I don't know the term 'natural'. The frequencies you give are those I know as the just scale, which is a modification of the Pythagorean scale.

Piano tuning is a bit of a black art: the low strings are tuned flat, and the high ones sharp - this is because the low strings don't vibrate as a simple string (they're too stiff) and the harmonics are sharper than they should be. Without stretched tuning the beats between high notes and the harmonics of low notes are most unpleasant.

Stu