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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?

Bob
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    Is there a typo in your final question? – Bill N Sep 13 '15 at 03:52
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    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). –  Sep 13 '15 at 04:00
  • I think that's what it is too, Chris. How does the bridge being there affect the sound? – Bob Sep 13 '15 at 04:36
  • Bill, I fixed it. – Bob Sep 13 '15 at 04:38
  • Can you tell whether you're lifting the string off the bridge? That would certainly drop the pitch, as the efffective length just increased. – Carl Witthoft Sep 13 '15 at 12:07
  • Bob: "one technique for lowering the pitch of the string on one side of the bridge is to lift the string on the other side." -- I'd imagine that the "string on the other side" is strung at some angle wrt. the "string on one side" (whose vibrations cause the sound). Now, holding the "string on one side" vertical for playing, the "string on the other side" is at a slight angle "downwards from the bridge". Therefore by lifting the "string on the other side" exactly upwards you may be shortening/releasing its tension towards the bridge (while lenghtening/tightening its other side). – user12262 Sep 13 '15 at 16:31
  • @user12262 But tension should be constant throughout the string. – Bob Sep 13 '15 at 18:50
  • @CarlWitthoft I don't know that I'm lifting it off the string completely, but I might be lift it off a bit. I'll check when I get it out. It's surprising to me that my finger could be used as a node, and that the existence of the bridge wouldn't dampen the sound. (Maybe it is) – Bob Sep 13 '15 at 18:52
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    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. – user12262 Sep 13 '15 at 19:43
  • @user12262 I suppose the tension isn't constant then due to friction with your finger? – Bob Sep 20 '15 at 00:57
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    @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.) – user12262 Sep 20 '15 at 07:29

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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.

Guill
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