The first person to make the connection between math and music was Pythagoras of Samos, a famous philosopher and cult leader who lived most of the time in southern Italy in 5th century BC. Among his claims to fame is the oldest known proof of what we call the "Pythagorean Theorem". If you have never heard of this guy, he is one of western civilizations strangest, but most influential thinkers. For Pythagoras, ratios were everything. He believed every value could be expressed as a fraction (he was wrong, but that is a whole different story). He also is the first to believe in the idea that mathematics is everywhere.
One bit of evidence of underlying rational numbers was in Greek music. At the time, music was not as complicated as it is today. The Greek octave had a mere five notes. Pythagoras pointed out that each note was a fraction of a string. Lets say you had a string that played an A. The next note is 4/5 the length (or 5/4 the frequency) which is approximately a C. The rest of the octave has the fractions 3/4 (approximately D), 2/3 (approximately E), and 3/5 (approximately F), before you run into 1/2 which is the octave A.
Many of the ancient Greek harps (kitharas) had six strings corresponding to these notes. (Kitharas, like all things preindustrial, were hand made and string lengths and count were never standardized, but six strings based on these simple ratios were probably popular choices.) Also, for you music experts, note that the scale is a "minor" scale, which we associate today with sounding sad or tragic. A perfect scale to accompany most Greek plays.
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