String Tension and Compliance

[James Collins]

An interesting corollary...

A question was raised about whether there was truth in the claim that extending the string length past the nut makes it easier to bend strings. It follows from the study of compliance where it was demonstrated that the amount of stretch in a string increased as the overall length increased. But the answer to the bending question is a bit surprising - it's both yes and no.

The "yes" part is easy to understand. If you increase the overall length of the string, there is more length to stretch. If you look at this characteristic another way you can see that it will take less force to bend a string the same distance as the overall length of the string is extended, since there is more overall stretch. But the downside is that you will have to bend the string further to bring it up to the same pitch. If you had to bend a string 10mm to raise its pitch one semitone, you might need to bend it 20mm to bring it up a semitone when you increase the overall string length by moving the tuner further from the nut.

But here's the surprising part - in order to bend the string up a semitone with the extended afterlength, not only will you need to bend it further, it will actually take more force than the same string with a shorter afterlength. I was not expecting that, but that's what I discovered using the compliance test rig. I strung up the rig with a string running to the nearest tuner to the nut and tuned it to A. I then suspended a weight on the centre of the string (a water bottle), and added a bit of water at a time thus bending the string a bit more each time until the string sounded Asharp (one semitone up). Then I weighed the water bottle. I moved the string to the outer tuner, adding 6 inches to the afterlength, and repeated the procedure. Here are the results:

Short Afterlength
Amount of deflection of string from A to A#: .576"
Weight to deflect string from A to A#: 993 grams

Long Afterlength
Amount of deflection of string from A to A#: .684"
Weight to deflect string from A to A#: 1185 grams

So, when I added more afterlength I had to bend the string further AND apply more force to raise it a semitone. So bending the string was easier if we are looking to bend the string a certain distance, but is actually harder if we want to bend it to a certain pitch.

I don't really understand why it would take more force to raise the pitch with the longer afterlength, but I am told by an engineer that it has to do with force vectors, and he pointed me to a physics problem of walker on a tightrope and talked about a free-body diagram showing all of the horizontal and vertical components of the forces involved. I'm afraid he lost me. But I can't argue with what I found using the test rig, even if I can't really explain it.

I don't think it has to deal with vectors. If you stretch the string 1" the force vectors are the same.

I think you could also look at it in a simple way. First think of the string as a spring that stretches and goes back to length.

According to Hooke's law: Fs = -kx is the equation for force on a spring where k is the spring constant (string material, length, diameter, etc) and x is the change in length from the equilibrium.

Imagine a spring with 1 coil every inch. Tension on the spring will be the same when the coils are the same distance. In order to bend to a new pitch you want to stretch each coil 0.1 inches. If you have a spring length 36", you have to increase the spring length by 3.6". Fs = -k * 3.6".

If the spring is otherwise the same but now 40" when you stretch the string it needs to stretch 4" for the same effect and the force equals -k * 4".

Force is proportional to the amount of stretch; and amount of stretch is proportional to the starting length.

Please note this is a little bit of an over simplification because the k will be different for the two string. The longer string will have a smaller constant. That is why for a shorter string it takes more force to stretch a given amount. -k*x when x is the same depends on the k.

Interestingly enough you seem to have independently copied Hooke's work on springs and you are basically measuring this constant for the strings.

Now I wonder if there is an optimum number of wraps for good intonation and good vibrato if the strings are click like on nylon trebles. Lots of after length would seem to make a string more resistant to going out of tune with incidental string bend with chords for example. But more string length is going to make it resistant to vibrato if I'm reading your results correctly.

 

[Turnaround]

I don't think it has to deal with vectors. If you stretch the string 1" the force vectors are the same.

I think you could also look at it in a simple way. First think of the string as a spring that stretches and goes back to length.

According to Hooke's law: Fs = -kx is the equation for force on a spring where k is the spring constant (string material, length, diameter, etc) and x is the change in length from the equilibrium.

Imagine a spring with 1 coil every inch. Tension on the spring will be the same when the coils are the same distance. In order to bend to a new pitch you want to stretch each coil 0.1 inches. If you have a spring length 36", you have to increase the spring length by 3.6". Fs = -k * 3.6".

If the spring is otherwise the same but now 40" when you stretch the string it needs to stretch 4" for the same effect and the force equals -k * 4".

Force is proportional to the amount of stretch; and amount of stretch is proportional to the starting length.

Please note this is a little bit of an over simplification because the k will be different for the two string. The longer string will have a smaller constant. That is why for a shorter string it takes more force to stretch a given amount. -k*x when x is the same depends on the k.

Interestingly enough you seem to have independently copied Hooke's work on springs and you are basically measuring this constant for the strings.

Now I wonder if there is an optimum number of wraps for good intonation and good vibrato if the strings are click like on nylon trebles. Lots of after length would seem to make a string more resistant to going out of tune with incidental string bend with chords for example. But more string length is going to make it resistant to vibrato if I'm reading your results correctly.

Thanks for this. If I interpret correctly, Hooke's work supports what I found and puts a formula to it.

 

[James Collins]

Thanks for this. If I interpret correctly, Hooke's work supports what I found and puts a formula to it.

Correct. Sorry for being so late to the party.

 

[LetItGrowTone]

Deserves a sticky.
Thank you.

 

[HaphAsSard]

What you are talking about is Adsr. Attack decay sustain release. Synths specialize in being able to alter all of those parameters at the turn of a dial, tap of a button or pad and I dare say there will be someone out there that has the electronic equipment to measure them.

As far as I've read, initials in "ADSR" refer to the duration in time of the respective phases, themselves identified by variations in amplitude. I referred specifically to variations in *pitch*, as found on plucked string instruments upon the attack.

I would suggest the shorter scale has faster attack, faster decay, less sustain and quicker release than a long scale which is why our long scales are perceived by our ears as having more depth of tone. Apologies in advance to all those that may be upset by my observations but tone is the key thing we all seek for is it not

Nor did I compare different scales, but identical (short) scale with either long or short afterlengths past nut and saddle. I was wondering if possible, minute differences in tone (feel aside), determined by such difference in afterlenghs, could account for a certain familiarity to be encountered by long-scale players when approaching a bass of the "short-plus" type (as defined above), which they might misinterpret as "more tension" from the longer total string.

Since I was all but ignored in this wondering e-loud of mine (except for an ultimately unimportant, passing comment I'd made about a Fodera gimmick) I stopped following the thread.