Guitar string tension: effect of total length - continued

[mythbuster]

TL;DR Summary

Equation needed for string tension and frequency which accounts not only for fixed points (nut,saddle) but endpoints of string (tuners, bridge).

This wiki page offers an incomplete equation for calculating tension over a length of string. Can you please help complete the equation?

https://en.wikipedia.org/wiki/String_bending

It is referenced here in this thread post #23:

https://www.physicsforums.com/threads/guitar-string-tension-effect-of-total-length.785548/

The equation does not include a variable for the overall length of the string. This is necessary because the mass per unit length of the string will be reduced as the string is brought to pitch because the string is elastic and it stretches, the equation does account for this. However, when there is more string beyond the fixed points (nut, saddle) to the end points (tuner, bridge) the elasticity of the string has more effect because those sections of the string are also stretched.

This is shown practically when there is less string beyond the fixed points with a floating Fender six point tremolo, such as when using a block which places the ball ends of the string closer to the bridge plate, or using locking tuners. The counterbalancing bridge springs need reduced tension when the overall length is reduced.

Also when the overall length is shorter there is less leverage, if that is the way to say it. Meaning that when overall length is shorter it takes more effort to bend the string a whole step higher, but less distance to reach that pitch. And that is desirable for many guitar players, so it would be helpful to understand the science behind this. Thank you for any help!

 

[pbuk]

Good question. Note that the change in mass per unit length you have mentioned is almost insignificant: almost all of the change in pitch in a bend is due to the change in tension.

As a quick guess, I would think that the correction for the total length of the string would be to multiply the $\cos \theta (T - EA)$ term by $\frac L {L_T}$ where $L_T$ is the total length between tuning post (or nut lock etc) and string anchor.

 

[mythbuster]

Thank you for the help! I found Hooke's law:

where

is the amount of deformation (the change in length, for example) produced by the force

and

is a proportionality constant that depends on the shape and composition of the object and the direction of the force.

Every study of string length and vibration I am finding in at *.edu sites only discusses a string with fixed endpoints and the effect of total length stretching, thinning the string is not compensated for. I have only found discussion of the "after-length" vs. "speaking length" in concert string instrument design, like a Cello tail piece, but no math to back it up.

 

[pbuk]

Thank you for the help!

No problem, but don't take it as correct, it was just a guess. I don't think there is much point checking the guess because of the last point below.

I found Hooke's law:

Hookes law will tell you how much lateral force you need to exert on a string to achieve a certain bend angle, but the situation is more complicated than that because you already need to exert a force in order to hold the string on the fretboard: a bend is therefore initiated by rotating the vertical down finger pressure and increasing it "a bit".

where

is the amount of deformation (the change in length, for example) produced by the force

and

is a proportionality constant that depends on the shape and composition of the object and the direction of the force.

If you are thinking you can calculate this constant (which is known as "Young's modulus") for a given string then forget it. The complexities of the physical process, particularly for a wire wound string (although we don't often bend wound strings), mean that this is always determined experimentally.

Every study of string length and vibration I am finding in at *.edu sites only discusses a string with fixed endpoints and the effect of total length stretching, thinning the string is not compensated for.

That's because these effects are insignificant; the only thing that really matters is the change in tension.

I have only found discussion of the "after-length" vs. "speaking length" in concert string instrument design, like a Cello tail piece, but no math to back it up.

That's because the math can only take you so far because of the approximations and effects that are ignored because they are almost impossible to predict mathematically, for instance saddle friction, fret shape (or finger pad for a fretless instrument), string curvature at the saddle, right hand position, pick/nail/pad (plucked) or bow tension/pressure/speed/angle...

And even if you could take all these things into account, they would only apply for one position on the fretboard and sounding action. In practice stringed instruments are designed (and set up) by a process of trial and error.

 

[Tele Savalas]

I can't believe I'm posting on a Physics forum, but I ended up here after a Google search, so here goes...

This question arose out of a thread on another forum asking why a particular model of Fender Stratocaster (the Eric Johnson model) felt 'stiffer' to play--i.e. harder to bend strings of the same gauge-- than other Strats. Numerous theories were proposed, but the one which seemed most connected to reality was the idea that the overall string length--i.e. the distance between the 'anchors' on each end (in this case, the tuner and the bottom of the tremolo block-- is marginally longer due to the way the string end is seated close to the bottom of the trem block on this model.

So I proposed a thought experiment in which a guitar is clamped to a fixed bench, with the distance between the nut and bridge (commonly referred to as 'scale length') the same as a standard Fender Stratocaster (25.5").

Now, imagine instead of a standard-length set of strings, you procure a set one mile long, and set up fixed anchor points a half-mile down the road in each direction on the line extending from the nut through the bridge. Finally, you lower the anchor points--or raise the guitar--to the point where there is enough break-angle to allow the guitar to be tuned to standard pitch.

The two questions which arise are: 1) how will the additional length affect the string tension?, and 2) does the break angle over the bridge and/or the nut change the tension?

My sense is the tension would be so stiff as to render the guitar unplayable, and that additional 'break-angle' would slightly increase that tension further, but I have no idea why.

Any ideas?

 

[pbuk]

For the same fret on the same string on the same scale length the change in pitch is determined only by the change in tension (except for some small effects which can be ignored). However how the effort to achieve that change in tension is achieved is perceived depends on many factors, some of which I have described above. Therefore:

1) how will the additional length affect the string tension?

It won't, otherwise the pitch would be different. However the longer the dead ends are the further you have to bend the string for the same change in pitch. With a string a mile long you wouldn't notice any difference before your finger comes off the fretboard.

2) does the break angle over the bridge and/or the nut change the tension?

No.