Even with equal-tempered tuning, standard piano tuning deviates from the mathematical pitch in the lower and higher octaves in a way which would show up as being out-of-tune using this visualization method: https://en.wikipedia.org/wiki/Piano_tuning#Stretch (Though the harmonics of lower notes should show up as being in tune with fundamentals of higher notes.)
You can experience this in a very un-subtle way with a guitar (especially if you lower the tuning). Plucking a string hard makes it go very sharp at first, before it settles to "the" pitch. Basically the only way to get an unchanging frequency from a stringed instrument is with a bow or some kind of external oscillator like feedback. Putting a guitar in tune has a lot to do with hard you plan to strike the strings after tuning is done.
