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How did Pythagoras connect mathematics with musical harmony?

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Rationalism Portrait of the French philosopher and mathematician René Descartes, oil on oak by Frans Hals, c. 1649; in the National Gallery of Denmark.

Pythagoras and his followers established a foundational link between mathematics and music by discovering that musical harmony is governed by simple whole-number ratios. Through empirical experimentation with instruments like the monochord, they observed that the pitch of a vibrating string depends on its length.12

Specifically, they identified that dividing a string into precise proportions produces harmonious intervals:
* Octave: A ratio of 1:2 (halving the string length).12
* Fifth: A ratio of 2:3.12
* Fourth: A ratio of 3:4.12

These proportions were central to the Pythagorean belief that numerical order underlies the natural world. By expressing musical relationships as mathematical ratios, the Pythagoreans pioneered a scientific approach to music theory. This concept of order later influenced broader philosophical ideas, such as the "music of the spheres," which posited that the movements of celestial bodies were governed by similar mathematical harmonies.12