A few songs in which I believe this to be the case are One and New York from the album X (Wembley Edition).
Is it just me, or does he sing out of key on these songs?
To start with, even the best singer doesn't hit every note perfectly right. As such, singing out of key is a gradual thing.
For the two songs you mention, I would not say that Ed sings out of tune. On contrary, his music has a quite produced feeling, and it wouldn't surprise me if some sour notes have had their pitch corrected. I wouldn't swear on this though, but it is a common technique in popular music.
So again, I don't hear anything particilarly out of tune here. Perhaps you are getting tricked by the falsetto singing style which kind of gives that feeling?
I personally didn't hear any notes that struck me as particularly off --you might need to specify particular notes for us to hear it --but I can hazard a guess as to why those might be in there, if they are.
For a long time, the trendy sound was a highly-produced, auto-tuned sound with computer perfected pitch. The pendulum is swinging back, and people now want something more organic sounding. So the decision may have been made to not auto-tune notes that were naturally just a little bit off.
In addition, many great singers of the past (Billie Holiday and Ray Charles being two particular examples) deliberately used imperfections in tone, rhythm and pitch. That gave them a richer and more nuanced palette of emotional colors for their songs.
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