Any complex waveform can be built by summing sine waves — this is the basis of additive synthesis

Any periodic waveform can be decomposed into, and reconstructed from, a sum of sine waves of specific frequencies, amplitudes, and phases. Additive synthesis exploits this in reverse: start with individual sine oscillators (frequency components) and add them together to build a target timbre. Sine waves are self-describing — they contain exactly one frequency component each. All other standard waveforms (sawtooth, square, triangle) are, in the ideal case, infinite sums of sinusoids; bandlimited versions truncate that sum. The practical implication: to produce a given timbre, choose the frequency, amplitude, and phase of each partial.

Examples

A sawtooth wave at 100 Hz ≈ sum of sinusoids at 100, 200, 300, 400 Hz… with amplitudes 1, 1/2, 1/3, 1/4… A 10-oscillator additive patch approximates this with the first 10 partials.

Assessment

Why can’t you represent a square wave exactly with a finite number of oscillators? What perceptual artifact appears when you add more and more harmonics?