I have a very difficult time understanding the relevance of time signature in music. Does a musician think in terms of time signature when composing a piece, or thinks in terms of melody?
When I play say a complex piece of music in my head I always go with melody, whether it's 3/4 or 4/4 never have a significance because I don't really hear that if I'm not composing for percussions. I guess it's the same for a composer too, he has a melody in his mind and tries to structure that into music. I understand that if a percussion accompanies that piece it needs to find the repetitive beat(and thus time signature is relevant here) but that's not always the case, for example if piece is for a piano only, or the melody is not repetitive.
We don't remember a piece of music by its time signature, we remember it by its melody. So is time signature something later added on paper to make the math right, or does it really have a musical significance?

