String instruments (mathematics)
String instruments, such as violins, guitars, harps, and cellos, rely on the physical principle of vibration to produce sound. When a string is impacted, it vibrates at multiple frequencies, creating sound waves that interact with the surrounding air. Each string instrument is designed with mathematical properties that influence sound production, including string tension, shape, and the spacing of frets. The pitch of a note is determined by the string's length, with deeper pitches corresponding to longer strings. Instruments can generate various notes either by using multiple strings or by altering the effective length of a single string. The mathematical relationship of frets on instruments like the guitar is based on scaling ratios that are connected to historical mathematical problems. Additionally, timbre, or the unique quality of sound produced by different instruments playing the same note, plays a crucial role in how we perceive music. Understanding the interplay of physics and mathematics in string instruments can deepen one's appreciation for their design and functionality.
Subject Terms
String instruments (mathematics)
Summary: The harmonics and timbre of wind instruments are described and computed using mathematics.
All stringed instruments exhibit a fundamental property of physics in that when impacted, they vibrate at numerous frequencies. The vibration of the string displaces the air around it, which—when impacted on the human eardrums—creates the sensation of sound. Some of the common instruments in the string family are violin, guitar, harp, mandolin, cello, and banjo. A modern violin has about 70 parts, and the overall design of such complex string instruments is inherently mathematical. Features such as string tension, area, and shape of the top plate, and spacing of frets all have mathematical properties that influence sound. For any string, at a given tension, only one note will be produced. To generate multiple notes from the instrument, many strings may be used to span the desired frequency spectrum (for example, harps) or the string may be forced to vibrate at different lengths, thereby changing the frequency (for example, guitars). On an equally tempered instrument like a guitar, the spacings of the frets, which help a player adjust string length, have to be scaled by the ratio 21/12. This problem is mathematically equivalent to duplicating a cube, which is one of the classic problems of antiquity. Mathematician Jim Woodhouse has studied violin acoustics using linear systems theory and mathematically modeled “virtual violins,” as well as related vibration problems like vehicle brake squeal.
Harmonic Series and Fundamental Frequency
When a string is plucked, struck, or bowed, it resonates at numerous frequencies simultaneously. The waves travel up and down the string. These waves reinforce and annul each other, which results in standing waves. The one-dimensional wave equation is used to model string instruments. A harmonic series is composed of frequencies that are an integer multiple of the lowest frequency. Fundamental frequency is the lowest frequency in a harmonic series. The musical pitch of a note is usually perceived as the fundamental frequency. The fundamental frequency (f) of a string can be computed as
where T is the string tension in newtons, m is the string mass in kilograms, and L is the string length in meters. The fundamental frequency is also known as the “first harmonic.”
Timbre
Timbre is the quality of a musical note and is what defines the character of a musical instrument. When two different instruments play the same note, the note could have the same frequency. The human ear distinguishes the source of the note because of timbre. Hermann Helmholtz was the first to describe timbre as a property of sound. When an instrument plays a certain note, the outputted sound consists of the fundamental frequency and its harmonics. These harmonics differ from instrument to instrument—what is known as “timbre.”
Bibliography
Hall, Rachel W., and Kresimir Josic. “The Mathematics of Musical Instruments.” American Mathematical Monthly 108, no. 4 (2001).
Mottola, R. M. “Liutaio Mottola Lutherie Information Website: Technical Design Information.” http://liutaiomottola.com/formulae.htm.
Rossing, Thomas. The Science of String Instruments. New York: Springer, 2010.