Harmonic entropy measures pitch-perception uncertainty; high entropy means the interval is not close to any simple integer ratio

Harmonic entropy (developed by Paul Erlich, based on Terhardt’s theory) quantifies the uncertainty in pitch perception for an interval. It uses the Farey series of integer ratios to model all candidate harmonic templates the auditory system might fit to a pair of tones. When two tones form a simple integer ratio (3:2, 4:3, 5:4, etc.), the Farey-series template has low uncertainty and harmonic entropy is low — the interval is tonally clear. When the ratio is complex or irrational, many templates compete, entropy is high, and the interval sounds tonally ambiguous or rough. Harmonic entropy captures a different component of dissonance than the Plomp-Levelt roughness model: it measures tonalness rather than roughness. Together, roughness (CDC-5 sensory) and harmonic entropy capture most of the perceptual complexity of consonance and dissonance.

Examples

Unison (1:1): harmonic entropy = 0 (perfectly determined). Octave (2:1): very low entropy. Perfect fifth (3:2): low entropy. Tritone (45:32 in JI): high entropy because no simple ratio is near. Microtonally detuned fifth (just under 3:2): entropy rises as the ratio drifts from the simple template.

Assessment

Explain the difference between roughness-based dissonance (Plomp-Levelt) and harmonic entropy. Give an interval that would have high roughness but low harmonic entropy, and one that might have low roughness but high harmonic entropy.