Measure the floor where you want the new sound-dampening wall, and allow 6.5 inches of thickness for the finished wall. This includes the lumber, which will take up 5.5 inches and two half-inch drywall panels on either side of the wall, which will add another inch.
Cut a 2-by-6 board to serve as a bottom wall plate, and attach it to the subfloor, using a framing nailer to secure it to the floor joists.
Measure and install a top wall plate directly over the bottom wall plate, nailing the 2-by-6 board to the ceiling joists.
Cut and install double 2-by-6 boards at the far ends of the wall. You now have an open frame in which you will build the staggered-stud wall.
Measure 16 inches from the end of the frame on one side, and install a 2-by-4 stud, lining the edge of it up with the edge of the bottom and top wall plates nearest you. Nail the stud to the bottom wall plate and to the top wall plate. The top and bottom plates are 2 inches wider than the stud, so there will be a 2-inch space on both plates on the far side of the newly installed stud.
Install another stud 16 inches from the first, once again aligning the stud with the edge of the bottom wall plate nearest you. Continue to install studs 16 inches apart until you run out of room at the end of the wall.
Walk around the stud wall to the other side directly opposite where you installed the last stud. On this side of the wall, you will install another row of studs aligned with the edge of the plates now closest to you. On this side, you will install one stud between each stud in the first row of studs. When you’re done, you’ll have two separate stud walls, with studs that stagger from one side of the wall to the other.
Install drywall panels on both sides of the wall. Because you’ve created airspace between the studs on either side of the wall, noises made on one side will fade in the air pocket, reducing the sound before it reaches the other side of the wall.