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Harmonic spectrum

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Approximating a square wave by sin ⁡ ( t ) + sin ⁡ ( 3 t ) / 3 + sin ⁡ ( 5 t ) / 5 {\displaystyle \sin(t)+\sin(3t)/3+\sin(5t)/5} {\displaystyle \sin(t)+\sin(3t)/3+\sin(5t)/5}

A harmonic spectrum is a spectrum containing only frequency components whose frequencies are whole number multiples of the fundamental frequency; such frequencies are known as harmonics. "The individual partials are not heard separately but are blended together by the ear into a single tone."[1]

In other words, if ω {\displaystyle \omega } {\displaystyle \omega } is the fundamental frequency, then a harmonic spectrum has the form

{ … , − 2 ω , − ω , 0 , ω , 2 ω , … } . {\displaystyle \{\dots ,-2\omega ,-\omega ,0,\omega ,2\omega ,\dots \}.} {\displaystyle \{\dots ,-2\omega ,-\omega ,0,\omega ,2\omega ,\dots \}.}

A standard result of Fourier analysis is that a function has a harmonic spectrum if and only if it is periodic.

See also

References

  1. Benward, Bruce; White, Gary (1999). Music in Theory and Practice. Vol. 1 (7 ed.). McGraw-Hill Higher Education. p. xiii. ISBN 978-0-697-35375-7.