Irrational c/m ratios place FM sidebands between harmonics, creating inharmonic spectra for metallic sounds

When the carrier-to-modulator frequency ratio is irrational (e.g., c/m = √2 ≈ 1.414, or π, or 1/e), the reflected lower sidebands do not land on harmonic positions but fall between the positive-frequency components. The result is a spectrum whose components are not related by simple integer ratios — an inharmonic spectrum. Inharmonic spectra are perceptually associated with metallic, bell-like, gong-like, or noise-like timbres. This is because in acoustic instruments, harmonic spectra arise from resonators with simple geometry (strings, pipes), while irregular resonators (bells, drums, metal plates) produce inharmonic spectra. FM’s ability to produce inharmonic spectra with only two oscillators makes it extremely efficient for percussive and metallic synthesis.

Examples

Bell patch: c=200Hz, m=280Hz (ratio≈1/1.4, near-irrational), I=10. The inharmonic components produce the characteristic shimmer of a bell. DX7 ‘E.Piano’ uses near-integer ratios; ‘Bell’ patches use non-integer ones.

Assessment

Distinguish between two FM patches: c/m=2/1 and c/m=2.1/1. Predict which will sound more ‘metallic’ and which more ‘organ-like’, and explain why.