Stack twelve perfect fifths on a low C and you ought to land on a C seven octaves up. You don’t quite. Twelve fifths overshoot seven octaves by about a quarter of a semitone, a gap called the Pythagorean comma, and no amount of careful tuning will close it, because it lives in the arithmetic: three-halves to the twelfth power is about 129.7, and two to the seventh is 128. A modern piano handles it by spreading the error evenly. Every fifth is tuned about two hundredths of a semitone flat, too little for most ears to catch, and in exchange every key works equally well. The major thirds pay more, about a seventh of a semitone sharp each, which is part of why a good choir singing on its own can settle into a sweeter chord than a piano can play. Every piano you have ever heard is slightly out of tune on purpose, so that it can be out of tune in the same way everywhere.