It might make you feel better to split the octave into two parts and eliminate the interval in the middle, i.e.
Do-re-mi-fa, pause, So-la-ti-do.
Each of those four note sections has the same intervals.
So arguably it's closer to being a "pi vs. 2pi" problem. (I am not really serious)
Isn't fa a half step down from being exactly between do and do'?
As far as the fencepost error, it helps to remember that musical intervals are multiplicative, rather than additive. An octave above do is f_Hz(do) • 2. Half an octave (fa#) would be f•sqrt(2). Generalized to a twelve tone scale, each half step is f_0•2^(1/12), dividing the octave into twelve equal multiplicative intervals.
I'm not saying fa is half way, I'm saying you can take the scale in two sections which are just transposed versions of the same intervals. This is not helpful, but it is true.
Pretty sure you can't get out of the fact that do appears twice by invoking multiplicative rather than additive logic:
two octaves played together contains 3 dos, 3 octaves contains 4, etc.
