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I enjoyed reading this interesting article, but I noticed that there are some very important variables that are not addressed: the most important is bow speed. The alternating sticking and slipping that is described is dependent upon the speed of the bow. The ideal speed will correspond to the speed of the vibrating string at that point. A bow placement that is closer to the bridge (node) will require a slower bow speed; as the bow placement distance from the bridge increases the bow speed will need to increase to keep up with the amplitude of the vibrating string further from the node. This third variable (in addition to the two you described in your article) creates an exponentially more complex interrelationship between the bow and the string. These variables are stable for a given string and frequency, but these change often when playing the violin (or in my case, the cello). There are still two more variables that must be accounted for: 1) Whenever the finger is depressed, the vibrating string length changes and the optimal placement of the bow must be changed accordingly. 2) Since the optimal speed of the bow is dependent upon the frequency of the vibrating string, the bow speed must be repeatedly adjusted to maintain the relationship to the frequency of the vibration. Take an oversimplified example where the optimal bow speed of A 110 is 1 foot per second. The bow speed for A 220 will need to increase to 2 feet per second, A 440 will be 4 ft/sec and A 880 will be 8 ft/sec. This can prove to be counterintuitive for a player because it means that very high frequencies are best executed with a very fast bow and close to the bridge, something that we normally avoid because of the node/speed relationship described above. The actual bow speed/frequency relationship is less than the mathematically correct example given because the amplitude of a long, low frequency string is much larger than that of a short, high frequency string. Even these variables are only the basis for the creation of the wide variety of qualities of sound that expressive players use to communicate emotion. More bow weight and slower speed, for example will produce a gripping sound while a faster, lighter bow over the fingerboard produces a purer, more intimate and affective sound.

It makes me smile to think of how wonderfully the human brain is able to constantly crunch all of these calculations and imbue an artist's intuition with a beautiful, expressive bow arm.