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How did string lengths reveal numerical ratios in musical harmony?

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Marble bust of Pythagoras, the ancient Greek mathematician and philosopher, showing a bearded man with a head covering and calm expression. — Pythagoreanism Greek philosopher and mathematician Pythagoras, portrait bust.

The Pythagoreans discovered that musical harmony is rooted in simple whole-number ratios by experimenting with the lengths of vibrating strings. By using a monochord—a scientific instrument featuring a single string and a movable bridge—they observed that specific fractional lengths produced consonant intervals:1234

Octave: A string one-half the length of the original produces a note an octave higher (ratio 1:2).12
Fifth: A string two-thirds the length produces a note a fifth higher (ratio 3:2).12
Fourth: A string three-fourths the length produces a note a fourth higher (ratio 4:3).12

These findings revealed that musical intervals could be expressed as numerical proportions, suggesting a hidden order in the range of sound. This empirical discovery was foundational to the Pythagorean belief that numerical relationships govern the cosmos, influencing later concepts such as the "music of the spheres". Today, these harmonies are understood through the physics of vibrating strings, where the number of waves fitting into a string length determines these simple ratios.521