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.

How so? That it goes from one note to the same note? That's because it's a real scale, not an integer scale.

It's because the diatonic scale has seven notes, not eight.

The octave interval is stupidly named by counting from 1: CDEFGABC': 8 notes!

For the next octave, the high C' has to be stupidly counted again: C'D'E'F'G'A'B'C".

To be fair, it's not just the octave. The interval between a C and a C is a unison. The interval between a C and a D is a second. Two octaves are a fifteenth (not a sixteenth). The entire thing is one-indexed.

It's perhaps better to think of it as the number of notes you have to count before you reach the second note, starting with the first. If you start with C, you have to count one note before you get to that same C. You have to count 8 before you get to the second C, and 15 before you get to the third.

A related problem in music is how rhythm is counted, e.g. in recording software. Each new bar in 4/4 starts on beat 1, 5, 9, 13, rather than 0, 4, 8, 12, which makes it surprisingly difficult to figure out where you are.