Well, I have a friend who seems to have perfect pitch. E.g. she can immediately sing just about any note you ask for, tell immediately what key a song is in, and notice (painfully) when anyone is a bit sharp or flat.   I envy that talent in most cases, but she is not always the best participant in song circles precisely because it DOES bother her a lot when people are off key, even if they're all singing the same notes relative to each other and singing the melody right, with everything being just a tad high or low.

I wouldn't ever claim to have perfect pitch, but I surprised myself a few years back when I picked up a guitar that had been stored, with strings loosened a lot, for a couple months. I didn't have a tuner or tuning fork or piano handy, so I tuned it entirely by ear. It took me what seemed an eternity - probably at least 5 minutes of tune-play-adjust-play-adjust until it finally sounded right to me. Shortly thereafter, I found my tuner, and when I checked the tuning against it, it was spot on, not just tuned relative to itself but exactly EADGBE too.   

Now, if I could just do that faster.

Genie