The "in-betwteen" notes, as you called them, McGrath, are called quarter tones, or semi tones. They are usually denoted as double sharps or double flats in Western music, but that is not really correct either. A double sharp in Western music is 2 half tones, or a full step. Eastern music uses totally different scales and modes, and the only way I have ever seen to notate it in Western style is to use funky key signatures such as 2 sharps and 1 flat........still not truly correct, I am sure.