The performance of complex polyrhythms—rhythms where the left and right hand move at different rates—is usually the province of highly trained individuals. However, studies in which hand movement is guided haptically show that even novices can perform polyrhythms with no or only brief training. In this study, we investigated whether novices are able to tap with one hand by matching different rates of a metronome while sawing with the other hand. This experiment was based on the assumption that saw movement is controlled consistently at a predictable rate without the need for paying primary attention to it. It would follow that consciously matching different stipulated metronome rates with the other hand would result in the spontaneous performance of polyrhythms. Six experimental conditions were randomised: single handed tapping and sawing as well as four bimanual conditions with expected ratios of 1:1 (performed with and without matching a metronome) as well as 3:4 and 4:3 (performed matching a metronome). Results showed that participants executed the saw movement at a consistent cycle duration of 0.44 [0.20] s to 0.51 [0.19] s across single and bimanual conditions, with no significant effect of the condition on the cycle duration (p = 0.315). Similarly, free tapping was executed at a cycle duration of 0.48 [0.22] s. In the bimanual conditions, we found that for a ratio of 4:3 (4 taps against 3 sawing cycles per measure), the observed and predicted ratio of 0.75 were not significantly different (p = 0.369), supporting our hypothesis of the spontaneous adoption of polyrhythms. However, for a ratio of 3:4 (3 taps against 4 sawing cycles per measure), the observed and predicted ratio differed (p = 0.016), with a trend towards synchronisation. Our findings show that bimanual independence when performing complex polyrhythms can in principle be achieved if the movement of one hand can be performed without paying much—if any—attention to it. In this paradigm, small rhythmic arm movements are possibly driven by an intrinsic timing which leads to spontaneous convergence on a cycle duration of around 0.5 s, while the movement of the other hand can be controlled consciously to occur at desired rates.