Room Acoustics - Best way to cover a glass door?

My small room (about 12x12) has a glass door (leading to a balcony) on the right side, between the speakers and the listening seats.  I have stand mounted mini-monitors and tube electronics.  I listen to acoustic jazz, blues and rock, with lots of female vocalists. Bass is not important to me; I live for soundstage, imaging and sweet midrange.

I am looking for a simple (and inexpensive) way to cover the door.  I plan to install hooks or curtain rod on the wall above it, and hanging some sort of material/curtain.  I am wondering if there is anyone with experience and recommendations on what material I should use.  I have been considering:

-something light- a simple sheet or cotton cloth from the fabric store.
-something medium- a fuzzy blanket or perhaps cloth of fleece, denim, corduroy,
-something heavy - room darkening curtains, possibly plastic lined
-Multiple layers of material?
-Free hanging or fixed on the bottom?
-Or what about blinds- vertical, horizontal, thin, wide, fabric wood.

I figure I will end up trying a couple things (ideally just 2), but any advice on where to start would be appreciated. Especially along the lines of "xyz is a really stupid idea" or "start with this". 

I am an engineer, so I tend to want to understand why something will work rather than just snake oil, but, ultimately it is all about the ears.

THANKS,

Bill