Musical Intervals

Physics: Consonant intervals correspond to simple frequency ratios: P5 = 3/2, P4 = 4/3, M3 = 5/4 (just). Equal temperament (12-TET): each semitone = 2^(1/12). Beat frequency = |f₁ − f₂| Hz, heard as amplitude modulation. Critical band ≈ 1/3 octave — intervals within one critical band sound rough (dissonant). Pythagoras noted consonance with string length ratios; Helmholtz (1877) explained it via roughness from beats between harmonics; Plomp-Levelt (1965): consonance peaks at simple ratios.