@skyscraper, I use ASC (Acoustic Science Corp., maker of Tube Traps) Wall Damp as constrained layer damping. It has a "tacky" surface like double-sided carpet tape, so holds together the two pieces of wood it is installed between. It was designed by Art Noxon to be installed between double layers of sheetrock in listening rooms, as a vibration killer. It absorbs and dissipates the energy fed into it, making it a great means of constructing non-resonant shelves. I have been in a room constructed with Wall Damp between two layers of sheetrock, and the walls were VERY dead.
Another product created to provide constrained layer damping is EAR Isodamp, the blue stuff you may have seen. It's not as tacky as ASC Wall Damp, so the two pieces of wood it is placed between may need to be glued together.
I'm not sure if 13-ply Baltic Birch and MDF are equally non-resonant (I suspect MDF is), but BB is much stiffer than MDF. I put my 60 lb. turntable (with four feet) on top of two layers of 3/4" MDF, the MDF supported on two cones in front and only one in the rear. The rear of the MDF bowed under the weight!