THEOREMSIn mathematics, a theorem is a statement that can be proved either "true" or "false" by a chain of reasoning. Two types of proofs employed by mathematicians are the Direct Proof and the Indirect Proof:
- A Direct Proof assumes that a theorem is true and then sets out to directly prove the truth of that theorem.
- An Indirect Proof, on the other hand, assumes that a theorem is true or false. However, in the midst of the proof a contradiction to the theorem is found, thus nullifying the validity of the original theorem.
It is important to note that only ONE contradiction is needed to disprove any theorem, thus nullifying its validity. Additionally, a theorem that is considered to be true must hold for ALL cases.
Here is an example of a Direct Proof:
THEOREM: TWO apples grouped together with another TWO apples results in a group of FOUR apples.
PROOF: Take two apples and place them on a table. Do the same with two other apples. There are four apples when you count them all.
Here is an example of an Indirect Proof:
THEOREM: TWO apples grouped together with THREE apples results in a group of FOUR apples.
PROOF: Take two apples and group them with three other apples. Now count them. (Notice, we are in the midst of the proof.) In counting them we find that the sum is 5...a contradiction!
By contradicting the theorem, our conclusion is not necessarily that (2 + 3) apples equals 5 apples, but that (2 + 3) apples is not equal to 4 apples. Remember, we are concerned with proving or disproving the original theorem.
Up until five years ago, I would routinely present my students with new theorems on technique on a fairly regular basis. However, the idea that I should be able to prove my theorems to them never entered my mind. I was continually seeking answers that made sense, and each time I made a new discovery I always managed to convince myself that my new idea was both logical and rational. Difficulty inevitably arose when I found (either weeks or months later) that my theorem could easily be contradicted. On many occasions I was required to swallow my pride and start from scratch.
In the scientific community, theorems must be subjected to the rigors of a proof before they are accepted as fact. However, within the nonscientific world of the musician, theories are often confidently asserted without a shred of evidence to support them. Following are some of the theories that have gained acceptance in musical circles over the years. Perhaps we can find some contradictions.
THEOREM: Your fingers do not lift while playing a passage.
PROOF: If this is true, then each of your finger tips should remain in contact with the surface of its respective key while playing the example below. However, it is impossible to precisely articulate the individual notes of the passage playing in this manner. The only remaining alternative is to lift your fingers before they descend. (Contradiction)
THEOREM: The articulation of the notes in a passage is produced by the transference of arm weight from note to note aided by the rotational motion of the wrist and the "larger muscles" of your arms and shoulders.
PROOF: Try using what are referred to as the larger muscles of your arms and shoulders to assist you in the realization of the above example. Do whatever you like…make circles with your wrist, pronate your wrist, swing your elbows in and out, lean forward with your upper torso…but do not move your fingers up or down. The notes of the above passage are nowhere near being cleanly articulated as I do this.
Another contradiction: I am effortlessly playing the above passage solely with my fingers. The individual notes are cleanly articulated. Additionally, I am grasping my right wrist with my left hand to prevent any interference from the "larger muscles" mentioned above. I am also leaning against the back of my chair as I do this.
THEOREM: Music and technique cannot be separated.
PROOF: Suppose that a composer has completed an extended piece for a huge xylophone which she has invented. Instead of using light mallets, the new instrument requires the use of ten-pound mallets to produce an optimum sound. She is familiar with two percussionists who are virtuosos on the traditional xylophone and must select one of them on short notice to premiere her piece. One man is a competitive weightlifter while the other man rarely exercises and is considerably weaker. Although weightlifting has nothing to do with music, it is logical to conclude that the stronger man—through his ability to manage the ten-pound mallets with relative ease—will produce a more controlled and technically polished performance than the weaker man. (Contradiction)
THEOREM: Playing legato builds technique.
PROOF: I played legato over a five year period with no improvement in my technical ability. (Contradiction)
THEOREM: Play with curved fingers.
PROOF: Horowitz played with relatively flat fingers much of the time. (Contradiction)
THEOREM: Play with flat fingers.
PROOF: When I saw Charles Rosen perform many years ago he played with extremely curved fingers. Bach is known to have played with curved fingers. (Contradiction)
THEOREM: Short people are better pianists than tall people because short people are more coordinated than tall people.
PROOF: To my knowledge, the claim that "short people are more coordinated than tall people" has never been scientifically proven. Extremely coordinated individuals exist within "short" populations as well as "tall" populations. Many great soccer and baseball players are short while some of the most coordinated people on earth are members of the United States’ N.B.A. where the average basketball player is approximately 6’6". If large random samples of short and tall people were taken, they would most likely result in two normal distributions reflecting the fact that the majority of both populations were of average coordination with a small percentage of both populations above average and a small percentage of both populations below average. (Contradiction)
THEOREM: Piano technique has nothing to do with the development or strenghthening of muscles. Rather, the ability to play a piece or passage with physical ease is due to the proper coordination of opposing muscles.
PROOF: The ability to perform any task is based on the coordination of muscles. However, kinesiologists—medical professionals who are experts in the science of human motion—maintain that coordination is dependent upon, and is enhanced by the proper development of muscles. Here are some examples demonstrating this fact:
I could go on and on. However, I think I have made my point. People simply are not in agreement as to how to teach technique and most theorems can easily be contradicted. So where does this leave us? I hope you will read on.
- An American League pitcher, capable of throwing a 95-mile-per-hour fastball, attempts the new task of throwing a heavier ball at the same speed. Although he understands how to coordinate the task of throwing a 95-mile-per-hour fast-ball, his muscles are not strong enough to coordinate the new task with the heavier ball.
- A woman capable of performing the task of walking bends incorrectly while lifting a heavy object and severely pulls a muscle in her back. The woman can no longer walk and must rest until the muscle can once again support the coordinated movements of that particular task.
- A stone mason capable of breaking 20 pieces of stone per hour with a sledge hammer attempts the new task of breaking 40 pieces per hour. He understands how to coordinate the new task through his experience of working at the faster rate for short periods of time. However, over a period of an hour he is incapable of coordinating the new task because of the incapacitating muscular fatigue that develops.
- A pianist capable of playing a physically demanding piece at half speed attempts the new task of playing the piece up to speed. She understands how to coordinate the new task through her experience of playing sections of the piece—which are shorter in duration—up to speed. However, she is incapable of coordinating the new task for the duration of the entire piece because of the incapacitating muscular fatigue that develops.