The carrier-to-modulator (C:M) ratio fixes where FM sidebands fall, making the spectrum harmonic or inharmonic

In FM synthesis the carrier-to-modulator frequency ratio (C:M) sets the spectral structure of the output: sidebands appear at the carrier frequency plus and minus integer multiples of the modulator frequency, so C:M determines where they land. When C:M is a simple integer ratio (1:1, 2:1, 3:1, …), sidebands fall on integer multiples of a common fundamental, producing a harmonic spectrum with a clear pitch; the ratio also selects which harmonics are present. Non-integer or irrational ratios place sidebands off the harmonic grid, producing inharmonic bell-like, metallic, or noise-like timbres with ambiguous pitch. Slight detuning from an integer ratio adds subtle inharmonicity, mimicking the stretched partials of real pianos and metal percussion. C:M is thus the primary lever for choosing between pitched and percussive FM timbres. A common misconception is that the modulation index alone controls timbre: in fact C:M fixes the sideband positions (the harmonic identity) while the index only sets their relative amplitudes. In a well-designed FM instrument the modulator is expressed as a multiple/divisor of the carrier (fM = fC · ratio), so transposing the carrier transposes all sidebands and preserves the timbre.

Examples

C:M = 1:1 → full harmonic spectrum; C:M = 4:1 (C=800, M=200 Hz) → sidebands at 200, 400, 600, 800, 1000, 1200 Hz (harmonic, pitched); an irrational ratio like 3:√2 (e.g. C=550, M=371 Hz) → inharmonic, bell-like. On the DX7, per-operator ‘Coarse’/‘Fine’ settings set the ratio; try Coarse between .5 and 4–5. In SuperCollider:

SynthDef(\fm, { arg freq=440, ratio=3, index=3;
 var mod = SinOsc.ar(freq*ratio, mul: freq*ratio*index);
 Out.ar(0, SinOsc.ar(freq + mod) * 0.2 ! 2);
}).add;

Assessment

Keeping the index fixed, change C:M from 1:1 to 1:2 to 1:3 and describe how the set of audible harmonics and perceived pitch identity shift. Compare patches at 3:1 versus 3:2.7 — which sounds more bell-like and why, in terms of sideband frequencies? For an instrument that plays different pitches at constant timbre, should the modulator be an absolute frequency or a multiple of the carrier, and why?