FM produces a harmonic spectrum only when the carrier-to-modulator frequency ratio is rational and small

A harmonic spectrum contains partials at integer multiples of a fundamental. In FM, this occurs only when c:m = N1:N2 where N1 and N2 are integers. An irrational ratio (e.g., √2:1) produces inharmonic sidebands, creating bell-like or metallic timbres. Even a rational but large ratio like 10:9 can sound quasi-inharmonic because reflected negative-frequency partials cluster near positive partials and create beating. Practical rule: for pitched harmonic sounds, keep both N1 and N2 small after simplification. For metallic or percussive sounds, use irrational or large-integer ratios.

Examples

c:m = 1:1 → fundamental + overtones. c:m = 1:2 → odd-rich spectrum. c:m = √2:1 → inharmonic metallic. FM bell: try c:m ≈ 1:1.4.

Assessment

Predict whether c:m = 3:2 produces a harmonic or inharmonic spectrum, and why. Propose a ratio for a bell-like FM sound.