String Tapers Too Near Machine Head?

[Chienmortbb]

I have a Fender Aerodyne and a Peavey Precision clone. On both basses the taper “bump” on the E string always seems to hit the machine head at  the “in tune” point. This means that there is more effect on the pitch per revolution of the head tuning up than tuning down. 
 

It happens on every make of string I have tried. These include Rotosound, Picato, Ernie Ball, Fender ( pre and post D’Addario), D’Addario and in the case of the Peavey various cheap brands. If you look closely at the picture you should be able to see what I mean. Are there any brands where there is either no taper or where the taper is   closer to the nut?

 

Edited November 6, 2019 by Chienmortbb

[itu]

GHS long scale +, and D'Addarios that fit 36" scale basses.

[Chienmortbb]

I will give those a try.