### Speed Walls

Let's assume that a person starts practicing a piece of music by first playing slowly, using mostly serial play, since that is the easiest way (let's ignore chords for the time being). As the finger speed is gradually increased, s/he will naturally hit a speed wall because human fingers can move only so fast. Thus we have mathematically discovered one speed wall, and that is the speed wall of serial play. How do we break this speed wall? We need to find a play method that has no speed limit. That is parallel play. In parallel play, you increase the speed by decreasing the phase difference. That is, speed is numerically proportional to the inverse of the phase difference. Since we know that the phase difference can be decreased to zero (which gives you a chord), we know that parallel play has the potential give you infinite speed and therefore it has no theoretical speed limit. We have arrived at a mathematical foundation for the chord attack!

The distinction between serial and parallel play is somewhat artificial and oversimplified. In reality, practically everything is played parallel. Thus the above discussion served only as an illustration of how to define or identify a speed wall. The actual situation with each individual is too complex to describe (because speed walls are caused by bad habits, stress and HT play), but it is clear that wrong playing methods are what create speed walls and each person has her/is own mistakes that result in speed walls. This is demonstrated by the use of parallel set exercises which overcome the speed walls. This means that speed walls are not always there by themselves, but are created by the individual. Therefore, there is any number of possible speed walls depending on each individual and every individual has a different set of speed walls. There are, of course common classes of speed walls, such as those created by stress, by wrong fingering, by lack of HS technique, lack of HT coordination, etc. It would be, in my opinion, very counterproductive to say that such complex concepts will never be scientifically or mathematically treated. We have to. For example, in parallel play, phase plays a very important part. By decreasing the phase to zero, we can play infinitely fast, in principle.

Can we really play infinitely fast? Of course not. So then what is the ultimate parallel speed limit, and what mechanism creates this limit? We know that different individuals have different speed limits, so the answer must include a parameter that depends on the individual. Knowing this parameter will tell us how to play faster! Clearly, the fastest speed is determined by the smallest phase difference that the individual can control. If the phase difference is so small that it cannot be controlled, then "parallel play speed" loses its meaning. How do we measure this minute phase difference for each individual? This can be accomplished by listening to her/is chords. The accuracy of chord play (how accurately all the notes of the chord can be played simultaneously) is a good measure of an individual's ability to control the smallest phase differences. Therefore, in order to be able to play parallel fast, you must be able to play accurate chords. This means that, when applying the chord attack, you must first be able to play accurate chords before proceeding to the next step.

It is clear that there are many more speed walls and the particular speed wall and the methods for scaling each wall will depend on the type of finger or hand motion. For example you can attain infinite speed with parallel play only if you have an infinite number of fingers (say, for a long run). Unfortunately, we have only ten fingers and often only five are available for a particular passage because the other five are needed to play other parts of the music. As a rough approximation, if serial play allows you to play at a maximum speed of M, then you can play at 2M using two fingers, 3M using three fingers, etc., serially. The maximum speed is limited by how rapidly you can recycle these fingers. Actually, this is not quite true because of momentum balance (it allows you to play faster), which will be treated separately below. Thus each number of available fingers will give you a different new speed wall. We therefore arrive at two more useful results. (1) there can be any number of speed walls, and (2) you can change your speed wall by changing your fingering; in general, the more fingers you can use in parallel before you need to recycle them, the faster you can play. Putting it differently, most conjunctions bring with them their own speed walls.