The Weight of Ukulele Strings

Ignoring for the moment all the tactile issues with different strings, I would like to focus just on the weight of the strings.

Let's start with a given frequency and a constant length of string (lets say the C string on a concert uke). To keep the frequency the same, the tension will need to change the same percent as the percent change in the mass of the string. So, if the string weight decreases 10%, then the tension must decreases 10%.

With so many fluorocarbon, nylon, and nylon-like strings available, does anyone know if any manufacturers have put out their "unit mass" or "mass per unit length" numbers for their strings? Or, does anyone have access to a lab scale that could measure the weight of a certain length of string? Data like diameter and breaking pressure don't help much across different types of strings.

It would be interesting to know which strings - of the many types out there - have the least mass and therefore the lowest tension.

Being a retired guy during Covid times means I have way too much time on my hands. But still, inquiring minds want to know.

You can calculate the weight when you know the density of the material. Density is mass per unit volume, and volume is squared radius times pi times length. So all you need to do is multiply density with volume and make sure your units are consistent.

https://www.daddario.com/globalassets/pdfs/accessories/tension_chart_13934.pdf

Start with the table on pages 5-10. Classical guitar strings (nylon, titanium...) are typical of most common ukulele strings.

-Wiggy

So the heart of what you are looking for is "linear density"; that is the mass per unit length of a string. This is not a number that most manufacturers bother to post anywhere. (You can measure this to pretty good precision by getting the mass of a long string and simply dividing by the total length of the string.) But... they usually do give you the number you need, if you will bear with me for a moment.

We are really talking about the physics of a string under tension. We can calculate the resonant frequency (f) of a string of a given length (L) under a given tension (T) if we know it's linear density (Greek letter mu.)

Now we don't know the linear density, since they don't give us that number most of the time, but we understand that the total mass will be the density of the string times the volume of the string. If we assume that the string is just a really long cylinder, we can calculate the volume by taking the cross-sectional area (pi * r^2) times the length (L) of the string. If we make this substitution, we find that the frequency looks like the equation in the box below.



So the frequency, assuming we keep the length the same, will depend on the square root of the tension divided by r^2. And the radius of the string is just 1/2 the diameter of the string, and that *is* something that a lot of manufacturers do tell us.

Now for the wrinkle, and where I think your question really comes in. The density of the string is still in there! How do we compare strings of different materials? Well, that is tricky but I suspect that for a given type of strings (say, fluorocarbon) we can assume that their densities are pretty similar. It depends much more strongly on the diameter of the string (since the radius is squared) than it does on the density (since that is NOT squared) so really just look at string "gauge" when comparing strings of the same material.

Thinner string, lower tension, in general. (If we decrease the radius in that denominator, we must decrease the tension in that numerator to compensate.)

By the way, this math also neatly shows why the tension tends to be higher on longer scale instruments. The materials are typically similar, so the density doesn't change by that much, and we need to keep the gauge of the string reasonable, so as the length in the denominator increases, the tension must increase as well.

Oh, and this also explains why there are often different sets of strings for different scale lengths. (Though not always. Some, like Oasis, just have one set of strings for all scale lengths.)

Yeah, I know that there was probably way too much overthinking in this, and most of us probably didn't care or didn't want to really look at the math, but I am a physicist and I did want to look at the math. I actually think about this stuff all the time when I play. And it was either type this up *or* do actual work, and I really needed a break from grading right now...

merlin666 said:

You can calculate the weight when you know the density of the material. Density is mass per unit volume, and volume is diameter times 2pi times length. So all you need to do is multiply density with volume and make sure your units are consistent.

Click to expand...

Yep, we do know the diameter of the strings, and if we knew the density of all the strings we could do this. However, since I don't think we do, I'm back to weighing the strings at a certain length to work this out. I think this would work.

Cluze said:

...

Now we don't know the linear density, since they don't give us that number most of the time, but we understand that the total mass will be the density of the string times the volume of the string. If we assume that the string is just a really long cylinder, we can calculate the volume by taking the cross-sectional area (pi * r^2) times the length (L) of the string. If we make this substitution, we find that the frequency looks like the equation in the box below.
...

Click to expand...

Now we can find out if the rectification is really working or just putting scratch marks on strings.

Cluze said:

... Now for the wrinkle, and where I think your question really comes in. The density of the string is still in there! How do we compare strings of different materials? Well, that is tricky but I suspect that for a given type of strings (say, fluorocarbon) we can assume that their densities are pretty similar. ...

Click to expand...

The type of information in your post is what I've been using to think about this. The question for me is not just for fluorocarbon vs nylon, but what about nylon vs sugar and all the other nylon-like strings. What if there there is a 10% difference or more for density in some of these? That 10% difference in tension might be meaningful for some folks.

It would be nice if the table that Wiggy pointed to contained ukulele string unit weight. But it doesn't. I'll spend some time with it to see if I can make some assumptions about D'Addario strings for other instruments like Wiggy mentioned.

I'll add this to all the other irrelevant information I keep wanting to collect!

Ed1: The 2nd column in the D'Addario table shows the weight per linear inch.

UW- Unit Weight. In all the charts and formulas in the brochure, unit weight is expressed in pounds per linear inch (lb/in)

Unfortunately, the table does not include string diameter. However, using their first example of "J4301." This is very likely from the EJ43 string set, and it would be string #1. Look up that (EJ43) set and you will find the diameters for string #1 written on the packaging. D'addario is good about that.

I hope this helps... -Wiggy

The length will be a constant for whatever scale you are measuring. IE: Tenor: 17" The only length that matters is the distance from nut to saddle.

Ed1 said:

Yep, we do know the diameter of the strings, and if we knew the density of all the strings we could do this. However, since I don't think we do, I'm back to weighing the strings at a certain length to work this out. I think this would work.

Click to expand...

Density of string materials is well known. I think nylon is around 1.14 and pvdf around 1.78 gut somewhere in between around 1.4 g per m^3.

merlin666 said:

Density of string materials is well known. I think nylon is around 1.14 and pvdf around 1.78 gut somewhere in between around 1.4 g per m^3.

Click to expand...

This is great. It made me look into pvdf and nylon. It appears that all pvdf is close to 1.78 and "composite nylon" has a higher density than nylon, which may or may not be true for some of D'Addario and Aquila strings. Aquila states its Nylgut is the same density as gut.

Although I am curious if the companies would give out the density of their strings, I'll put that aside for now. I'll use the numbers above (with various diameters) to answer some of my questions.

Thanks again.

When you come to a fork in the road, take it!

While I wasted a lot of time with the formulae (a good education though), I decided to put that all aside and grab my old spread sheet that was basically the "Ukulele String Comparison Spreadsheet" that was once available here with some Mya Moe (Worth Strings) added in. Using that spreadsheet and the D'Addario pdf from above, here are some thoughts - both new and old.

It appears that Aquila Nylgut and D'Addario Nyltech are the same. I think this was mentioned in the past.

D'Addario has the best info! It would be nice if they would include Ukulele strings in its charts. Regarding the concert sets, the Pro-Arte clear nylon has the least tension, followed by the t2 titanium, rectified clear nylon, and then the black nylon.

The tension of the two GHS and one Worth fourocarbon concerts is more than any of the non-fluorocarbons, despite being of a smaller diameter. I dont have any other string data from other companies, so this is a very small f/c sample.

Finally, I know the above has very little to do with the "playability" of strings. There have been lots of posts on this in the past. I will, however get one each of the D'Addario's mentioned and try them on one uke sometime in the future. This is a different version of my wanting to get seven sets of the same strings and put them on my seven concerts to listen to them without string differences being involved.

'Nuff said.

Y'all do realize that you don't know the string diameter, right? Ever change strings? Ever notice that they won't stay in tune until they "stretch". In the box string diameter is not string diameter tensioned on a uke. In the box string diameter is also not string diameter tensioned on a uke and then relaxed, as plastic deformation has occurred!

sopher said:

Y'all do realize that you don't know the string diameter, right? Ever change strings? Ever notice that they won't stay in tune until they "stretch". In the box string diameter is not string diameter tensioned on a uke. In the box string diameter is also not string diameter tensioned on a uke and then relaxed, as plastic deformation has occurred!

Click to expand...

Yes the math gets complicated if you include elasticity coefficients and diameter changes as a result of tension, needs some calculus for that. I think that's the reason why Aquila does not publish official gauge information.

Hello,
I hope that the formula we at Aquilastrings employ can be easier
Mimmo

i need to lie down now...

mimmo said:

Hello,
I hope that the formula we at Aquilastrings employ can be easier
Mimmo

View attachment 132143

Click to expand...

I was hoping you would weigh in on this. Any different numbers for string density for the Titanium or Sugars?

I think the formula you use helps a lot. If one wants to compare string tension at a certain frequency on one ukulele, all that is needed is the diameter and the string density.

For example, The D'Addario nylon concert (old EJ65C) had less tension than the Martin fluorocarbon (600) at the C string but more tension at the A string. So overall it's a wash.

Too much thinking --- my brain hurts.:wallbash:

mimmo said:

Hello,
I hope that the formula we at Aquilastrings employ can be easier
Mimmo

View attachment 132143

Click to expand...

Hi Mimmo. Thanks for your input. I hope y'all are back in production. Even if it's on a limited basis.

well guys,
Titanium is a kind of nylon whose density is 1.08. The name is just a brad name. No Titanium inside. the color is just a brand color. Sugar bioplastic has the density of gut and nylgut/new nylgut/supernylgut =1.30.
ciao
Mimmo