Wind Instruments

A version of the following will appear in the 2011 volume "Encyclopedia of Mathematics in Society".

Wind instruments convert the energy of moving air into sound energy --- vibrations that are perceptible to the human ear.  Under this definition, wind instruments include the human voice, pipe organs, woodwind instruments such as the clarinet, oboe, and flute, and brass instruments like the trumpet.  The nature of this vibration and of the associated resonator tube are responsible for the unique timbre of each type of wind instrument.

The sources of vibrations

In the human voice, the flow of air from the lungs causes the vocal cords (also called vocal folds) in the larynx to open and close in rapid vibration.  This periodic stopping of the air stream creates oscillatory pulses of air pressure, or sound.  The frequency of this vibration and thus the pitch of the resulting sound are determined by the length and tension of the cords.  The singer controls these factors using the musculature of the larynx. 

The rapid open-close vibration of the vocal cords is present in many wind instruments.  In brass instruments such as the trumpet, trombone, French horn and tuba, the lips of the performer form a small aperture that opens and closes in response to air pressure.  These are sometimes called lip-reed instruments.  In single-reed instruments like the clarinet and saxophone, a thin cane reed vibrates in oscillatory contact with a specially shaped structure (the mouthpiece) to bring about the open-close effect.  .  The oboe and bassoon utilize two cane reeds held closely together with a small space between them that opens and closes in response to flowing air, controlled by the muscles of the lips. 

A third important mechanism for converting the energy of moving air into vibration utilized in the flute and the so-called flue pipes of the pipe organ.  In these instruments vibration occurs when flowing air passes over an object with a distinct edge that splits the airstream.  The resulting turbulence gives rise to oscillatory vibration. In the modern flute, the performer’s lip muscles actively control the interaction between the airstream and the edge.  In the recorder and other whistle-type instruments, as well as flue pipes of the organ, this is controlled by the mechanical design of the instrument alone.

Tube resonators and overtones

With the exception of the human voice, all of the wind instruments described above are constructed with a tube resonator enclosing a column of air that functions in much the same way as the vibrating string.  Oscillations in air pressure inside the tube reflect from the ends, resulting in significant feedback with the primary vibrating medium. In the Harmonics entry in this volume, the relationship between the vibration frequency and length of a string fixed at both ends is explained.  In that idealized setting, changing the string length by small integer factors (1/2, 1/3, 1/4,  for example) results in frequency changes that are recognizable as musical intervals (an octave, an octave plus a fifth, two octaves, respectively).  The resonating air column in wind instruments behaves similarly to a vibrating string.

An important performance practice on most wind instruments is overblowing. Not to be confused with simply playing overly loudly, overblowing refers to the fact that changes in the airflow can cause  the resonating air column to vibrate at an overtone above its fundamental frequency.  This allows performers on modern instruments to achieve a large range of pitches (often two octaves or more) from a relatively compact resonating tube.  Instruments with cylindrical tubes open at both ends (e.g. the flute) overblow at the octave, as do conical instruments that are closed at one end such as the oboe and saxophone.  On the other hand, cylindrical tubes closed at one end (such as the clarinet) overblow at the twelfth:  an octave plus a fifth.   The relative weakness of the overtone at the octave and other even-numbered overtones account for the particular timbre of the clarinet.

Altering the tube length in performance

Just as the length of a vibrating string determines the frequency/pitch of the vibration, the length of the resonating air column accounts for the pitch of notes played by a wind instrument.  In reed instruments, the resonating tube is perforated along its length with holes.  By systematically covering some of these holes but not others, the performer effectively changes the length of the resonating column.  This in turn causes the vibrating reed assembly to assume the frequency of the air column.   Most brass instruments have secondary lengths of tubing which are brought into play by mechanical valves by which the performer alters the length, and hence the fundamental frequency of the vibrating air column.  The exception to this is the slide trombone, which features a concentric tube arrangement by which the outer tube can move to lengthen the air column resonator.

FURTHER READINGS:

Sundberg, Johan. The Science of Musical Sounds. San Diego: Academic Press, 1991

Wood, Alexander. The Physics of Music, seventh edition. London: Chapman and Hall,  1975