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Which string-length ratios produce the octave, fifth, and fourth?

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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.

In the philosophy of Pythagoreanism, the discovery that musical harmony is rooted in simple whole-number ratios was a foundational insight into the order of the cosmos. By experimenting with the lengths of vibrating strings, the Pythagoreans identified the following ratios for the primary musical intervals:12

Octave: A string length ratio of 1:2 produces the octave.12
Fifth: A string length ratio of 2:3 (two-thirds the length) produces the fifth.12
Fourth: A string length ratio of 3:4 (three-fourths the length) produces the fourth.12

These simple ratios demonstrate the Pythagorean belief that numerical proportions and harmony are fundamental to the structure of reality. When strings are divided according to these ratios, the resulting intervals are perceived by the human ear as the most fundamental and harmonious sounds.312