At a workshop being taught by Jack Sobon at Hancock Shaker village, he told us the reason he picked the 13/16" size for his peg "holes" was that he got a deal at a hardware store close out sale on a whole big box of 13/16" bits. That was the reason he's using that size.
He had us split our pegs from straight grain red oak stock and then trim them to size on a shaving horse with a draw knife.
His shaving horse had a series of different size holes in the seat, for us to use to 'test' the size of our eight sized tapered pegs.
Good luck, Jim