Additive Synthesis
Additive synthesis is probably the simplest synthesis algorithm to understand. The fundamental idea is that all sounds are made up of an infinite number of sine waves at different volumes mixed together. Of course, when synthesizing you can only mix a finite amount of sine waves, otherwise you would be able to synthesize any sound with additive synthesis! That said, you can approximate any sound with additive synthesis, even to the point where it's indistiguishable!
But, additive synthesis isn't limited to just sine waves; You can use other kinds of waveforms as well. Interestingly, additive synthesis is actually the exact same thing as mixing! This means when you mix the two waveforms together, you are able to hear both of them playing together. Of course, the more waveforms you add together, the more difficult it becomes to pick them out, and the less it sounds like multiple "instruments" or "voices".
Fast Fourier Tranform
TODO
(Electric) Piano Example
TODO