I’ve known this for a while, but I’ve never seen it explained so clearly as it is in the link I’ll provide:
It is impossible to have a perfectly tuned instrument. A perfect fifth is 3/2 of its partner note. A perfect octave is 2 times its partner note.
In other words, if you start with C, multiply its frequency by 3/2 to get to the next higher G.
If you start with C, multiply its frequency by 2 to get to the next higher C.
But if you take perfect fifths to their logical conclusion, you realise that they’re incompatible with perfect octaves.
C to G is 3/2. To get to the next C following perfect fifths, you have to multiply by 3/2 12 times.
So, from C, following perfect fifths, to a (much higher) C is (3/2) ^ 12.
But that is 531441/4096. ~ 129.746337891
By following perfect octaves (powers of 2), the nearest perfect octave to that is 2^7, 128.
Perfect fifths and perfect octaves are inherently incompatible.