Shaded pole ac motor voltage conversion?

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I've got a few Vornado fans which are 120v 60hz but now i am in singapore which is 230v 50hz. I've done a quick fix to get them running here which is to splice a fan speed controller (which i assume is a snubbed TRIAC) which works but makes the shaded pole motors on the fans hum a bit.

So i was wondering if i could just replace the fan speed controller with a series capacitor instead? I remember enough about electronics that because the load is mostly inductive the capacitor would have reactance which would create voltage across it but i cant remember how to do the math or if i would need to change capacitance for each speed setting of the fan (fan changes speed by energising additiinal coils on the motor)
 
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