discrete spectrum and harmonics

A musical note is not a single frequency but a stack of them — a fundamental pitch plus overtones at two times, three times, four times that rate — and the particular recipe of overtones is what makes a violin sound different from a flute playing the same note. A Fourier series makes this concrete: a periodic signal has energy only at a discrete ladder of frequencies, the integer multiples of one fundamental, and the list of strengths at those frequencies is its spectrum.

Because the harmonics in a Fourier series are indexed by an integer n, the allowed frequencies are f_0, 2 f_0, 3 f_0, ... where f_0 is the fundamental frequency — no frequencies in between are present. Plotting the coefficient magnitudes (|c_n|, or sqrt(a_n^2 + b_n^2)) against frequency gives not a smooth curve but a set of isolated vertical spikes, which is why it is called a line spectrum or discrete spectrum. The n-th spike is the n-th harmonic: n = 1 the fundamental, n = 2 the second harmonic (one octave up), and so on. Periodicity is exactly what forces this discreteness — being periodic means repeating, and only frequencies that fit a whole number of cycles into the period can survive.

This discrete picture is the working language of acoustics, vibrations, and power electronics. Timbre is harmonic content; total harmonic distortion measures unwanted high harmonics in an amplifier or power line; an organ pipe's character is its spectrum of overtones. The contrast to keep in mind: a periodic signal has a discrete (line) spectrum, but a one-shot, non-repeating pulse spreads its energy over a continuous band of frequencies — that is the continuous spectrum of the Fourier transform, the limit you reach when the period grows to infinity and the spectral lines merge into a smear.