FM bandwidth grows with modulation index, so raising the index brightens the sound

An FM tone’s sideband series is infinite in theory, but high-order sidebands become negligible, giving a finite effective bandwidth. Chowning’s rule of thumb is BW ≈ 2·fM·(I+1), where fM is the modulator frequency and I the modulation index (equivalently Carson’s rule BW ≈ 2(D+M) with D the peak frequency deviation, since I = D/M). At I = 0 only the nearest sidebands exist, so the bandwidth collapses to ~2·fM — the same spread as amplitude modulation and a near-pure tone. As I rises, more sidebands gain significant amplitude, the spectrum widens steeply, and the timbre brightens; because bandwidth depends on both index and modulator frequency, raising fM at fixed I also widens the spectrum. At high I, sidebands extend below 0 Hz and fold back (reflection), making the spectrum asymmetric but still harmonic. This is why an envelope on the modulation index produces time-varying brightness: high I at the attack yields a bright transient, decaying I a mellower sustain — the basis of the DX7 electric-piano ‘pluck’, and why FM packs a rich spectrum far more cheaply than building it additively.

Examples

fM = 200 Hz: I = 1 → BW = 800 Hz (narrow); I = 3 → 1600 Hz; I = 5 → 2400 Hz with spectral asymmetry from reflections. A 500 Hz carrier with a 300 Hz modulator: I≈0 → BW≈600 Hz (AM-like); I = 5 → BW = 3600 Hz (about 24 discrete components). Classic DX7 piano: index envelope starts high, decays.

Assessment

Compute the effective bandwidth for a 400 Hz carrier with a 250 Hz modulator at index 3 (and again for fM = 300 Hz at I = 4). If the index decays from 4 to 1 over one second, describe the perceptual change and explain why raising the index turns a near-sine into a dense spectrum rather than merely a louder sine.