage (from m. 70, fourth eighth note, to m. 71, third eighth note). Table 1 presents the range of notational values used in both layers, with the corresponding IOIs (in msec) and tempi (in bpm).

The superposition of the two layers can thus be interpreted as resulting in an implied polymetric structure of 3/4 against 4/4 in which the quarter-note beats are interleaved due to an eighth note displacement (417 msec) of the beats of one layer in relation to those of the other layer. Although there are no coinciding beats, the two layers could be partially integrated at the eighth and sixteenth pulse levels.

Experimental Goals

The features of the source materials made it possible to explore the influence of a set of musical (within-subjects) variables on beat induction and mensural determinacy. The three specific questions that guided the experimental design were:

1. Does style-specific expertise (passive and active exposure to Western twentieth-century music) influence participants' tapping performance?
2. What is the relative degree of mensural determinacy of the implied quarter-note beat for each rhythmic layer, and how does it change when layers are combined?
3. What is the influence of musical parameters (within-subjects) and participants' characteristics (between-subjects) on participants' ability to synchronize with the implied beat?

Because the population sampling method was convenient rather than random, the main analysis was limited to a relatively small set of within-subjects variables or "primary" musical parameters (global temporal structure, accent, and texture). However, the features of the source materials afforded an exploration of the influence of a number of "secondary" musical parameters (global pitch structure, articulation, contour, and register). Participants' characteristics also allowed for post-hoc exploration of the influence of a set of between-subjects variables (gender, formal musical training, main instrument, and piano skills). These findings are reported as possible avenues for future research.

Participants and Between-Subjects Variables

Thirty participants (women = 13; men = 17), were recruited from a graduate program in music and from a semi-professional choir. The basic requirement was that participants be musically active and have received formal musical training equivalent to at least one year of undergraduate studies. Because the rhythmic sequences used in this experiment were relatively complex, it was thought that the task would be too difficult for nonmusicians. All participants except one were right-handed, and no participant reported hearing problems.

Prior to the experiment, participants completed a questionnaire on their musical training and activities, experience with twentieth-century music, and familiarity with the source materials. Six participants were amateur musicians and the remaining participants were music students and/or professional musicians; participants also identified themselves as primarily engaged in one of five areas of specialization, i.e., composition (2), ethnomusicology (1), musicology (5), theory (9), and performance (13), the latter including all amateurs, whose primary musical activity was choir singing. Prior knowledge of polyrhythm and/or polymeter was not a requirement; however, 19 participants reported having experience with performing polyrhythms and these participants were asked to provide some details about learning and performance strategies (not included in this report). Four participants reported no familiarity with polyrhythm or polymeter; the concepts were explained to them prior to the beginning of the experimental tasks. Only three participants knew about or had heard Carter's 90+ before, but none reported having performed or studied the piece extensively, and only two participants identified themselves as very familiar with the composer's music.

Participants varied widely in SSE, which was calculated in hours per week spent performing, analyzing, and/or listening to twentieth-century Western art music (M = 9.84, SD = 8.23). In addition to gender, three between-subjects variables were selected for exploratory analyses of the influence of specific aspects of musical training on participants' ability to extract a regular beat: formal musical training, main instrument, and piano skill level. Participants' formal musical training, as represented by degree, varied from the equivalent of one year of undergraduate studies to a completed PhD (no degree = 6, bachelor's = 5, master's = 18, and PhD = 1); this data was transformed into two groups: "no degree/bachelor's" (11) and "master's/PhD" (19). Participants' main instruments were combined into three roughly balanced groups based on register and/or sound production: "piano/percussion" (9), which included only one percussion player; "high melodic" (14), which included flute, alto/soprano voice, and clarinet; and "low melodic" (7), which included double bass, cello, and bass/tenor voice. Reported skill level at the main instrument ranged from beginner/intermediate (5) to advanced (15) and expert (10), but this information was not used for statistical analysis. Finally, participants also varied in their reported skill level at the piano: "none/beginner" (10), "intermediate" (11), and "advanced/expert" (9).

Manipulation of Independent Variables (Source Materials)

Each layer was transcribed using the notation software Finale 2006, converted into MIDI files, and manipulated using the sound editing software Cubase SX and samples from the Garritan Personal Orchestra VST. 7 All files were converted in .wav format and normalized to peak amplitude. The temporal structure of the notated score was preserved, but the excerpt was manipulated in various ways to allow for a comparative analysis of the influence of musical parameters on perceived temporal structure, for a total of 18 stimuli (see Table 2 for a descriptive list of the experimental stimuli; sound files for the stimuli can be accessed at http://hdl.handle.net/1811/85871):

1. Texture: ten stimuli featured segregated layers (L vs. R) and eight stimuli presented the two layers simultaneously (L+R);
2. Pitch (timbre and frequency): eight stimuli used the reverberated teak stick sample (indeterminate pitch) and ten stimuli used the reverberated Steinway piano sample (determinate pitch);
3. Accent (attack velocity): notated accents in L were preserved in five and suppressed in seven stimuli; in R, accents were added to attacks coinciding with the implied quarter-note beat when it corresponded to an event onset in four stimuli; all accents were produced by increasing attack velocity based on the standard MIDI scale (flat velocity = 70, accented notes = 85–110);
4. Articulation (duration): five of the ten stimuli with determinate pitch featured a reversal of articulation, from staccato (45% of quarter-note value) to legato (100%) and legato to staccato; tenuto accents were performed at 125% of the notated value, without changing the IOI;
5. Contour and register: in two L stimuli, the contour was collapsed to a range of one octave by transposing pitch outliers down or up an octave; one L+R stimulus featured register reversal.

All sound files included the entire excerpt, and the exact duration of the stimuli varied depending on whether one or both layers were present (L and L+R = 64.167 sec; R = 62.917 sec). Sound files were performed stereophonically. For the segregated layers conditions (L or R), each layer was piped through both channels equally; in the combined layers condition (L+R), stereo separation was created by panning one track 33% to the right and one track 33% to the left to avoid incidental noise. 8

The 18 stimuli were presented in four blocks of trials based on texture and pitch. Blocks 1 and 2 presented the segregated layers conditions (L vs. R), first with indeterminate pitch or pitch "off" (block 1), and then determinate pitch or pitch "on" (block 2). Blocks 3 and 4 presented the combined layers conditions (L+R), first with pitch off (block 3) and then pitch on (block 4). This design allowed for within- and across-block comparisons. Blocks 1 and 3 featured a balanced design, which allowed for hypothesis testing; the unbalanced design in blocks 2 and 4 only afforded exploratory analysis.

Apparatus and Procedure

The experiment took place in a private room in the music department. Participants were sitting at a desk and the principal investigator (this author) was sitting about 5 feet away on the left side of the desk. All sound files were placed on an interface controlled by a program created with Max/MSP and run with a MacBook 1.83 GHz Intel Core Duo. The sound files were played through standard quality headphones (SENTRY HO268) and participants were instructed to adjust the volume to a comfortable level. Participants controlled all manual operations and tapping was performed using the right arrow key on the computer keyboard.

Before beginning the experiment, participants signed a consent form and the principal investigator read an introductory statement about the experimental procedure. This was followed by a short practice session with a stimulus not included in the experiment. During the experiment, all instructions were given verbally and participants could ask for clarifications at any time as well as provide verbal comments in between listening and tapping tasks. For the listening-only part of the task, participants were instructed to move or not with the music as they would normally do. For the listening-and-tapping part of the task, participants were instructed to tap to the "underlying pulse or beat," which might involve not tapping at all if they felt there was no underlying pulse, or altering the speed or duration associated with the beat if they felt that it was altered, as it may occur in actual music.

In the first three blocks, participants heard each stimulus twice, first without tapping and then tapping along. 9 Although participants were instructed to first listen without tapping, many used this first listening as a "practice" session, trying out a given underlying pulse or beat by tapping on the desk, conducting with one or both hands, foot tapping, or a combination of these. For the last block, participants heard each sound file three times. After the first listening, they were asked to describe the texture. The main purpose of this first listening was to establish a common vocabulary that could then be used by the principal investigator to subsequently direct the participants' attention to each layer. For the second and third listenings, participants were instructed to focus successively on each one of the two contrasting layers and tap along "its main underlying pulse or beat." When a participant had identified more than two layers, which occurred only for L, the principal investigator instructed the participant to focus on the combination of identified layers that corresponded to L.

It was assumed that order of presentation could have an effect. Thus, presentation order was counterbalanced based on attended layer (blocks 1, 2, and 4), and was further varied systematically across all participants within layers, whenever applicable (blocks 2, 3, and 4). In addition, in blocks 1 and 3 (indeterminate pitch condition), unaccented stimuli were presented first to avoid a carry-over effect of accentual structure. It is assumed that this counterbalancing accounted for possible presentation order interactions with within-and between-subjects variables, but because of the unbalanced design in blocks 2 and 4, and the small number of cases for each presentation order group (up to 24 orderings in block 4), this assumption was tested only in the first block of trials (L-first vs. R-first).

Three types of data were gathered during the experiment: (1) participants' verbal comments were recorded using a SONY ICD-P320 digital voice recorder; (2) observations of participants' body movements during listening-only and listening-and-tapping were noted by the experimenter; and (3) tapping responses (timepoint series measured in milliseconds) were recorded by the computer program and saved as text files. All results presented here are derived from the tapping responses. The time-points text files were first imported into Excel and used to generate inter-tap interval (ITI) series for each tapping response. ITI series were then plotted as two-dimensional graphs and used for visualization of participants' tapping behavior. A computer script processed the data sets to identify taps that were synchronized with underlying pulses, and the criterion used for synchronized taps was ± 100 msec (e.g., Drake, Penel, & Bigand, 2000; Snyder & Krumhansl, 2001). 10 Cut-off points across stimuli were then normalized.

The experiment lasted between 1 and 2 hours depending on the number of verbal comments made by each participant. Participants were instructed to take a short break between the second and third blocks of trials.

Performance Measures and Analysis

Two basic types of performance measures were used for statistical analysis, i.e., tapping variability and phase synchronization. Variability measures in pulse-finding experiments are typically based on the standard deviation (SD) of an ITI series, i.e., the root-mean-square (RMS) deviation of the ITI values from their mean, and exclude false alarms. The measure used here was the coefficient of variability (CV), which is calculated by dividing the standard deviation by the mean ITI (SD/mean ITI), to allow for comparisons across tempi. To take into account local as well as global factors, all taps were included, whether or not they were actually synchronized with the implied beat (as determined by score analysis). Therefore, the observed CV performance in this study is not directly comparable with previous findings. The inclusion of all taps in the CV measure means that it is more sensitive to tapping fluctuations (yielding generally higher CV values than other tapping studies), which makes it possible to take into account participants' overall tapping behavior, but can also hamper statistical significance due to larger variance.

There were two different, but closely related, phase synchronization measures used in this study. Tapping synchronization, Tsyn, corresponds to the ratio of participants' synchronized taps, ST, to the total number of taps (Tsyn = STn/Tn); beat synchronization, Bsyn, corresponds to the ratio of participants' synchronized taps to the total number of available quarter-note beats, Q (Bsyn = STn/Qn). To differentiate between the interleaved quarter-note beats, phase synchronization measures are further designated by the labels L and R, whenever necessary; for example, TsynQL refers to tapping synchronization with the left-hand phase of the quarter-note beat and BsynQR refers to beat synchronization with the right-hand phase of the quarter-note beat. While both of these measures reflect the influence of the implied beat on participants' tapping behavior, Bsyn is a more objective measure of mensural determinacy as it is less sensitive to extraneous taps, including tapping with subdivisions of an implied beat. In most cases, results based on both types of synchronization will be reported.

It was hypothesized that SSE as represented by passive and active exposure to twentieth-century music would be reflected in participants' tapping behavior, with more exposure resulting in less tapping variability. To test this effect, a Pearson's correlation was applied using overall CV value and the number of weekly hours spent listening, performing, and/or analyzing twentieth-century music. Hypothesis-driven and exploratory statistical analyses were also conducted within and across blocks of trials. Analyses within and across blocks 1 and 3 tested the influence of the primary musical parameters (i.e., global temporal structure as represented by layer, accent, and texture) on participants' tapping behavior using CV, TsynQ, and BsynQ. Analyses within blocks 2 and 4 explored the influence of primary (layer and accent) as well as secondary (articulation, contour, and register) musical parameters. Analyses across blocks 1 and 2 as well as across blocks 3 and 4 explored the influence of global pitch structure (off vs. on), while those across blocks 2 and 4 explored the influence of texture (segregated vs. combined layers).

The two pulses used for statistical analysis correspond to the two different phases of the same beat period of 833 msec, i.e., the left- and right-hand phase of the quarter-note beat (designated as QL and QR, respectively). To assess the degree of mensural determinacy of each layer independently and in combination, phase discrimination analyses were also conducted both within and across blocks of trials using BsynQ only. In these analyses, phase dominance (QL vs. QR) was determined based on statistically significant differences in participants' synchronization performance to the L- and R-phase of the quarter-note beat (BsynQL or BsynQR); lack of statistical significance was interpreted as representing phase ambiguity (QA). In the context of this experiment, phase dominance is understood as a manifestation of mensural determinacy whether layers are presented independently or in combination. On the other hand, phase ambiguity in segregated layers indicates a low degree of mensural determinacy; in combined layers, a lack of statistically significant difference between the two phases of the quarter-note beat may result from an integrated percept which may or may not be mensurally determinate, depending on whether the tapping phase is stable or not (phase shifting).

Repeated measures univariate analyses of variance (ANOVA) were used within and across blocks to assess significant main effects and interactions of within- and between-subjects variables, and paired-samples t tests were used to determine significant differences between sets of trials, whenever necessary. The significance level for all statistical analyses was based on a 95% confidence interval (p < .05), and trends are reported based on a 90% confidence interval (p < .10); significance levels are given in full whenever possible. Missing cases were excluded from each separate analysis and the number of valid responses is provided for each result. Pairwise comparisons and post-hoc tests were performed using a Bonferroni adjustment for multiple comparisons, and Greenhouse-Geisser adjustments for sphericity were applied whenever necessary. The choice of adjustments was guided by the repeated measures design, with a preference for a more conservative adjustment because of the relatively small sample size.

Finally, post-hoc exploratory analyses of between-subjects variables (i.e., gender, formal musical training, main instrument, and piano skill level) were conducted within each block only, using all three measures (CV, TsynQ, and BsynQ). Given that the sample was convenient rather than random, it should be noted that generalizations are applicable only for the population represented by the sample, i.e., graduate music students, amateur musicians, and professional musicians with at least one year of formal musical training who are actively engaged in musical activities (including performance, analysis, and composition).

Overall Tapping Performance and Influence of Style-Specific Expertise

Table 3 presents descriptive statistics for the experiment overall, including tapping variability (CV), and tapping and beat synchronization (TsynQ and BsynQ) for each block of trials. CV varied widely across participants, with individual participants' means for the whole experiment ranging from 0.088 to 0.880 (M = 0.379, SE = 0.034). Mean synchronization percentages were consistently low overall, but exhibited a high degree of variance across participants. Within blocks, individual participants' TsynQ performance ranged from 15.5 to 70.8%, while BsynQ performance ranged from 5.7 to 70.5%. The observed discrepancies between these two synchronization measures suggest that a number of participants might have been tapping sporadically (i.e., heard some passages as not having a regular underlying periodicity), at slower or faster consonant rates (e.g., half- or eighth-note level), or partially consonant rates (e.g., dotted quarter-note level), with TsynQ < BsynQ suggesting a faster tapping rate, and TsynQ > BsynQ suggesting slower and/or intermittent tapping. Although individual participants achieved the highest tapping synchronization percentages in block 1 (segregated layers with pitch off), this is also where the largest variance across participants (within each synchronization measure) was observed.

As shown in Figure 2, there was a significant negative correlation between CV and style-specific expertise, r(26) = -.406, p = .032. Participants who had more weekly exposure to twentieth-century music exhibited more stable tapping behavior than participants who had less weekly exposure. However, there appears to have been a number of outliers as well as a ceiling effect at about 10 hours per week, which suggests that there might be other between-subjects variables that influenced participants' tapping performance. 11

Fig. 2. Participants' overall coefficient of variability (CV) against style-specific expertise (SSE), measured in hours weekly devoted to musical activities with twentieth-century music. The distribution of participants' individual overall CV performance is consistent with the hypothesis that SSE contributes to more regular tapping.

Primary Musical Parameters

Exploratory Analysis of Secondary Musical Parameters

EFFECT OF ATTENDED LAYER AND REGISTER (BLOCK 4)

Block 4 presented combined layers conditions with pitch on, which afforded focal attending. Within each condition, the manipulation of musical parameters was similar to that of block 2, i.e., there were variations in accent (off/on), articulation (staccato/legato), and L contour (as written/collapsed). One notable difference between blocks 2 and 4 was the consistent presentation of R with accents off; this was done to mimic more closely the source materials. In addition, the combination of layers allowed for the manipulation of their relative registral position (low/high). The articulation of each layer was also modified to facilitate focal attending. In particular, the sparser L-high was presented with legato articulation to make it more audible, and the denser R-low featured staccato articulation in an effort to alleviate its interference with L-high. Given the number of variables and the necessary limitations of one experimental session, within-layers and between-layers conditions were not balanced, resulting in a 2 × 4 factorial design with attended layer (L/R) and condition (each of the four stimuli) as within-subjects variables, but a number of confounds. This design allowed for an investigation of the relative salience of each attended layer as well as an exploration of the effect of different combinations of musical parameters on participants' tapping performance.

Fig. 3. Interaction of attended layer and condition for BsynQ in block 4 (estimated marginal means are expressed as fractions). The significantly lower percentage of quarter-note beats that were synchronized to in the L-attend condition for C4 is consistent with a detrimental effect of the register reversal on participants' tapping synchronization performance.

A univariate ANOVA was performed with attended layer (L-attend/R-attend) and condition (C1 through C4; refer back to Table 2 for details) as within-subjects variables. Attended layer had a significant main effect on CV and TsynQ, but not on BsynQ. Participants' tapping performance was more stable when attending L than R (.306 as compared to .462), F(1, 26) = 11.977, p = .002, and a larger proportion of participants' taps were synchronized with the quarter-note beat when attending L (30.0 as compared with 23.6% for R-attend), F(1, 26) = 8.980, p = .006. The specific combination of variables featured in each condition also had a significant main effect on TsynQ, F(3, 78) = 4.453, p = .012. Participants' tapping synchronization performance was significantly poorer for C4 (23.3 as compared with 29.1% for C3, p = .036); there was also a trend toward significance in the same direction for BsynQ, F(3, 78) = 2.843, p = 0.56. Finally, although condition did not have a main effect on CV, F(3, 78) = 1.943, p = .164, its effect was not negligible as participants' tapping was significantly less stable with C4 than C1 (M = 0.086, SE = 0.026, p = .017). This finding was also reflected in a significant interaction of attended layer and condition for both TsynQ and BsynQ, F(3, 78) = 3.602, p = .043, and F(3, 78) = 4.459, p = .013, respectively. Although participants synchronized with a generally higher number of quarter-note beats (BsynQ) in the L- than R-attend condition, the number of quarter-note beats synchronized to for C4 was significantly lower in the L-attend condition (15.5 as compared with 22.4% in the R-attend condition; see Figure 3). 16 These results suggest that register reversal had a detrimental effect on participants' synchronization with QL, which could be due to a greater interference of R when presented in the low register and/or a beat induction bias in favor of the low register.

Table 11 presents the results of the phase discrimination analysis (paired-samples t tests) for each condition within block 4. When comparing participants' downbeat synchronization within attended-layer conditions, it can be observed that the quarter-note beat was mensurally determinate only in four out of eight trials, all of which involved the L-attend condition. For three of the L-attend trials (C1, C2, and C3), participants synchronized significantly more with QL than QR, t(27) = 3.283, p =.003 (C1), t(27) = 3.467, p =.002 (C2), and t(28) = 2.881, p =.008 (C3). However, when attending to L in the high register, participants synchronized significantly more with the upbeats (QR), resulting in R-phase dominance, t(28) = 2.083, p =.047 (C4). In contrast, when participants tapped in the R-attend condition, their tapping consistently gave rise to phase ambiguity, t(26) = 1.286, p =.210 (C1), t(28) = .469, p =.643 (C2), t(29) = .553, p =.585 (C3), and t(29) = 1.083, p =.288 (C4). These results are consistent with the suggested bias in favor of the low register for beat induction.

Table 11. Phase discrimination (paired-samples t tests) for each attended-layer condition (L-attend/R-attend) within block 4 (QL = L-phase dominance; QR = R-phase dominance; QA = ambiguous phase); condition 1 (C1) is most closely related to the composer's notated version.

	ATTENDED LAYER (L/R)	ACCENT (ON/OFF)	ARTICULATION (STA/LEG)	CONTOUR (ASW/COL)	REGISTER (LOW/HIG)	BsynQL vs. BsynQR (MEAN %)	Phase Profile
C1	L	on	staccato	as written	low	27.0 > 10.8*	QL
	R	off	legato	-	high	22.9 > 19.6	QA
C2	L	off	legato	collapsed	low	24.4 > 15.0*	QL
	R	off	legato	-	high	21.0 > 20.1	QA
C3	L	off	legato	as written	low	25.6 > 13.9*	QL
	R	off	staccato	-	high	22.4 > 21.4	QA
C4	L	on	legato	as written	high	15.2 < 21.7*	QR
	R	off	staccato	-	low	19.2 < 21.3	QA

Note: * = p < .05

Exploratory Analysis of Between-Subjects Variables

A series of two-way univariate repeated measures ANOVAs was conducted to test the influence of a number of between-subjects variables on all performance measures as well as possible interactions with the previously tested within-subjects variables (excluding those testable only through paired-samples t tests). Given the irregular temporal structure of the two layers and the added difficulty associated with focal attending, it was hypothesized that participants with more formal musical training (as represented by degree, i.e., no degree/bachelor's vs. master's/PhD) would exhibit lower tapping variability and higher synchronization percentages. The general characteristics of the source materials, which presented superposed contrasting rhythmic strands in two different registers of the piano, also prompted the exploration of the influence of participants' main instrument (piano/percussion, low melodic, high melodic) and self-reported piano skill level (none/beginner, intermediate, advanced/expert). The effect of gender was investigated from a purely exploratory perspective. Results will be reported separately for each between-subjects variable across all experimental blocks. Because of the sampling method (convenient sample) and the increased probability of error associated with repeated analyses with the same variables, these results should not be taken as conclusive. However, they are suggestive of the many factors that need to be taken into consideration when studying listeners' rhythmic percepts with performance tasks (such as tapping studies) using musical stimuli.

Mensural Determinacy and the Influence of Musical Parameters

Mensural determinacy was assessed by comparing participants' synchronization performance in each of the two phases of an interleaved quarter-note beat, as represented by each layer (L and R), across segregated and combined layers conditions. Table 14 presents a summary of the results of the phase discrimination analyses for each layer and condition in the four experimental blocks. Perceived mensural determinacy is understood as being represented by the phase profile associated with participants' spontaneous tapping, with significant differences in synchronization between the two phases of the quarter-note beat resulting in phase dominance (QL or QR = higher mensural determinacy), and lack of significant differences resulting in phase ambiguity (QA = lower mensural determinacy). The influence of some of the musical parameters on the mensural determinacy of each layer can be observed in the changes in phase profile from one set of conditions to another. Overall, the left-hand layer was generally perceived as being more mensurally determinate than the right-hand layer, with intensity accents and global pitch structure having a differential effect depending on layer and texture condition. As shown in Table 14a, when presented alone, the left-hand layer was perceived as being mensurally determinate regardless of pitch condition (off/on). In contrast, when presented alone, the right-hand layer was perceived as being mensurally determinate only when the quarter-note beats were accented and pitch was off (see Table 14b); when pitch was on, the right-hand layer was perceived as being mensurally ambiguous, pointing to an interaction of global temporal structure with global pitch structure that resulted in higher synchronization to the upbeat, an effect that remains to be systematically investigated. In the first experimental block with segregated layers and pitch off, higher degree of mensural determinacy for the left-hand layer also manifested in a presentation order effect, with participants who heard and tapped to the left-hand layer first achieving higher synchronization performance than participants who heard and tapped to the right-hand layer first. This finding suggests that memory could play a role in the interpretation of metric structure in complex rhythmic sequences.

The superposition of the two layers in two different pitch conditions provided further insight into the influence of global temporal structure and streaming (both mensural and textural) on beat percepts in stratified textures. When the two layers were combined in the pitch-off condition, the textural integration of the two segregated layers appeared to promote R-phase dominance, with the accented right-hand phase of the quarter-note beat being perceived as mensurally determinate regardless of L-accent condition (see Table 14b). This result is not entirely surprising given the generally faster pace of events in the right-hand layer; it also points to a regularizing effect of the more mensurally determinate left-hand layer on the fluctuating series of events presented by the right-hand layer, not unlike that of the left-hand part in the ragtime pieces used by Snyder and Krumhansl (2001). On the other hand, when layer-specific attending was made possible by global pitch structure, attending to the left-hand layer led participants to perceive that layer as mensurally determinate, except when the right-hand layer was presented in the low rather than high register. Surprisingly, when participants attended to the left-hand layer in the high register, their tapping resulted in R-phase dominance, providing further evidence of the role of the low register in beat induction (Hove et al., 2014).

In this experiment, mensural determinacy and the effect of various musical parameters, including accent and pitch, were only assessed globally. A more specific interpretation of the different structural factors leading to mensural determinacy in these two rhythmic sequences would require a closer analysis of the source materials, taking into account participants' synchronization percentages in the different experimental conditions. Nonetheless, the contrasting structure of the two layers affords some preliminary observations, especially in terms of the influence of first- and second-order periodicity on beat induction. The lower degree of mensural determinacy suggested by the ambiguous phase profile of the right-hand layer, which is characterized by frequent changes in the rate of surface (first-order) events, is consistent with previous findings that pulse finding appears to depend mostly on first-order relationships (e.g., Demany & Semal, 2002). The higher level of mensural determinacy of the left-hand layer and the observed differential effect of accents also provides further evidence of the influence of accent on the perception of second-order periodicity (Jones & Pfordresher, 1997); this is consistent with the finding that an unmetered series of acoustic events can be perceived as more regular if streaming (via loudness or pitch distance) gives rise to second-order periodicity (Carson, 2007).

Tapping Profiles and the Role of Style-Specific Expertise

Figures 4 and 5 present two contrasting examples of participants' tapping profiles for the four trials featuring segregated layers with pitch off (block 1), which were found to give rise to the highest mean tapping synchronization percentage across participants. These tapping profiles provide useful illustrations of the influence of accent and streaming on tapping behavior as well as of the challenges associated with an investigation of spontaneous beat percepts in complex rhythmic sequences.

Fig. 4. Participant 30's tapping profile for the four trials in block 1 (S1 = L-off; S2 = L-on; S4 = R-off; S5 = R-on). Beat synchronization percentages (Bsyn) for the beat and measure pulse levels along with the mean inter-tap intervals (ITI) for each trial are given below the graph (QL/QR = quarter note; DH = dotted-half note; W = whole note). The left-hand tapping contours (S1 and S2) are representative of mensurally determinate tapping behavior, with a change of tapping period associated with the addition of accents at the whole-note pulse level (S2); the right-hand tapping contours (S4 and S5) are more variable, with temporary beat percepts that are only partially consonant with the implied beat structure.

The tapping profile of Participant 30 shows a clear contrast in tapping behavior depending on layer and accent condition. This participant's tapping to the more mensurally determinate left-hand layer resulted in relatively flat contours (S1 and S2), while the tapping to the unaccented right-hand layer (S4) was characterized by fluctuations in tapping period that closely match the written-out rubato, but still exhibited some mensural determinacy (e.g., inter-taps 42-55, M = 542 msec, SD = 52.36), with a tapping period that is partially consonant with the dotted-half note pulse (triplet quarter = 556 msec). The tapping contour of the accented right-hand layer (S5) also suggests some influence of the underlying short-long pattern created by the irregular subdivision of the durational pattern of a quarter note followed by a half note at the beginning of this excerpt (e.g., inter-taps 1-4, 5-8, and 13-15; see Appendix A, m. 68, beat 4, to m. 71). On the other hand, while the tapping period of the unaccented left-hand layer (M = 847 msec) is closely-related to the notated quarter-note beat (833 msec), that of the accented left-hand layer is more attuned to the whole-note "downbeat" (M = 3.336 as compared with 3.333 sec), resulting in a deceptively low beat synchronization percentage (18% of quarter-note beats synchronized to as compared to 74% for the whole-note pulse level).

Fig. 5. Participant 7's tapping profile for the four trials in block 1 (S1 = L-off; S2 = L-on; S4 = R-off; S5 = R-on). Beat synchronization percentages (Bsyn) for the beat and measure pulse levels along with the mean inter-tap intervals (ITI) for each trial are given below the graph (QL/QR = quarter note; DH = dotted-half note; W = whole note). Although this participant's mean tapping period is fairly homogeneous across layer and accent conditions, the left-hand layer is perceived as more mensurally determinate than the right-hand layer, especially when the latter is unaccented.

In contrast, the tapping contours of Participant 7 (see Figure 5) are very similar across both layer and accent conditions, as attested to by the narrow range of their associated tapping periods (M = 818–844 msec). In fact, this participant's tapping behavior was remarkable both in its consistency and accuracy, with an overall coefficient of variability of merely 0.069 and beat synchronization percentages similar to those found in previous tapping studies using actual music as source materials (e.g., Snyder & Krumhansl, 2001). This tapping profile is also representative of the overall finding that the left-hand layer (S1 and S2) was perceived as more mensurally determinate than the right-hand layer, the latter exhibiting mensural determinacy only when it was accented (S5) or combined with the left-hand layer (not represented in this figure). The beat synchronization percentages displayed in Figure 5 also suggest that when tapping to the unaccented left-hand layer, this participant's beat percepts were more attuned to the quarter-note beat (BsynQL = 91%) than the whole-note pulse level (BsynW = 74%).

The exceptional tapping performance of Participant 7 provides an opportunity for a preliminary exploration of the communication process at play between a mechanical performance of the composer's score and a listener whose tapping behavior is representative of style-specific expertise.

Fig. 6. Mapping of Participant 7's taps to L-off (pitch off), mm. 68–71 (red = false alarm; green = hit; square = surface event synchronized to; arrow = tap without surface event). The three false alarms at the beginning of the trial appear to be induced by displaced quarter-note beats.

Figures 6 and 7 offer a particularly effective representation of this communicative process and point to perceptual mechanisms that would require further investigation. In these figures, the first 14 taps of Participant 7 to each of two versions of the left-hand layer are matched to the notated score, allowing for closer observation of the interaction between tapping behavior and rhythmic structure. Taps synchronized with events are shown in boxes while taps that are not synchronized with a surface event are marked with arrows. In the first trial, which presented the unaccented left-hand layer with pitch off, the only "false alarms" (red boxes) were induced by a displaced periodicity equivalent to the quarter-note beat at the very beginning of the excerpt. Synchronization to these surface events led this participant to anticipate the whole-note pulse 56 (m. 69, red arrow), an error that was quickly corrected, as shown by successful synchronization to the following three implied quarter-note beats. Synchronization to the quarter-note beat is once more led astray, as shown by a second anticipated whole-note downbeat (pulse 57, m. 70, red arrow), possibly due to the influence of a series of offbeat events that exhibit a consonant but displaced period equivalent to the eighth-note pulse level, thus providing a suggestive example of the influence of first-order periodicity on beat tracking. Nevertheless, successful synchronization to sounded and implied quarter-note beats is once again quickly re-established and sustained for the remaining duration of this trial.

On the second trial with accented whole-note downbeats (see Figure 7), this participant was able to avoid synchronization with the displaced series of quarter-note beats, and synchronized with almost all of the quarter-note beats in this passage, even in the absence of corresponding sound events (the lower beat synchronization percentage for S2 shown in Figure 5 results from this participants' decision to stop tapping before the end of the trial, where the left-hand layer is particularly sparse; see Appendix B, mm. 85-88). This remarkable performance is suggestive of the role of tempo memory and intensity accents on beat tracking in irregular rhythmic sequences. Furthermore, within a conceptual framework that understands pulse finding and beat tracking as the product of a communication process between the composer and a listener, the displaced series of quarter-note beats in the very beginning (mm. 68-69), the presentation of a regular series of quarter-note beats a few measures later (mm. 70-71), and the presence of accented downbeats equivalent to a consonant whole-note pulse level suggest that in this passage, the composer may have been writing for listeners who have the necessary style-specific expertise to recognize beat clues, and would derive enjoyment from a playful manipulation of their temporal expectations.

Fig. 7. Mapping of Participant 7's taps to L-on (pitch off), mm. 68–71 (red = false alarm; green = hit; square = surface event synchronized to; arrow = tap without surface event). In the second trial with accents on, this participant successfully avoided synchronizing with the initial series of displaced quarter-note beats.