Would it be possible to have a song dictionary?

If you consider the first 15 or so notes of a song, that comes to a finite but very large number of possibilities. Would it be possible to take the first few notes of existing songs and arrange them in some sort of dictionary?

What prompts this question is that I heard a tune in a dream. I have no musical ability. I am certain that I lifted the tune from somewhere, but I have no idea from where. It would be nice to be able to look it up and find out where it came from. I would think that there would be some general benefit to this. For example, there have been a few cases where song writes were sued for plagiarism for stealing someone else’s song. It would be convenient to run a standard check on a song to make sure that nobody thought of it already.