The premise is that the strings on the bass side of the instrument are under more tension than those on the treble side. Sometimes the premise is that it's the treble side strings that are under greater tension. The premise is usually stated in the context of rationalizing asymmetrical bracing or other structural features. In either case this is a myth for all modern plucked steel strung instruments. Violin and viola (and some cello) string sets generally do have the higher pitched strings under greater tension. And the same is likely the case for historical plucked instruments strung with plain gut strings. It is certainly true that identical strings of the same scale length, tuned to different pitches will be under different tensions. But the days when the availability of strings was so limited that instruments needed to use two or more strings of the same gauge and material for different notes are long over.
Last updated: January 21, 2019
Sometimes the path of scientific investigation is long and difficult. But every now and again you luck out and find that someone else has already been where you want to go and done the work for you. In this case it is the D'Addario String Co. that has done the work. For some reason they seem to be unique among string manufacturers in that they publish the nominal string tension of every string they make. It is a simple matter to look up this data for each and every string set they produce.
If you do that for the strings of any modern plucked steel strung instrument (guitar, mandolin, bass guitar, etc.), you'll notice that the tension of each string in a set is pretty much the same. The tension isn't identical mind you, but it is very close across strings in a set. Certainly no side predominates when it comes to string tension. That this should be the case should make sense. A set that varied in tension significantly across the strings would feel strange to the player, with some strings feeling stiffer than others. Also if tension was uniformly and substantially higher on one side than on the other, this would tend to bend the neck more on one side than the other which could on some instruments (those with a lot of strings and long thin necks) result in a playability problem.
Strings sets for modern plucked instruments from other manufacturers all exhibit this uniformity of string tension across strings, too. There are bound to be odd exceptions of course, based on individual preferences, but as a general rule all strings in a set for any modern steel strung plucked instrument will be under the same tension when tuned to pitch.
Things are different for the higher pitched bowed instruments (violin, viola and, to some extent, cello). Here you'll find the higher pitched strings are under greater tension than those of lower pitch. The issue here is how the strings behave under the bow. Playability for these instruments is enhanced with the use of higher tension on the higher pitched strings. If you take a look at the tension of string sets for the violin family you'll see that this tension differential is highest for violins, less for violas, and usually non-existent for the cello and the bass.
Another case where the treble strings will be under higher tension is for antique plain gut strung instruments. The pitch of a string is a function of its length, tension and mass per unit of length. Assuming that all strings on an instrument have the same length, lowering tension or increasing the mass of a string will lower its pitch. Mass can be increased by simply making the string fatter, but at some point this is no longer practical and so modern instrument strings will wind metal wire over the lower pitched strings to increase mass. But in the days before overwound strings were available the only way to achieve the pitch of the lowest strings on an instrument was to lower the tension. So a number of antique instruments with gut strings will also have the higher pitched strings under greater tension than the lower pitched strings.
For more on string tension and pitch see the page on String Tension in the Technical Information section of this site.