No. Don’t glue or hard fasten a single layer to the joists. Think drum skin.
If you don’t have 1/2” to spare for resilient, hard fasten one layer of 5/8 type X drywall, then add another with Green Glue in between. HD stocks it. Code will likely require this fire separation anyway. Green glue does not dry, forming a viscoelastic dampening layer, similar to how acoustic drywall works. These are well tested and well documented methods.
As Bill has suggested, homasote on the floor above works. Underpad and carpet is very effective as a damp ending layer as well.
If you blow in cellulose (again every bit as good as rock wool and likely 1/4 the price) it will flow around wires and duct etc. HD does rent equipmemt but you’ll have to play a bit to approximate dense pack. In any case, fiberglass would be your last choice over cellulose or rock wool.
I posted progress and sound testing results of my theatre build here:
https://www.avsforum.com/threads/measured-before-after-results-with-green-glue.2929668/
We are 100% happy with the results..and the basement is open to our first floor via an open stairwell. That build used rock wool, resilient and 2 layer drywall with Green Glue. My film studio (2nd floor, commercial space) , about 900 square feet, used dense pack cellulose 16” in the floor truss cavity. The entire ceiling of the main floor (4500 sq/ft) used 5/8” type X and cellulose which gave us the commercial fire rating. That was full of ducting, wire etc. cellulose was cheap and required far less labour. That studio was extremely quiet along with double walls, isolated HVAC etc.
Are there any HVAC connections between the spaces? If so, they will require some work first before you even think of ceiling work.