FM and PM produce identical output at high sample rates; PM is preferred for digital implementation

Frequency modulation (modulating instantaneous frequency) and phase modulation (modulating instantaneous phase) are related operations. At sufficiently high sample rates, numerical integration in FM closely approximates continuous integration, making FM and PM spectrally identical. At low sample rates, integration error causes their partial amplitudes to diverge. Because the PM formula s(t) = AC·sin(2πfCt + I·sin(2πfMt)) requires no integration — just two table lookups and an addition — it is far easier to implement digitally. Yamaha DX-series synthesizers implement PM internally while labeling it FM.

Examples

Yamaha DX7 uses PM. At 96 kHz sample rate FM ≈ PM spectrally. At 22 kHz they diverge audibly. SuperCollider PM: SinOsc.ar(carrier, SinOsc.ar(mod) * index).

Assessment

Write the PM formula. Explain in one sentence why synthesizer manufacturers prefer PM over true FM for digital implementation.