Why do we need a Mathematical Theory?
Any discipline can benefit from a basic mathematical theory if a valid theory can be formulated. Every field that was successfully mathematized has inexorably advanced by leaps and bounds. This is because once the theory is correctly formulated, the powerful mathematical tools and conclusions can all be applied with great certainty. Below is my first attempt at such a formulation for piano. As far as I know, it is the first of its kind in the history of man. Such virgin territory has historically yielded enormous benefits very quickly. I was surprised myself by how many useful, and sometimes hitherto unknown, conclusions we can draw from some very rudimentary theories, as you will see. Whatever math I use below is truly simple math. At this early stage, we can do a lot with the simplest concepts. Further advances are obviously possible by application of higher mathematics. I will also discuss some of those possibilities.
There is little question that the art of piano playing suffers from a total lack of mathematical analysis. In addition, no one doubts that speed, acceleration, momentum, force, etc., play critical roles in piano play. No matter what genius lies behind the artist, the music must be transmitted, through flesh and bone, and via a mechanical contraption consisting of wood, felt, and metal. Therefore, we are dealing with not only a mathematical, but also a total scientific approach that involves human physiology, psychology, mechanics and physics that all tie together to represent what we do at the piano.
The need for such an approach is demonstrated by the fact that there are many questions for which we still have no answers. What is a speed wall? How many are there? What causes them? Is there a formula for overcoming speed walls? What are pianists doing when they play harsh versus sweet, or shallow versus deep? Is it possible to teach two different pianists to play the same passage in exactly the same way? Is there any way to classify different finger motions like there is for the horse's gaits? We answer all of these questions below.
The advantages of a mathematical theory are obvious. For example, if we can mathematically answer the question of what a speed wall is (or what they are -- if the theory predicts more than one!), then the theory should immediately provide us with possible solutions on how to break the speed wall(s). Today, no one knows how many speed walls there are. Just knowing how many there are would be a terrific advance. It may be important to prove mathematically that no two pianists (or one pianist) can ever play the same piece exactly the same way. This is because, in that case, listening to someone else play may not be harmful because you cannot imitate it exactly anyway (assuming that exact imitation is not desirable), and trying to teach a student to imitate a famous artist exactly is then proven to be impossible. This will clearly affect how teachers teach students using examples of recordings from famous artists.
Until quite recently, chemists scoffed at physicists who were able to apply equations to lots of things but couldn't even come close explaining simple chemical reactions. Biology and medicine also initially developed in their own ways, with little math and using methods that were far removed from fundamental science. Medicine, biology, and chemistry, all started initially as pure art. Now, all three disciplines are intensely mathematical and rely on the most advanced scientific principles. The ensuring accomplishments in these fields are too numerous to discuss here. One example: in chemistry, the chemists' most basic foundation, the periodic table of the elements, was explained by physicists using quantum mechanics. As a result of becoming more scientific, all three disciplines are enormously successful and are advancing in leaps and bounds. The "scientification" of any discipline is inevitable; it is only a matter of time, because of the enormous benefits that can follow. The benefits of scientification will also apply to music.
So how do we apply the exact science of mathematics to something that is perceived as art? Certainly, in the beginning, it will be crude, but refinements are certain to follow. Already, piano technicians know that the piano itself is a marvel in the use of basic physics in its design. Piano technicians must be familiar with an enormous amount of science, math, and physics in order to ply their "art". A mathematical theory of piano playing must start with a scientific approach in which each item under discussion is clearly defined and classified; see "The Scientific Method" in Chapter Three. Once this is accomplished, we search for all relevant relationships between these objects. These procedures comprise the essence of Group Theory. It is elementary! Let us begin.