Real string tension measurement

Maths and the oud

This got me thinking about long-lost maths and scrawling on notepads instead of the work I should be doing so I thought, just for fun, I'd chip in.

It should be possible to calculate the tension in situ too, without having to take the string off the oud. My school maths and physics are pretty rusty, but I'll give it a go. I know there are engineers and ex engineers on this forum too, so perhaps they can correct me or complete the mess I start.

Say the string has a length L.
1) find the centrepoint of the string by locating the octave harmonic.
2) use your fishing scale (newton metre) to pull the string sideways by a known amount, say 2cm - you can call this distance the deflection (d).
3) your newton metre will now register a force pulling the string back into line which you can call the restorative force (R)

This is where my maths gets shaky and the right answer is one of two options:

The easy way I suspect is wrong but I'm trying it anyway:

I think the force R divided by the string tension should be the same as the deflection (d) divided by half the string length. This is because they should both be equal to the tangent of the angle of deflection (you don't actually need to know this angle). The deflection distance and half the string length form the opposite and adjacent of a right angled triangle.

In other words, tan theta = 2d/L

Likewise, the restoring force (R) and the string tension at rest (T) form two components, at right-angles to each other, of a vector which is the deflected string tension. Again, tan theta = R/T. This is the bit I'm not so sure about, but it does seem to make sense.

So if R/T =2d/L then T=RL/2d

ie to get the string tension in newtons, multiply the deflection distance in cm by the restorative force in newtons, multiply this number by 2 and divide by the string length.

This seems a bit easy to be true, and I have a feeling it might be ignoring the effect of string elasticity on tension, in which case, option 2 is a lot more complicated and I can't work it out.

BUT, there should be a relationship between the resting tension of the string and the following four variables:

1. The frequency of the string at rest (or actually, the octave above the frequency, as you would be working with half the string length)
2. the frequency of the displaced (stretched) string (again, it will actually be the octave)
3. The string length (or half the string length)
4. The restorative force registered on the fishing scale as you pull the string sideways
5. The distance of sideways displacement of the string.

(The standard formula relating frequency (f) and string length is

f=1/2L.square root (T/mu)

where f is frequency, L is length, T is tension and mu is the mass per unit length of the string)

The formula will be quite complicated with quite a bit of trigonometry and logarithms and I'd love to try and work it out but I think it surpasses my abilities. However, once you have the formula, measuring string tension would be a piece of cake

Hi DaveH, thanks for your response!

when I find time I can do an experimental comparison of your calculation with the experiment.

Fact is at least that the device I designed does measure the real longitudinal tension (i.e. along the string) as it would sit on the lute (oud) simulating the situation as close as it can be: one nut and the string tied with a knot.

So what you measure (as precise as the scale you use is) here is the tension you're looking for without relying on any theory that might be right or wrong...

As far as comfort is concerned you can do the measurement when you change the strings anyway - they are then really set into their proper tension.

My gut feeling says that your method of calculation can only be possibly correct as long as the string is in a regime of linear elasticity (that means Hooks law) which is difficult to tell for mqaterials like nylon or nylgut or even worse, gut. Where the string sounds good you might well be in a place where the elasticity of the string changes with the tension.

Now, wenn you pull the string sidwards you might change the elasticity (you certainly change the pitch and the tension, but that would be allright in linear case). Also you would have to make sure not to detune the string by pulling it - that's what you do when you want to lower the pitch a bit...

Anyway, thanks, and when I have some experimental results I'll let you know

best regards

Oh well. At least I got to get in touch with my nerdy side a little.

I think the discrepancy is either because the scale isn't that accurate for low forces (don't know how big the fish you catch are!) or, as you mentioned before, the string isn't behaving completely elastically so the extra extension when you pull it sideways isn't producing a proportional restorative force (ie not obeying hook's law). OR my equation is completely screwy - as I say I'm not a physicist mathematician or an engineer.

There's an often quoted tuning adage with both lutes and, I think, ouds from the pre-tuning fork era. According to this, you tighten the string "until it's about to break". Personally, I would find it hard to tell this point until just after I'd reached it, which wouldn't be much use. But I think the point being made, in modern materials science terms, is that the strings sound best when the elastic limit is being reached, which would back up the second possibility. Incidentally, with gut, there are two main types of chemical bonds in the collagen and elastin which account for its mechanical properties: hydrogen bonding, which is weaker, and disulphide bridges, which are very strong. There are therefore two phases to its extension, one where the hydrogen bonds are being stretched and a second where they're broken and the much stronger disulphide bridges kick in. This probably accounts for the unique acoustic properties of gut (my knowledge of organic chemistry is slightly more recent than my knowledge of mechanics, but still less than perfect).

I don't think a few mm error in finding the octave point will make much difference - you could just as well do it by measuring. I don't think using the third harmonic would change much - the balance may behave more accurately but the extra tension will likely make for even more error due to the reducing elasticity. Likewise if you use the midpoint but just pull it further sideways. You could try it with a more sensitive balance, if you have one to hand.

I'd appreciate if someone with more knowledge could comment though and perhaps suggest a way it might be possible to measure string tension in situ indirectly. It would be a useful thing to be able to do. Would using the vibrating frequency be a better bet?A quick google search finds plenty of devices for doing this on tennis rackets, but I imagine these operate much more within the elastic limit of the string.

Thank you for the last contributions.
One very simple reason why the method you proposed seems to yield to large values seems to be that by pulling the string sidewards (at any point by any measurable amount of length) you also inevitabely increase the longitudinal tension (as the to large measurement shows).

This is independent of the issue whether the string reacts with an unchanged or changed elastic constant. Of course the latter seems to inscrease anyway at the upper range of possible string tension, be it nylon or gut.

If one wanted to take into account the length of the string behind the nut, things get really complicated, as every string has here different ammounts of length.

I will check it experimentally whether it really makes any big difference.

In reality things are even more complicated because of the sharp angle of the strings on the nut so that the tension behind the nut could be quite a lot bigger than the tension of the vibrating string (as certainly is the case with renaissance or baroque lutes, especially with wound strings.) What it certainly changes is the force on the peg. Also a string could break between peg and nut though it hasn't reached its breaking point on its vibrating side.

Actually I don't see theoretical reasons why this should matter, as we don't measure the force per length but rather just the force along the string. Theoretically all strings of the same linear density (mass per unit length) should vibrate the same fundamental frequency when pulled with the same tension on the same vibrational length. Maybe I'm wrong here? Of course, the longer the string is, the more windings on the peg you need on the peg to achieve the same tension because the length strech spreads over the total length. But the strech per length unit should be independent of the string length ond only depend on the tension.

Als I measure the tension on the vibrating side of the string, so eventual increase of tension on the peg side doesn't really matter in the frequency equation.


BTW the only fishing lines I ever pulled with this scale are oud strings

PS: I just checked the fishing scale with a wieght that showed 259 g on my digital kitchen scale und 0,26 kg on the fish scale.

This makes less than 5% error which is small compared to the other difference.

Does the distance behind the bridge affect the tension after the bridge?

interesting Rojaros
I have a question and i guess with the method you described you can answer me (if you have the will )
does the distance behind the bridge have any effect on the tension generated after the bridge? The only parameter that can change the distance before the bridge is the bowl length. let's say you have 2 bowls, one is 49cm and the other is 52cm, you put the bridge to have a 60cm stringlength. When you put similar strings (let's say Pyramid oud strings) on both ouds the 52cm bowl would generate more tension?
I think with your method of string tension measurement we can have an answer about this issue. If you do 2 measurements, at different places of the metalic thing that you have do you generate different tensions after the metalic block?
Thanks !

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