The trick to this method is that the drywall panels are hung on a pair (or more) of wooden "cleats" made from a sliced 2x4. The upper cleat on the back of the drywall is isolated from the lower cleat on the wall behind by inexpensive closed-cell foam tape. No part of the outer and inner walls touch directly. In practice, very little sound is transmitted through the foam, and the walls achieve a very high degree of soundproofing. The weight of the drywall keeps it in place so surprisingly well, that I use only two cleats: one near the top and one across the middle.
Overall, this method is fairly easy. It's not nearly as quick as using resilient channel, because it involves splitting a 2x4 lengthwise. (In either method, you will want to use foam tape to add extra soundproofing, so this extra step isn't a tradeoff, unless you choose to buy the resilient-channel pre-taped. The parts list is very small - drywall, a table saw or bandsaw, one 2x4 for every 4x8 drywall panel, nails, drywall screws, foam strips, and some pipe insulation. Surprisingly, this method requires much less precision than you would think, because some mistakes are in a sense self-correcting. Of course, the DIY version of this method does assume skill and confident use of limb-shearing power tools to do a potentially-dangerous "rip cut". If you don't have a woodworker's confidence with this step, find someone who can do it for you. A great recommended alternative is to have the lumberyard cut the wood for you upon purchase. In the section on ripping the wood, I'll tell you what to say to get the cut we want.
Since soundproofing carries with it a lot of myths and misconceptions, this Instructable will start with a little soundproofing theory before heading into the steps.
Step 1: Soundproofing Theory
First I'd like to 1. clear up some basic soundproofing theory, and 2. explain some typical alternative methods.
"Soundproofing" (as distinguished from acoustical treatment) is concerned with blocking sound.
You block sound with 1. mass, and 2. isolation.
Putting foam or other absorptive things on the inside of the wall is not soundproofing, but acoustical room treatment, which is used to reduce echoes and resonances and prevent excess sound buildup within a room. Soundproofing and room treatment are completely different things. Room treatment improves the sound within the same room. Soundproofing is about preventing sound from getting out or in. Definitely, putting foam on the walls can help keep your neighbors from calling, but it's not because of "soundproofing" -- it's because you have treated your room and in effect "turned down the volume" like they asked you to do.
Soundproofing a wall involves the principles of mass and isolation. A heavy wall will soundproof better than a lighter wall of the same construction. BUT isolation is the trick that will let you achieve the same results with a lighter wall.
A traditional wall involves panels (drywall, siding, stucco) connected to a frame, with drywall attached to the inside. There is usually fiberglass insulation inside the wall. It can typically block about 30dB of sound if constructed tightly. The way it works is mainly through the mass of the wall (drywall + frame) with some absorption by the insulation (effectively "turning down the volume") in the little "room" between the panels.
You can make a better wall by finding ways to isolate one panel from the other. In the staggered stud method, the vertical studs are staggered in depth so that the front and back panels are screwed into different sets of studs. However, they are still attached to the same top and base board of the frame, so some sound will travel straight through.
The ultimate method involves "double wall" construction. You essentially create a room inside a room with no part of the inside wall touching any part of the outside wall. Double walls can block in the range of 55-60dB of sound. A disadvantage is that the extra wall thickness can eat up a lot of space within a small room.
In between these two extremes, there is the resilient channel method, which involves attaching springy metal strips to the studs of the outside wall, then screwing drywall into a flange on the strips in such a way that the wall can flex against the resilient channel without touching the outside wall's studs. When constructed properly, these walls can block into the 50dB range. Other implements can be added to the resilient structure to get into the high 50dB range, such as foam tape and varieties of "isolation clips" that are clever ways to attach the channel to the studs without screwing directly. Of course, the cost of these adds up quickly.
My method is also in between the single and double wall construction, and it is similar to the resilient channel method in creating a "springy" wall that will flex. Unlike the standard resilient channel method in which the channels are screwed into both the stud and the drywall, in my method there is no direct mechanical path from the outside wall to the inside wall. So my performance should be most comparable in soundproofing capability to resilient channel methods with isolation clips. (Exactly how close we come remains to be verified by testing however, so stay tuned for future revisions of this instructable.) I will say, however, that it's "pretty darned good".
Of course, if you're running a commercial operation or otherwise have the funds, definitely look into a well-documented industry-standard soundproofing method like resilient channel or double-wall construction. They've been studied and measured thoroughly, and there's little mystery about why they work and how well they work. But if you're desperate and/or short on funds like many musicians, this method could be just what you need to get excellent results without a lot of investment.
For more information on soundproofing, especially technical aspects of soundproofing and studio acoustics, I recommend F. Alton Everest's books Master Handbook of Acoustics and Sound Studio Construction On a Budget. For non-technical soundproofing advice (mixed with a lot of product hype) see Soundproofing.org, (which I am compelled to point out I have no affiliation with and am generally critical of, even though, quite coincidentally, they are local to me.)