Quietest fan for room

Dyson I think makes the quietest fan available but cost $200.00+.
Checking out a fan noise level in a store like Home Depot is not a good idea. The white noise level is probably 60 to 80 DB to start. If you have a DC to AC converter ask if you can take the display model to your car, that may give you an idea of how quiet it is.
Some fans will have a DB rating so check the manufacturers web site for info. Also the less a pitch or angle on the blade the quieter it will be.

The cheapest AND quietest way is to mount PC fan onto the chassis of hot component(I believe amp). You will need to ID available DIN for fan and measure the voltage between terminals in order find proper fan. Than it's only mechanics to mount the PC fan or couple onto the chassis.

Plan B: If you have wife or GF, you can get pheasant feather fan and give it to her while listening:-)

An analog,all tube fan sounds much better than the digital SS ones....and make sure you upgrade the power cord!

A fan in another room which sucks the air out of your music room is the way to do it.
Create a vent or vent shaft to your music room. Say some PVC near the ceiling or in the ceiling and suck the hot air out. The vents over the equipment would probably be best.
Small silent computer fans at the ends of the PVC will not be noisy.
My two cents.
Add in some large block of Ice in tubs near your seating position.

large block of Ice in tubs

Options are truly unlimited... How about some freon cooling system?