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 te

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.