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I understand how length of a string on any string instrument affects it's pitch, but I don't understand how a short wind instrument produces a high pitch.

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Wind instruments work by setting up sanding waves in the air column inside them. Shorter instruments have shorter air columns and thus standing waves with shorter wavelengths resulting in higher pitches.

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When working with waves the wavelength and frequency (pitch) are inversely related. Sound waves have the relation frequency times wavelength equal the speed of sound. Wind instruments are using the longitudinal dimension of air in the instrument as a medium like a "string". In both the string and wind instruments, shorter wavelengths in the excited medium (string or column of air) emit higher frequencies. The exact notes are determined by the choice of valves causing different resonances, much like the action of fingers on the string changing the effective length of the strings.

Here's video of a lab experiment that uses the frequency-wavelength relation to the speed of sound to calculate that speed. It also has useful diagrams of the types of resonances that occur in a tube.

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