I have experienced this quite often and have only just properly realised that it's quite a common phenonemon on my dream exits.

I will wake from a dream. The dream was non-lucid. The dream didn't feature music in any way, but when I wake a have a tune in my head that is always an original tune, but just a short cycle that repeats itself.

Anyone else experience this? It's quite strange that the music wasn't perceivable in the dream itself, but on waking I know that it was an inherent part of the dream.

Incidentally, I like to write music, but I've never woken with a tune that was so good I considered it to be worth recording.