Just to add: harmonics are sine waves, in essence.

A pure sine wave has only one harmonic.

You need to add up a bunch of sine waves of different frequencies (multiples of the fundamental) to get something as, in a way, unnatural as a square (or sawtooth) wave.

That's why they're great building blocks for subtractive synthesis too. Pass them through a resonant filter, and you get something interesting.

They are also very easy to make digitally (a square wave is just on-off-on-off..).