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The principle of Frequency Modulation (FM) synthesis is very simple; we just use one waveform to modulate the frequency of another.  Becuase of the very complex harmonics generated by FM, as any communications engineer will know, the resulting FM waveform has a rich harmonic content that can be exploited for music synthesis.  In fact, the analogue synthesizer already does FM of a sort, with the LFO modulating the pitch of the VCO.  The only real difference is that an FM synthesizer will use one oscillator to modulate the frequency of another, where the fundamental frequency of both oscillators is of the same order.

In summary:

• FM synthesis was developed by Yamaha in the early 1980s;
• it was made famous in the classic Yamaha DX7 synth;
• FM synthesis is particularly strong in producing 'bell' like timbres;
• FM synthesizers are notoriously hard to program - made of the sounds can be very dissonant;
• the technology gave rise to the 1st generation PC sound cards, notably the Yamaha OPL4 chipset.