# String length vs. tension?

Discussion in 'Luthier's Corner' started by Tim Barber, Jul 29, 2003.

1. ### Tim Barber

This is something that has been bugging me for a while. There are several headstock and string anchor designs that are intended to make the total string length (tuner to ball end) greatest for the lowest pitched string. The conventional wisdom being that longer string=tighter tension for a given pitch. This is true if the vibrating length of the string increases: the tension would have to be increased to achieve the same frequency of vibration (pitch). The Novak fanned-fret system works, because the actual sounding length of the lower strings is greater, so the tension/pitch ratio actually does change. However, for a fixed open string length and a given mass of string, a change in tension would have to result in a change of pitch as well. So I don't understand how increasing the total length of a string by lengthening it beyond the witness points at nut and bridge can result in a tension change without affecting the open pitch.

Or am I missing something basic?

2. ### l0calh05t

Oct 14, 2001
Cottbus, Germany
no, you aren't missing anything, you are absolutely correct

3. ### geshel

Oct 2, 2001
Seattle
Yup.

As near as I can figure, the difference that these designs would make would be in flexibility, by providing more string to stretch when it's vibrating. However this is dependent on smooth movement over the nut or saddle, which isn't necessarily given.

4. ### pilotjones

Nov 8, 2001
US-NY-NYC
HOORAY!!!!

You are absolutely correct. The length beyond the nut or bridge saddle is meaningless with regards to string tension.

A fairly new member, who got it right the first time instead of following the "urban legend." You made my day!

5. ### pilotjones

Nov 8, 2001
US-NY-NYC
Man, Taylor beat me while I was typing.

6. ### Tim Barber

Happy to be of service!

7. ### geshel

Oct 2, 2001
Seattle
Could you explain this, please? Vibrate the wood how? Why does it result in floppiness (lower tension, I presume you mean)?

8. ### Tim Barber

Not sure I'm following. Changing the break angle over the bridge will change the downward component of the force which is transmitted to the top of the instrument through the bridge, but it won't change the physics of the string. As I understand it, the string's linear mass/density (µ), vibrating length between nodes at nut and bridge (L), tension (T), and fundamental pitch (f) are interrelated. You can't change one without affecting at least one of the others.

i.e.:

decrease in T = decrease in f
decrease in L = increase in f
increase in µ = decrease in f etc. if other variables remain constant.

Flexibility of the string is probably another issue. I'm sure that differing string construction and materials could well result in differing amplitudes of string vibration at a given frequency. Greater flexibility could well be experienced by the player as "less tension" but I don't think it's the same thing.

BTW I'm not trying to challenge others who have far more experience, just trying to figure it out for myself. And partly going on dim 30-year-old memories of high school science class.

Maybe the only way to be sure is to build a test rig and do some measurements.

9. ### pilotjones

Nov 8, 2001
US-NY-NYC
Timmy

Everything you put down is right regarding length-tension-frequency-unit weight (linear density) is correct. Also, the fact that what you do with the string beyond its vibrating length is insignificant - when you follow the assumptions of a "perfect" node at each end of the string.

The things not yet considered are the things that the assumptions of the equations don't take into account. These are:

- the string may move over the bridge somewhat, especially during plucking. If this happens, the string would feel looser as you pluck it. And the effect would be increased if the amount of string on the far side of the saddle is increased (and hence its longitudinal stiffness is decreased). Personally, I don't think that this happens, otherwise we would see it in rapid wear of the bridge saddles.

- the bridge may deflect somewhat, especially during plucking. I am thinking that maybe this is normal behavior on a stand-up bass, especially the way it is constructed - saddle point floating several inches above the body, free to "swing". Again, the effect would be increased if the amount of string on the far side of the bridge is increased.

10. ### geshel

Oct 2, 2001
Seattle
Because the frequency of vibration is determined (for the most part) solely by the properties of the string between the bridge and the nut.

There might be subtle effects due to the string portion on either side of those barriers, but it's not much.

It's like fretting the note. It's still "long", you just made the vibrating part shorter. The nut does the same thing. The string could the 8 feet long between the nut and the tuner, and the tension wouldn't need to change to keep it at the same pitch.

11. ### boomgungun

Oct 31, 2008
ya but my question is not the frequency, it's the overall tension. In your case, assuming if you release the nut like u release a fret, and the tension doesn't change, which is an ideal frictionless and roller break points situation. Then shouldn't the tension determined by the entire length rather than just the fretted part (playing length) ?
If u're saying fretting the note does not change tension, then why should moving the nut position change anything?
If we change the nut position, it changes the pitch, so then we've to change the tension to return to its pitch, then does that change the tension of other portions? could the tuner-nut and bridge-tailpiece have a different tension now?

12. ### geshel

Oct 2, 2001
Seattle
Pitch is determined by the mass, length, and tension of the vibrating part of the string.

If you don't change one of those three things, pitch doesn't change. Make the string longer past the nut, the vibrating part is still the same length as before, so for the pitch to be the same, the tension has to be the same.

Really, there were a million threads on this back when this one was fresh in 2003. I'm assuming you ran a search, otherwise how did you find this one. So. . .read the others too, as they go into this in enormous detail.

13. ### boomgungun

Oct 31, 2008
I understand that string tension is determined by cofactors such as string gauge, scale length, and the pitch.
imagine u've a string without nut and bridge, then stopping the string in two points in the middle, making it sound a certain pitch, now if u keep the tension the same, but with a much longer string, stopping at the same distances in the middle, would the vibrating length now have a different pitch? perhaps lower? this seems confusing cuz now the entire length seems to matter in tension if I were to keep the stopping pitch the same... this is why I wonder if the friction and compression play a role in this..
thanks anyway.

14. ### geshel

Oct 2, 2001
Seattle
The pitch would be the same. The tension, mass, and length of the vibrating part are exactly the same as they were before.

15. ### boomgungun

Oct 31, 2008
Ah! So the tension is the same throughout the string regardless of angles and length of the entire string? Thanks!

16. ### geshel

Oct 2, 2001
Seattle
In the ideal case. But for the most part, even in the real world, yep.

17. ### boomgungun

Oct 31, 2008
ah, u cleared up my clouds, thanks for the fast response. somehow I thought angle might affects the tension of a string too...

18. ### TheJoshinator

Sep 23, 2012
Nope - the tension in a string is evenly distributed, equivalent to hanging a certain amount of weight on the end of it; no matter what point you pick on the string, or what angle you have it running around a corner, it has the same force pulling on it. Thus, the length of the string beyond the nodes has no bearing on its tension between them.

In addition, what you were saying before about having a different tension if you move the nut back has nothing to do with the length of string behind the nut. The greater tension on a 35" scale than a 34", for example, is due to the fact that you're lengthening the vibrating section but tuning it back up to the same pitch.

19. ### boomgungun

Oct 31, 2008
you're right TheJoshinator, it only affects the pitch of the before and after length, but not their tension. somehow I was thinking of the mean of all 3 pitches in all 3 sections rather than just considering the pitch in the mid section.

20. ### boomgungun

Oct 31, 2008
http://www.noyceguitars.com/Technotes/Articles/T3.html "Also, as the bridge and nut are not frictionless carriages for the string, tension can get 'stored up' in the lengths of string between nut and machine and bridge and tailpiece and, with playing, this stored tension in the vibrating length and thus the pitch"

btw, That is what I meant when I said "unless the friction at the breakspoints of both at the nut and bridge are so much that the "tensions" don't even out even after time for settling or
(work hardening). Maybe with a sharp break point angle, ie. very high bridge and low nut&saddle, then the tension could be hold up with less in the after length than the playing length" But this seems to be not the case since they even out quite easily..