You can use a similar technique by building a double floor (see illustration below) that rests on half-inch-thick rubber mats (look for these from specialty suppliers). The double floor with 3/4-inch plywood on top and bottom, and fiberglass insulation between the 2 x 4 studs, supports the inner room, which also uses studded walls with drywall on both sides and sound-deadening fiberglass insulation within the wall cavities. A 6-inch air gap between the walls of the inner room and the outer room is sufficient to greatly reduce sound transmission. Carefully compute the dimensions of the inner room, taking into account the airspace between the inner and outer walls, as well as the actual thickness of the inner and outer walls, to come up with a consistent air gap of 6 inches. The dimensions of the inner room will, consequently, be much smaller than that of the outer room. The studded walls of the outer room should also use drywall on both sides.
Specialty sound-deadening insulation products are available for use within the walls and include such brands as Owens Corning QuietZone and Roxul Safe ‘n’ Sound.
Since you’ve constructed an inner room and an outer room, it will by nature have two doors, which will be very effective in keeping sound from escaping. You should consult a specialty sound contractor for details on sound-deadening ventilation techniques that can be applied to the hot and cold air supply and return ducts as well as methods of wiring that prevent sound leaks to the outside.
It is beyond the scope of this article to go into precise carpentry techniques of floor, wall, joist, stud and header construction, but abundant material is available on the Web and in books.