A piano that has just been tuned well is out of tune on purpose. On paper an octave is a doubling: the A above middle A should vibrate exactly twice as fast. But a piano string is a stiff steel wire, not the perfect string of the textbook, and the stiffness pushes its overtones a little sharp of where the arithmetic puts them. Tune every octave to an exact doubling and each high note clashes faintly with the sharp overtones of the note below it; the piano sounds slightly sour. So tuners listen instead of counting, and widen the octaves a hair, until the treble sits sharp and the bass flat of the theoretical scale. In 1938 O. L. Railsback measured all 88 notes on pianos tuned by skilled technicians and plotted the drift: close to the paper in the middle of the keyboard, bending away at both ends. The curve still carries his name. I like that the right answer was found by ears long before anyone drew it, and that there is no single right answer: each piano gets its own, because each set of strings is stiff in its own way.