# Spectrum

The spectrum of audible sound is the interval from roughly 20 to 20000 Hz. Any particular audio signal whose total duration is finite has a power spectrum which gives the distribution of its (finite) power over the audible spectrum. This will be a continuous function of frequency that describes the power density, or power per unit of frequency.

A periodic sound has infinite duration (and infinite total power) but it may be represented by an audio signal describing a single period.  The power spectrum can then be identified as the (period averaged) power of each of its harmonics. Such a spectrum is called discrete, although one can also assign discrete spectra to inharmonic sums of sinusoids (whose components are note tuned to any fundamental frequency).  When one encounters a discreet spectrum one can assign it an (ill-defined) continuous spectral envelope, made by drawing a smooth curve through the points of the spectrum.

From any audio signal one can also calculate a series of short-time power spectra,  each being the power spectrum of the sound windowed into a short interval of time, typically between 10 a 100 milliseconds.  The attainable frequency resolution varies inversely with the length of time window chosen. The short-time spectrum of a sound is associated with the sound’s pitch, loudness, and timbre, but not in any simple way.     