OK, I'm going to do something I've never done here publicly and state my credentials: MIT - B. Architecture + Acoustics, (e-mail me privately if you want to know the year ;-)
Acoustical treatment/construction solutions will depend on which of two categories the problem fits. (1) Transmission from one space to another (one of the spaces can be the outdoors) and (2) reflection/reverberation within a space. Sometimes both are issues, but they still require vastly different approaches/solutions.
Eliminating transmission (the issue Grant is dealing with) is accomplished through MASS+STIFFNESS and/or ISOLATION. I emphasize this, because stuffing a wall cavity full of fuzz (Fiberglas, or other ABSORBTIVE material) is futile, unless of course your termites have been complaining about the sound reverberating in their little ears!)
'Staggered studs' will physically separate the wall surface in one space from the wall surface in an adjoining space and is an excellent solution (for 2x4 or 2x6 studwalls) if the ceiling in ONE of the spaces is less than 9 or 10 feet. If both spaces have higher ceilings, then even with staggered studs, the air inside the studspace will still couple the two surfaces together enough to cause induced vibration in the 'receiving' space (whether not the studspace is stuffed with fuzz ;-)
Having isolated the two surfaces from one another, further attenuation (if necessary, which it often is with 2x4 studs, not so much with 2x6's because they're stiffer) can be achieved by adding mass. The most straightforward way to do it is to add another layer of drywall -- first on the 'transmission' side of the wall, and then, if necessary, on the receiving side.
In the days before drywall it was difficult (OK impossible) to add more mass to a (plastered) wall surface. Plaster has much less mass than drywall, and it doesn't like to be laid on too thick (like I'm doing now!) In those days, a (literally) curtain of sheet lead (yikes!) was often threaded between the staggered studs to eliminate the coupling problem. It's really unnecessary to do that today, but the technique (they don't use LEAD!) is still very effective (along with resiliant, or 'floating' floors) for vertical isolation (rooms above/below other rooms.)
While on the subject of 'resiliant', it IS possible (but not practical or cost effective in residential applications IMO) to 'float' a wall surface in front of another wall surface by hanging a second sheet of drywall (in front of the first) on resiliant (springy) metal clips designed absorb the energy of the sound wave against the wall surface. In a home, it's a real PITA because a 1/2"+ space must be maintained at all the edges of the 'floating' wall surface and filled with caulking or foam rubber, etc.
If you have an existing (partition) wall (between two spaces) and want a quick-and-easy solution to reduce the sound transmission through the wall, my recommendation would be to just add another layer of 5/8" drywall to both sides and see how it goes. TIP: using lots of x-tra drywall screws creates a 'stressed-skin' effect in the added drywall layer, which increases the stiffness of the wall.
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