acta psychologica ELSEVIER Acta Psychologica91 (1996) 131-151

Spatiotemporal variability in cascade

a,b A. (Tony) A.M. van Santvoord a,*, Peter J. Beek a Department of Psychology, Faculty of Human Movement Sciences, Free University, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands b Center for the Ecological Study of Perception and Action, University of Connecticut, Storrs, CT, USA

Received 14 March 1994; revised 3 June 1994; accepted 28 July 1994

Abstract

Three expert jugglers and three intermediate jugglers performed a three-ball cascade pattern under spatially and temporally constrained conditions. In the spatially constrained conditions the balls were thrown to a specific height, whereas in the temporally constrained conditions the balls were thrown to the beeps of a metronome. The experiment was conducted to examine the hypothesis that juggling represents a spatial clock in that jugglers attempt to set up an invariant time base for the hand movements by throwing the balls consistently to a fixed height. Specifically, two expectations following from this hypothesis were examined: (1) that the spatiotemporal variability of the produced patterns would be less when juggling to an externally specified height than to an externally specified beat, because throwing to a height would be more in line with what jugglers actually do, and (2) that in both conditions the space-time trajectories of the balls would be less variable than the space-time trajectories of the hands. Examination of the observed patterns in terms of a set of theoretically motivated variables confirmed the second expectation. At the level of the individual variables the first expectation was not confirmed by the data: the spatiotemporal variability of the patterns was very similar under the two conditions. However, at the level of ensemble variables, the variability of the ball loop time (defined as the time that the ball was carded by the hand plus the subsequent flight time) was smaller when juggling to a height than when juggling to a beat, while the variability of the hand loop time (defined as the time that the hand carded a ball plus the time that it moved empty) was the same. These results were largely independent of skill level; only a few differences between expert and intermediate jugglers were found. The implications of the findings with regard to the development of a theory of perceptual-motor control in which spatial and temporal variables are linked in a task-specific manner are discussed.

* Corresponding author. Fax: +31 20 4445867, Tel: +31 20 4448532.

0001-6918/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0001-691 8(94)00044- I 132 A.A.M. van Santvoord, P.I. Beek /Acta Psychologica 91 (1996) 131-151

PsyclNFO classification: 2330

Keywords: Motor control; Motor variability; Motor learning; Dynamical systems theory

I. Introduction

How the spatial and temporal properties of movements are controlled in the execution of actions is a fundamental problem in 'motor control'. Broadly speaking, two types of accounts may be distinguished: those referring to processes internal to the human motor system and those addressing the behavioral level of the movements themselves. To the former class belong inferential structures, such as cognitive constructs (i.e., motor programs, schemata) and neurophysiological mechanisms (i.e., CPGs, neural circuits), which are deemed responsible for the spatiotemporal regularities observed at the behavioral level. The latter class consists of a variety of non-inferential constructs varying from the lawful regularities formulated in the human performance literature (e.g., Fitts' law, Fitts, 1954), to the lawful regularities between perceptual variables and movement identified by exponents of the ecological approach to perception and action (e.g., Turvey, 1990), and the (conceptually related) mathematical models proposed by exponents of the dynamical approach to movement coordination (e.g., Haken et al., 1985). In the present article, which examines spatiotemporal patterns in cascade juggling, the concern is with the behavioral level per se. No inferential constructs will be proposed to account for the observed relations. The approach adopted is a descriptive one that seeks to identify general principles reflecting the organization of the human action system in accomplishing specific task goals. A theoretically important and challenging issue in understanding the space-time properties of movement is that of the spatiotemporal variability of performance. Every movement, simple or complex, is characterized by a certain variability in its spatial and temporal dimensions. With regard to this general theme, the study of discrete targeting movements has led to the identification of what is generally known as the speed-accu- racy trade-off in movement: Movement is more accurate at low speeds and less accurate at high speeds. As noted by Newell et al. (1993), early investigations of the speed-accu- racy trade-off focused almost exclusively on spatial error (e.g., Beggs and Howarth, 1972; Fitts, 1954; Keele, 1968; Woodworth, 1899), but by now the effects of movement speed on both spatial and temporal variability are well documented (e.g., Hancock and Newell, 1985, Newell, 1980; Newell et al., 1979, Newell et al., 1980, Newell et al., 1982; Schmidt et al., 1979). A general conclusion from this work is that the degree of spatial and temporal variability of movement in general and the exact form of the speed-accuracy trade-off in particular are a function of the task and the nature of the task instructions (e.g., Carlton, 1994; Newell et al., 1993; Zelaznik et al., 1988). Clearly, the prevailing impression that constant error shifts and speed-accuracy functions are largely task-dependent poses a problem for the formation of an encompassing theory of variability in motor control. As a prerequisite for the development of such a theory it seems important, if not necessary, to investigate the task-specificity of spatiotemporal variability in a variety of different task domains, including both discrete and continuous A.A.M. van Santooord, PJ. Beek /Acta Psychologica 91 (1996) 131-151 133

'T 1

h

F q I D Fig. 1. Schematic representation of juggling three balls in a figure-eight pattern, momentary situation: ~, is the angle of release; Vo is the velocity of the ball at the moment of release; h is the height to which the balls are thrown relative to the point of release; F is the base width of the flight parabola, and D is the width of the elliptical hand movement.

perceptual-motor actions. Experimental studies are called for that examine how the space-time patterns within each of these domains are adapted when relevant task constraints are manipulated. In the present study, space-time patterns in cascade juggling are examined and compared under both spatially and temporally constrained conditions. In the spatially constrained conditions jugglers throw the balls to a fixed height (indicated by target markers), while in the temporally constrained conditions jugglers throw the balls in synchrony with the beeps of a metronome. The basic question is: Are the spatiotemporal properties of a juggler's performance affected by these task manipulations, and, if so, how? In the temporally constrained condition, does the temporal variability decrease and the spatial variability increase, whereas the obverse is true in the spatially constrained condition? Our expectations with regard to the space-time relations that will be observed under these conditions are based on our previous analyses of cascade juggling. Cascade juggling is a complex, cyclic activity in which the hands move along more or less elliptical trajectories, one evolving clockwise and the other anticlockwise (see Fig. 1). The objects (hereafter referred to as 'balls') are released at the inside of the ellipses and caught at the outside. The balls fly to the other hand along a parabolic trajectory. In the simplest version, in which two hands juggle three balls, the two parabolic flight paths i intersect at a point close to the points of release. As a result, the balls travel in the figure-eight pattern that is characteristic for the cascade. Fig. 1 defines the essential spatial variables of juggling: the height (h) to which the balls are thrown relative to the points of release, the angle of release (9,), the base width (F) of the parabolic flight trajectories, i.e., the horizontal distance between throwing and

We reserve the term 'path' for the curve in space traced out by the ball or the hand and the term 'trajectory' for the curve in space traced out by the ball or the hand and its time history. 134 A.A.M. van Santvoord, PJ. Beek /Acta Psychologica 91 (1996) 131-151 catching of the same ball, and the width (D) ('amplitude') of the elliptical hand trajectories, i.e., the horizontal distance between throwing and catching by the same hand. Essential temporal variables of the task are the flight time of a ball between the hands (TF), the time that a hand is filled with a ball between a catch and a throw (time loaded: TL), and the time a hand moves empty between a throw and a catch (time unloaded: T v). A relevant additional variable is the velocity of the ball at release (V0), because it determines, together with the angle of release and the acceleration due to gravity (g), the parabolic flight trajectory. Obviously, these variables are related. The basic spatial equation of juggling can be derived from the parabolic flight path of the balls, which is described by F = VxoTr, 2Vro = gT F, and 8h = gT2F, and reads F cot 9r = 4-h (1)

assuming that catching and throwing occur at the same height and that air friction is negligible (Beek, 1989a). Both F and D are associated with the catch of a particular ball. However, D is related to the movements of one hand and connects two consecutive ball flights, whereas F is related to a ball flight and connects the movements of the two hands. The basic temporal equation of juggling was identified by Claude Shannon (cf. Horgan, 1990; Raibert, 1986) and can be derived as follows (see Beek, 1992; Beek and Turvey, 1992). For a periodic (in which T F, T L, and T e are assumed to be fixed), let T" denote the duration of a complete N-ball, H-hand juggling cycle, that begins and ends with the same event, e.g., catching ball i in hand j. Since there are N balls, the total loop time for hand j during T * is equal to N(T L + Tu), and since there are H hands, the total loop time for ball i during T * is equal to H(T L + TF). Hence, averaged over time, N(T L + T v) = T* = H(T L + TF), or TL+T r T B N T L + T v T H H' (2)

where T H = T L + T u is the average cycle time of the hands and T B = T L + T F is the average cycle time of the balls. The spatial and the temporal constraint on juggling are connected through the velocity of release, which determines the height to which the balls are thrown, and in combination with ~Pr and T r, the horizontal distance over which they travel, F = VxoTr. The velocity of release, however, does not uniquely determine T F because a ball need not be caught at the same height at which it was thrown. Tr, in turn, does not uniquely determine T L + T u, because there are still many combinations of T L and T v that satisfy Shannon's equation. The 'motor problem' of juggling, to use the apt term of Bernstein (1967), is to a considerable extent a problem of appropriate timing (cf. Austin, 1976; Beek, 1989b; Beek and Turvey, 1992), but the solution to this problem is partly, and perhaps primarily, spatial. By throwing the balls consistently to a specific height jugglers may set up an invariant time base for the hand movements. Juggling may be viewed in this regard as creating and sustaining a spatial clock (Beek, 1989a). A.A.M. van Santooord, P.I. Beek / Acta Psychologica 91 (1996) 131-15l 135

From a naive point of view, it is reasonable to assume that when jugglers are invited to juggle to a height or to the beeps of a metronome, they will attempt to satisfy the task goal as closely as possible, and that therefore spatial measures will be more tightly controlled than temporal measures when juggling to a height and that the obverse is true when juggling to a beat. It follows from the notion of juggling as a spatial clock, however, that in both cases spatial and temporal variables will be closely interwoven. The prevailing space-time relations might in fact be so strong that they will militate against the naively expected effects of the task manipulation. If the interpretation is correct that jugglers throw the balls consistently to a specific height to set up a time base for the hand movements, it might even be expected that juggling to an externally specified height will be accomplished with less spatiotemporal variability than juggling to an externally specified beat. Moreover, it might be expected that, in both conditions, particularly the throws will be tightly controlled, so as to secure reproducibility of the zeniths, and that the space-time properties of the flight trajectories will be less variable than those of the hand trajectories. In the context of an examination of these two expectations, it is important to compare the performance of expert and less trained jugglers under the suggested task constraints, because it might well be that the effects of the proposed task conditions on the spatiotemporal variability of performance will interact with skill level.

2. Method

2.1. Subjects

Three expert jugglers (who could juggle 5 or more balls in a cascade pattern) and three jugglers of intermediate skill (who could not juggle more than 3 balls in a cascade pattern) participated in the experiment. The ages of the jugglers, who were all right- handed, varied between 21 and 36 years. One of the authors (AvS) was the only participant who was aware of the theoretical background of the experiment. All participants (except AvS) were paid for their services.

2.2. Procedure

The subject was placed in front of a 16 mm high speed motion picture camera (Teledyne-type DBM 55, Teledyne Camera Systems, Arcadia, CA) at a distance of about 5 m. The focus of the zoom lens of the camera was adjusted so that the juggler and the entire ball trajectories were in view of the camera. A flashing light with a fixed frequency was placed in view of the camera so that the actual frame rate could be checked against the nominal frame rate (64 Hz). The gravitational vertical was defined by a plumb line suspended from the ceiling. Proper lighting conditions for filming were created with the help of four 2000 Watt stage-lamps. The equipment of the subject consisted of three so-called 'stage' balls (diameter 7.3 cm, weight 130 g). The subject was invited to juggle the balls in a cascade fashion, either under a spatial or under a temporal constraint. 136 A.A.M. van Santvoord, P.I. Beek /Acta Psychologica 91 (1996) 131-151

In the spatially constrained conditions the subject was invited to throw the balls to a specific height as indicated by two targets. The target markers consisted of two small plastic balls (diameter 3.1 cm) that were suspended from the ceiling with a translucent fishing line with a distance of 48 cm between them. The height of the markers, which could be adjusted by shortening or lengthening the fishing line, was measured relative to the hands of the subject when standing in upright position with the upper arms flush along the body and the elbows flexed at an angle of 90 degrees, so that the forearms stretched out horizontally. Each subject performed four trials of juggling, each to a different height (25 cm, 50 cm, 75 cm, and 110 cm). In the temporally constrained conditions the subject's task was to juggle to the beeps of a metronome. The beeps were produced by a computer program which generated (ambient) tones of 50 ms duration at a pitch that alternated between 220 Hz and 275 Hz with a fixed time interval between them. The electronic signal that produced the tones was also fed to the camera where it triggered an electronic device that marked the arrival of a pulse in the leading edge of the film. Thus, a frame-based measure for the temporal locations of the beeps could be obtained by visual inspection of the film. Also under this constraint each subject performed four trials of juggling, each to a specific beat. The time intervals were 301, 430, 521, and 628 ms. These time intervals correspond (approximately) to juggling to the four externally specified heights when the proportion of time that the hand carries a ball to the hand cycle time is 3/4. Before each trial the subject was allowed a few cycles to practice accommodating the task constraints in question. When the subject reported that a satisfactory pattern was achieved, the movie camera was started. During each trial three complete juggling cycles were recorded, amounting to a total of 18 (3 × 3 balls × 2 hands) hand cycles and 18 flight trajectories. Recording of one trial of an expert juggler failed (in the 25 cm height condition), while two trials were recorded for a duration shorter than three complete juggling cycles (amounting to 12 and 15 hand cycles and flight trajectories, respectively). In all statistical analyses the group mean of the 25 cm height condition was entered as an estimate of the value of the missing trial, so as to remain on the conservative side with regard to finding significant differences between experts and intermediates.

2.3. Data acquisition

After professional development and further processing, the films (Kodak 7251, Ektachrome high-speed daylight film, 400 ASA) were projected onto the opaque screen of a film motion analyzer (NAC type MC OF) by means of a 16-mm projector (NAC type RH 160F), connected to a personal computer. The position of the projector was adjusted so that the plumb line was aligned with the vertical axis of the screen. Frame by frame, the x- and y-coordinates of the centers of the balls were digitized, fed into the computer, and stored for later analysis. The actual frame rate during a trial was estimated by averaging the number of frames between the onsets of the flashing light over the digitized trial. The actual frame rates varied between 61.1 Hz and 65.5 Hz over the recorded trials with the exclusion of two trials that were recorded at 104.3 Hz. The actual frame rates provided the time basis for the subsequent computation of the A.A.M. van Santvoord, P.J. Beek /Acta Psychologica 91 (1996) 131-151 137 duration of the various time components in juggling. From the film recordings made during the temporally constrained conditions also the frame numbers were collected that were marked by a pulse, indicating that a beep was produced.

2.4. Data reduction

Using a combination of fitting procedures and inter- and extrapolation routines on the raw displacement data (i.e., the trajectories of the balls) the spatial and temporal locations of the key events of juggling, i.e., catching, ball release, and the balls reaching their respective zeniths, were identified. The inaccuracy of measuring the temporal locations of these events was one frame at most, which corresponds to a maximal error of 16 ms. The inaccuracy of the derived spatial estimates was 1-3%, depending on the selected focus of the zoom lens.

2.4.1. Spatial variables From the spatial locations of throwing and catching and the zeniths of the ballflights the spatial variables defined in Fig. 1 were calculated: the height of juggling (h), i.e., the vertical difference between a point of release and the zenith following it; the base width of the flight parabola (F), i.e., the horizontal difference between a point of release of a ball and the interception point of that bali; the width of the elliptical hand movement (D), i.e., the horizontal difference between a point of release of a hand and the point of the subsequent catch by that hand. The angle of release (~r) was calculated with the help of fitting procedures to the parabolic ballflights. For each recorded trial the means of these variables and their coefficients of variation (CVs, defined as the standard deviation divided by the mean) were computed (n -- 18) for later analysis. 2

2.4.2. Temporal variables From the temporal locations of throwing and catching the following temporal variables were calculated: time flight (TF), the time between a throw and a successive catch of the same ball; time loaded (TL), the time between a catch and a successive throw by the same hand; time unloaded (Tv), the time between a throw and a successive catch performed by the same hand; hand cycle time (Tn = TL + T v) and ball cycle time (TB --TL + Tr). The within-trial means and CVs were computed for each of these variables.

2.4.3. Dynamical variable The velocity of the ball at release (V0) and its horizontal and vertical components were calculated with the help of fitting procedures.

2 At this point it should be realized that the CV of the angle of release is an arbitrary quantity, because its magnitude depends on the definition of the angle in question. Although it cannot be meaningfully compared to the CVs of physically non-arbitrary spatial measures, such as h and F, the CV of ~Pr was included in the analysis to examine how it was affected by the experimental manipulations. 138 A.A.M. van Santvoord, P.J. Beek / Acta Psychologica 91 (1996) 131-151

3. Results

3.1. Attaining the task goals

3.1.1. Spatially constrained conditions To determine how well the subjects juggled to an externally specified height, the within-trial means and standard deviations of the difference as well as the absolute difference between the vertical location of the zeniths of the ballflights and the target heights were calculated, which are collected in Table 1. The fact that the mean differences were negative on most trials implies that, on average, the balls were thrown lower than the target markers. Two-tailed t-test on the values within each trial revealed that the mean error differed significantly from zero on 14 trials (as indicated in Table la). A repeated-measures analysis of variance (ANOVA), involving a 4 (Height) × 2 (Skill) factorial design, revealed no significant effects for the mean error and a significant main effect of Height, F(3,12) = 12.4, p < 0.001, 3 for the standard devia- tion of the mean error, as it increased systematically with increasing height. Also when applied to the within-trial means and the standard deviations of the absolute difference the ANOVAs revealed no significant effects for the means and a significant effect of Height, F(3,12)= 17.19, p < 0.01, for the standard deviations. Hence, in terms of absolute error the four height conditions were performed equally well.

3.1.2. Temporally constrained conditions To determine how well the subjects juggled to an externally specified beat, the within-trial means and standard deviations of the difference as well as the absolute difference between the beep-beep interval and the interval between the successive throws were calculated. The obtained values are collected in Table 2. In most cases the means are close to zero indicating that the rhythm of juggling closely matched the rhythm of the metronome. Two-tailed t-tests on the within-trial differences revealed that the rate of juggling was significantly faster than the rhythm defined on the metronome in five trials (see Table 2a). Although four of these were from the expert group, repeated-measures ANOVAs, involving a 4 (Beat)× 2 (Skill) factorial design, per- formed on the within-trial means and the standard deviations disclosed no significant effects for the difference nor for the absolute difference. Hence, also the four temporally constrained conditions were performed equally well.

3.2. Spatiotemporal patterns

The kinematic properties of the spatiotemporal patterns that were produced by the expert and intermediate jugglers under the spatially and temporally constrained task conditions are summarized in Tables 3-6, which collect the within-trial means and

3 All p values that are reported in this article for repeated measures ANOVAs are based on conservative adjustments of the degrees of freedom for potential sphericity violations according to the Huynh-Feldt convention. A.A.M. van Sanwoord, P.J. Beek / Acta Psychologica 91 (1996) 131-151 139

Table 1 Within-trial means and standard deviations of the vertical difference (a) and the absolute vertical difference (b) (in ram) between the zeniths of the ballflights and the target heights for the expert (El-E3) and intermediate (11-13) jugglers in the spatially constrained condition (regardless of hands) Targetheight(cm) 25 50 75 110 Subject Mean SD Mean SD Mean SD Mean SD (a) E1 -13 a 16 9 35 -38 a 48 -72 a 57 E2 12 a 14 -20 a 28 - 75 a 40 - 206 a 70 E3 1 19 52 a 23 -42 a 32 I1 13 33 - 12 47 -26 a 30 - 22 58 12 - 31 a 22 10 47 -29 68 -97 a 70 I3 -60 a 18 -31 a 25 -7 38 23 61

(b) El 18 10 28 22 52 31 76 51 E2 14 12 23 25 78 34 206 70 E3 16 10 52 23 43 30 I1 29 19 39 27 34 21 50 36 12 33 20 36 31 60 42 98 69 13 60 18 33 22 31 21 48 41 ap<0.01.

Table 2 Within-trial means and standard deviations of the difference (a) and the absolute difference (b) (in s) between the intervals of the metronome beeps and the intervals between successive throws for the expert (El-E3) and intermediate (I1-I3) jugglers in the temporally constrained condition (regardless of bands) Beatinterval (mD 301 430 521 628 Subject Mean SD Mean SD Mean SD Mean SD

El -0.020 b 0.012 0.001 0.023 -0.004 0.040 0.007 0.029 E2 0.003 0.050 - 0.048 a 0.076 - 0.005 0.048 - 0.001 0.042 E3 - 0.001 0.019 -0.039 b 0.021 -0.031 b 0.026 0.005 0.019 I 1 -- 0.001 0.026 0.000 0.058 -- 0.006 0.031 -- 0.034 b 0.046 12 -- 0.001 0.014 0.000 0.017 0.004 0.020 0.000 0.046 13 -- 0.001 0.022 -- 0.003 0.032 0.001 0.032 0.000 0.031

(b) E ! 0.021 0.012 0.018 0.013 0.031 0.025 0.024 0.017 E2 0.048 0.009 0.076 0.046 0.041 0.025 0.032 0.025 E3 0.017 0.008 0.039 0.021 0.032 0.025 0.0t5 0.013 I 1 0.021 0.013 0.049 0.028 0.027 0.015 0.045 0.035 I2 0.011 0.009 0.013 0.011 0.016 0.013 0.038 0.024 13 0.018 0.012 0.028 0.014 0.026 0.017 0.023 0.019 a p < 0.05; b p < 0.01 140 A.A.M. van Santvoord, P.L Beek / Acta Psychologica 91 (1996) 131-151

Table 3 Within-trial means and coefficients of variance (CV) of the spatial variables in the spatially constrained condition Target height(cm) 25 50 75 110 Subject Mean CV Mean CV Mean CV Mean CV Angle of release: tpr (rad)

El 1.304 0.03 1.455 0.02 1.487 0.01 1.503 0.01 E2 1.327 0.03 1.400 0.01 1.432 0.01 1.450 0.01 E3 1.453 0.01 1.501 0.01 1.509 0.01 I1 1.338 0.03 1.453 0.02 1.485 0.01 1.482 0.01 I2 1.361 0.02 1.444 0.02 1.476 0.03 1.491 0.02 13 1.336 0.03 1.488 0.01 1.465 0.01 1.526 0.02

Height: h (m)

El 0.278 0.07 0.521 0.09 0.740 0.05 1.052 0.05 E2 0.243 0.06 0.473 0.05 0.581 0.05 0.854 0.07 E3 0.423 0.05 0.747 0.04 0.893 0.03 I1 0.222 0.12 0.365 0.09 0.543 0.09 0.801 0.09 I2 0.358 0.05 0.515 0.07 0.739 0.08 1.003 0.08 13 0.266 0.08 0.451 0.06 0.670 0.06 0.998 0.05

Base width of ball flight: F (m)

El 0.277 0.11 0.218 0.19 0.226 0.16 0.273 0.24 E2 0.235 0.14 0.317 0.09 0.314 0.11 0.396 0.10 E3 0.188 0.11 0.199 0.12 0.211 0.17 I 1 0.202 0.13 0.168 0.25 0.181 0.27 0.273 0.26 I2 0.278 0.08 0.243 0.20 0.266 0.53 0.300 0.43 13 0.246 0.14 0.142 0.14 0.259 0.21 0.170 0.67

coefficients of variation of the spatial and temporal variables of interest for the two types of task constraints.

3.2.1. Measures of central tendency To establish the effects of the experimental manipulations on performance, we first focus attention on the magnitude of the within-trial means, even though they have no immediate bearing on the question under investigation. Repeated-measures ANOVAs were performed on the within-trial means of each of the spatial variables of Eq. (1) (~0r, F, h) and each of the temporal variables of Eq. (2) (Tr, TL, Tu). The ANOVAs in question had a 2 × 4 × 2 × 2 factorial design with Skill (expert vs. intermediate) as the only between-subject factor and Pattern Size (25 cm/301 ms, 50 cm/430 ms, 75 cm/521 ms, 110 cm/628 ms), Task Constraint (beeps vs. targets) and Hand (left vs. righ0 as within-subject factors. The factor Pattern Size requires some explanation in that it collapses at each level the values attained in the spatially constrained condition with those in the temporally constrained condition, under the assumption that the size of the A.A.M. van Santooord, P.J. Beek /Acta Psychologica 91 (1996) 131-151 141

Table 4 Within-trial means and coefficients of variance (CV) of the spatial variables in the temporally constrained condition Beat interval (ms) 301 430 521 628 Subject Mean CV Mean CV Mean CV Mean CV Angle of release: ~r (rad)

E1 1.436 0.02 1.489 0.01 1.4% 0.02 1.519 0.01 E2 1.373 0.01 1.414 0.01 1.455 0.02 1.472 0.02 E3 1.406 0.01 1.432 0.01 1,463 0.01 1,504 0.01 11 1.425 0.03 1.486 0.02 1.513 0.02 1.498 0,02 12 1.393 0.03 1,446 0.01 1.468 0.01 1.491 0.02 13 1.433 0.01 1.473 0.01 1.501 0.02 1.484 0.02

Height: h (m)

E1 0.381 0.06 0.656 0.04 0.833 0.09 1.035 0.06 E2 0.408 0.07 0.594 0.06 0.625 0.07 0.895 0.06 E3 0.345 0.05 0,432 0,06 0.579 0.07 0,873 0.05 I1 0.309 0.12 0,528 0.11 0.670 0.10 0.818 0.12 I2 0.393 0.09 0.553 0.09 0.769 0.06 1.021 0.07 13 0.380 0.03 0.544 0.05 0.695 0.06 0,935 0.08

Base width ofball flight: F (m)

El 0,187 0.24 0.199 0.21 0,233 0,38 0.206 0.37 E2 0.321 0.10 0.364 0.09 0.288 0.20 0.341 0.19 E3 0.218 0.08 0,231 0.12 0,242 0.12 0,228 0.24 11 0.175 0.23 0.166 0.31 0,156 0.48 0.232 0.36 12 0.256 0.19 0.267 0.09 0.306 0.19 0.317 0.36 13 0,198 0.14 0.196 0.21 0.182 0.36 0.311 0.35

patterns produced in the temporally constrained condition approximately matched those attained in the spatially constrained condition. Visual inspection of the data collected in Tables 3-6 revealed that this was indeed the case. As expected, the factor Pattern Size had a significant effect on all variables: the relevant statistics were: F(3,12) = 177.72, p < 0.0001 (~Pr), F(3,12) = 4.87, p < 0.05 (F), F(3,12) = 221.06, p < 0.0001 (h), F(3,12) = 240.74, p < 0.0001 (TF), F(3,12) = 212.22, p < 0.0001 (TL), and F(3,12) = 4.41, p < 0.05 (Tu), which indicates that the subjects adhered to the task instructions. The factor Task Constraint had a significant effect on all variables except F, h and Tu: F(1,4) -- 8.21 p < 0.05 (q~r), F(1,4) = 11.96, p < 0.05 (Tr), and F(1,4) = 13.87, p < 0.05 (TL). These variables were larger when juggling to a beat than when juggling to a height. In four cases, there was also a significant Pattern Size × Task Constraint interaction: F(3,12) -- 11.95, p < 0.001 (~r), F(3,12) = 5.48, p < 0.05 (h), F(3,12) = 4.41, p < 0.05 (Tr), and F(3,12)= 5.01, p < 0.05 (Tu), which occurred because the previous effect of Task Constraint was predominantly located in the smaller Pattern Size conditions. The factor Skill had no effect on any of these kinematic variables. Finally, 142 A.A.M. van Santvoord, P.I. Beek /Acta Psychologica 91 (1996) 131-151

Table 5 Within-trial means and coefficients of variance (CV) of the temporal variables in the spatially constrained condition Target height (cm) 25 50 75 110 Subject Mean CV Mean CV Mean CV Mean CV Time flight: TI: (in s) El 0.437 0.04 0.592 0.03 0.706 0.03 0.874 0.02 E2 0.435 0.03 0.604 0.02 0.666 0.03 0.798 0.03 E3 0.555 0.02 0.743 0.02 0.805 0.02 I1 0.415 0.06 0.532 0.05 0.641 0.03 0.772 0.04 I2 0.495 0.02 0.599 0.03 0.737 0.03 0.849 0.04 I3 0.453 0.05 0.577 0.03 0.679 0.03 0.866 0.03

Time loaded: Tt (in s) El 0.307 0.06 0.427 0.05 0.629 0.08 0.961 0.04 E2 0.257 0.06 0.312 0.06 0.378 0.05 0.503 0.08 E3 0.449 0.03 0.864 0.04 0.878 0.04 I l 0.277 0. l I 0.412 0.09 0.559 0.06 0.818 0.05 12 0.366 0.05 0.561 0.05 0.759 0.07 0.904 0.06 I3 0.295 0.07 0.470 0.05 0.674 0.05 1.012 0.03

Time unloaded: Tv (in s) El 0.190 0.10 0.253 0.09 0.263 0.10 0.262 0.07 E2 0.203 0.06 0.296 0.06 0.318 0.08 0.364 0.09 E3 0.221 0.09 0.206 0.11 0.242 0.12 11 0.183 0.15 0.219 0.09 0.242 0.10 0.240 0.16 12 0.207 0.09 0.212 0.09 0.236 0.1 ! 0.266 0.12 I3 0.203 0.10 0.226 0.10 0.227 0.10 0.236 0.12

the factor Hand was involved in a significant Pattern Size × Hand interaction effect on tpr, F(3,12) = 4.79, p < 0.05, which occurred because on average the right hand had a larger tpr than the left hand in the smallest Pattern Size condition. In sum, the size of the produced patterns increased with increasing target height as well as with increasing beat interval. Across the two experimental conditions, the size of the produced patterns was very similar, except in the smallest Pattern Size condition. The shortest beat interval led to larger patterns on average than the smallest height instruction. This effect occurred because in these conditions the corresponding propor- tion of time that a hand carried a ball to the hand cycle time was significantly smaller on average than the nominal value of 3/4 on the basis of which the beat intervals were selected.

3.2.2. Measures of variability Turning to the experimental hypotheses, we first examined the degree of variability of the spatial variables of Eq. (1) and their patterning. The within-trial coefficients of A.A.M. van Santvoord, pal. Beek / Acta Psychologica 91 (1996) 131-151 143

Table 6 Within-trial means and coefficients of variance (CV) of the temporal variables in the temporally constrained condition. Beat interval(ms) 301 430 521 628 Su~t Mean CV Mean CV Mean CV Mean CV Timeflight: T e (in s)

E 1 0.504 0.04 0.677 0.03 0.770 0,04 0.885 0.02 E2 0.571 0.03 0.679 0.03 0.707 0,02 0.828 0.03 E3 0.506 0.03 0.565 0.03 0.664 0.03 0.818 0.02 I 1 0.491 0.05 0.622 0.07 0.742 0,04 0.799 0.07 I2 0.520 0.04 0.651 0.03 0.764 0.03 0.878 0.03 13 0.525 0.03 0.618 0.04 0.712 0.04 0.841 0.04

Time loaded: TL (in s)

El 0.334 0.06 0.618 0.04 0.787 0.07 1.023 0.05 E2 0.336 0.06 0.471 0.13 0.856 0.04 1.070 0.05 E3 0.384 0.05 0.607 0.06 0.817 0.06 1.085 0.04 I1 0.404 0.10 0.664 0.08 0.804 0.07 0.994 0.08 12 0.379 0.08 0.648 0.06 0.806 0.04 1.010 0.07 I3 0.367 0.04 0.670 0.04 0.861 0.06 1.050 0.04

Time unloaded: T v (in s)

E 1 0.225 0.07 0.244 0.07 0.250 0.12 0.249 0.09 E2 0.269 0.07 0.294 0.10 0.188 0.12 0.200 0.11 E3 0.209 0.08 0.178 0.08 0.171 0.10 0.184 0.11 I1 0.188 0.10 0.193 0.21 0.222 0.16 0.200 0.16 12 0.220 0.10 0.219 0.07 0.239 0.05 0.248 0.17 13 0.226 0.10 0.189 0.13 0.188 0.15 0.213 0.08

variation of these variables were compared in a similar ANOVA as for the means with an additional within-subject factor Variable (~pr, F, h), which had a significant effect, F(2,8) = 98.73, p < 0.0001. Post-hoc comparisons revealed that the within-trial CVs of h were significantly smaller than those of F (p < 0.05). There was also a main effect of Skill, F(1,4)= 9.62, p < 0.05, which occurred because the CVs of the variables of interest were on average smaller in the expert jugglers than in the intermediate jugglers. A significant Variable × Skill interaction, F(2,8) = 6.69, p < 0.05, revealed that the main effect of Skill was primarily due to the fact that the CVs of F were smaller in the expert jugglers than in the intermediate jugglers while the average CVs of the other two variables were about the same in the two groups. The third simple effect was found for Pattern Size, F(3,12)= 8.73, p<0.005: CVs tended to increase on average with increasing pattern size; again, a significant Variable X Pattern Size interaction, F(6,24) = 11.84, p < 0.005, occurred because also this effect was located predominantly in the CVs of F. No effects, however, were found in which the factor Task Constraint was involved. 144 A.A.M. van Santvoord, P.J. Beek / Acta Psychologica 91 (1996) 131-151

0.1

0.0

-0.1

-0.2 , ~ ~ , ,' ~, ~,~ 8 .~. -0.3 "O i 1 (J,;2 .'~ -0.4

-o.s

,i -0.6

-0.7 I I -0.4 -0.3 -0.2 -0.1 0.0 0.1 02 0.3 0.4 horizontal displacement (m) Fig. 2. An example of the ball trajectories of a trial in the spatially constrained condition (50 cm) from an expert juggler (E3). The solid circles indicate the positions of the targets. The open circles represent the mean distance of the events to their mean spatial location for the zeniths ( two circles), the throws (bottom two circles) and catches (middle two circles). (Position data filtered at 10 Hz).

These results show that the produced parabolic flight paths were characterized by an increasing variability from throwing to catching: the throw (as indexed by h) was less variable than the horizontal position of the catch (as indexed by F), regardless of skill level and the prevailing task constraints (height vs. bea0. It could be argued that this finding was to be expected, because, with a fixed projection point and a fixed velocity of release, small differences in the angle of release would result in a continuously increasing variability along the parabolic flight path. However, as neither the projection point nor the velocity of release were fixed, variations in the variables and the angle of release could, in principle, compensate one another (as indeed was found by Stimpel (1933) in his study of throwing to a target). To further examine the relative variability of the throws, zeniths, and catches, a measure was developed for characterizing the spatial dispersion of these key events. When portrayed in a plane, the spatial locations of each key event during a trial are represented by two clouds of points, one corresponding to the right hand and the other to the left. The spatial dispersion of each of these clouds was defined as the mean absolute Euclidean distance of each point relative to the mean of the cloud (see Fig. 2). For each type of event (throws, zeniths, and catches) the within-trial dispersion was calculated for the horizontal and vertical dimension separately. These values were subjected to a 2 × 3 × 2 × 4 × 2 × 2 repeated-measures ANOVA with the factors Skill, Event (throws, zeniths, catches), Direction (x and y), Pattern Size, Task Constraint and Hand. A main effect of Event occurred, F(2,8)= 48.98, A.A.M. van Santvoord, P.J. Beek / Acta Psychologica 91 (1996) 131-151 145 p < 0.0001, due to the fact that the average dispersion of the spatial locations of the throws was smaller than that of the zeniths of the baliflights, which was smaller than that of the catches. The mean values were 1.8 cm, 2.7 cm, and 3.5 cm, respectively. Post-hoc comparisons revealed that these values differed significantly from one another (p < 0.05). A significant Direction × Event interaction occurred, F(2,8) = 42.66, p < 0.0001, because the dispersion of the catches was larger than that of the zeniths in the horizontal, but not in the vertical dimension. In addition to these effects, Pattern Size was a significant main effect, F(3,12)= 31.28, p < 0.0001, because the degree of spatial dispersion of the key events of juggling increased systematically with the height of juggling (and, hence, decreased systematically with the rate of juggling). Post-hoc comparisons revealed that the average values for the two largest pattern sizes were both significantly different from all others, while those for the two smallest pattern sizes were about the same. A significant Pattern Size X Event interaction occurred, F(6,24) = 7.36, p < 0.001, which revealed that this increase was stronger for the catches and the zeniths than it was for the throws. Also Pattern Size x Event x Direction proved to be a significant source of variation, F(6,24) = 9.29, p < 0.0001, because the previous effect was also a function of direction. Furthermore, two two-way interactions were found in which the factor Skill was involved. Direction x Skill was a significant effect, F(1,4) = 8.58, p < 0.05, due to the fact that the dispersion in the x-direction was smaller than in the y-direction in the experts, while they were about the same in the intermediate jugglers. Also Pattern Size x Skill was a significant effect, F(3,12) = 3.83, p < 0.05, as the average increase of dispersion with increasing pattern size was stronger in the intermediate jugglers than in the experts. Finally, a three-way interaction Pattern Size X Skill × Hand, F(3,12) = 7.23, p < 0.01, occurred because the increase of spatial dispersion of the events associated with the hands with the size of the pattern was stronger for the right hand than for the left in the experts, but very similar for the two hands in the intermediate jugglers. The results of this additional analysis suggest that, irrespective of the task demands, each hand is attracted to a fixed projection point from which the balls are released; these points appear to be the anchor points for the hand movements (cf. Beek, 1989a). From this stable projection point, jugglers seem to primarily control the height of the ballflight; the control of the distance over which is thrown is only of secondary importance, but increases with skill. Further support for the latter observation comes from an analysis of the within-trial CVs of the horizontal and vertical components of the velocity of release. A similar ANOVA as before on these component velocities with Direction (horizontal vs. vertical) as an additional factor revealed a significant main effect, F(1,4) -- 55.79, p < 0.01, for Direction: the CVs of the horizontal release velocities were much larger on average than the CVs of the vertical release velocities (0.238 vs. 0.036). Also Skill was a significant main effect, F(1,4)= 9.33, p < 0.05, as the CVs were smaller on average for the experts than for the intermediate jugglers. Again, the factor Task Constraint was not a significant main effect, nor was it involved in a significant interaction effect. To conclude the analysis of the spatial variability a comparison was made between the spatial variability of F (the distance between throwing and catching of the same ball) and D (the distance between throwing and catching by the same hand). An ANOVA with Variable (F and D) as the additional within-subject factor revealed that 146 A.A.M. van Santvoord, P.J. Beek /Acta Psychologica 91 (1996) 131-151

the within-trial CVs of F were significantly smaller than those of D (0.22 vs. 0.32), F(1,4) = 46.67, p < 0.01, and that the degree of variability increased significantly with Pattern Size, F(3,12)= 14.67, p < 0.001. It also revealed that the added factor inter- acted significantly with Pattern Size, F(3,12) = 7.87, p < 0.01, due to the fact that the CVs of D increased more strongly with increasing pattern size than the CVs of F. Finally the effect of Skill was significant, F(1,4) = 12.10, p < 0.05, because the CVs were on average smaller in the expert jugglers than in the intermediate jugglers. In combination with the previous finding that the variability of F was larger than that of h, this result confirms the expectation that the spatial properties of the flight trajectories are more tightly controlled than those of the hand trajectories. Our next concern was the relative variability of the temporal variables of Eq. (2), which, averaged over time, had to be satisfied. By collecting the within-trial CVs of TF, T L, and T v in an ANOVA with Time Event as an additional factor, it was possible to compare the behavior of these variables under the task conditions imposed. It was found that the factor Time Event was a significant main effect, F(2,8) = 112.99, p < 0.0001, indicating the existence of significant differences between the CVs of Tr, TL, and T v. Post-hoc comparisons revealed that the mean values, which were 0.03, 0.06, and 0.10, respectively, differed significantly from one another. In addition to this main effect, a significant Time Event × Pattern Size interaction was found, F(6,24) = 4.64, p < 0.01, because on average the CVs of TL increased with increasing pattern size, while the CVs of T v decreased, and the CVs of Tr remained roughly invariant. Again, no effects of Task Constraint were found. This result is consistent with the suggestion from the analysis of the variability of the spatial measures that the throws and hence the flights of the balls are more tightly controlled than the trajectories of the hands, which holds in the temporal domain as well. To compare the temporal variability of the events associated with the hands and that of the events associated with the balls, a similar ANOVA was performed on the within-trial CVs of Tn and T s with Cycle Time as an additional factor. The theoretical motive for comparing these two quantities is that they define the numerator and the denominator of the left hand side of Eq. (2). A significant effect of Cycle Time was found, F(1,4) = 90.64, p < 0.001, because on average the CVs of hand cycle time were larger than the CVs of ball cycle time. In addition, a significant Task Constraint × Cycle Time interaction occurred, F(1,4) = 16.30, p < 0.05, which resulted from the fact that the CVs of ball cycle time were on average larger in the temporally constrained condition than in the spatially constrained condition (0.036 vs. 0.031), while the average CVs of hand cycle time differed only marginally in the two task conditions (0.048 vs. 0.047). Finally, a significant Skill × Task Constraint × Cycle Time interaction was found, F(1,4) -- 18.38, p < 0.05, which, according to post-hoc comparisons, was due to the fact that the CVs of hand cycle time remained about constant across the two task conditions in the experts (0.042 vs. 0.044), while they were larger in the temporally constrained condition in the intermediate jugglers (0.054 vs. 0.049). These results support both our expectations. The fact that variability of the ball cycle time was smaller than that of hand cycle time confirms the expectation that the spatiotemporal variability of the events associated with the balls is smaller than that of the events associated with the hands, while the fact that this effect was more pronounced in the spatially A.A.M. van Santvoord, P.I. Beek /Acta Psychologica 91 (1996) 131-151 147 constrained condition is consistent with the expectation that juggling to a height is accompanied with less (spatio)temporal variability than juggling to a beat.

4. General discussion

The goal of the present experiment was to examine the spatiotemporal characteristics of cascade juggling and to test the validity of the portrayal of (cascade) juggling as a spatial clock. A set of conceptually motivated variables was selected to assess the space-time patterns that were produced by expert and intermediate jugglers while juggling three balls in cascade fashion to either an externally specified height or to an externally specified beat. Specifically, it was expected that the observed spatiotemporal variability would be less when juggling to an externally specified height than to an externally specified beat. Furthermore, it was expected that in both conditions the spatiotemporal variability of the flight trajectories would be less than those of the hand trajectories. To what extent were these expectations confirmed by the results? To begin with the latter expectation, a number of clear indications were found that, under both task constraints, the jugglers primarily attempted to throw the balls as consistently as possible, both in space and in time. Spatially, the dispersion of the points of throwing was found to be smaller than that of the zeniths of the ballflights, which in turn was smaller than that of the points of catching. Although this finding may have been due in part to the chain of causation from throwing to catching, the variability of time flight was smaller than that of time loaded, which in turn was smaller than that of time unloaded. Furthermore, the variability of ball cycle time was smaller than that of hand cycle time, suggesting that the focus of control is on the temporal component of Shannon's equation relating to the events associated with the balls rather than those associated with the hands. In addition, an analysis of the variability of the horizontal and the vertical components of the velocity of release suggested that the height to which the balls are thrown is more tightly controlled than the horizontal distance over which the balls are thrown. Finally, the variability of the horizontal distance between throwing and catching by different hands proved to be less than that of the horizontal distance between throwing and catching by the same hand. Each of these variability measures was a function of increasing pattern size, except for the CVs of the horizontal and vertical component of the release velocity and, hence, the CVs of time flight and height, which remained invariant across the four pattern sizes. Collectively, these results support the notion of a spatial clock in showing that the focus of control is the space-time trajectory of the balls, which are thrown from a stable spatial projection point and caught in such a way that, of all relevant time components, variations in flight time are minimal. This finding was very robust in that it was independent of Task Constraint, Pattern Size, Skill, and Hand. With regard to the factor Skill, it is worth noting that the differences between expert and intermediate jugglers resided predominantly in the variability (CVs) of spatial variables such as the angle of release, the base width of the flight parabolas, and the width of the elliptical hand loops, and not in the temporal variables and variables associated with timing, such as height and the velocity of release. With regard to the effect of Hand, it can be remarked that 148 A~4.M. van Santvoord, Pal. Beek /Acta Psychologica 91 (1996) 131-151 only a main effect of hand was found for the CVs of the angle of release, which were smallest in the right hand. A few interaction effects involving Hand occurred, but these appeared to be of minor importance. Generally speaking, the absence of any main effects of Task Constraint on the variability measures of the individual variables indicates that the spatiotemporal variabil- ity of performance when juggling to a height and when juggling to a beat are very similar, which, as such, disconfirms our hypothesis. However, as the analysis was not restricted to the behavior of individual variables, it became clear that the spatiotemporal properties of the observed patterns are tightly interwoven and that there was at least one subtle difference in performance under the two task conditions. A comparison between the CVs of hand cycle time and of ball cycle time revealed not only that these differed significantly, but also that they varied as a function of Task Constraint. Particularly, it was found that the variability of the ball cycle time was smaller in the spatially constrained condition than in the temporally constrained condition, while the variability of hand cycle time was about the same in the two conditions. As flight time is directly related to the height of juggling, this finding is consistent with the expectation that the spatiotemporal variability of performance would be less when juggling to an externally specified height than when juggling to an externally specified beat, albeit at the level of ensemble variables only. Nevertheless, the effect in question is essential because hand cycle time and ball cycle time are the key quantities in Shannon's equation of juggling which implies that they cannot vary independently. It is important to realize, in other words, that the variables of juggling are so intertwined, that the effects of a manipulation of task constraint could only appear at the level of the ensemble variables of ball cycle time and hand cycle time and not at the level of the individual variables. Given the severe physical constraints of (cascade) juggling, it is remarkable that such a task-specific effect of instruction could indeed be demonstrated. Thus, the overall conclusion of this study ought to be that the conceptualization of (cascade) juggling as a spatial clock is appropriate in that jugglers minimize the variability of the flight time of the balls by throwing them as consistently as possible and modulating the catching to reduce timing errors. Therefore, juggling to an externally specified height seems to be more in line with what jugglers are actually doing than juggling to an externally specified beat. This finding, which was already anticipated in earlier publications (Beek, 1988, Beek, 1989a, Beek, 1989b; Beek and Van Santvoord, 1992), both complements and qualifies previous work on the temporal organization of cascade juggling. It suggests that the temporal patterning in cascade juggling should be appreciated in relation to the way in which jugglers set up and satisfy the spatial constraints that allow this temporal patterning to occur. Quite literally, the perceptual- motor problem of juggling is a spatial-temporal one that is solved primarily by exploiting the invariance of the gravitational gradient. By introducing a spatial constraint in the form of two intentionally selected targets in the work space, and by orchestrating the flow of the juggled balls around these selected regions in space, jugglers conve- niently create and sustain a timekeeper for the beat of the skill that provides the benchmark for the timing and sequencing of the subactions of juggling. As has been suggested before (Kelso, 1981; Kugler et al., 1980), the temporal properties of coordi- nated movements are not prescribed a priori by some internal entity but emerge from the A.A.M. van Santvoord, P.I. Beek /Acta Psychologica 91 (1996) 131-151 149 dynamics of the act as defined over the actor-environment system. The notion of time as an emergent property is inherent to the notion of a spatial clock; the fact that it was supported by the present experiments is therefore of interest for the understanding of the timing of movements in general. The theoretical implications of the present experiment, however, are not limited to the timing of movements and the need to develop a theory of the control of action in which time and space are truly integrated. It raises at least two other related issues of theoretical importance, viz. the issue of task-specificity and the issue of proper aids for perceptual-motor learning, that we briefly comment on before closing. Probably, the space-time relations in cascade juggling are rather specific to the activity in question. If it is true that animals and humans attempt to match their dynamic properties with those of their environment so as to achieve specific task goals, then the prevailing style of organization will be a function of the selected goals, and thus be highly task-specific. From this perspective, it is more than conceivable that there are other tasks and activities than juggling that also demand accurate control of movement kinematics, but in which the space and time are integrated in a different manner. Rather than restricting the study of the spatial-temporal properties of the control of action to a number of well-documented experimental tasks, it appears to be due time to examine space-time control in the context of a variety of complex actions, so as to be able to eventually delineate the generic principles and strategies that are brought to bear in solving perceptual-motor problems. An example of such a generic principle seems to be that perceptual-motor workspaces are often organized around stable fixed points, such as the projection points in juggling. Oscillatory movements are often not perfectly har- monic but have characteristic phases at which the variation of multiple trajectories is minimal. For example, analogous observations of such 'anchoring' have been reported for maximal extension in flexion-extension oscillations of the forearm, both in bimanual oscillation (Byblow et al., 1995) and in rhythmically tracking an oscillating stimulus (Byblow et al., 1994). It is reasonable to assume that such an in-depth analysis of space-time control in the context of specific skills will lead to the kind of insight needed to be able to adequately constrain and guide the acquisition of these skills. The suggestion implicit in the present experimental results is that, at least during certain stages of the learning process, jugglers will benefit more from spatial feedback than from temporal feedback, even though the aim is to achieve proper space-time relations. In retrospect, therefore, the finding that novice jugglers did not learn quicker with the aid of a metronome than without it (Beek and Van Santvoord, 1992), is understandable. As argued by Zanone and Kelso (1992): The degree to which information is useful for an actor is the degree to which it can actually penetrate and, thus, change the dynamics of the task space.

Acknowledgements

The research reported in this paper was supported in part by a grant from the National Science Foundation (SBR 94-22650) awarded to M.T. Turvey, University of Connecti- cut, USA. The authors are grateful to Wiero Beek, James Cauraugh, Claire Michaeis, 150 A.A.M. van Santvoord, P.I. Beek / Acta Psychologica 91 (1996) 131-151

Pier van Wieringen, and Howard Zelaznik for their very helpful comments on an earlier version of this article.

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