My bathroom wall is a little rust

I just recently purchased a duplex that has a main unit up top and a basement apartment on the bottom. As of right now, the ceilings in the basement apartment are sheetrock and are only 6'6" high. There is one room where the sheetrock has been torn down with the floor joists exposed which adds a little height and gives the illusion of more height. I will be living in the basement and wanted to know the best way to sound proof the space between the joists from the upstairs. The upstairs has hardwood floors. Any suggestions? Thanks.

hello all... first of all I have zero experience or knowledge doing anything like this.... currently the tv area of this room has this wood built-in that can only fit a pretty small tv. I would like to remove the wood built-in and expand the tv space into one large open insert area within the wall. As you can see in the pictures this area is not a part of the actual wall of the room but this custom built in work that expands out from the wall. From my limited knowledge I'm thinking I can pull out the wood tv built in and then carve into this wall which from my guess is hollow inside to the desired open area indicated in the picture with the red rectangle. And then install some drywalls on the borders to close off the cut areas. Any ideas of what needs to be done to accomplish this? thank you very much!