The universe is built on harmonies. The Pythagoreans had it right when they married mathematics, music, and the cosmos. Just as mathematical patterns underlie the musical scales and intervals most pleasing to the ear, they also describe the probability waves at the heart of quantum theory.

More than 2500 years ago, according to ancient sources, Pythagoras applied his discoveries in music theory to the behavior of the celestial objects. Pluck a string, and the pitch of the note it produces is related to the length of the string. When the lengths of two strings are a simple ratio—1/2, 2/3, and so on—the notes together sound harmonious. Pythagoras believed that the universe itself hummed with its own harmony—beautiful, divine, and inaudible to human ears. He characterized the intervals between successive orbits, such as those of Mercury and Venus, as either tones or half-tones (the smallest intervals in the musical scale), adding up to seven whole tones that form a perfectly harmonious natural order.

When the French physicist Louis de Broglie proposed the wave nature of electrons and other matter particles, he may very well have had musical harmonics in mind. De Broglie showed how the different energy levels of Niels Bohr’s atomic model emerged naturally by describing electrons as standing waves of various frequencies, thereby generalizing the wave theory of light to material particles. Just as a guitar string can be plucked in certain ways to produce particular sounds, electrons in de Broglie’s scheme are forced to oscillate in particular patterns, corresponding to certain frequencies and energy levels.