It all depends on whether you fan motor is strict AC or AC/DC.
Most DC motor (except those using perm magnet as poles, i.e. model car motor) can run with AC power with reduced efficiency, so are also called AC/DC motor. They have VERY strong torque on power up and thus make ideal motor for electrical tools like drill, grinder, blender and saw. Their rated speed mostly depends on input voltage and number of poles, affected very little by AC frequency. More poles, more torque but lower RPM, principle apply to AC motor as well.
Strict AC motor using different principle to run. They have very weak torque especially during power up. In fact they can't even start rotating without some external aids. AC Motor RPM depends on AC frequency and the number of magnetic poles. For example, a 2-pole AC AC motor accepting 50Hz will have theoretical RPM of 3000RPM (50Hzx60s), and realistic RPM closes to 2900RPM. For 4-pole AC motor with same AC power the RPM will be 1500RPM theoretically. AC motors mostly use in task when torque is not important, because they are cheaper and last longer than DC motor. AC motors are used in everywhere outside powered tools.
Luckily, only bigger fans use AC motor. 12cm fans are small and they happen to use mostly DC motor which accept AC power as well. You are cut the voltage with transformer to silence a loud 12cm fan. I'm using an 0-250V adjustable transformer for my loud 120x120x38mm NMB fan, dropped the voltage from 220V to 150V, it's now inaudible. The transformer itself costs more than then fan though. An alterntive will be to go for external DC route. Get a adjustable DC transformer which let you choose among 7.5V, 9V and 12V, and get a powerful DC 12cm fan. It's probably cheaper and easier to do than AC fan route.
Correct me if I'm wrong.