I don't know, guys, I think you've all missed the main point here. Most single phase motors are designed to operate at a single frequency. They are, in effect, tuned devices. The second phase which has to be fabricated somehow inside the motor (usually with caps or caps and coils) are tuned to their design frequency, either 50 or 60Hz.
You can find variable frequency AC sources including inverters but the problem is in the motor. It simply doesn't want to run anywhere but at its design speed.
With the exception of a universal motor, single phase motors have to struggle like crazy simply to start from a full stop. Trying to do it with an off-design frequency may be the stuff lab experiments are made of, but, in the world where these machines have to do something useful, it just doesn't work.
I know there are obscure exceptions. But that's not what this is about.