Harmonic spectrum

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

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

See also

  • Fourier series
  • Harmonic series (music)
  • Periodic function
  • Scale of harmonics
  • Undertone series

References

  1. ^ Benward, Bruce and Saker, Marilyn (1997/2003). Music: In Theory and Practice, Vol. I, p.xiii. Seventh edition. McGraw-Hill. ISBN 978-0-07-294262-0.
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