One oscillator (the modulator) wobbles the frequency of another (the carrier). Fast enough, that wobble stops being vibrato and becomes timbre — sidebands appear around the carrier, spaced by the modulator's frequency. Integer ratios give harmonic tones; non-integer ones give bells and metal. The index controls how many sidebands, so it acts as a brightness knob. Give the index its own envelope and you get FM's signature attack.
Bandwidth via Carson's rule: 2·(deviation + mod freq).
play with your keyboard (a s d f …) or click the keys