How to put up MDF wall without studs
There are many ways to build what you're proposing, but the one that jumped to mind first is akin to a panel/frame assembly.

Start with a post (~1.5"x3.5"/ ~38mm x 88mm, or larger), firmly attached to the floor and ceiling. This could be a bracket that wraps around the post, it could be a bolt at one end that mates with a hole, or it could be a couple of L brackets. At the doorway, make sure the posts are extra solid.

Butt a panel of mdf to the post and secure it as drawn. Caulking/sealant in the gaps will probably help reduce sound transmission. Set the next post plumb/ in line and repeat.



Depending on the gaps at the floor and ceiling, you'd probably want either a foam weatherstripping or caulking/sealant and possibly a strip of timber (~3/4"/18mm) on both sides to make it look finished.

Note, the fancy version of the post is a little harder to make airtight and more bothersome both in terms of milling and assembly. It is a little cleaner looking, though.