FM synthesis builds complex spectra by using one oscillator's frequency to modulate another's

Frequency Modulation (FM) synthesis, invented by John Chowning, feeds one sinusoid (the modulator M) into the frequency input of another (the carrier C), letting just two oscillators generate spectra that additive synthesis would need many partials to build. Three parameters govern the result: carrier frequency C (pitch), modulator frequency M, and the modulation index I (the ratio of frequency deviation to M). Sidebands appear at C ± kM for all integers k, with amplitudes set by Bessel functions of I: a small I gives a near-pure tone, a large I gives a dense spectrum. The C:M ratio sets harmonicity — simple/integer ratios overlap sidebands onto a common harmonic series (harmonic, pitched tones), while inharmonic ratios give bell and metallic timbres. Because it is computationally cheap and can imitate bells, electric pianos and brass, FM became the first mass-market digital synthesis method — the Yamaha DX7 (1983) sold roughly 300,000 units and used chained 6-operator algorithms.

Examples

C:M = 1:1, index ≈ 1: a slightly brighter sine; index ≈ 5: an organ-like harmonic tone. C:M = 1:2.41: an inharmonic metallic bell. A modulation-index envelope from high to low mimics the attack-decay brightness of a plucked string. The DX7 chains multiple FM operators for richer timbres.

Assessment

What C:M ratio gives a bell-like (inharmonic) sound? How does raising the modulation index from 0 to 5 change the number of significant sidebands and the audible brightness, and which parameter controls brightness independently of pitch? Why is the DX7 historically significant?