Sound Absorbing Drapes

Properly treating a room is the best bang for the buck all most every time.  If your aim for less reverb the material is key.  Most synthetic fibers are at least moderately reflective so natural fibers like cotton, hemp etc. are preferred to reduce liveliness.  Good luck, let us know how it works out.  

IF windows are involved, heavy fabric vertical blinds can be effective, and still allow some daylight when partially open, then easily more light, easier than opening draperies, or fully drawn open, that's good in front of sliding doors.

draperies: fullness is the term that means the amount of fabric that makes the folds, more fullness means more fabric like prior OP mentioned.

I worked with professional video and sound companies for 35 years and they all used a heavy velvet.  We hung it mostly off of trusses, but if you can find all or most  of the studs on your wall, you can use threaded black pipe and flanges. I’d suggest keeping it a a few inches away from the wall.

All the best.

Would it ever make sense to use a mass loaded vinyl inside the fabric for the drapes? 

I had Martin Logan Reqests in about 8 ft in front of a 4x6 ft window centered I between.  I have 2 2x6 by 1:2 in thick fiberglass covered panels that I found really didn’t make much difference.  Fast forward 28 years 

I replaced them with a pair of focal Sopra 3.  Out come the panels   I found they focused the sound stage and better defined the instruments location. However they shrank the depth and I lost that alive quality.  I have ordered sheer shades  the are like fabric venicsion blinds. When rolled down they are about 2 in thick, however the fabric is thin but you can adjust the inside vane to control the light. I’m hoping to kind of get the best of both worlds. Sparkling sound will a little more focus.