The only difference between that detail and a typical residential floor-ceiling assembly is that the drywall in the fire-rated assembly is designed to stay in one piece after exposure to fire but it would provide no increase in sound separation. Air borne sound abatement detailing generally uses mass, resiliency, and absorption. In a residential ceiling the mass is provided by gypsum drywall, the resiliency is provided by resilient channels and the absorption is provided by thermal or acoustic insulation. Structure borne sound abatement usually consists of resilient coverings or light weight concrete toppings on the floor above. Ceilings should be easier than walls because there is plenty of room for insulation, less perimeter joints, and it is easy to hang a ceiling resiliently but then there is the foot traffic above to contend with. At walls the challenge is to provide reasonable sound separation without making the wall too thick which leads to many products from resilient layers in drywall to Green Glue between layers. The kind of insulation doesn't usually matter unless it is sprayed which seals cracks very well. Cracks create major sound leaks in field build enclosures.

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The cheapest method of reducing sound transfer (it will not be sound proof) after eliminating connecting leaks like utility and HVAC penetrations and put two layers of drywall on the ceiling, is to suspend the drywall ceiling on some kind of resilient system. The cheapest system is 1/2" deep resilient metal channels. I use this system on all ceilings below a habitable floor. Here is a link to resilient metal channels at Home Depot (the channel is shown upside down). Be careful to not screw the ceiling boards to the joists. It also helps to use a rubber like resilient layer below the underlayment or finish wood flooring in the room above. Of course, wall to wall carpeting is also helpful not necessarily for music but the sound of foot traffic. Filling the floor cavity with insulation absorbs some sound but more importantly it prevents sound from traveling horizontally from a crack above to a crack below. Any kind of insulation will have essentially the same effect but mineral wool would probably be the most cost effective if installed tight to the joists. Spray foam has the advantage of sealing cracks but I doubt it is more effective. If there are recessed lights in the ceiling the housings must be rated for insulation contact (IC) and the permitted lamp output may be reduced. Of course, recessed lights would be a ceiling penetration so you should avoid them altogether.

I would also add that you want the heating ducts and plumbing pipes wrapped with insulation as well. Right now is the PERFECT time to do it because everything is exposed. Metal (like duct work) LOVES to transmit noise (telephone WIRES are METAL...ahem...just saying). Wrap those puppies and you will have removed a HUGE source of sound from multiple areas of the house (there is NOTHING like hearing the details of someone's day via the duct work from two stories above...at 1am). I've always liked the 'double stuffing' of the ceiling. If you double stuff the joists (two layers of mineral wool instead of one) then you can probably go with the lower priced product. So long as their densities are the same and the material make up are the same then you should get awfully close to the same ratings. Use the resilient channels throughout...you can't think of the bedroom as being one area and the rest another...if one part 'leaks' noise then the other part will too. Think of the entire space as a bathtub. If you have a hole in one end of the bathtub, it will leak because the water will find it's own level. Noise kinda does the same thing. The other option, if you don't want to use the resilient channels (because you want the ceiling height maintained) then you can look at Green Glue liquid acoustic barrier. There *MIGHT be an application where they apply the liquid/foam to the joists and then mount/screw the drywall to the joists. Green Glue also produces drywall/foam additive that can be used INSIDE the holes for the screws (you drill the pilot hole, you fill it with the foam/liquid and then drive the screw into place). This acts like a mini-resilient channel. And then use the acoustic drywall. And remember your SOLID CORE door for the bedroom. That way you reduce every ounce of noise moving from the basement area into the bedroom area. You should be fine using ALL of the options open to you. You can never have too much noise abatement. You really can't. Silence is silence. Once you achieve it, it is difficult to figure out which bit was 'overkill'.