Why does Western music have exactly 12 notes?

Western music has 12 notes because of a near-miss in arithmetic. After the octave, where one note vibrates twice as fast as another, the next simplest and sweetest interval is the perfect fifth, a ratio of 3 to 2. If you start on one note and keep stacking perfect fifths, folding the results back into one octave, something almost magical happens at the twelfth step: you land very nearly on your starting note, seven octaves up. Twelve fifths give you twelve different notes, and that is the chromatic scale.

But "very nearly" is not "exactly". Twelve pure fifths overshoot seven octaves by a small gap called the Pythagorean comma, about a quarter of a semitone. Tune an instrument with pure fifths and one interval is left so sour that musicians nicknamed it the "wolf".

The modern fix is a clever compromise called equal temperament: every fifth is squeezed by a tiny amount, about 2 cents, so the circle closes perfectly and every key sounds equally in tune. It has dominated Western music since the 18th century.

The system carries Pythagoras's name, and a legend says he discovered it by listening to blacksmiths' hammers. The legend is false, and the tuning actually goes back to ancient Mesopotamia.