The 7 note scale you suggest (do re mi fa so la ti do) is comprised of different intervals (2 2 1 2 2 2 1) in the 12-fold equal tempered scale. There are infinite ways of exploring an octave in music, but unfortunately listener demand for such exploration is near infinitesimal.
12th root of 2, to the 12th = 2 :D The collection of 7 intervals I provided, 2 2 1 2 2 2 1, which are a differential representation of "(do) re mi fa so la ti do", sum to 12. Those intervals are linear within the log2 scale you identified as having a 12th root of 2 basis, or in other words, are the major diatonic (7 note) scale which are a subset of the 12-tone equal tempered scale. The laws of logarithms can help explain why these intervals are additive, whereas the semitone basis (12th root of 2) is multiplicative.
