Percussion instruments and math
Percussion instruments are a diverse category of musical tools characterized by their sound production through vibrations initiated by striking various surfaces, such as tubes, rods, or membranes. They are considered some of the oldest musical instruments in human culture, with historical artifacts like the bianzhong bells from ancient China illuminating their significance in music theory. Mathematically, percussion instruments present a unique study due to their overtones, which do not follow the harmonic series typically found in string instruments. For instance, the frequencies of vibrating bars, such as those seen in xylophones, vary with the square of their length, leading to distinct inharmonic overtones that contribute to their characteristic sound.
Drums, a commonly recognized type of percussion instrument, utilize vibrating membranes stretched over a cylinder, resulting in a complex mode of vibration and a distribution of overtones that differ from other instruments. Research in spectral geometry poses intriguing questions, such as whether the shape of a drum can be discerned by analyzing its sound frequencies. Overall, the intersection of percussion instruments and mathematics reveals a rich field of study that enhances understanding of both musical acoustics and mathematical principles.
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Percussion instruments and math
Summary: The vibrations that emanate from percussion instruments vary mathematically based on the type of instrument.
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 string and wind instruments—the resonant overtones typically do not follow the harmonic series. In the last half of the twentieth century, 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 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
where the constant α 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 feature 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 the Hubei Provence. The inscriptions on the bells make it clear that octave equivalence and scale theory were known in China as early as 460 b.c.e.
Membranes
Drums are perhaps the most common percussion instrument. Consisting of vibrating membranes (called 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 (called “nodes”) that arrange themselves in concentric circles and diameters of the drum head.
An important question in the study of spectral geometry asks: “Can one hear the shape of a drum?” In other words, can mathematical techniques be used to work backwards from the overtone frequencies to determine the shape of the drumhead that caused the vibration? The answer, as it turns out, is “not always.”
Bibliography
Cipra, Barry. “You Can’t Always Hear the Shape of a Drum.” In What’s Happening in the Mathematical Sciences. Vol. 1. New Haven, CT: 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 114, no. 3 (2003).
Rossing, Thomas, D. The Science of Percussion Instruments. Singapore: World Scientific Publications, 2000.
Sundberg, Johan. The Science of Musical Sounds. San Diego, CA: Academic Press, 1991.