Percussion instruments are characterized by vibrations initiated by striking a tube, rod, membrane, bell, or similar object. Percussion instruments are almost certainly the oldest form of musical instrument in human history. The archeological record of percussion instruments, in particular the bianzhong bells of ancient China, give clues to the history of music theory. From a mathematical point of view, percussion instruments are of special interest because unlike other types of instruments such as strings and winds, the resonant overtones typically do not follow the harmonic series. (See Harmonics entry in this volume.) In the last fifty years, a question of great interest in applied mathematics has been the famous inverse problem: Can one hear the shape of a drum?
Rods and Bars
Some percussion instruments produce a distinct pitch by the vibration of a rod or bar. Examples included the tuning fork (a U-shaped metal rod suspended at its center), a music box (a metal bar suspended at one end) and the melodic percussion instruments such as the xylophone and marimba (suspended at two non-vibrating points or nodes along the length of metal or wooden bars). Like vibrating strings, the frequency of the bar’s vibration, and hence the pitch of the musical sound it produces, are determined by its physical dimensions. In contrast to the string, in which the frequency varies inversely with the length, the vibrating bar has a frequency that varies with the square of the length. The resonant overtone frequencies fn of the vibrating bar are related to the fundamental frequency f1 by the formula
fn = a(n+1/2)2 f1,
where the constant a is determined by the shape and material of the bar. In contrast with the harmonic overtone series of vibrating strings
fn = n f1,
these inharmonic overtones give percussion instruments their distinct metallic timbre. The overtones of vibrating bars decay at different rates, with rapid dissipation of the higher overtones responsible for the sharp, metallic attack, while the lower overtones persist longer. The bars of the marimba are often thinned at the center, effectively lowering the pitch of the certain overtones, in accord with the harmonic series.
Bells
Like the vibrating bar instruments, the classic church bell possesses highly non-harmonic overtones. These are typically tuned by thinning the walls of the bell along the circumference at certain heights. A distinctive feature in the sound comes from the fact that apart from the fundamental pitch, the predominant overtone of the church bell sounds as the minor third above the prevailing tone. This accounts for the somber nature of the sound.
The bianzhong bells of ancient China were constructed in a manner that produced two pitches for each bell, depending on the location at which it was struck. In the 1970s, a set of 65 such bells were discovered during the excavation of the tomb of Marquis Yi in Hubei Provence. The inscriptions on the bells make it clear that octave equivalence and scale theory were known in China as early as 460 BCE.
Membranes
Drums are perhaps the most common percussion instrument. Consisting of vibrating membranes (the drum heads) stretched over one or both ends of a circular cylinder, drums exhibit a unique mode of vibration which accounts for their characteristic sound. Mathematical models of vibrating drumheads provide a fascinating application of partial differential equations. The inharmonic overtone frequencies are distributed more densely than for vibrating strings or rods. Further, each overtone is associated with a particular vibration pattern of the drum head. These regions can be characterized by the non-vibrating curves (nodes) that arrange themselves in concentric circles and diameters of the drum head.
An important question in the study of partial differential equations asks “Can one hear the shape of a drum?” That is, can mathematical techniques be used to work backwards from the overtone frequencies to determine the shape of the drum head that caused the vibration? The answer, as it turns out, is “not always”.
FURTHER READINGS:
Cipra, Barry. ”You Can’t Always Hear the Shape of a Drum” in What’s Happening in the Mathematical Sciences, volume 1. New Haven: American Mathematical Society, 1993.
Jing, M. “A Theoretical Study of the Vibration and Acoustics of Ancient Chinese Bells.”
Journal of the Acoustical Society of America. (v.114/3, 2003).
Rossing, Thomas, D. The Science of Percussion Instruments. Hackensack, NJ: World Scientific, 2000
Sundberg, Johan. The Science of Musical Sounds. San Diego: Academic Press, 1991
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