Go to Home Depot and demo the fans, buy the best $20 fan you can find. Geez, do I have to do all the thinking around here?
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:-)
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.
Agree on a ceiling fan, though none of mine are in my listening rooms. They do work exceptionally well. If possible for you to implement, that's my recommendation.
I've been around the Dyson, it seems quiet as some have mentioned. I wonder about their power, but I'll defer to others on that. The price, like most Dyson products, isn't low, but one may see value there if it suits the bill.
Last year, I purchased a Vornado at Costco for $50. It's incredibly quiet. So much so that my wife commandeered it during the heat wave last week after I suggested it when she complained about the noise of one of our typical $20 fans preventing her to sleep well. Bonus, it's made in the USA. If the other two options don't work for you, I'm sure you'll find this fan provides the solution.
I have two of those Vornados and believe me, they are not quiet, at least in the bedroom. The one I have in the living room is further away from me than the one in the bedroom but it still interferes with the music.
They do, however, a great job at moving the air around. I've been using them all year round.
All the best,