These results also provide the mathematical basis for explaining the well known trick of alternating fingers when playing the same note many times. One might think at first that using just one finger would be easier and offer more control, but that note can be played repetitively faster by playing parallel using as many fingers as you can for that situation, than playing serially.
The need for parallel play also singles out trills as a particularly difficult challenge to play fast because trills must in general be executed with only two fingers. If you tried to trill with one finger, you will hit a speed wall at, say, speed M; if you trill with two fingers, the speed wall will be at 2M (again, ignoring momentum balance). Does mathematics suggest any other way of attaining even higher speeds? Yes: phase truncation.
What you can do is to lower the finger to play the note but then raise the finger only sufficiently to reset the repetition mechanism, before playing the next note. You may need to raise the finger by only 90 degrees instead of the normal 180 degrees. This is what I mean by phase truncation; the unnecessary part of the total phase is truncated off. If the original amplitude of finger travel for the 360 degree motion was 2 cm, with a 180 degree truncation, the finger now moves only 1 cm. This 1 cm can be further reduced until the limit at which the repetition mechanism stops working, at about 5 mm. Phase truncation is the mathematical basis for the fast repetition of the grand and explains why the rapid repetition is designed to work with a short return distance.
A good analogy to gaining speed in this way is the dribbling action of a basketball, as contrasted to the swinging action of a pendulum. A pendulum has a fixed frequency of swing regardless of the swing amplitude. A basketball, however, will dribble faster as you dribble closer to the ground (as you reduce the dribble amplitude). A basketball player will generally have a hard time dribbling until s/he learns this change in dribble frequency with dribble height. A piano acts more like a basketball than a pendulum (fortunately!), and the trill frequency increases with decreasing amplitude until you reach the limit of the repetition mechanism. Note that even with the fastest trill, the backcheck is engaged for a correct trill, because the keys must always be depressed completely. The trill is possible because the mechanical response of the backcheck is faster than the fastest speed that the finger can achieve.
The trill speed is not limited by the piano mechanism except for the height at which the repetition stops working. Thus it is more difficult to trill rapidly with most uprights because phase truncation is not as effective. These mathematical conclusions are consistent with the fact that to trill fast, we need to keep the fingers on the keys and to reduce the motions to the minimum necessary for the repetition mechanism to work. The fingers must press "deeply into the piano" and just lifted sufficiently to activate the repetition mechanism. Furthermore, it helps to use the strings to bounce the hammer back, just as you bounce the basketball off the floor. Note that a basketball will dribble faster, for a given amplitude, if you press down harder on it. On the piano, this is accomplished by pressing the fingers firmly down on the keys and not letting them "float up" as you trill.
Another important factor is the functional dependence of finger motion (purely trigonometric, or hyperbolic, etc.) for controlling tone, staccato, and other properties of the piano sound relating to expression. With simple electronic instruments, it is an easy task to measure the exact finger motion, complete with key speed, acceleration, etc. These characteristics of each pianist's playing can be analyzed mathematically to yield characteristic electronic signatures that can be identified with what we hear aurally, such as angry, pleasant, boisterous, deep, shallow, etc. For example, the motion of the key travel can be analyzed using FFT (fast Fourier transform), and it should be possible, from the results, to identify those motion elements that produce the corresponding aural characteristics. Then, working backwards from these characteristics, it should be possible to decipher how to play in order to produce those effects. This is a whole new area of piano play that has not been exploited yet. This kind of analysis is not possible by just listening to a recording of a famous artist, and may be the most important topic for future research.