Spectral Formula of a Waveform

For the curious: the spectral formula of a waveform gives a simple relationship between the partial number and its amplitude (remember, we know its frequency from the Fourier theorem).

For example, the formula for a square wave is an amplitude of 1/n, when n is odd, or 0, when n is even.

Using the formula, we can compute the following partial strengths for a square wave:

1st partial = 1/1

2nd partial = 0

3rd partial = 1/3

and so on.

There are some other simple waveforms whose formulae are well known:

Sawtooth wave: 1/n when n is even or odd.

Triangle wave: 1/n2 when n is odd, 0 when n is even. (Note how "fast" the amplitudes of a triangle wave drop off—it’s said to be a "not very bright" timbre.)

Surprisingly, as we mentioned in Section 1.4, symmetrical waveforms (like the square and triangle) have no even partials.