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When I built my first studio I did not stagger my double wall studs for soundproofing my recording studio. This was a small mistake, but nonetheless a mistake. I am going to give you two reasons why staggering your studs is important with recording studio design.
1) Structural Integrity
I had asked Rod Gervais, the author of Home Recording Studio: Build It Like The Pros about why he recommends staggering studs in a double wall system. His answer was that overtime studs can bend and if they bend inward they could actually touch each other thus short circuiting the system. He did admit, although this is rare he has seen it throughout his decades of building studios.
Staggered Stud Diagram (16" OC)
2) Weaker Soundproofing Due To Coupling
I asked JH Brandt, a well respected and experienced studio designer the same question. Should you stagger studs? He also said it is a best practice in double wall design and the reason is that when studs are not staggered the drywall becomes a diaphragm that vibrates similar to a speaker cone. This pushes the air in the air cavity, which then starts to vibrate the other wall sympathetically. Once this process gets going it creates a bit of a feedback loop as well. This will weaken the system as a whole because sound can vibrate through the system more easily.
Now, damping materials like MLV or Green Glue can help reduce the vibrations of the drywall, but simply staggering your studs, even by a few inches will reduce or eliminate the sympathetic vibration between your walls.
3) A Note On R Value and Insulation
One of the reasons we did not stagger our studs in my studio is because we needed an R30 insulation in the walls for our climate. However, R Value is additive, meaning if you need R30 you can use two insulations with an R15 value or one R20 and R10. You get the picture. This means if you don't have the space for R30 you can use two different insulations in each stud wall to make it work. Another option is to use a 2x6 exterior wall and a 2x4 interior wall and this will give you more space.
Conclusion
Having two well respected studio designers teach me how I was wrong is honestly pretty cool. They both gave great and different answers for why you should stagger your studs. I hope these answers help you decide to stagger your studs when building a studio and give you the understanding on a deeper level to know why it is important.