Making the FM modulation index a time-varying function produces dynamic, evolving spectra

If the modulation index I is held constant, the FM spectrum is static. When I is made to vary with time — for example, following an amplitude envelope shape — the spectrum evolves continuously. As I increases, bandwidth grows; as I decreases, the spectrum narrows. Each individual partial follows its own Bessel function curve, so individual components do not change monotonically even as overall bandwidth grows. This mechanism is the core of FM’s power for natural-sounding synthesis: a simple envelope controlling I produces the same kind of attack-transient spectral evolution that natural instruments exhibit. The technique is far simpler than additive synthesis (which would require separate envelopes for each partial) yet produces comparably complex spectral evolution.

Examples

Brass patch: I rises rapidly with the amplitude envelope on attack (bright, many harmonics) then sustains — mimicking how brass instruments have richer spectra at loud dynamics. Setting I to follow the inverse of amplitude gives woodwind behavior (high harmonics recede as note sustains).

Assessment

Why does a single envelope controlling the FM modulation index produce different trajectories for each individual harmonic component, even though the index itself follows a simple curve?