The carrier-to-modulator frequency ratio sets FM sideband placement and harmonicity

In FM synthesis the modulator’s output is added to the carrier’s frequency argument, producing sidebands in pairs at carrier ± n × modulator frequency (n = 1, 2, 3…), with amplitudes following Bessel functions of the modulation index. Three parameters control the timbre: carrier frequency (pitch), modulation index d/fM (how many and how strong the sidebands are — higher index gives a denser, brighter spectrum, index near zero gives a near-sine), and the carrier-to-modulator ratio C:M. The ratio governs harmonicity: integer or simple ratios place all sidebands on harmonics of a common fundamental, yielding harmonic, pitched timbres; non-integer ratios scatter sidebands off the harmonic grid, yielding inharmonic bell-like or metallic clangorous sounds. Raising the modulator frequency spreads the sidebands further apart. Because a few oscillators can produce a spectrum equivalent to many harmonics, FM is far cheaper than additive synthesis of the same spectrum.

Examples

Carrier 500 Hz, modulator 100 Hz (ratio 5:1): first sidebands at 400 and 600 Hz. Ratio 1.41 or 2.41: inharmonic bell. SuperCollider: {SinOsc.ar(800 + SinOsc.ar(200, mul: MouseY.kr(0,400))) * 0.2!2}.play; — MouseY sweeps the index, MouseX the modulator frequency. Pure Data on Bela: [osc~ 440] carrier, [osc~ 880] modulator (2:1), [*~ 200] sets the index swing.

Assessment

With carrier 500 Hz and modulator 100 Hz, list the first three sideband frequencies. Describe the timbral change as the ratio moves from 1:1 to 1:1.41, and what happens to the spectrum as modulator amplitude rises from 0 to 500.