[gtrubetskoy]

The piano keyboard (usually) has 7 octaves. Frequency doubles every octave, therefore if the lowest key is C, then the highest C all the way on the right is

   C*2^7

You can also follow fifths and arrive at the same note after 12 fifths. A fifth of a note is 3/2 of its frequency. Thus, starting from the same C, you arrive at

   C*(3/2)^12

But...

   (3/2)^12 = 129.7, while 2^7 = 128.

And that's (roughly) the problem that Bach's temperament addresses by ever so slightly adjusting the frequency so that in any key it sounds "right".

[dontreact]

I don't follow this. The answer contained in the link seems more relevant as it has to do with the fact that the harmonics of real metal strings are not ideal and so the detuning at both ends compensates for this.

It seems like you are talking about what would happen if you tuned a piano using something other than equal temperament. Why is the iterated fifths relevant at all in this case?

[jedimastert]

Generally, Pythagorean tuning uses perfect fifths, as it's the easiest to hear and to match using harmonics. That example is just showing that if you using consecutive intervals (such as fifths) then other intervals (such as octaves and thirds) won't line up.

[taco_emoji]

This doesn't address the question at all. You're talking about how it's tuned "out of tune" within an octave, not at the extremes of a piano keyboard. In any temperament, C1 should have exactly half the frequency of C2, but that's apparently not how pianos are actually tuned.