Establish the area with which you have to work. A soundproof room is actually two rooms, one inside the other with an amount of dead space between them. The ideal space will have a concrete floor, like an unfinished basement, to minimize noise transmitted through floor vibration. Prepare the space by installing mineral insulation (thick, heavy and green, not light and pink) between the joists above the music room.
Build the first wall using 2-by-6-inch floor plates, framed for one door. Add two beads of heavy black caulking beneath the plate to reduce vibration. Use standard 2-inch studs on 8-inch centers, but offset the studs to line with outside and inside edges of the floor plate. Cap these walls 6 inches below the insulated ceiling joists. Use 2-inch studs to build ceiling joists for the music room. There should be a 2-inch gap between the top of the room and the bottom of the joists.
Weave mineral insulation between the staggered studs until all walls are covered. Drywall inner and outer sides of all walls as well as the interior ceiling. Fill the ceiling with mineral insulation above the drywall. Mount a solid wood door to open inward into the room. Add weather stripping to tightly seal the door when it is closed.
Build conventional stud frame walls 4 to 6 inches outside the inner room, framed for one door that will open outward. These walls can go to the ceiling joists. Add floor-to-ceiling mineral insulation prior to adding drywall to the outside only. Run thick beads of caulking the entire height of the wall, about 8 inches apart. Let this caulking dry. It will act as a spacer to create dead air before you add a second layer of drywall to the outside. Attach a solid door and weatherstrip the same as the first.
Address electrical details. Every opening in either wall is a route for sound to escape. Surface-mounted electrical service is recommended with supply coming through conduits in the wall. Pack open space around the conduit with insulation and caulk around it to provide a seal.
Consider how to ventilate your room. As with electrical, any wall opening will allow sound to escape. A centrally located ceiling vent connected to a high volume ventilation fan on the outer wall can supply air by way of a length of flexible vent hose. This method requires one or two floor level vents exhausting into the inter-wall space. Your installation may have additional drawbacks or solutions.