AC induction motors Do Not Work That Way.
Here's what's happening inside an induction motor. It's a cylinder, with 3 field windings around the edge. It uses either 3-phase power, or single-phase with an extra "start" winding to get it spinning in the right direction.
The three windings make a magnetic field which rotates around the cylinder at line frequency, so 60 cycles/second (3600 RPM) or 50 cycles (3000 RPM).
It's like there's an imaginary spinning magnet with 2 poles: north and south.
Yes, you could use 6 windings and have the imaginary spinning magnet have 4 poles. That would rotate half the speed.
The rotor (spinning part) is nothing but a hollow cylinder that catches the rotating magnetic wave and is dragged along by it. It resembles a hamster wheel and is actually called a squirrel cage. There are no permanent magnets in it, this all happens via eddy-current induction, hence the name of the motor.
By nature, this has a small amount of "slip". So a motor whose field spins at 3600 rpm is rated for 3450 RPM. The slip varies slightly by load (if you've ever worked in a wood-shop, you know the sound of a motor "loading up") but that is the one speed it goes, assuming the input frequency is 60 Hz.
The entire motor is carefully tuned to be efficient (avoid making internal heat which would have to be actively cooled) when fed a sine-wave.
Now you know the score, you can see where altering the 60Hz waveform will do absolutely nothing to change the speed of the motor, but break the sinewave it needs to be efficient (coolable). Also, you were getting a lot of inductive back-EMF at your switch, which I'm amazed didn't fry it.
And now you know how to control the speed of the motor: feed it a sinewave at a different frequency. Yeah, that's harder. And not really thinkable until silicon power switching got pretty good. Which is what a Variable Frequency Drive is: the "magic bullet" to solve this problem. While you're doing VFD, you might as well also do proper 3-phase, to increase efficiency, allow reversibility and avoid the hokey "start winding" found on single phase motors. By the way, an interesting work on this is Don Lancaster's "Magic Sinewaves".