Discussion in 'Luthier's Corner' started by pilotjones, Nov 16, 2003.

1. ### pilotjonesSupporting Member

Nov 8, 2001
US-NY-NYC
In a thread in the Basses forum, several technical questions came up regarding string tension. In response to this, I have made up a spreadsheet.

First let me say, I invite everyone to check my math and derivations. I will then make any corrections to it.

EDIT- this following paragraph now refers to the second sheet of the spreadsheet.
Second, let me just note that the first column represents a 35" E string, with string length beyond nut and bridge neglected, so that the results are as if the string were clamped hard at these points. The second column uses these extra lengths in the equations, under the assumption that the string is free to slide over these points. The third column is a bit of a hack to simulate plucking the portion of the string on the headstock.

Third, thanks to JP for the last sheet, which is his fret calculator with a few revisions.

EDIT- this following paragraph now refers to the first sheet of the spreadsheet.
The first part of the sheet is for figuring what will be the tension required to bring a given string, at a given scale length, to a given pitch. This should be useful to anyone playing with different scale lengths. EDIT- It also contains sections for figuring any of the four factors, given the other three.

EDIT- this following paragraph now refers to the second sheet of the spreadsheet.
The second part of the sheet attempts to see how much the pitch should increase due to the plucking deflection. This is something not accounted for by the "ideal" string equation. It also calculates the finger's plucking force.

EDIT- this following two paragraphs not entirely accurate, due to corrections that were made in later revisions.
The results in the second part are not what I expected. It predicts that if the string is clamped at bridge and nut, the transient "going sharp" of the string due to its initial deflection is only .115 cents; and that if the string is allowed to move over the bridge and nut this is only slightly reduced to .104 cents. The most surprising thing to me is that both of these values are so low. As Sheldon D. pointed out, a string generally goes by several cents when first plucked; so either I my formulas are off, or this is caused by some other mechanism.

Another surprising result was that the clamped-string and free-string plucking forces were virtually the same.

So, please check it out and comment, correct, etc.

Pete

EDIT: 2003.11.17.1200: spreadsheet has a problem (see below), will fix and repost.
EDIT: 2003.11.18.2310: deleted this attachment; new version posted below.
EDIT: 2004.11.02.0745: Added notes to bring up to date.

2. ### geshel

Oct 2, 2001
Seattle
Nice work. Something isn't adding up though (I say this intuitively, not looking at the math yet).

If I pluck behind the nut, and stretch the main portion of the string, I can get a semitone (50 cents) with about 0.2" of deflection.

This is purely due to increased tension in the nut-tuner length.

Any increase in tension there is the same as the increase between tuner-bridge (assuming a frictionless nut, but if anything there's more as I'm pulling on that part of the string).

So, there is the same % increase in tension due to that deflection.

Thins get tricky because the length of vibrating string also increases, which has a counteracting effect. And since the "instantaneous frequency" depends on the position etc etc, it's all really non-linear and I'm sure the math would fill a small closet. Anyway - I'd hazard to claim that the real difference due to post-nut/bridge length would be more significant.

3. ### geshel

Oct 2, 2001
Seattle
Edit: it's really more like a 1/4-tone (oops, that's really 50 cents) with that deflection at the mid-point. Still, a lot more than .1 cents though.

4. ### pilotjonesSupporting Member

Nov 8, 2001
US-NY-NYC
Hey, Taylor, I thought you'd be up for the peer review.

If you put in .2 deflection, 1.5in between nut + tuner, and pluck it at .75in (halfway), the sheet yields 30 cents sharp - not too far from your quarter-tone (50 cents) estimate. Hopefully this is because I got the part where I compensate for the increased vibrating length right. To be more specific, I do use the increased length to generate a new tension, which results in a higher pitch. I do not however increase the effective length between nut and bridge for purposes of the string equation, since to my mind that implies that the wave is moving along this new bent/deflected/increased-length path, which it is not; I believe the wave to still be moving along the straight nut-to-bridge path.

5. ### pilotjonesSupporting Member

Nov 8, 2001
US-NY-NYC
BTW another unexpected result I got was that in the derivation, the k of the spring equation drops out. I had fully expected it to remain, which would have caused the core size and and composition to enter into it; as it is, they are uninvolved, assuming linear stress-strain, etc.

EDIT: as corrected below, the k does enter into it.

6. ### geshel

Oct 2, 2001
Seattle
Actually - that doesn't describe the experiment I was doing. I was putting a .2" deflection in a total string length of 36"+, and getting a 50 cent pitch increase.

When I thought about it, plucking behind the nut was unnecessary: how much does the pitch change when you bend a note fretted at the 12th fret? That's pretty close to the same thing, and it's a lot more than a couple cents.

7. ### geshel

Oct 2, 2001
Seattle
Oh, OK: you're using the formula

Which I don't think is at all the case? It depends on the mechanical properties of the string, doesn't it? (how far it will strech when subjected to an increase in tension).

Certainly in the "gross" case the above formula is not true (double the tension when the string is pulled to twice the length ). But I don't think it's even correct for minor deviations.

Nov 8, 2001
US-NY-NYC

Nov 8, 2001
US-NY-NYC

10. ### pilotjonesSupporting Member

Nov 8, 2001
US-NY-NYC
Corrections are almost done. Will post today or tomorrow. It involved an entire page of derivations. Thanks a lot, mister-two-last-names!

11. ### pilotjonesSupporting Member

Nov 8, 2001
US-NY-NYC
Here's the new rev.

- Core size does enter into it. Increasing the core size increases the amount of initial sharpness of the note. It also slightly increases the required pluck force.

Personally I'm more comfortable with the result that core size matters. Seems correct unless someone finds anything else...

- allowing for string slip over nut and bridge results in a bout a 10% reduction in how far the note goes sharp.

EDIT - attachment removed in favor of updated version that follows in later post

12. ### geshel

Oct 2, 2001
Seattle
Cool - that looks more like what I'd expect. Still hard to guess how the stretching will affect harmonics etc (longer length important in that case - though the length diff is pretty darn small).

13. ### pilotjonesSupporting Member

Nov 8, 2001
US-NY-NYC
I think predicting how the stretching affects harmonics is thesis work, since I think that any analysis of them in a non-idealized manner would fall into that category.

I believe it is a general trait, however, that the longer a string is, the closer its harmonics are to being in tune. This is due to it being closer to the idealized state of length/thickness approaching infinity, and also to the idealized state of deflection angle at the ends approaching zero.

14. ### pilotjonesSupporting Member

Nov 8, 2001
US-NY-NYC
I have attached a newer version. In it I have added a simpler first sheet, which allows you to find any of the four variables, given the other three.

This can be used to figure out string guages for an equal-tension set, would be useful for figuring Novax-type fretboards, etc.

The sheet for figuring pitch increase due to deflection is moved to the second sheet.

15. ### pilotjonesSupporting Member

Nov 8, 2001
US-NY-NYC
Re-attaching the document, which was lost in the forum changeover.

EDIT: 2004.11.02 removed attachment in favor of updated version in next post.

16. ### pilotjonesSupporting Member

Nov 8, 2001
US-NY-NYC
Updated version.

Scans of derivations removed due to new limitations on attachment size.

Note frequency sheet updated.

17. ### Suburban

Jan 15, 2001
lower mid Sweden
Well, Peter, a gigantic job, as usual. I haven't been able to go thru it properly, but it seems good enough.

The most important factor, which is usually left out(!), is the core of the string. It is the core that carries the tension, whereas the windings add weigh.

I'm surprised that you are surprised that clamped-string and free-string are so close. The thing is, that the tension variation between the two is rather minimal. And if you talk Real Life, the difference between a clamped nut and a 10 degree backslant peghead is really - nothing. What could have more impact would be possibility to rotate around the clamping point, but that would not be very significant either.

My point is, that 'sharp when pucked' is insignificant! It's there, sure, but has a very miniscule impact on what we want as output: music.

String tension also has a miniscule impact on the music, but does impact the playing comfort.

18. ### Corey YGuest

Jun 3, 2010

This attachment doesn't work anymore. Any chance of a re-up? I need some calculations for an odd fixed bridge design and I think this would greatly help.

19. ### pilotjonesSupporting Member

Nov 8, 2001
US-NY-NYC
Sure thing. Lets give this a shot.

I just saved out this version 1.4 as .xls from a non-Excel application, so would someone please download this and open it up and let me know whether it works properly. Thanks.

File size:
112.5 KB
Views:
182
20. ### octaedro7

I dunno if this helps but I opened it on OpenOffice Calc and everything seems to work, although I just looked at it on the surface...