No, not directly anyway. You'd have to make a simple html page that points to both the gif and music file. If you know how to use Flash, you could try using a flash animation. And it might end up being smaller than an animated gif.
A gif is a picture format. midi is a sound format. If you wanted to make a sig with sound, then you could make it a shockwave or flash object... but I don't know if those are allowed.