Tapping to Bach: Resonance-Based Modeling of Pulse | CiteSeerX

Music Perception Fall 2003, Vol. 21, No. 1, 43–80

© 2003 BY THE REGENTS OF THE UNIVERSITY OF CALIFORNIA ALL RIGHTS RESERVED.

Tapping to Bach: Resonance-Based Modeling of Pulse PETRI TOIVIAINEN

University of Jyväskylä JOEL S. SNYDER

Cornell University We studied the process of pulse finding by using mechanical performances of composed music in which it is fairly challenging to find a pulse. Human participants synchronized to short musical excerpts that varied in the amount of musical information present, the amount of syncopation, and the metrical position of the first note. We quantified the period and phase of tapping, the amount of nonsynchronized tapping, the time to start tapping, the variability of the intertap interval, and the deviation of taps from the intended musical events. The amount of musical information and the amount of syncopation were predictors of pulse-finding ability (metrical appropriateness of tapping, time to start tapping, variability of tapping, and deviations from the beat), and the metrical position of the first note of excerpts biased participants to tap with the corresponding phase. In addition, the degree to which participants tapped in consensus correlated positively with pulse-finding ability. We modeled participants’ behavior by using a resonance equation to calculate the strength of all metrically appropriate periodicities. The summed activation strength across the periodicities correlated with pulse-finding ability. Our results demonstrate that musical pulse finding is a useful behavioral paradigm for modeling the influences of stimulus features on complex sensorimotor synchronization. Received August 29, 2002, accepted July 3, 2003

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synchronization is a ubiquitous phenomenon that occurs in many different organisms and in many different physiological systems (Glass, 2001; Pikovsky, Rosenblum, & Kurths, 2001; Strogatz & Stewart, 1993). Sensorimotor synchronization, the synchronization of the motor system with an external periodic stimulus, is a well-studied example that IOLOGICAL

Address correspondence to Petri Toiviainen, University of Jyväskylä, Department of Music, P. O. Box 35(M), FIN-40014 University of Jyväskylä, Finland. (e-mail: [email protected]) ISSN: 0730-7829. Send requests for permission to reprint to Rights and Permissions, University of California Press, 2000 Center St., Ste. 303, Berkeley, CA 94704-1223. 43

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occurs in fireflies (Buck & Buck, 1968), katydids (Sismondo, 1990), and humans (Néda, Ravasz, Brechet, Vicsek, & Barabási, 2000; Mates, Müller, Radil, & Pöppel, 1994). On the time scale of around 300–1200 ms, sensorimotor synchronization allows people to entrain motor behavior with an external periodic source (Engström, Kelso, & Holroyd, 1996; Mates et al., 1994; Peters, 1989). This entrainment is especially important in forms of human behavior such as marching in formation, dancing, and ensemble music making. In addition, interpreting the temporal structure of music from the listener’s vantage point may require processes similar to those underlying sensorimotor synchronization. Although recent studies have described human sensorimotor synchronization using uniform isochronous stimuli such as a series of tones from a metronome (Engström et al., 1996; Madison, 2001; Mates et al., 1994; Repp, 2000, 2001; Semjen, Schulze, & Vorberg, 2000; Semjen, Vorberg, & Schulze, 1998; Thaut, Miller, & Schauer, 1998), less is known about human sensorimotor synchronization in a more ecologically valid context. The current study seeks to understand the cognitive and computational processes underlying musical pulse finding, an ecologically valid form of sensorimotor synchronization that requires complex periodicity detection. Pulse finding (or beat finding) in music refers to the subjective sense of periodicity that often results from music, enabling listeners, dancers, and musicians to synchronize motor output with music and with each other. Pulse finding is a two-stage process consisting of the perceptual process of beat induction by which the subject determines the appropriate period and phase of tapping, as well as the sensorimotor process of synchronization to the perceived beat. Music theorists also believe that this behavior is the basis for more abstract percepts such as meter (Cooper & Meyer, 1960; Lerdahl & Jackendoff, 1983), the sense of regularly alternating accents that helps instill a particular dance with a basic movement character. For example, a polka has an accent on every other pulse, whereas a waltz has an accent on every third pulse. A complementary way to describe meter is in terms of multiple pulse trains that are simultaneously active in the mind of the listener (Lerdahl & Jackendoff, 1983). For example, if two simultaneous pulse trains are in phase with each other but have periods in a 2:1 relation, every other fast pulse will coincide with a slow pulse. Alternatively if the pulse trains are in phase but have 3:1 period relations, every third fast pulse will coincide with a slow pulse. These points of coincidence correspond to relatively strong accents in the meter (Figure 1). An important difference between musical pulse finding and simpler forms of sensorimotor synchronization is that pulse finding requires a more complex form of periodicity detection. When synchronizing with a metronome (i.e., equally spaced uniform tones), it is necessary only to compute the approximate period between successive sounds and the relative phase with

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Tapping to Bach

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Fig. 1. Two different meters, represented as the superposition of phase-locked mental pulse trains. The first meter represented shows the superposition of two pulse trains in a 2:1 period relation, resulting in alternating strong (S) and weak (W) accents on every other temporal position (above). Similarly, the second meter represented shows the superposition of two pulse trains in a 3:1 period relation, resulting in alternating S and W accents on every third temporal position (below).

which to tap. Music, on the other hand, rarely contains equally spaced sounds, instead usually containing sounds separated by time intervals that are approximately related to each other by simple integer ratios such as 2:1 and 3:1 (Fraisse, 1982). Therefore, to synchronize with music, a listener must determine which sound events occur on the primary pulse and which do not. The fact that people are fairly consistent with each other in choosing the tapping period and phase suggests that there are important perceptual principles that guide pulse-finding behavior (Drake, Penel, & Bigand, 2000; Snyder & Krumhansl, 2001). In addition to the problem of processing patterns of different-sized time intervals is the problem that music is typically created by biological systems that have systematic and stochastic sources of timing variability even in sequences meant to be isochronous. Despite the seeming complexity of pulse finding, people are able to synchronize with music extremely rapidly (Snyder & Krumhansl, 2001) and flexibly (Drake, Penel et al., 2000; Repp, 1999a, 1999b), a behavior that may be extremely important for some types of improvisatory music such as jazz.

Behavioral Studies of Musical Pulse and Meter Despite the development of various computational models of musical pulse finding in recent years (Desain & Honing, 1999; Eck, 2000; Gasser,

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Eck, & Port, 1999; Large, 2000; Large & Kolen, 1994; Large & Palmer, 2002; Parncutt, 1994; Scheirer, 1998; Todd, 1994; Todd, O’Boyle, & Lee, 1999; Toiviainen, 1998), important questions remain about human pulsefinding abilities. In particular, little is currently known about the musical features that affect pulse finding. The available studies that used musical and nonmusical stimuli suggest that pulse finding may be characterized by a preferred pulse period near 500 ms (Fraisse, 1982; Parncutt, 1994; van Noorden & Moelants, 1999), impressive sensitivity to changes in the period and phase of the stimulus (Repp, 1999a, 1999b, 2000, 2001; Semjen et al., 1998, 2000; Thaut et al., 1998), and a slight motor anticipation of the beat (Aschersleben & Prinz, 1995; Mates, Radil, & Pöppel, 1992; Vos, Mates, & van Kruysbergen, 1995; Wohlschläger & Koch, 2000). The main behavioral method for studying pulse finding requires subjects to tap the perceived pulse of musical excerpts (Drake, Jones, & Baruch, 2000; Drake, Penel, et al., 2000; Parncutt, 1994; Snyder & Krumhansl, 2001; van Noorden & Moelants, 1999). An advantage of using a tapping paradigm instead of, for instance, a rating task to study pulse finding is that one can study multiple aspects of the behavior by using a single task. These aspects include period and phase of tapping, time to find a pulse, variability of the intertap interval, and deviations from the beat. Second, pulse finding measures real-time sensorimotor processing without the potential for poststimulus attention or memory influences. Third, pulse finding is an ecologically valid task because moving in time with music is prevalent in many types of musical situations across cultures. A disadvantage of this method is that it can be difficult to disentangle perceptual and motor processes, a problem shared with other tapping paradigms (Wing & Kristofferson, 1973). Pulse finding is similar to synchronization tapping (e.g., Semjen et al., 1998), in which subjects attempt to tap in exact synchrony with an isochronous stimulus. In contrast to synchronization tapping, however, there is no objectively correct period and phase of tapping in musical pulse finding. Instead, there can be several combinations of tapping period and phase that are musically appropriate according to the meter. Thus, a central issue is what cognitive principles govern how people find and follow a pulse in music and what computational processes underlie these principles. A recent study of pulse finding using piano ragtime music (Snyder & Krumhansl, 2001) attempted to determine the types of information subjects use in this behavior. Removing melodic and harmonic information by equalizing the pitch of all notes in the music generally had little effect on participants’ ability to find and follow a pulse. However, removing the temporally regular part normally played by the left hand of the pianist had detrimental effects on many aspects of pulse finding. This probably occurred because participants had to rely on the highly syncopated right-

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hand part that frequently contained tones on weak metrical positions. We wish to further explore the importance of syncopation and other musical features, in particular pitch-based ones, by using a different type of music, baroque counterpoint, to be described below. A second behavioral issue we will explore is the importance of metrical structure in musical pulse finding. As described above, meter is the sense of regularly alternating strong and weak points in time that define a sense of complex periodicity in music. During the process of pulse finding, a subject must not only determine the appropriate tapping period but also the appropriate tapping phase. A likely strategy is for subjects to tap at perceived strong metrical positions at which tones are likely to occur in Western music (Palmer & Krumhansl, 1990). Such evidence of metrical tone distributions supports the idea that strong metrical positions correspond to points of high temporal expectation (Desain, 1992; Jones, 1976; Jones & Boltz, 1989; Large & Jones, 1999; McAuley, 1995). Tapping on strong metrical positions would therefore maximize the number of tones occurring in time with taps, providing feedback for maintaining synchrony. Thus, metrical organization may play an important role in determining the phase with which subjects tap once a pulse period is established. It must be noted, however, that intentions and instructions may also play a role. For instance, a participant may intend to tap either on the beat or in syncopation with it. It is unknown how well subjects can detect metrical structure during real-time pulse finding. In other words, when listening to the beginning of a piece of music, it is unclear how well subjects can begin tapping at the first beat of the notated measure. This is particularly important because not all music begins on the first beat of the measure. Studies of iterative tone patterns suggest that subjects organize serial patterns using the first presented tone as a reference point (Garner & Gottwald, 1968; Preusser, Garner, & Gottwald, 1970). In a repeating tone pattern HHHLHLLL . . . where H and L correspond to high and low tones, respectively, subjects can subjectively hear the pattern organized as HHHLHLLL or as LHLLLHHH. The starting point of the first iteration plays an important role in biasing the probability of hearing one or the other organization. We attempted to extend this finding to short musical patterns that do not repeat iteratively, but instead contained subjective periodicities (i.e., pulse and meter). If the starting point within the measure has a strong influence on how subjects metrically organize music and the metrical structure is sufficiently ambiguous, then they should exhibit a bias to tap at the metrical position that corresponds to the starting point of the stimulus. A final behavioral issue we wished to address was how to quantify pulse finding in individual subjects and in samples of subjects. We applied dependent measures similar to those used in a previous study that quantified the period and phase of tapping with respect to the metrical structure, the time

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to find a pulse, the variability of the tapping period, and the deviation of taps from the intended beat (Snyder & Krumhansl, 2001). In addition, we attempted to quantify the degree to which subjects exhibited similar period and phase of tapping within our sample. We expected that this index of behavior within our sample might help predict difficulty of pulse finding on other dependent variables.

Computational Approaches to Pulse Several classes of models exist for sensorimotor synchronization with simple and complex signals. A complete model of pulse finding must account for multiple aspects of behavior. These include the initial extraction of one or more periodicities from the musical signal, the determination of which periodicity is most appropriate for tapping, the generation of motor outputs that correspond to the period and phase of the most appropriate periodicity, and the continual adaptation of period and phase of motor output to compensate for timing variability in the stimulus and in the synchronizing mechanism. Determining which model is best at this complex task depends on how accurately the model predicts human behavior and the plausibility of the proposed mechanisms with respect to neurobiology and computation. One source of difficulty in modeling musical pulse perception is the rhythmic complexity of music. This refers to two phenomena generally found in the temporal structure of music. First, the periodic components of music that evoke a pulse percept are not strictly periodic: there are always extraneous and missing events. Therefore, the temporal structure of music is usually periodic only in the statistical sense. Second, music is never performed with accurate timing. In other words, there are temporal fluctuations present in every musical performance. For instance, musicians use ritardandi and accelerandi to express the composer’s musical ideas. Furthermore, even with a constant tempo, the timing of performers is inaccurate with respect to the ideal durations of the tones. Models of musical pulse finding have been based on various formalisms. These include rule-based systems (Desain & Honing, 1999; Longuet-Higgins & Lee, 1982; Steedman, 1977), statistical approaches (Palmer & Krumhansl, 1990; Brown, 1993), optimization approaches (Parncutt, 1994; Povel & Essens, 1985), control theory (Dannenberg, 1984; Dannenberg & MontReynaud, 1987), connectionist models (Scarborough, Miller, & Jones, 1992), information theory (Tanguiane, 1993), linear signal processing (Todd et al., 1999; Scheirer, 1998), Bayesian estimation (Cemgil, Kappen, Desain & Honing, 2000), multiagent models (Dixon, 2000, 2001; Dixon & Cambouropoulos, 2000; Goto & Murayoka, 1995, 1999), and oscillators (Eck, 2000; Gasser et al., 1999; Large & Kolen, 1994; McAuley, 1995; Toiviainen, 1998, 2001).

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With respect to the type of input, these models can be separated into those that take as input musical score data with simple integer ratios between note onsets, and those that accept performance data containing expressive and random temporal variations. The first category contains the models of Steedman (1977), Longuet-Higgins and Lee (1982), Palmer and Krumhansl (1990), Brown (1993), Povel and Essens (1985), and Parncutt (1994); the remainder of the models listed above perform beat tracking, that is, are intended to synchronize to performed music containing both random and intentional temporal fluctuations, and often after having been given the proper initial beat period and phase. Because of its rhythmic complexity, a musical piece usually evokes a number of different pulse sensations, each of which has a different perceptual salience. For a given piece of music, the most salient pulse sensation can vary between listeners. According to experimental literature (Fraisse, 1982; Parncutt, 1994; van Noorden & Moelants, 1999), there exists, however, a region in which the preferred pulse periods tend to accumulate: the range of most salient pulse sensations lies between 67 and 150 events per minute, corresponding to periods of 400–900 ms, the greatest salience being in the vicinity of 500 ms. This period corresponds roughly to the characteristic oscillation period of human limbs. The phenomenon of preferred pulse periods is taken into account in the models of Todd et al. (1999), Parncutt (1994), and Toiviainen (2001). Models of pulse finding can be separated according to whether they use audio input or higher level symbolic representations, such as musical instrument digital interface (MIDI), which contains note onset times, pitches, and amplitudes of notes. Models based on audio input usually have a preprocessing stage that extracts note onset times from the audio information. The extraction of other features of notes, such as pitch and amplitude, cannot yet be performed reliably in the general case. Therefore, a model that uses a symbolic MIDI input allows us to investigate aspects of pulse finding that may not yet be tackled with audio-based models. These aspects include, for instance, the effect of pitch and amplitude information on pulse perception. Although some evidence indicates that pitch information present in music may affect the perception of pulse (see, e.g., Dawe, Platt, & Racine, 1993, 1994, 1995; Tekman, 1997, 1998; Thomassen, 1982), most models of pulse finding developed to date rely on purely temporal information only, that is on note onset times and durations. One notable exception is the MIDI-based multiple agent model by Dixon and Cambouropoulos (2000). In this model, the salience of notes is determined by their duration, pitch height, and amplitude. Experiments with performed classical piano sonatas showed that this model performed significantly better in pulse finding than a similar model that did not use pitch and amplitude information. The model of Dixon and Cambouropoulos (2000) is noncausal and is therefore not aimed at accounting for the temporal evolution of a pulse percept.

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Musical Characteristics of BWV 805 Before introducing the behavioral and computational experiments, we describe the music from which we created our stimuli, the last of four organ duettos for solo performer by J. S. Bach (1685–1750), BWV 805 (Williams, 1980). The set of duettos, BWV 802–805, was published as part of the Clavier-Übung (keyboard practice), Part III. The translation of the introduction to this published set reads, “For music lovers and especially connoisseurs of such work, to refresh their spirits, composed by Johann Sebastian Bach, Royal Polish and Electoral Saxon Court Composer, Capellmeister, and Directore Chori Musici in Leipzig” (David, Mendel, & Wolff, 1998). BWV 805 lasts about 3 min played at 75 half-note beats per minute (i.e., 800 ms per half note). It is in 2/2 meter and in the key of A minor (Am), with 108 total measures, mostly grouped in four-measure phrases. The music is modestly ornamented and highly imitative, with nearly every phrase of the piece that appears in one voice later appearing in the other. Furthermore, much of the piece is invertible between the two parts. In other words, at most points in the music, the left-hand (low in pitch) and right-hand (high in pitch) parts could be switched in register without harmonic difficulty. Another feature relating the two parts is that when note density is high in one part, it tends to be low in the other. For the purposes of this study, BWV 805 proved convenient for the following reasons. At a general level, the two-voice texture and the highly structured form made analysis and manipulation of the music relatively simple. Rhythmically, the music is diverse with many different durational patterns occurring in both voices, with varying amounts of note density and syncopation. The rhythmic diversity of the composition was a convenient property because it allowed the study of the effect of temporal structure on pulse finding. One disadvantage of this music is that it is quite difficult for nonmusically trained subjects to find and sustain a consistent pulse, as indicated by pilot testing. Thus, we decided to take advantage of the conveniences of BWV 805 while using musically experienced individuals as subjects.

Method PARTICIPANTS

There were 16 participants (5 male, 11 female; mean age = 19.6 years). Each participant had at least 8 years of musical experience playing an instrument or singing (mean = 12.1 years), with some experience in the preceding 2 years. Fifteen of the 16 participants were right-handed. No participant reported a hearing problem or a movement disorder. On scales

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from one to seven, participants reported an average familiarity with the music of J. S. Bach of 4.5 and an average familiarity with BWV 805 of 1.7. MATERIALS AND STIMULI

The experiment was run using a MAX 3.5.9-9 object-oriented interface on a 350-MHz Macintosh G4 computer. The interface played all stimuli by sending MIDI information to Unity DS-1 1.2 software via Open Music System (OMS). Unity converted the MIDI information to audio and sent the audio information out of the computer to a Yamaha 1204 MC Series mixing console, to which AKG K141 headphones were attached. We downloaded a MIDI version of BWV 805, Bach’s fourth organ duetto, from the Classical MIDI Archives (http://www.prs.net/bach.html#800). This website lists the creator of the MIDI file we used as D. Jao. Before creating our stimuli, we adapted the original MIDI file using Digital Performer 2.61-MTS in the following ways. First, we set the tempo to a half-note interbeat interval (IBI) of 800 ms, or 75 beats per minute, with all notes quantized to the nearest eighth note. Second, we made the MIDI file metronomic such that all IBIs were equal throughout the piece. This was done because following tempo changes and other timing deviations was not a topic of this study, and dealing with metronomic stimuli facilitated the analysis. Third, we set all MIDI note-on velocities (which determine acoustic amplitude) to 110 and all note-off velocities to 64. Last, we played the MIDI file with the Church Organ timbre from the Unity DS-1 library. The equalization of note-on velocities was not considered a limitation, because church organs are not velocity-sensitive and thus produce the same acoustic amplitude regardless of the velocity with which a key is pressed. We used the modified MIDI file of BWV 805 to create stimuli for the experiment. We took eight short excerpts, each consisting of eight measures. The first four excerpts began and ended on the first eighth-note beat of the measure, whereas the last four began and ended on the third eighth-note beat of the measure. The eight excerpts were from the following sections (denoted by measure number and eight note beat): (1) 18:1–26:1, (2) 54:1– 62:1, (3) 73:1–81:1, (4) 99:1–107:1, (5) 37:3–45:3, (6) 45:3–53:3, (7) 63:3–71:3, and (8) 87:3–95:3. At the beginning of each excerpt, there were notes in both the left-hand and right-hand parts so as not to bias participants toward listening preferentially to one voice or the other. The excerpts (Figure 2) formed four pairs (Excerpts 1 & 2, 3 & 4, 5 & 6, and 7 & 8) such that the left-hand part of one excerpt in the pair contained a pattern similar to the right-hand part of the other excerpt in the pair, and vice versa. To illustrate, the left-hand part of Excerpt 1 contained a pattern similar to the right-hand part of Excerpt 2, and the right-hand part of Excerpt 1 contained a pattern similar to the left-hand part of Excerpt 2. For each of the eight excerpts, we created three versions. One was an exact copy of the excerpt containing both voices, the whole version. The other two are the left-hand part only and the right-hand part only. The whole, left-hand, and right-hand versions of the eight excerpts yielded 24 stimuli. To avoid order effects, we selected four random orders of the 24 stimuli. To generate four additional stimulus orders, we took the reverse of each of the four original orders. Thus, we generated eight total blocks of the 24 stimuli, four of which are the reverse of the other four. We generated two Latin squares for each set of four stimulus orders (i.e., original and reverse) such that we presented each stimulus order an equal number of times in each serial position, across participants. PROCEDURE

We tested each participant in a single experimental session that lasted approximately 1 hour. For the tapping trials, participants read instructions directing them to “tap the beat of the music with the index finger of your dominant hand on the keyboard the way you normally tap your foot while listening to music” and to “find the most comfortable beat but do not begin tapping until you have found the beat mentally.” As practice, participants tapped twice each to whole, left-hand, and right-hand versions of an excerpt from BWV 803, a

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duetto from the same set of pieces as BWV 805. Next, each participant tapped a pulse to either the four original orders or the four reverse orders of the stimuli. Therefore, each participant tapped to four blocks of the stimuli, each block containing all 24 stimuli, for a total of 96 tapping trials. Before each new block of stimuli, participants tapped to the two

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Fig. 2. Excerpts used in the experiment. (a) Excerpts 1–4 beginning on the first eighth-note metrical position. (b) Excerpts 5–8 beginning on the third eighth-note metrical position. Square brackets indicate the start and the end of each excerpt.

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final stimuli in that block. These two preblock trials were for warmup, and we did not analyze the data from them. Between stimuli, the computer played 17 monotonic piano tones (all middle C) for 6–10 s with randomly selected interonset intervals (100–1000 ms) to degrade the memory representation for the interbeat interval of the musical stimuli. There was a variable time interval from the end of these piano tones to the beginning of the next stimulus, ranging from 1600 to 3200 ms. Between blocks, participants rested for a couple of minutes while the experi-

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Fig. 2. Continued

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menter saved the data from the previous block of stimuli and started the next block. Before debriefing, participants filled out a questionnaire concerning their music and dance experience, their familiarity with the stimulus materials, and general demographic information.

Results and Discussion DEFINITION OF PULSE-FINDING MEASURES

To provide a multifaceted quantitative picture of tapping behavior, we derived multiple performance measures that captured the period and phase of tapping, the time to find a pulse, variability of tapping, and tapping distance from the beat. For each performance measure, we determined a value for the left-hand, right-hand, and whole versions of excerpts for each participant. We used only tap times that occurred after the first note of the music and no later than 99 ms after the final note of the music. We first determined the tapping mode, defined by the period and phase of tapping, as follows. We calculated the period of the taps by first quantizing each tap to the nearest eighth-note metrical position, and then determining the number of eighth notes (eighth note = 200 ms) between the taps. We adopted a criterion for determining the period based on the following considerations. There are well-known synchronization constraints at tempi outside the range of 300–1200 ms intertap interval (Fraisse, 1982). Within this range, only periods of two and four eighth notes correspond to metrical beats in 2/2 meter. We also analyzed tapping with a period of eight eighth notes even though it falls outside of this range, because it corresponds to the measure length and might therefore be salient to subjects. Taps with these periods will subsequently be referred to as metric. Correspondingly, we defined tapping with periods other than two, four, and eight eighth notes as ametric. In other words, according to our classification scheme, participants could tap with periods of two, four, or eight eighth notes, or ametrically. The phase corresponded to the closest eighth-note metrical position for each metric tap. Therefore, when tapping with a period of two eighth notes, there were only two possible phases within each cycle of tapping. However, with periods of four and eight eighth notes, there existed four and eight phases, respectively. For each period, we numbered the phases with respect to the first eighth-note beat of the measure. Table 1 shows one measure in 2/2 meter at the eighth-note level, illustrating each of the 15 possible modes of tapping, along with the observed proportion of tapping in each mode. For example, mode 2_1 means tapping with a period of two eighth notes and in the first phase (i.e., at the odd eighth-note metrical positions). To illustrate ametric tapping, the last three lines of Table 1 show examples in which all periods are other than two, four, or eight eighth notes.

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TABLE 1 Definition of Tapping Modes And Proportion of Tapping in Each Mode Mode

2_1 2_2 4_1 4_2 4_3 4_4 8_1 8_2 8_3 8_4 8_5 8_6 8_7 8_8 Ametric

Eighth-Note Metrical Positions* Mean Proportion of Taps

12345678 12345678 12345678 12345678 12345678 12345678 12345678 12345678 12345678 12345678 12345678 12345678 12345678 12345678 12345678 12345678 12345678

.17 .01 .45 .01 .31 .01 .00 .00 .00 .00 .00 .00 .00 .00 .03

*Underline indicates tap.

The remaining measures were all computed from the unquantized data, that is, the tap times before they were quantized to the nearest eighth note. To index the number of beats it took to find a pulse, we defined beats to start tapping (BST) as the time in milliseconds from the first note of the music to the first metric tap, divided by the halfnote duration (800 ms). Thus, BST was the number of half-note beats from the first note to the first metric tap. The first metric tap was defined as the first tap preceded by an intertap interval of either two, four, of eight eighth notes. The next performance measure indexes the variability of the intertap interval. We derived the variability measure, the coefficient of variation (CV), by dividing the standard deviation of the intertap intervals by the period of the most prevalent metric tapping mode for that participant. Finally, we calculated the deviation from the closest sounded eighth note within the participant’s current tapping mode for each metric tap. We defined the coefficient of delay (CDel) as the mean deviation of taps from the beat, divided by the mean of all metric intertap intervals. A negative value indicated anticipation, whereas a positive value indicated delay. The coefficient of deviation (CDev) was the mean absolute deviation of taps from the beat, divided by the mean of all metric intertap intervals. We normalized these performance measures by the tapping interval in order to control for the possibility that participants would tap at different rates.

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Petri Toiviainen & Joel S. Snyder PULSE-FINDING PERFORMANCE

A three-factor repeated-measure analysis of variance (ANOVA) using the univariate MANOVA syntax in SPSS 10.0.1 was used to analyze each performance measure, with start position (first eighth note and third eighth note metrical positions), stimulus version (whole, left-hand, and right-hand), and excerpt (1-8) as the three factors. Note that excerpt was nested within start position because Excerpts 1–4 began on the first eighth note of the measure, and Excerpts 5–8 began on the third eighth note of the measure. For significant main effects of the version factor, we computed post-hoc contrasts with a Bonferroni correction for multiple comparisons between the whole and left-hand versions, and between the whole and right-hand versions. This allowed us to test whether having both left-hand and righthand parts together (i.e., the whole version) facilitated performance compared to when only one was present (i.e., the left-hand and right-hand versions). Post-hoc tests were not performed for excerpt because it is extremely difficult to interpret differences between individual pairs of excerpts, especially given that there were three different versions of each excerpt. Instead variability across excerpts is largely explained by the interactions between excerpt and version, and by using a factor analysis that included stimulus features. The performance measures included the proportion of tapping in the most prevalent tapping modes (2_1, 4_1, 4_3, and ametric), BST, CV, CDel, and CDev. For each of these performance measures, we used the mean value across the four presentations of each of the 24 stimuli for each participant. Possible sphericity violations were corrected by adjusting the degrees of freedom using the Greenhouse-Geisser epsilon. The top and bottom of Figures 3–5 show the main effects for start position and version for each of the performance measures. To obtain the values in the graphs, we calculated the mean values across participants for each level of the main effects (start position and version). The numbers used to calculate the values in the graphs were the same submitted to the ANOVAs: the means across the four presentations of the stimuli for each participant. PROPORTIONS OF TAPPING

Figure 3 displays the mean proportion of tapping across participants for the four most prevalent tapping modes, showing the main effects of start position and version. The most prevalent tapping mode was 4_1 (45% of taps), tapping with a period of four eighth notes (i.e., 800 ms) and in the first phase. The next most prevalent tapping mode was 4_3 (31% of taps), followed by 2_1 (17% of taps). Three participants who primarily tapped on 2_1 accounted for most of the tapping in that mode. The other 13 participants mostly tapped with a period of four eighth notes. This corresponds well to the preferred and

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Fig. 3. Mean proportion of taps across participants (+SE) for tapping modes 2_1, 4_1, 4_3, and Ametric, shown for excerpts starting on the first and third eighth-note metrical positions (top), and for whole, left-hand (LH), and right-hand (RH) versions of the excerpts (bottom). The numbers used to calculate these values were the means across the four presentations of the stimuli for each participant.

optimal tapping range observed in previous studies (Fraisse, 1982; Peters, 1989). No other metric tapping modes were observed for more than 1% of the time. Only a small proportion of taps were ametric (3% of taps), indicating proficient overall pulse-finding performance. This distribution of tapping modes indicates that participants coordinated their period and phase of tapping to the music as predicted by the metrical structure and by tempo constraints, with 93% of taps occurring in modes 2_1, 4_1, and 4_3.

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For mode 4_1, there was a significant effect of start position, F(1, 15) = 18.90, p < .001, with more tapping on 4_1 for the stimuli beginning in the first metrical position. There were no effects of version, F(2, 30) = 1.54, e = .712, ns, or excerpt, F(6, 90) = 1.00, ε = .638, ns. Mode 4_3 showed a complementary pattern to 4_1 with significantly more tapping on 4_3 for stimuli beginning on the third position, F(1, 15) = 20.71, p < .001. There were no effects of version, F(2, 30) = 0.87, ε = .771, ns, or excerpt, F(6, 90) = 0.67, ε = .515, ns. Thus, the manipulation of start position strongly biased participants toward tapping in the corresponding phase, consistent with findings in the serial organization literature (Garner & Gottwald, 1968; Preusser et al., 1970). Mode 2_1 showed no effects of start position, F(1, 15) = 0.80, ns, version, F(2, 30) = 2.15, ε = .552, n.s., or excerpt, F(6, 90) = 1.29, ε = .348, ns. The lack of effect for start position is not surprising because both start positions (first and third eighth-note beats) fall in the 2_1 tapping mode (see Table 1). For ametric tapping, there was an effect of start position with slightly more ametric tapping for excerpts beginning in the third position, F(1, 15) = 6.46, p < .025, suggesting that there is an advantage for the more metrically appropriate starting position. There was also an effect of version, F(2, 30) = 5.35, ε = .600, p < .05, with more ametric tapping for the left-hand version than the whole version, t(15) = 2.51, p < .05, and marginally more for the right-hand version than the whole version, t(15) = 2.20, p < .10. This suggests that the two musical parts together reinforced a pulse with better tapping performance when both parts were present. Excerpt showed no effect on ametric tapping, F(6, 90) = 1.99, ε = .521, ns. However, version and excerpt showed a significant interaction, F(12, 180) = 6.26, ε = .151, p < .01. For Excerpts 1, 3, 5, and 8, the left-hand versions had the most ametric tapping, while for Excerpts 2, 4, 6, and 7, the right-hand versions had the most ametric tapping. The stimuli that caused the most ametric tapping were those with sparse and irregular rhythmic structure (see Figure 2). BEATS TO START TAPPING (BST)

As shown in Figure 4, participants exhibited extremely rapid pulse finding. Participants usually began to tap between two and three half-note beats (1600-2400 ms) after the excerpt began. For BST, there was no effect of start position, F(1, 15) = 3.20, ns. However, version showed a significant effect, F(2, 30) = 14.04, ε = .602, p < .005, with participants taking less time to find the beat on the whole versions than on the left-hand version, t(15) = 3.09, p < .05, and than on the right-hand version, t(15) = 3.99, p < .005, again suggesting a facilitation effect for the two parts being present together. Excerpt did not significantly affect BST, F(6, 90) = 0.78, ε = .667,

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59

Fig. 4. Mean half-note beats to start tapping, BST, across participants (+SE) shown for excerpts starting on the first and third eighth-note metrical positions (top), and for whole, left-hand (LH), and right-hand (RH) versions of the excerpts (bottom). One half-note beat is equal to 800 ms. The numbers used to calculate these values were the means across the four presentations of the stimuli for each participant.

ns. The interaction between version and excerpt was again significant though, F(12, 180) = 5.97, ε = .202, p < .01. The versions of excerpts that had the highest BST again generally corresponded to those with more temporally irregular structure. In particular, for Excerpts 1, 3, 5, and 8, the left-hand versions had the highest BST, whereas for Excerpts 2, 4, 6, and 7, the right-hand versions had the highest BST. TAPPING VARIABILITY, DELAY, AND DEVIATIONS

Figure 5 displays CV, CDel, and CDev for the three stimulus versions. For CV, values were around 3% of the tapping period (24 ms for an 800ms tapping period). Start position showed a significant effect with more

60


Fig. 5. Mean timing coefficients across participants (+SE), CV, CDel, and CDev, shown for excerpts starting on the first and third eighth-note metrical positions (top), and for whole, left-hand (LH), and right-hand (RH) versions of the excerpts (bottom). A coefficient of .01 is approximately equal to 8 ms. The numbers used to calculate these values were the means across the four presentations of the stimuli for each participant.

tapping variability in the stimuli beginning on the third position, F(1, 15) = 9.37, p < 01. As with ametric tapping and BST, it appears that there is an advantage for beginning in the more metrically appropriate start position. Version was also significant, F(2, 30) = 8.49, ε = .695, p < .01, with more variability in the left-hand version, t(15) = 2.84, p < .05, and the right-hand version, t(15) = 3.04, p < .05, than in the whole version. We observed no effect of excerpt, F(6, 90) = 1.40, ε = .573, n.s. As with ametric tapping and BST, the interaction between version and excerpt was significant, F(12,

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61

180) = 3.57, ε = .331, p < .025. For Excerpts 1, 2, 3, 5, and 8, the left-hand versions had the highest CV, whereas for Excerpts 4, 6, and 7, the righthand versions had the highest CV. There was an overall positive tap delay of around 2% of the tapping period (16 ms for an 800-ms tapping period) but no significant effect of start position on CDel, F(1, 15) = 0.35, ns. Version was significant, however, F(2, 30) = 3.79, ε = .974, p < .05. This significant difference was due to more delay on the whole version than on the right-hand version, t(15) = 2.64, p < .05, with no difference between the whole and left-hand versions, t(15) = 1.12, p = .28. Excerpt showed no significant effect, F(6, 90) = 1.09, ε = .571, ns. The occurrence of an overall tap delay is inconsistent with previous studies of tapping to metronomes that demonstrate anticipatory tapping of 30–70 ms (Mates et al., 1994), but is consistent with studies of tapping to music (Snyder & Krumhansl, 2001) and other patterns more complex than a metronome (Wohlschläger & Koch , 2000). Another factor besides pattern complexity that could affect tap delay is the instrument timbre, which in this case was an organ with onsets less sharp than typical laboratory sine tones. For CDev, participants were off the beat by around 3% of the tapping period (24 ms for an 800-ms tapping period), with stimuli beginning on the third eighth note having slightly higher values, F(1, 15) = 11.52, p < .005. Additionally, CDev showed a significant effect of version, F(2, 30) = 9.07, ε = .765, p < .005, with higher values on the left-hand version, t(15) = 4.12, p < .005, and marginally higher on the right-hand version, t(15) = 2.24, p < .10, than on the whole version. Excerpt was also significant for CV, F(6, 90) = 4.04, ε = .634, p < .01. Version and excerpt again interacted, F(12, 180) = 8.72, ε = .181, p < .005. For Excerpt 2, the whole version had the highest CDev; for Excerpts 1, 3, 5, and 8, the left-hand versions had the highest CDev; and for Excerpts 4, 6, and 7, the right-hand versions had the highest CDev. In summary, these results suggest that the metrical position of the first note (first vs. third eighth note) of excerpts biased participants to tap in the corresponding phase. Additionally, tapping in a phase that was metrically less strong according to the musical notation (i.e., 4_3 rather than 4_1) resulted in slightly less stable performance across measures of tapping behavior. Participants tapped more ametrically, took more time to find the beat, had higher tapping variability, and tapped farther from the beat (i.e., more ametric tapping, larger BST, higher CV and CDev, respectively). Excerpts containing both the left- and right-hand parts afforded more stable pulse-finding performance than when only one part was present. However, whether more stable performance occurred with the left-hand or the righthand versions depended on whether the particular excerpt had a more temporally regular left- or right-hand part. We return to this point later when we quantify important features of the stimuli. The use of multiple perfor-

62


mance measures of tapping behavior provided converging evidence for greater difficulty while synchronizing to stimuli with less musical information, and with more rhythmic irregularity. Two main limitations to this experiment remain, the use of only one piece of music and the use of musically experienced participants. We used BWV 805 mainly because it allowed a clean manipulation of starting position and an evaluation of the role of the left- and right-hand parts by themselves. We expect that other pieces in the same meter (2/2) and pieces in other meters would show similar effects of start position, although this would most likely depend on a number of factors such as the clarity of the downbeat. In addition, the effect of removing melodic voices or parts would likely generalize to other pieces of music, especially when the removed aspect carried strong cues to pulse (Snyder & Krumhansl, 2001). The fact that we were only able to test musically experienced participants on our stimulus set suggests that pulse finding is in part a skilled behavior that requires training. Methodologically, it suggests that when selecting stimuli for experiments with participants who are not musically experienced, the stimuli must be made relatively simple. We expect that participants who are not musically experienced would show similar effects of our manipulations given the appropriate stimulus difficulty level, although this is clearly an empirical question. We do not think that our musically experienced participants’ behavior in response to the stimulus manipulations were the result of musical training and were not particularly adaptive according to our pulse-finding measures. CONSENSUS

In addition to studying the tapping accuracy of each individual with respect to the presented stimuli, we wanted to quantify the degree of mutual tapping consensus within the group of participants. To this end, we developed a measure of consensus that is based on the tapping times of all participants for a given stimulus. The measure is based on Shannon’s entropy (Shannon, 1948), a measure used for estimating the degree of ordering and thus information content in various systems, including spike trains of neural assemblies (Rieke, Warland, Van Steveninck, & Bialek, 1997). Such an entropy-based consensus measure does not rely on any assumed metrical grid and can thus be used for measuring the degree of mutual tapping synchronicity even in the absence of any external rhythmic stimulus. The consensus measure for a given stimulus is calculated as follows. Let tk, k = 1, . . . , N denote the set of all tap times by all participants to the stimulus. We estimate the tapping probability density, p(t), by p(t) =

1 Nσ √2π

N

Σe k=1

2

2

-(t - tk) /2σ

(1)

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63

The probability density p(t) is thus a sum of Gaussian kernels placed at each tapping time. The value of the standard deviation σ of the Gaussian kernels affects the smoothness of the probability distribution: a large value of σ results in a smooth distribution and vice versa. We used the value of σ = 50 ms, which is close to the observed tapping accuracy (as measured by CV, Cdel, and Cdev), and was found to be a good compromise between smoothness and resolution of the estimated probability density. From p(t), we obtain the consensus measure S from the differential entropy according to S = 1 – H(p) = 1 + ∫ p(t) ln(p(t))dt;

(2)

the constant has been included in order to make the values positive. Stimuli for which the tapping times are clustered close to each other have tapping probability density functions with high and narrow peaks, resulting in a low entropy and thus a high consensus value. On the other hand, stimuli for which the tapping times are temporally more diffused have tapping probability density functions with less pronounced peaks, resulting in a higher entropy and thus a lower consensus value. Figure 6 depicts the tapping density functions for Stimuli 10, 12, and 13, which received the highest, the 12th highest, and the lowest consensus values, respectively. To evaluate the relationship between consensus and the other derived performance measures, we calculated the consensus values for each of the 24 stimuli. Consensus correlated positively with the proportions of 2_1, r(22) = .64, p < .005, and 4_1, r(22) = .62, p < .005, tapping modes, and negatively with the proportion of the 4_3 mode, r(22) = -.44, p < .05; ametric, r(22) = -.70, p < .001; BST, r(22) = -.62, p < .005; CV, r(22) = -.61, p < .005; and Cdev, r(22) = -.68, p < .001. This pattern shows a clear association of consensus, with more tapping in metrically appropriate modes and more stable overall performance. It thus seems that consensus, the degree to which subjects tap at the same time points within the stimuli, correlates strongly with overall pulse-finding difficulty. We examined next the effect of excerpt, version, and starting point on the consensus values. The average consensus value for the whole versions, S = 0.087, was higher than that for both left-hand (S = 0.071) and righthand (S = 0.074) versions, but the effect was not significant, F(2,21) = 2.13, n.s. The average consensus value for stimuli beginning on the first metrical position, S = 0.084, was higher than that for the stimuli beginning on the third metrical position, S = 0.071, this effect being marginally significant, F(1,22) = 4.07, p < 0.1. No effect of excerpt on consensus was found. PULSE FINDING AND MUSICAL VARIABLES

To determine the relationship between musical features and pulse-finding performance, we calculated values on four musical variables for each

Fig. 6. Estimated tapping densities p(t) for three stimuli. (a) Stimulus 10 (highest degree of consensus); (b) Stimulus 12 (12th highest degree of consensus); (c) Stimulus 13 (lowest degree of consensus).

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65

of the 24 stimuli. The choice of these variables was based on general musictheoretical knowledge of features that are important in the formation of musical pulse and meter. The first musical variable, onsets, is the number of all note onsets at the eighth-note level divided by the total number of eighth-note metrical positions in the stimulus. The second variable, direction changes, is the total number of melodic direction changes divided by the total number of eighth-note metrical positions in the stimulus in which a direction change could occur (i.e., all but the first and last positions). We assign the direction change to the second of three notes defining the event (e.g., D4 in the case of C4 D4 C4). The third variable, syncopation, is determined as the number of instances where there is no note onset on metrical position one or five (i.e., the two strongest metrical positions). The final variable, long notes, is the total number of onsets with duration greater than an eighth note that occur on eighth-note metrical positions one, three, five, and seven. Because of the large number of performance variables and musical variables, we performed a factor analysis that included all variables together (using principal component analysis and Varimax rotation). This analysis revealed three factors that accounted for 59%, 19%, and 12% of the total variance, respectively. Multiple variables loaded positively on Factor 1, including syncopation, ametric taps, BST, CV, and CDev. Loading negatively on this factor were number of onsets, number of melodic direction changes, 2_1, and consensus. This suggests that tapping performance, as measured by ametric taps, BST, CV, CDev, and consensus, was associated with the amount of syncopation in the stimuli, such that abundant syncopation led to less stable tapping performance. Furthermore, tapping on 2_1 was associated with a high number of note onsets and melodic direction changes. For Factor 2, 4_1 loaded positively whereas 4_3 loaded negatively, reflecting the trade-off between tapping in the 4_1 mode and tapping in the 4_3 mode. Lastly, CDel loaded positively on Factor 3, individually.

Model of Beat Induction MODEL DESCRIPTION

To model the probability of tapping in each mode, we developed a dynamic system for measuring the accentuation strength of each tapping mode. In the model, each tapping mode is associated with a state variable, subsequently referred to as the resonance value. The resonance value is taken to represent the accentuation strength of the respective mode and thus the probability of tapping in that particular mode. The temporal evolution of the resonance value depends on several factors of the input stimulus and

66


their relationship with the particular mode. More specifically, the resonance value of a mode and thus the probability of tapping in that particular mode gets high when (1) the stimulus contains a large number of notes whose onsets coincide with the temporal grid representing the mode; (2) these notes are perceptually salient, the degree of salience depending on both temporal and pitch-related factors, described below; and (3) the period is close to the preferred tapping period (400–600 ms). Although amplitudes of notes have been found to contribute to some extent to the accent structure of music and thus may in general affect the perception of pulse, amplitude information was ignored because organ music does not contain any significant amplitude variations. Temporal factors affecting the temporal evolution of resonance include the onset times of the tones and the interonset intervals following the onsets. At each tone onset, tapping modes that coincide with the onset start to increase their resonance. The rate of increase depends on several factors that are described below. This increase extends to the next tone onset and its amount thus depends on the duration of that particular tone. In general terms, tapping modes for which there are many tone onsets occurring have high resonance values. Furthermore, the resonance values depend on the duration of the tones so that tones of long duration give rise to higher resonance values than do tones of short duration (cf. Parncutt, 1994). Besides purely temporal factors, pitch information may have an effect on the perception of pulse. Musical analysis of the stimuli used in the experiment led us to add two pitch-related factors to the model. The first one was associated with minor second intervals in the melodies. More specifically, tones that were preceded by a tone either a minor second below or a minor second above were weighted more than other tones. From a music theoretical point of view, this seems justified because in tonal music a minor second interval often leads to a tone that is stable in the tonal hierarchy and thus may be perceived as stronger than other tones. This is formalized in Bharucha’s (1984, 1996) melodic anchoring principle. The other pitchrelated factor that was included in the model was the presence of melodic closure. In a three-tone sequence, where the second interval was in an opposite direction and had a smaller size than the first interval, the last tone was given a stronger weight than other tones. Again, from a music theoretical point of view, this seems justified because the tones that form a closure are perceived as more stable than other tones (Krumhansl, 1995; Narmour, 1990). Both the minor second factor and the closure factor were associated with weight parameters that determined the relative importance of each factor. Figure 7 displays an example of the location of pitch-based accents used by the model. The last factor included in the model was the primacy factor. This factor gave a higher weight to the modes that coincide with the first note of the

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67

Fig. 7. Pitch-based accents in the first four measures of the right-hand part of Excerpt 3. Abbreviations: closure (c), minor second (s).

stimulus. Again, the primacy factor had a weight parameter that determined its relative importance. The resonance dynamics are modeled with a discrete damped system driven by an external force. Let the tone onset times be denoted by tk, k = 1, 2, . . . , where tk tk+1. The resonance value of tapping mode i at the onset of tone k, rki, is determined by i

rki = f k-1 – rki -1 (tk – tk-1) / τ,

(3)

i

where f k-1 is the driving force of the mode at the onset of tone k – 1 and τ is the time constant; the parameter τ determines the length of the temporal integration window. With the absence of any external force, the resonance value decays approximately by the factor of 1/e ≈ 0.37 during an interval of τ. In the present model, the value τ = 3 sec was used. i The driving force f k is modeled to consist of three multiplicative factors depending on (1) the interaction between tone k and mode i, (2) the properties of tone k, and (3) the properties of mode i. The driving force is thus expressed as i

i

f k = Ak BkC

i

(4)

i

The interaction term, Ak, equals unity if the onset of tone k coincides with mode i and equals zero otherwise. Therefore at each tone onset, modes that coincide with the onset receive a positive driving force and thus increase their resonance, the amount of increase depending on the other factors. Modes that do not coincide with the onset of tone k receive no driving force, resulting in a slight decay of their resonance. The term Bk models the durational accent of tone k as well as the melodic accents caused by closure and minor second movement, as described above. Bk is expressed as Bk = dk + ηcck + ηssk,

(5)

where dk denotes the durational accent of tone k and is defined as dk = 1 – e-(t

k+1

- tk)/τd

(6)

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where τd = 0.5 s. This expression approximates the perceived durational accent as a function of duration as observed by Parncutt (1994). The term ck denotes the presence of melodic closure at tone k. It equals unity when tone k exhibits closure, and zero otherwise. Similarly, sk denotes the presence of a minor second movement, equaling unity when tone k is preceded by a tone either a minor second above or a minor second below. The terms ηc and ηs are model parameters determining the relative importance of melodic accents caused by closure and minor second movement, respectively. i The third term of Equation 4, C , depends on the period of tapping mode i and the phase of the mode with respect to the first tone of the stimulus. It is expressed as i

i

C = w(T ) (1 + ηppi) i

(7)

i

where T is the period of mode i, and w(T ) is defined as i

i

*

w(T ) = 1 – ln(T /T )2,

(8)

*

where T = 500 ms. This expression approximates the resonance curve obtained by van Noorden and Moelants (1999). According to Equation 8, i * w is maximal when T = T . Thus, tapping modes whose periods are in the vicinity of this value receive the highest resonance. The dependence of the weighting factor on the period of the tapping mode is depicted in Figure 8. The second factor of Equation 7 models the primacy effect. In this factor i p equals unity for the modes that coincide with the first tone of the stimulus, and zero for the other modes. The term ηp is a model parameter determining the strength of the primacy effect. For modes that coincide with the 1 0.9

Weight w(T)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2

0.4

0.6

0.8

Period T (secs)

1

Fig. 8. Weighting of resonance values as a function of oscillator period.

1.2

69

Tapping to Bach

first tone of the stimulus, the driving force is thus multiplied by a factor of 1 + ηp. To obtain predictions for the proportions of tapping in each mode, the resonance values for each mode are averaged across time, after which the average resonances are normalized so that their sum equals unity. The dynamic behavior of the model and thus the resonance values given for each tapping mode depends on the values of the weights ηs, ηc, and ηp. To examine the effect of these weights, we carried out simulations with ηs, ηc, and ηp having the value of either zero or unity, thus leading to eight parameter value combinations. The normalized average resonances of each mode were compared with the proportions of tapping in each mode by calculating the correlation coefficient and the root mean squared error (RMSE) across all tapping modes and all stimuli. Because participants did not use most of the tapping modes, the distribution of proportions of tapping is skewed, which may artificially inflate the correlation coefficient. Therefore, we made another comparison between the model data and the tapping data by including only the three most prevalent tapping modes, 4_1, 4_3, and 2_1. The correlation and RMSE values for each parameter value combination are shown in Table 2. As can be seen, the value of the primacy weight, ηp, has the most significant effect on all the measures: increasing the value of ηp from zero to unity increases the correlation and decreases the RMSE value. This result is in line with the result that the tapping mode was largely determined by the starting position of the stimulus. The best results in terms of all the four measures are obtained with the parameter value combination ηp = 1, ηc = 1, and ηs = 0, suggesting that, TABLE 2 Correlations (r) and Root Mean Square Errors (RMSE) Between the Tapping Data (Proportions of Tapping in Each Mode) and the Model Data (Average Resonance Values of Each Mode) Obtained with Different Weight Values ηs, ηc, and ηp ηp

ηc

ηs

1 1 1 1 0 0 0 0

1 0 1 0 1 1 0 0

0 0 1 1 0 1 0 1

r

All Modes RMSE

0.88*** 0.85*** 0.85*** 0.83*** 0.73*** 0.70*** 0.67*** 0.66***

0.086 0.095 0.091 0.097 0.113 0.118 0.123 0.124

Prevalent Modes r RMSE

0.78*** 0.72*** 0.66*** 0.62** 0.26 0.19 0.12 0.11

0.175 0.191 0.187 0.196 0.231 0.240 0.245 0.248

NOTE—The r and RMSE values are given separately for all tapping modes and three most prevalent tapping modes (2_1, 4_1, and 4_3). The rows are ordered by the values of r. df = 22; * p < .05; ** p < .01; *** p < .001

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besides the primacy effect, the presence of closure had an effect on pulse finding, whereas the presence of minor second movement did not. This parameter value combination was used in subsequent analyses. Further optimization of the parameter values was not carried out, because it is not evident which measure of fit should be used in the optimization procedure. PULSE FINDING AND RESONANCE VALUES

0.2

Proportion of Taps in 4_1

r = -.51

0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.18

0.2

0.22

0.24

0.26

Resonance 2_1

0.7

0.28

0.3

r = .93

0.6 0.5 0.4 0.3 0.2 0.1 0 0.1

0.15

0.2

0.25

Resonance 4_3

0.3

0.35

Proportion of Aperiodic Taps



Figure 9 shows scatterplots associating resonance values with the proportions of tapping in the modes 2_1, 4_1, and 4_3 as well as ametrically. As can be seen, the resonance values of modes 4_1 and 4_3 have a high positive correlation with the proportions of tapping in the respective modes. The resonance values of the mode 2_1, however, fail to correlate signifi-

0.9

r = .83

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1

0.2

0.3

0.4

Resonance 4_1

0.12

0.5

0.6

r = -.78

0.1 0.08 0.06 0.04 0.02 0 4

6

8

10

Total resonance

12

Fig. 9. Resonance values for modes 2_1, 4_1, and 4_3 plotted against the proportion of tapping in the corresponding mode, for each of the 24 stimuli. For modes 2_1, 4_1, and 4_3, stimuli starting on the first eighth-note metrical position are plotted as closed symbols whereas stimuli starting on the third eighth-note metrical position are plotted as open symbols. Total summed resonance across all tapping modes is plotted against proportion of ametric tapping, for each of the 24 stimuli. Correlation coefficients are displayed for each plot.

14

71

Tapping to Bach

cantly with the proportions of tapping in the respective mode. The total resonance of the model correlates significantly with the proportion of ametric tapping. Figure 10 displays scatterplots associating the model’s total resonance with BST, CV, CDel, and CDev. As can be seen, all these measures have a negative correlation with the total resonance, indicating that high resonance values are associated with rapid pulse finding and accurate tapping. To compare the human pulse-finding performance on the 24 stimuli with the representations of the resonance model, we performed another factor analysis. For each of the 24 stimuli, we included all performance measures and the following model representations: the average resonance values as a proportion of the average total resonance for modes 2_1, 4_1, and 4_3. These averages were computed over each excerpt, giving one value for each excerpt. The analysis revealed three main factors, accounting for 44%, 31%, and 9% of the total variance, respectively. Factor 1 had positive weights for ametric, CV, and CDev, BST, and resonance of 2_1, with negative weights

1.6

0.04

r = -.56

1.4

0.036

1.3 1.2

0.03

1 6

8

10

Total resonance

0.03

12

0.028 4

14

8

10

Total resonance

12

14

r = -.82

0.05 0.045

0.02

CDev

CDel

6

0.055

r = -.55

0.025

0.015

0.04 0.035

0.01 0.005 4

0.034 0.032

1.1

0.9 4

r = -.63

0.038

CV

BST (1/2 notes)

1.5

0.03 6

8

10

Total resonance

12

14

0.025 4

6

8

10

Total resonance

12

Fig. 10. Total summed resonance across all modes is plotted against BST, CV, CDel, and CDev, for each of the 24 stimuli. Correlation coefficients are displayed for each plot.

14

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for consensus, proportion of 2_1, and total resonance. Therefore, this first factor reflects an association of the model’s overall representational strength (i.e., total resonance) with more stable pulse-finding performance and faster tapping. Factor 2 had positive weights for resonance of 4_3 and proportion of 4_3, with negative weights for resonance of 4_1 and proportion of 4_1. Therefore, the second factor reflects the strong primacy effect in tapping behavior and the model’s ability to predict the tapping mode accordingly. Factor 3 had a strong positive weight for CDel and a strong negative weight for total resonance. The fact that the model accounts for overall performance is noteworthy, as some of the performance measures (e.g., CV and CDev) are not directly observable from the model because, due to the metronomic quality of the stimuli, the model was exactly synchronized with the beat.

General Discussion We investigated the ability of human subjects to find and follow a pulse in brief excerpts from a single piece of contrapuntal music, and we examined the musical features that allowed them to perform this behavior. In addition, we developed a novel application of a resonance model that allowed a probabilistic approach to predicting several aspects of human behavior in pulse finding. Most importantly, we used the concept of internal resonance to predict subjects’ general ability to find and follow a pulse. These modeling results suggest that despite the unitary output of such models, the internal representation of periodic components in the signal (i.e., resonance values) can predict behavior in a sample of subjects. PREDICTING TAPPING BEHAVIOR WITH RESONANCE VALUES

The performance measures obtained from the tapping experiment were compared with the model’s internal representations of the periodic components in the stimuli. More specifically, the proportions of tapping in each mode with each excerpt as well as the measures of tapping accuracy were compared with the average resonance values of each tapping mode and the average total resonance, computed over each excerpt. It was found that the resonance values predicted tapping behavior in terms of overall tapping performance. In particular, the average total resonance correlated significantly with the measures associated with the accuracy of tapping and the time to start tapping. More specifically, excerpts that yielded a high total resonance value were associated with low variability of intertap intervals (CV), small deviation of tapping times from the beat (CDev), and a low number of beats to start tapping (BST). Therefore, the total resonance value predicted overall tapping performance, which is notable because, thanks to

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the metronomic quality of the stimuli, the model was exactly synchronized to the beat and these performance measures were thus not directly observable. The model was able to predict the proportions of tapping in modes 4_1 and 4_3 in the sense that the resonance values of these modes correlated significantly with the respective proportions of tapping. In the tapping experiment, the choice of tapping mode was to a great extent determined by the primacy factor. In other words, participants frequently started to tap in a mode that coincided with the first note of the stimulus and were reluctant to switch to another mode during the presentation of the stimulus. In the simulations, this phenomenon was clearly seen as the strong effect of the primacy weight on the model’s performance: increasing the value of the primacy weight resulted in an increase of the correlation between the resonance values and proportions of tapping in modes 4_1 and 4_3. Besides purely temporal factors, the model used in this study takes into account various pitch-related factors. More specifically, the driving force depends on the presence of minor second movement and melodic closure. If the respective weight values were changed, the effect of these factors on pulse finding could be modeled. Increasing the weight of melodic closure resulted in a better performance, suggesting that melodic closure may affect pulse finding. Although direct comparisons between the present model and previous models were not made, this result suggests that including appropriate pitch-related features in a pulse-finding model may result in a better performance. Compared with previous models of pulse finding, the resonance approach used in the present model provides a relatively simple means for predicting a variety of performance measures. In particular, it accounted for measures related to tapping accuracy and time to start tapping (i.e., variables CV, Cdev, and BST) better than other methods published to date. Predicting the tapping mode was more intricate, but this is a problem that many approaches seem to have difficulty with. A feature that distinguishes the present model from most other models is that it does not rely on temporal factors only, but takes into account pitch-related features as well. This holds also true for the multiple agent model by Dixon and Cambouropoulos (2000), which is, however, noncausal and thus not intended for modeling the temporal evolution of a pulse percept. As the number of pitch-related features included in the present model was limited, further work should be carried out to investigate the role of other nontemporal parameters in pulse finding. The present model expects musical score data as input, and contrary to models by Dannenberg (1984), Dannenberg and Mont-Reynaud (1987), Scheirer (1998), Cemgil et al. (2000), Dixon (2000, 2001), Eck (2000), Large and Kolen (1994), Large (2000), Large and Palmer (2002), and Toiviainen (1998, 2001), is not intended for performed music that contains

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temporal fluctuations. This limitation could be avoided by using a bank of adaptive oscillators (Large & Kolen, 1994; Toiviainen, 1998) with a range of different periods and phases that synchronize to the musical performance. As the stimuli used in the present experiment were metronomic and the focus of the study was not on beat tracking, adaptive oscillators were not included in the model. A further limitation of the model is that it works only with symbolic representations of music, such as MIDI, and fails to operate with audio input. For a purely temporal model, it is relatively easy to extract the necessary information from an audio signal by, for instance, onset detection. Dealing with pitch-related factors is, however, more demanding with audio input, especially with polyphonic music. CONSENSUS AS A MEASURE OF POPULATION BEHAVIOR

In addition to deriving performance measures that described the relation between the tapping data and the rhythmic structure of the stimuli, we investigated the mutual consensus of tapping between the participants. To this end, we developed a function for computing the degree of tapping consensus between the participants. We compared the obtained consensus values with the other derived performance measures and found that a high degree of consensus was associated with a high proportion of tapping in metrically appropriate modes, that is, in 4_1 and 2_1. In other words, when participants tapped in a mode suggested by the musical score, they tended to be mutually well synchronized. Furthermore, as may be expected, a high degree of consensus implied stable performance, as indicated by low values of the variability of intertap intervals (CV), the deviation of tapping times from the beat (CDev), and the number of beats to start tapping (BST). In other words, when participants tapped mutually at the same points, they also tended to tap accurately with the beat of the stimulus and started to tap early. This suggests that the measure of consensus used in this study is a strong predictor of pulse-finding effortlessness. Comparison of the consensus values with the musical variables obtained from the stimuli showed that the degree to which participants tapped mutually at the same points depended on the melodic and rhythmic structure of each stimulus. More specifically, a high degree of consensus was associated with a high number of note onsets, a high number of melodic direction changes, and a low degree of syncopation. This suggests that frequently and regularly occurring note onsets give rise to a pulse percept that is salient and accurate. MUSICAL FEATURES INFLUENCING PULSE FINDING

As was evident in an earlier study of piano ragtime music (Snyder & Krumhansl, 2001), syncopation in the excerpts we used caused much difficulty in tapping performance. In particular, a factor analysis showed that

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high amounts of syncopation (i.e., few notes on metrically strong positions) loaded strongly on the same factor with the time to find a pulse, the amount of ametric tapping, the variability of tapping, and the deviation of taps from the intended temporal location. This suggests that, once induced, the sense of pulse requires continual reinforcement from the musical surface in order to guide synchronization behavior optimally. In further support of this conclusion is the fact that the density of musical events also predicts tapping performance. In other words, excerpts with few notes onsets and few melodic direction changes did not enable participants to strongly synchronize their tapping with the perceived pulse. The resonance model of pulse applied in the current study accounts for these effects by assuming a decay function for each periodic mode in the absence of note onsets in phase with the respective mode. The finding that different musical styles (i.e., piano ragtime and baroque counterpoint) each showed a detrimental effect of syncopation for pulse-finding performance indicates that this is an important general feature of rhythmic structure. CONSTRAINTS ON THE DISTRIBUTION OF TAPPING MODES

It is remarkable that the participants used very few of the possible tapping modes with periods of 400, 800, and 1600 ms. Around 93% of all taps were in modes 2_1 (period of 400 ms and including the downbeat), 4_1 (period of 800 ms and including the downbeat), and 4_3 (period of 800 ms and including the upbeat), whereas 3% of taps were ametric (inconsistent period). It is likely that our manipulation of start position biased participants to tap more in mode 4_3 than would be normally observed. These tapping modes are the most metrically appropriate according to the scored meter (2/2), which specifies that there are two beats per measure with a primary beat period of 800 ms. Besides meter, tempo most likely constrained the likelihood of tapping in particular modes. The dominant modes contained periods of 400 and 800 ms that correspond to the maximal region of pulse sensation (Fraisse, 1982; Parncutt, 1994; van Noorden & Moelants, 1999). Other tapping modes that are in theory metrically appropriate with periods of 200 and 1600 ms fall outside the region (300– 1200 ms) that subjects are best at sensorimotor synchronization (Engström et al., 1996; Mates et al., 1994; Peters, 1989). At tapping rates faster than 300 ms, there is a large increase in tapping variability (Peters, 1989), whereas at rates slower than 1200 ms, subjects no longer synchronize but instead react to tone presentations (Engström et al., 1996; Mates et al., 1994). SERIAL ORGANIZATION AND METRICAL ORGANIZATION IN PULSE FINDING

Another behavioral finding of the current study was the fact that the start position of the excerpts within the notated measure had a major influ-

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ence on tapping phase. Specifically, participants usually tapped at the metrical position of the first tone of the excerpt (the first or third eighth note of the measure). This finding suggests that subjects may perceptually organize periodic stimuli with a strong bias for perceiving the opening tone of a stimulus as a relatively important event. With subsequent metrical cycles, participants seemed to continue perceiving the metrical position corresponding to the starting position as an appropriate tapping phase. Such an interpretation is consistent with findings from perceptual organization studies (Garner & Gottwald, 1968; Preusser, Garner, & Gottwald, 1970) in which the preferred serial organization of iterated patterns of tones depended to a strong extent on the starting position within the iterated pattern. Importantly, these earlier studies did not rely on synchronization tapping responses. Therefore, it seems likely that the present findings did not result from an artifact of the response type. Nevertheless, future perceptual experiments of metrical organization could test this hypothesis by using a nontapping paradigm. Importantly, our results suggest a profitable line of research in understanding the factors that influence synchronization with complex periodic stimuli that have relatively ambiguous period and phase structures. Music is an example of this stimulus type because there are often multiple possible tapping periods within a comfortable range (Fraisse, 1982) and multiple phases within these periods in which tone onsets occur. Based on the observed influence of syncopation, it is reasonable to predict that subjects would most likely choose tapping modes that maximize the number of taps that co-occur with note onsets. In addition, our results suggest that subjects may apply the reasonable assumption that the first tone of a stimulus most likely corresponds to a phase in which note onsets often occur in subsequent metrical cycles. In addition to these findings, it is also possible that the relative perceptual salience of events at different metrical positions has an influence on tapping mode. These include features related to duration-based (Povel & Okkerman, 1981; Parncutt, 1994) and pitch-based (Dawe et al., 1993, 1994, 1995; Tekman, 1997, 1998; Thomassen, 1982) accents. Although most models of pulse finding and meter take at least some of these issues into account, the large number of possible influences on tapping mode suggest that an explanatory model will have to address issues of basic sensory processing, stimulus feature integration, and selective attention to stimulus features. These issues are in addition to the problem of determining the most appropriate type of mechanism, which has been the main point of debate among computational experts thus far.1 1. This work was supported by grants from the Academy of Finland and the Fulbright Foundation to the first author. The second author was a predoctoral trainee on NIMH training grant T32 MH 19389, “Multidisciplinary Training in Developmental Psychology.” We thank Dr. Edward W. Large and his colleagues for helpful comments on this manuscript.

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