musicians use numbering systems that are actually far more confused than anything discussed here. how many notes in an OCTave? "do re mi fa so la ti do" is eight, but that last do is part of the next octave, so an OCTave is 7 notes. (if we count transitions, same thing, starting at the first zero do, re is 1, ... again 7.
the same and even more confusion is engendered when talking about "fifths" etc.
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.
don't you mean 11-fold? ... oh wait, they aren't even consistent
They sum to 12
actually they multiply, 12th root of 2, to the 12th
You can blame the Romans for that, as they practiced inclusive counting. Their market days occurring once every 8 days were called nundinae, because the next market day was the ninth day from the previous one. (And by the same logic, Jesus rose from the dead on the third day.)