I never thought that my Bachelor Thesis would ever come in handy. Thank you for this question!
No, you cannot keep up two unrelated rhythms in a stable coordinated fashion when tapping for your finger for instance.
Let us start with (and skip over) the oldest paper that I have about it. Haken, Kelso and Bunz (1985) described in a mathematical model how you would coordinate two rhythmic movements (tapping fingers) and how you change phases. In-phase means you tap at the same time, whereas anti-phase means you tap with one whilst going up with the other and vice versa. I will not describe these formula, but now you know they exist.
From now on, I will refer to "two agents" instead of fingers. I do this, because the coupling and rhythm is not limited to body parts. Even different people start to get synchronized during rhythmic movements. Richardson (2007) showed that people, without explicit instructions, started to synchronize their movements in a rocking chair. However, also during walking/running, people start to synchronize (Nessler, 2012).
What research over the years has shown that (stability of) rhythmic movements rely on three factors to be coordinated.
- First, you need to be coupled, i.e. you must be able to see/feel the other agent's rhythm. When you don't perceive a rhythm, their is nothing to adapt to (See also Richardson, 2007; Nessler, 2012; Meerhof & De Poel, 2014).
- Second, the phase of the movements: Moving in-phase is very easy and you can do that from low frequencies up to very high frequencies. Moving in anti-phase is a little more difficult. The agents will be stably coordinated at low frequencies, however, as you increase the frequency it will be more and more difficult to keep it up, ultimately leading to a synchronization period, after which you'll move in an in-phase pattern (try it out; Meerhof & De Poel, 2014).
- Third, as I already said just now: The frequency of movements. At low frequencies movements are easier to coordinate in a stable fashion than higher frequencies (Konvalinka, 2011)
In this last study, two participants in different rooms listened to a beat (of either 96, 120, 150 Hz, which could differ between participant) for eight seconds, after which they had to continue that beat by tapping their index-finger. Following, depending on the condition, they (1) kept hearing the computer-generated beat, (2) listened to their own generated beat, (3) listened to the beat of the other subject while the other heard his own generated beat, or (4) the two participants hear the beats generated by the second participant. There were many many results given all these conditions. What it all came down to was that the participants' own beat-frequency moved toward the other's, and that the rhythm became less stable, when they listened to the beat of the other participant. When the participants heard each other, the frequencies moved to each other and ultimately produced a very stable beat (they synchronized).
This paper thus showed that, whenever you are coupled (which fingers always are, it's called proprioception) you tend to synchronize. It is thus impossible to keep rhythm when you perceive a completely unrelated beat.