The FM modulation index (deviation ÷ modulator frequency) sets sideband amplitudes and thus brightness

In FM synthesis the modulation index I is the ratio of frequency deviation d (how far the modulator swings the carrier from its unmodulated frequency) to the modulator frequency: I = d/fM. Because deviation is proportional to the modulator’s amplitude, I is directly proportional to modulator amplitude for a fixed modulator frequency. The index — not the raw modulation amplitude — is the correct timbre knob: at constant amplitude, doubling fM halves the spectral spread, so keeping I constant while scaling fM preserves the partials’ amplitude structure. I determines the amplitude of each component in the output spectrum. Near zero, energy stays at the carrier and the tone is a near-sine; as I rises, more and stronger sidebands appear, brightening and thickening the sound. At certain higher indices the carrier component itself shrinks and can vanish entirely — energy is redistributed, not added as loudness. The index controls spectral shape, not gain or volume. In practice I is driven by an envelope on the modulator, giving FM sounds their characteristic time-varying, brass-like brightness.

Examples

I ≈ 0.1: near-sine / near-AM, only the two nearest sidebands matter. I ≈ 1: narrow, mellow spectrum. I ≈ 5: many significant sidebands, bright, carrier amplitude markedly diminished. An envelope sweeping I from high to low over the attack (e.g. 3 → 0.5 over 1 s) mimics the bright attack of a plucked string or brass decaying to a rounder tail.

Assessment

Explain what the modulation index is and how modulator amplitude relates to it. Compute I for d = 800 Hz at fM = 400 Hz, then at fM = 800 Hz, and say why the timbre differs at fixed amplitude. In a 2-operator patch, sweep I from 0 upward and describe what happens to (a) brightness, (b) audibility of the carrier, (c) the number of distinct sidebands.