The Difficulties of Polyrhythms and Golf

by Richard J. Jagacinski, PhD

Over the years, I have enjoyed thinking about playing the piano and playing golf at least as much as actually performing these activities. Such enjoyment is very fortunate given the time constraints imposed by my teaching and research. What occupies my thoughts is the question of what makes these rhythmic skills so difficult. For example, consider the task of performing three beats of a musical piece with the right hand, while simultaneously performing two beats with the left hand, a 3:2 polyrhythm. The task of each individual hand, when performed in isolation, is extremely simple. In contrast, the difficulty of doing the two together is considerable and quite incommensurate with their individual difficulties. Accounting for this surprising degree of difficulty is a challenge for theories of perceptual-motor coordination.

A Correlational Analysis of Polyrhythmic Tapping

There are two basic strategies that one might use to cope with the difficulty of performing a polyrhythm. One strategy is to make the two hands as independent as possible so that they are effectively working in isolation. Then each hand has a simple task of performing evenly spaced taps. A second strategy is to combine the tasks for the two hands so that they effectively become one integrated task. The empirical challenge is to distinguish which of these strategies people actually use.

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.

Implications for Perceptual-Motor Compatibility

Research on the perception of tonal patterns has revealed that, sometimes, the patterns may be heard as an integrated whole, and, sometimes, they may be heard as two parallel streams. This latter phenomenon, auditory streaming, occurs when there are large pitch separations between two subsets of tones and when the rate of presentation is relatively fast. What if performers must synchronize their tapping to a set of tones forming a 3:2 pattern, and the tones form either an integrated or streamed percept? These conditions can be achieved by presenting the 3-tone subpattern at a high pitch and the 2-tone subpattern at a low pitch, and by making the pitch difference either small or large. If performers use an integrated motor pattern to coordinate their two hands, one would expect better performance with an integrated auditory percept than with a streamed percept. Consistent with this expectation, in the previously mentioned experiment with moderately skilled piano players, 3:2 tapping was less variable when the pitches were close together than when they were far apart. In an experiment led by Stu Klapp with musically naive performers, the performance benefit of an integrated percept was even more marked. These effects are believed to be due to the simpler structural correspondence between the integrated tonal percept and the integrated motor pattern, an example of temporal perceptual-motor compatibility.

Implications for Training

These results suggest that practicing the 3:2 pattern with an integrated tonal percept may be easier than practicing with a streamed tonal percept. The integrated nature of the motoric pattern also suggests that effective rhythmic practice should involve the two hands together. In other words, whole-task practice should be better than part-task practice for each hand separately. To test this hypothesis, Zeke Martin, Stu Klapp, and colleagues compared whole-task training of both hands together with part-task training of each hand separately in a 3:2 polyrhythmic tapping task. Whole-task training with integrated tones was found to be superior to part-task training with streamed tones. The latter condition was designed to encourage independent performance of the two hands, but was unsuccessful.

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.

A Correlational Analysis of a Golf Swing

Hitting a golf ball is another rhythmic task that most people find quite difficult. I am no match on the golf course for my 83-year-old father, but I do enjoy thinking about the temporal structure of a golf swing. For many golfers, the rhythm of a stroke refers to the pattern of speeding up and slowing down during the swing. It can be distinguished from tempo, which typically refers to the overall speed of the stroke.

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.

Stability and the Shapes of Abstract Trajectories

An additional aspect of the golf study was that 65-year-old golfers were found to swing somewhat faster than 20-year-old golfers. This result is intriguing because older adults typically perform more slowly than younger adults. It is also intriguing that there are so many articles in golf magazines that give advice on how to slow down one's swing, rather than on how to speed it up. Our working hypothesis is that part of the golf swing may be unstable, and that older golfers may have more difficulty than younger golfers in controlling this instability.

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.