One method for answering this question is based on a style of analysis developed by Alan Wing, Dirk Vorberg, and their colleagues. If a person repeatedly performs a rhythmic 3:2 pattern, each repetition will exhibit random, unintended temporal variations from the desired pattern of intertap intervals. Independent components of this temporal variation can be identified statistically and are assumed to correspond to independent aspects of an internal pattern generator. Different internal pattern generators can be identified from different patterns of intertap covariation.
For example, the 3:2 polyrhythmic pattern consists of a simultaneous tap by both hands, followed by a right, left, right sequence: (Right1, Left1)---->Right2-->Left2-->Right3---->(Right1, Left1)--> etcetera.
If a person makes his or her hands perform independently after the initial simultaneous tap, then the last interval performed by the right hand (Right2 to Right3) should be uncorrelated with the interval performed by the left hand (Left1 to Left2). In contrast, if a person integrates his or her hands into a single pattern, an interval occurring early in the pattern (Left1 to Right2) should be uncorrelated with an interval occurring late in the pattern (Left2 to Right3).
Beth Marshburn, Stu Klapp, Mari Jones, and I analyzed 3:2 tapping by moderately skilled piano players. The pattern of correlations indicated performers were not making their hands perform independently, but were integrating their hands into a single pattern. This result converges with conclusions drawn by Diana Deutsch, Jeff Summers, Jeff Pressing, and their respective colleagues. If integration is the preferred method of coordinating the hands, then it has certain implications for perceptual-motor compatibility and for training.
One can find conflicting recommendations in the piano literature for single-handed and dual-handed practice of polyrhythms. The present results suggest that dual-handed practice may be more effective for establishing a polyrhythm, given that it is an integrated pattern.
Although most golf instructors urge players to maintain a constant rhythm, individual instructors differ in how they believe rhythm is related to tempo. Some instructors recommend keeping tempo constant, lest variations disrupt rhythm. Others recommend varying tempo depending on the demands of the shot. This latter recommendation suggests that tempo is behaviorally independent of rhythm.
As will be reported in the June issue of the Journal of Motor Behavior, Neil Greenberg, Min-Ju Liao, and I tested this hypothesis by attaching an accelerometer to the clubhead of an eight-iron. Moderately skilled golfers were required to hit a plastic golfball into a target net about 15 feet away. We partitioned the accelerometer signal into a backswing, downswing, and follow-through. Then we analyzed the temporal structure of shot-to-shot variability in a manner similar to the previously described analysis of polyrhythmic tapping. We were particularly interested in whether we could detect some hierarchical structure among the backswing, downswing, and follow-through that would suggest that tempo could be varied independently from rhythm, that is, that components of the swing were some proportion of the overall swing duration. We found little evidence for such a structure. Rather, it appears that the backswing, downswing, and follow-through are approximately independent intervals that are chained together to create the overall swing. Given this temporal structure, it would be very difficult to vary tempo without also varying rhythm. The results converge with findings by B. Abernethy, R. J. Neal, and colleagues and suggest that golf instructors who recommend a con-stant tempo are offering good advice.
Partial support for this hypothesis comes from comparing the shapes of repeated swing trajectories in an abstract representational space. The space is formed by plotting the accelerometer signal against successively delayed versions of itself. The resulting graph shows that, during the backswing, the trajectories representing multiple swings form a tight bundle. However, part way through the downswing, they rapidly come apart, and then stay apart for the remainder of the swing. The swing may be unstable in the region in which the trajectories separate. However, converging measures of stability are still necessary to test this hypothesis.
The stability of polyrhythmic performance can also be investigated by examining abstract trajectories. In this case, the cyclic movements of the two hands can be crossplotted in an abstract state space. Lieke Peper, Peter Beek, and I are investigating the hypothesis that the topological shapes of the abstract trajectories of polyrhythms are related to their stability. As suggested by dynamical systems theory, the maintenance of stable performance may be a key factor in determining task difficulty. Analyses of trajectory shape may give additional insight into the integrated nature of polyrhythmic coordination.
Although I know that not all of my colleagues would agree, I expect that as theory evolves in this area, it will be strongly influenced by both correlational analyses of variability and analyses of trajectory shapes. For now these approaches provide different insights into what makes some rhythmic tasks so difficult, and they have strong implications for training.