How important is 2/3 to the Greeks?

I came across this interesting notion this week, that western music is greatly dependent on the fraction 2/3.

When Pythagoras was developing his scale, he first took a “monochord”, which basically is a piece of string with a specific length, and played a note with it. Then he took 2/3 of his monochord, and played another note, as shown in the picture below. 

Interestingly, he found that note, played together with the first note, very pleasing, and thus named the relationship of the two pitches “perfect fifths”. The basics of western music, lies on the three chords: the first, the fourth and the fifth, which the fifth is the “perfect fifths” of the first and the first is the “perfect fifths” of the fourth. Therefore, the base of western music is simply bounded by the fraction 2/3, fancied by Pythagoras and the human ear.