I found a difference too. Stringing through the body had no effect on the string tension as measured with a digital tension meter, but the compliance showed some change - the longer the amount of string between the nut and the tuner, or between the saddle and the anchor point, the more compliant the string was. Compliance in this context is the degree of side to side movement for a given force under set string tuning. The longer the "after length" (as they call it in the violin family) the more compliant the string.
Do you recall what the change in compliance was?
At the very best, with a string that was perfectly free to move along it's entire length from tuner to ball end, the change in compliance would be about 3 to 4 percent. Strings are not free to move like this however. There is friction to overcome before the afterlength can take part in the overall stretching of the string when it is deflected. There is also the additional resistance of the string to flowing over a break point, especially at the bridge plate hole when strung through body. The bend radius is very small there compared to the bend radius over a barrel type saddle. Also, there is a total of 90 degrees of bend when strung through body, compared to top stringing, which can be as small as 10 degrees.
So I would guess that an increase in compliance might be on the order of about 1 or 2 percent?
More importantly, could such a difference be noticed?
On an upright bass, the afterlength has more freedom to stretch. The bridge can be seen to move noticably when a string is depressed to the board, stretching the afterlength along with it. Unlike an electric bass saddle, the upright bridge top has some freedom of movement. This, plus a relatively long afterlength, can make a noticable difference.
One thing that I would like to measure, is the effect of an unset witness point on compliance. Also the effect of curls or kinks in the string. If someone were to move a string from top strung to thru body, there would be new opportunities for bends and curls within the active length of the string. There would be a new one at the saddle if witness point was not set, and curls from the tuning post wrap that moved into the afterlength at the nut. If not ironed out, these might account for an increase of compliance after restringing.
I suspect that it is mainly all these collateral changes that account for much of what might be noticed when making a setup change.
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