A waveform and its set of harmonics are two equivalent descriptions of the same sound

Fourier analysis establishes a two-way equivalence: the nature of a musical tone is defined by the numbers and amplitudes of the harmonics it contains, and any given set of harmonics gives a given waveform — conversely, given a set of harmonics you can derive a unique waveform. Time-domain shape and frequency-domain spectrum are therefore two descriptions of one sound. This duality is the conceptual bridge between subtractive synthesis (shaping a waveform) and additive synthesis (building a spectrum from sines): changing one necessarily changes the other.

Examples

Adding odd harmonics at decreasing amplitude builds toward a square wave in the time domain; filtering that square reshapes both its spectrum and its visible waveform.

Assessment

Explain the bidirectional relationship Fourier analysis gives between a waveform and its harmonic set, and how it links additive and subtractive synthesis.