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You can use a round tenon. Typically you would use a dividers or a simple gauge and turn the tenon until the dividers/gauge just slip over the tenon. Remove a little bit of wood at a time until it fits. Some folks will use an open end wrench as a gauge. Or you could make one from a scrap of plywood with a notch of the correct width cut in it.
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Personal preference here: square mortise and tenon.
"I tried being reasonable..........I didn't like it." Clint Eastwood
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Square is easier to make tenon, harder for mortise. I suggest staying cylindrical, drill a 1 7/8 with Forstner, turn the tenon close but slightly larger than needed, and then continue to sneak up to perfect fit by the cut and try, repeat method.
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If you are scared of turning it too small and have a sloppy fit. Use sand paper to get it to the final thickness. You could drill an 1 7/8" hole in a piece of scrap to use to test fit.
Mel
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If you are worried about getting a solid fit, you can drill your socket hole in a piece of scrap, then cut the scrap in half and use it as a gauge as you turn the tenon.
Ralph
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You may get more input if you post this in the wood turning section.