√2

 The ancient musicians and mathematicians KNEW about and were fluent with the √2.  

They proved that to us with the ancient Babylonian cuneiform tablet known as YBC 7289 from ~1800 BC.

Giving the result of √2 in sexagesimal (1, 24, 51, 10), proving they had the modern TRITONE.

Or from the ancient Indian Vedic perspective ~1000 BC - In the Baudhayana sutra it appears as: The measure is to be increased by its third and this [third] again by its own fourth less the thirty-fourth part [of that fourth]; this is [the value of] the diagonal of a square [whose side is the measure].

 Either way you look at, the ancient folks had √2 down pat. So they knew the octave, and how to equally subdivide it.  The mathematical formula for this musical TRITONE interval would be the geometric mean of √(AB), where A=1, and the next octave B=2, thus 1.414 = √2.

look what happens when you cut pythagoras' epogdoon in half = √2