Why is every modern piano tuned slightly out of tune on purpose?

A modern piano is slightly out of tune on purpose because a keyboard with 12 notes per octave cannot make every interval pure in every key, so the tuning spreads the error evenly. Pure intervals are simple ratios: a 3:2 perfect fifth, a 5:4 major third. That system is called just intonation. Equal temperament gives up on purity and divides the octave into 12 identical semitones instead, every interval modified by the same rule from its pure state. Each step multiplies the frequency by about 1.059463, the twelfth root of 2, so twelve steps double it exactly.

The cost is not shared fairly. Fifths and fourths come out almost indistinguishable from pure ones, but thirds and sixths drift further. A pure major third is about 386 cents; the equal-tempered one is 400. That 14-cent gap, about a seventh of a semitone, is large enough to be clearly audible.

The reward is freedom. Equal temperament fits the keyboard and makes every key equally usable, which is why it has dominated Western music since the 18th century, and why composers could modulate anywhere.

Not everyone accepts the deal. Barbershop quartets always sing without instruments, because a fixed-pitch instrument in equal temperament makes perfectly pure chords impossible. When four voices do lock into pure tuning, their overtones reinforce each other so strongly that listeners hear a fifth note nobody is singing, an effect a modern keyboard can't produce.