Barrel taper after the threaded shank, will depend on the cartridge being chambered.
IMHO, the taper shouldn't start for at least 3 inches after the shoulder. This makes for a much stiffer barrel/receiver lock up and of course, not lonely gives you some meaty area to grip in the blocks but provides a very good bedding area ahead of the recoil lug or collar.
That being said, I once built a 30-06 and a 6.5-06 on Rem 700 actions with the taper starting right at the receiver. It wasn't an extremely sharp taper for the first 6 inches but enough to give a very sleek appearance and shave off a bit of weight. The rifle was built as a light weight sheep hunting rifle. The contour was slimmer than a factory barrel but not by much. It shot quite well for the first 3 shots and started to heat up pretty quickly after that.
Take a look at Remington Model 7 rifles, with pencil thin barrels. Their taper starts right at the receiver.