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Double Bass Tailpiece Length & Strings Tension

I used to be utterly confused by the question of string length vs. string tension. I talked to some pretty good people about it, and although most of them knew and understood the answer, they were not able to explain it as a good teacher would explain to student. My turn to try? :hmm:

If you shorten the tailpiece so that you now have greater string length between the bridge and the tailpiece, there is little to no change in string tension within practical parameters. Think for a moment of a Koto, a Japanese instrument that has multiple bridges. While playing, the musician actually moves the bridges back and forth to change the pitch of the various strings.

So, instead of thinking about moving the tailpiece away from the bridge, think about moving the bridge away from the tailpiece. You still have essentially the same result: the bridge to tailpiece and bridge to nut proportions change in relation to each other. Whatever pitch a given string is tuned to will go up because you have made the vibrating string length shorter.

If you move the tailpiece back and increase the string length, and if you tuned the string up to keep the same tension, the pitch would go up as the tension goes up. However, since you want to keep the same pitch, you'd tune the string back down and the tension would go back to what it was before.

I think this game can be played only until the afterlength begins to exceed the vibrating string length, which would be a fairly unlikely scenario on acoustic instruments. But I also think alterations to tailpiece length, tailpiece weight, tailpiece suspension, and string afterlength can have an effect on the sound; they're just due to other factors.
 
As I alluded to before, this is a Creationist vs. Darwinist arguement which I think will never be settled. On one side you have the godless, evidence based fools, who insist that afterlength and the size and geometry of the tailpiece do not affect string tension or downforce. They have a false and shallow belief that tension and downforce are calculable solely from break angle over the bridge, active string length, and bridge height. I was one of them for most of my life, but now I pray for their souls.
We're all insufficient. What explains it is mathematical chaos theory:
Math.com Wonders of Math
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(look especially how the waveforms change with successive runnings of the functions, which is quite similar to how strings behave on an instrument as a note is repeated)
:D
 
I believe he's talking about one of those fretted instruments... :(

You mean like this?
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