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I have a zither with movable bridges, and one technique for lowering the pitch of the string on one side of the bridge is to lift the string on the other side. Because the length of the string is remaining the same, the tension must be decreasing, but why would raising the string on one side lessen the tension in the string? As far as I can tell, pulling upwards on the string can only serve to increase tension, which should increase pitch.

So my question is, given a string held down at two ends, held upwards in the middle by a bridge, why does pulling upwards on one side of the bridge lower the pitch on the other side of the bridge?

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    $\begingroup$ Is there a typo in your final question? $\endgroup$
    – Bill N
    Commented Sep 13, 2015 at 3:52
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    $\begingroup$ I've never seen a zither up close, much less played one, so I'll just leave this as a comment. It sounds like you're just increasing the effective length of the string. Sure, there's a slight increase in tension (frequency goes as the square root of tension), but there's a significant increase in the length allowed to vibrate (frequency goes as the reciprocal of length). $\endgroup$
    – user10851
    Commented Sep 13, 2015 at 4:00
  • $\begingroup$ I think that's what it is too, Chris. How does the bridge being there affect the sound? $\endgroup$
    – Bob
    Commented Sep 13, 2015 at 4:36
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    $\begingroup$ Bob: "But tension should be constant throughout the string." -- Think of a fishing line being suspended from (the tip of) a fishing rod, with the sinker swimming pretty far away. Now, if someone grabs this suspended fishing line (very gently, with two fingers) right in the middle and lifts it straight up just a little -- What happens? The distance from fishing rod tip to finger tips is reduced; and the tension of the fishing line in this section and also along the fishing rod is reduced! (But there's additional pull on the sinker.) At least that's my suggestion; you will know best. $\endgroup$
    – user12262
    Commented Sep 13, 2015 at 19:43
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    $\begingroup$ @Bob: "I suppose the tension isn't constant then due to friction with your finger?" -- Yes, that's what I imagine. It should be interesting to compare the pitch you get by pulling upwards with your fingers with the pitch you'd get by pulling upwards (with similar force, direction, point of contact ...) with a suitable very smooth hook. (p.s. What a shame that PSE Mathjax still doesn't seem to support pstricks commands for drawing.) $\endgroup$
    – user12262
    Commented Sep 20, 2015 at 7:29

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Without a diagram, I am not sure I have the correct picture, but as I understand it, the string is initially divided in two parts by the bridge. If you raise the string off the bridge by pulling at the 1/4 mark (from left), then the length of the remaining string (1/2 L) will increase to 3/4 L, so the frequency of this section will decrease. See Chris White's comment.

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