What B strings taper right after they pass the nut

[Delberthot]

I just went to change the strings on my new (2nd hand) bass collection bass and snapped the B string.

The tuner is really close to the nut but the string didn't taper until well after this point so I had to wrap the full thickness of the B string round the tuner and ended up snapping the string.

i was using Elites strings which funnily enough were on it previously but that one did taper shortly after passing through the nut.

My question to all you guys and gals is - what brand of string has a B that tapers shortly after passing the nut?

[Figjam]

my ernie ball slinkies B string starts tapering about an inch past the nut, and thats on a 34" scale.

[chalie]

I have this kind of problem on few basses I got and I don't want to find tipical strings for just these basses since good strings are deserve to use on all basses so I do a small modification on the way you string those basses.
I insert a 0.250 dia x 1/2" aluminium sleave to the B and have it slide down to the ball-end side before doing your regular stringing.
This will have the taper side 1/2" away from the existing point but you better have your own measure how long shoud the sleave be. but I think most case would be 1/2". You have to mind the placement over the nut as well not that the taper is on that too.