For sound attenuation, I have often used Roxul- mineral wool, it's greenish in color and I believe actually has 3 desirable propeties. 1.) Fire Rated and is listed in the UL Fire Resistance Directory, for use in firestop assemblies to pack out the annular space in through floor and through wall penetrations, like pipe, conduit etc. 2.) R value for insulation 3.) Acoustical/Sound Attenuation.
I think this material is more common in commercial or multi unit residential buildings, any good drywall supply house should be able to get it.
One thing with sound attenuation, is fasteners, I believe that is where most sound is transferred, especially with metal studs, maybe not so much with wood framing, even when joint compound covers the fastener head, less fasteners, less sound transfer, some research on that may help too.
With some research, you should be able to pinpoint a wall type that suits your needs to dampen sound better than what you have, typically on a set of building design drawings, there are details of all the specific wall types for the building, and they vary depending on fire ratings, sound attenuation/acoustical and room usage.
See the link below, another similar material would be thermafiber, either or, check out the website, Roxul is a quality product, with some good specifications for sound absorption.
ROXUL