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The simple waveforms in an FM synth are produced by something called an ‘operator’. This is how FM works: it modulates the frequency of one waveform with another, and in so doing creates complexity in the resulting waveform. If the vibrato rate were to be increased sufficiently, we would perceive timbral rather than pitch changes in the resulting sound. This modulates the pitch of the note at a frequency that’s independent of the note’s frequency, changing its waveform accordingly. When playing a single note on our imaginary guitar, you can add vibrato with your left hand. When playing a chord on this guitar, each note of the chord would be a partial of the resulting waveform changing notes would change the partial content and thus the timbre of the sound produced.
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The difference between modulation and simply mixing sine waves together is easiest to understand using the analogy of an imaginary guitar whose strings produce sine waves. Rather, it uses one simple waveform to modulate the frequency of another, with the result then being used to modulate another simple waveform, which in turn can modulate another, and so on. But FM doesn’t just mix together those simple waveforms (although it can). FM synthesis harnesses this principle in order to generate complex waveforms from much simpler waveforms.