I'm not sure about putting anything between the flooring and subfloor. Insulate with Roxul between the bays and cross fir the ceiling below. There is heavy vinyl sheet soundproofing you can put on the bottom of the joists between the joists and firing strips.

Thanks. I was wondering if insulation (what type/R value?) between the 16" high floor trusses (24" on center) would provide sufficient noise reduction for the bamboo flooring on the 2nd floor. We will initially be using the downstairs as unfinished area, (screened windows, covered space). I planned to insulate between the floor trusses and then appropriately rated gypsum/drywall (due to moisture) will be installed, taped, and finished. Later, we will likely convert the 1st level space to finished living space as time and budget allows.

The building supplier Did the materials take off and they sent Roxul R-30 insulation batts 24" width for floor truss and roof truss/attic insulation. So, per your suggestion, I already have that, now just need the additional, heavy vinyl sheet soundproofing. Do you have a recommendation on brand, thickness or another spec/rating I should look for?

Would this same technique help with external sound reduction/sound proofing walls? The 2nd story of our addition is basically a "great room" and has 4 perimeter(exterior) walls. We framed with 2x6s rather than 2x4s for strength and to provide more flexibility regarding techniques for sound reduction, such as high R rated insulation.

The cork underlaymant sounds great for the ceiling. I would go for that. Synthetic stuff is probably going to be similar in price. I was talking about this stuff: https://www.acoustimac.com/soundproofing-massloaded-vinyl-1lb50sf

but always go with natural(er) materials if you can.

You could also double up the drywall and go two layers on the ceiling below.

You can definitely use roxul in the walls. I saw it done on all of the interior walls of a project I'm currently on.

Cross firring reduces the contact area between the joists and ceiling below reducing thermal and sound transfer.

Crossfir is this:

They did it for a number of reasons here but sound was one of the benefits.

Better to always use materials that are slightly vapor permeable in the outside walls so moisture doesn't build up on alternating sides as seasons change

After further research: a method for helping with sound insulation of the downstairs ceiling: one to 2 layers of 1/2-5/8" gypsum (if 2 layers, those gypsum boards should have an insulation material/vinyl or specific type of glue between), and the gypsum board is screwed into a metal channel that attaches to a metal bracket (nicer if the bracket has a rubber piece on the bottom) that attaches to the under side of wooden floor trusses. The purpose of the channel and bracket system is to separate/deaden the vibration/sound traveling from the floor above the wood structure (through the subfloor and joists/floor trusses). The insulation I checked out includes Roxul Safe n Sound or Corning acoustic insulation which can be placed in single or double layers in the floor joist/floor truss cavity. The bracket system might be similar to the reason for/purpose of cross fir, along with the heavy vinyl sound insulating material mention in the intial response to my question. The vinyl with cross firing most likely is less cost than the metal bracket/channel system, but I have not yet compared price.