Harmonic Series

A harmonic series is a set of vibrations whose frequencies are all integral multiples of one fundamental frequency. If F₁ is the fundamental frequency then members of the harmonic series may be found by F_n = n*F₁, where n = 1, 2, 3, etc.

Periodic Complex Waves

Any set of sine waves whose frequencies belong to a harmonic series will combine to make a periodic complex wave, whose repetition frequency is that of the series fundamental. The individual components may have any amplitude and any relative phase, and those determine the shape of the complex waveform.

Any periodic waveform of period P may be built from a set of sine waves whose frequencies are from a harmonic series with a fundamental frequency equal to 1/P. Each sine wave must have just the right amplitude and relative phase. These values can be determined from the shape of the complex waveform.

Nonperiodic Complex Waves

Any set of sine waves whose frequencies do not belong to a harmonic series will combine to make a complex wave that is not periodic, and will generally sound impure or unsteady in one way or another.

Any nonperiodic waveform may be built from a set of sine waves, but their frequencies will not belong to a harmonic series. Each component must have the right amplitude and relative phase, which can be determined from the shape of the complex waveform.