Spring 1992 • 29 by G . William Troxler Montpelier, M D
In the preliminary article of this eight-part series (DPN January 1992), Bill explained how to identify the relationships among musical tones and understand scales. He also familiarized us with some basic terms like octave, diatonic, chromatic, interval, fundamental, and solfege. Now he takes a look at how strings vibrate and produce sound, and some characteristics of that sound.
• Vibrating Strings Physicists have spent a great deal of time describing the movement of vibrating strings. They have impressive formulas to explain what most dulcimer players already know: the more tension applied to a string, the higher its pitch. Under equal tension, heavy gauge wire will produce a pitch lower than that of light gauge wire. If the tension on a string remains the same while its length increases, the pitch will drop. These are observable physical behaviors of strings. Yet to really understand how harmony works and how strings produce music, we must turn to physics for some not-so-obvious explanations. Pluck a dulcimer string and imagine it vibrating back and forth through its resting position until it runs out of energy. Figure 1 shows the string's position of maximum travel as the loop position and the ends of the strings as the nodes. This description is satisfying, but too simple. Actually, a string moves in very complex ways. Once plucked or struck, a string vibrates as though it were not only its full length, but also half its length, a third of its length, a quarter of its length, a fifth of its length and so forth on out to infinity! Here's another way to think of this complex motion. A single real string produces a sound that is the combination of an infinite number of "ideal" strings. One ideal string sounds as if it were as long as the entire string. Another ideal string sounds as if it were half the length of the actual string. Still another ideal suing sounds as if it were one third the length of the actual suing. Ad infinitum, these ever-shorter imaginary strings add tones to the acoustic energy radiating from the real string. The actual movement of the real string is the sum of the individual movements of these ideal strings. Figure 1 shows how the individual movements would appear and how the sum of the movements would look if we could take a stop-action photo of the string in motion.
• Overtones How does all this affect the sound the string produces? Well, when a suing is shortened by one-half, its pitch increases by a factor of two; that is an octave. Another way to think of this is that the half-length suing vibrates twice during the same time the fulllength suing vibrates once. So, when a dulcimer string is plucked, the resulting sound will include the fundamental tone plus the tone an octave higher; but mat's not all. When a string is shortened to one-third its original length, its pitch increases by 3/2. continued on page 31
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