The first subdivison of the octave works out quite nicely by using sexagesimals. The value of the √2 = 1;25 to the first sexagesimal place or 1 5/12. Going 1/12 to either side of it gives 4/3 and 3/2 whose product is 2.
The ratios are 1, 4/3, 3/2, 2 which fit the proportion 6:8:9:12. Early music theory contains the subdivision of the octave into forths and fifths. Aristoxenus who is a revisionist of the Pythagorean theory discribes some rather complicated subdivisions of the octave with fractions of a tone. The lyre, an early stringed instrument, was known in ancient Mesopotamia and early music theory was most likely passed on by tradition.
Supplemental: √2 was known to 3 sexagesimal places in ancient Babylon. Theon of Alexandria published a method for extracting square roots arithmetically. Nicomachus, a Pythagorean, wrote a Manual on Harmonics and an Introduction to Arithmatic which contained a study of superparticular numbers. Two other ancient string instruments are the pandura and the monochord.