Electrical frequency of motor with different stator and rotor poles

If you have a synchronous machine that has a properly designed, the rotor will be designed so that the rotor pole faces occupy most of the rotor circumference. If you then put that rotor in a machine that has a stator with a different number of poles, the pole faces will cover parts of more than one stator pole of vice versa. If you then turn the rotor, it will generate voltage, but is will not have a sinusoidal waveform. It would probably be possible to predict what you would get, but it not be easy. You could rectify whatever you get and have a useable power source, but it would not be an efficient machine.

If you have an induction motor, you could probably take out the rotor and mount two permanent magnets in the rotor to make a permanent-magnet synchronous generator. That could make a somewhat useful generator regardless of the number of stator poles, but it would have the shortcomings described above.

synchronous machines

I mention synchronous machines because it's explicitly stated in the question and this answer specifically address it for synchronous motors.

_{For plain ordinary induction motors, the rotor poles adjust automatically to the number of stator poles via the process of induction i.e. there is no such physical commodity as the number of rotor poles taken out of context and not relative to the stator. Put another way, any rotor (that is physically the right dimensions) will work in any induction motor irrespective of the number of stator poles. This is not true of synchronous machines....}

Consider a simple rotor (i.e. just a bar magnet); it has two poles, north and south then, consider that the stator has 4 poles. The rotor must align with one of the south and one of the north ends of the stator field but that is impossible if the stator sequence is N, S, N, S because consecutive north and south are at right angles and not 180°. And, if the stator was N, N, S, S then it's only a 2-pole-machine.

Now consider that the rotor has 4 magnets at right angles (4 poles). This could still work because two north poles on the rotor can align with two south poles on the stator etc. but, the bar magnets have to be arranged to produce N, S, N, S i.e. it isn't as simple as two bar magnets at right angles. This works and stator poles = rotor poles (but both having a N, S, N, S sequence.

Then, consider what happens when you put a 4 pole rotor into a stator with 6 poles - in no way can all the norths on the four rotor poles line up with all the souths produced at any instant by the stator. This is because there is a physical angle misalignment. Take it to more extreme cases and you'll find that the number of poles on the rotor has to equal the number of poles on the stator. It's impossible to make a synchronous motor when stator poles don't equal rotor poles and, both have to be N, S, N, S etc..

But why go to all the trouble of making multiple stator poles and then make a simple and ineffective (compared to the more sophisticated stator) rotor with half or a third or a quarter of the poles?

What's the motivation? None as far as I can tell.