Serial, Parallel Play
The finger motions for playing the piano can be classified at the most basic level as serial or parallel. In serial play, each finger is lowered in succession in order to play. A scale is an example of something that can be played serially. In parallel play, all fingers move together. A chord is an example of parallel play. As we shall see later, a scale can also be played parallel.
Serial play can be described by any oscillatory function such as a trigonometric function of hyperbolic function. It is basically characterized by an amplitude (the distance you move the finger up and down) and a frequency (how fast you play). Except for chords and rapid rolls, most slow pieces can be played serially, and beginners tend to start by playing serially. In parallel play, there is a well defined phase relationship between different fingers. Therefore, we must now discuss phase in some detail.
Phase is a measure of where the finger is, in relation to the other fingers. Suppose that we use the trigonometric function (sine, cosine, etc.) to describe finger motion. Then in its rest position, the finger is, say, at zero degrees in phase space. Since we know how pianos should be played, we will build some of that knowledge into our definition of phase. Since lifting the fingers off from the keys is in general not the correct way to play, we will define the zero of the phase as the upper rest positions of the keys. Thus the zero phase position of the black keys will be higher than the zero phase position of the white keys by the extra height of the black keys. Furthermore, we assume that if you lift the fingers off the keys, this extra motion does not count as far as the phase is concerned. These conventions are in accord with good technique and also simplify the mathematics. Then the phases of this motion can be defined as: finger depressing half way down = 90 degrees; depressing to bottom position = 180 degrees, rising half way up = 270 degrees, and rising back to the original position = 360 degrees, which is also zero degrees again. Now in parallel play, if the second finger starts its motion when the first finger is at 90 degrees, the third finger starts off when the first finger is at 180 degrees, etc., then this parallel play will play notes 4 times faster than serial play at the same finger speed. In this case the phase difference between fingers is 90 degrees. If you were to decrease the phase difference to 9 degrees, the notes will be played 40 times faster -- this example illustrates the power of parallel playing for speeding up your play. In a chord, the phase difference is zero.
Serial play can be defined as parallel play in which the phase difference between successive fingers is about 360 degrees or larger, or in which the phases are not inter-related. Hand motion helps both serial and parallel play, but in different ways. It helps serial play by adding to the amplitude. But it affects parallel play in a most important way by helping you to control the phase. With these simple definitions, we can start to generate some useful results.