Pythagorean Scale is important because, an extremely important number called "Pythagorean Limma" is hidden in this scale. Pythagoras analyzed the % distances between the above consecutive notes and found them as under. (Refer to following table)
Thus, Pythagoras found that the distances between S-R, R-G, G-M, P-D, and D-N as 12.5%, and that between M-P and N-S' as 5.3497942 %.
He considered this distance as 'Limma' meaning 'left over' in Greek. This means that when we go on adding 12.5% from S, the 'left over' distance ending in P is the same as that ending in S'. This is the 'Pythagorean Limma'.
It's value is 1.053497942, rounded off to 1.0535, because; as shown above, 1.4238281 x 1.0535 = 1.5; and 1.8984375 x 1.0535 = 2.
This measurement of 1.053497942 (or 1.0535) appears in the fundamental construction of 22 Shrutis.
Pythagoras was a great Mathematician, Musicologist, and an Astronomer. To him, number was everything. He considered 4 sets of numbers. For him...
He saw a great unity in all creations in nature, and felt that music was a 'Melody of Spheres (Planets)'. He trusted that musical notes and the orbital distances between planets would have a relation.
Thus, his musical octave gave us the extremely important 'Pythagorean Limma', of 5.3497942, rounded off to 5.35%.