Working out the string tension on a Tenor Guitar.

[Allen]

I've been asked to build a few tenor guitars tuned with linear tuning in GCEA like a Tenor Ukulele but in steel strings. I've been trying to figure out the string tension so as to be able to brace the top accordingly.

Nick gave me his plans for a Regal Tenor and it turns out to be almost identically sized to my Baritone body, so I'm going to use that but not ladder braced and with a 22" scale length instead of the 20+" on the Regal. Going with a conventional pinned bridge instead of the tail piece as well. So the plan was a reference, but not using it per say.

On RyCooder's site there is a section on Tenor Guitar tunings, and string size recommendations.

I chose the following sizes.

G - .025" wound
C - .020" plain
E - .015" plain
A - .011 plain

Now, to the math. D'Addario has a pdf download that they call a tension chart of their strings. You need to work out the tension of the string using this information and following formula.

UW- Unit Weight. In all the charts and formulas in the brochure, unit weight is expressed in pounds per linear inch (lb/in).
L- Scale Length. This is the vibrating length of the string. This is determined by measuring the distance from the nut to the bridge of the instrument in inches (in).
F- Frequency or pitch. This is the pitch at which you will be tuning the string expressed in cycles per second (Hertz).

To calculate the tension of a string in pounds use the formula below, inserting the three variables described above:
T(Tension)= (UW x(2xLxF)2)/386.4

If you know what tension you want the string to have, you can calculate the string unit weight. You can then use the charts in this guide to locate a string with approximately the same desired unit weight.
UW(unitweight)= (Tx386.4)/(2xLxF)2

From this I've worked out the following using the medium string selection with a low G instead of the re-entrant option.

G String - .025" 80/20 Brass Round Wound
G String Tension = (.00012339unit weight x (2 x 22" x 196hz)2) / 386.4
G String Tension = 23.7498210783 lbs

C String - .020" Plain Steel
C String Tension = (.00008861unit weight x (2 x 22" x 261.6hz)2) / 386.4
C String Tension = 30.3827059939 lbs

E String - .015" Plain Steel
E String Tension = (.00004984unit weight x (2 x 22" x 329.6hz)2) / 386.4
E String Tension = 27.1281810328 lbs

A String - .011" Plain Steel
A String Tension = (.00002680unit weight x (2 x 22" x 440hz)2) / 386.4
A String Tension = 25.9961109731 lbs

So, from this data, assuming it's correct, the total string tension on this instrument is going to be approximately 105 lbs.

[nnickusa]

mcdonaldstrings.com has an on-line calculator Allen. Just plug in the numbers, and it'll give you the tensions, to at least double check your results....

Given my very limited eperience, I have little more to offer, but look forward to seeing how yours turn out.

Cheers,
Nick

[Allen]

I have tried Graham's calculator before going this route.

I picked the 22" bouzuiki and put the strings I wanted in. As it's a double course I halved the total string tension. Problem is that it gives a total string tension of just over 55lbs so half is at about 27lbs. for the total of all 4 strings.

Something wrong there, as that number just can't be right. That's why I went the long hand version.

[jeffhigh]

I used the calculator, but used guitar at 55.8 scale and the notes you have specified and just added up the first four strings and came up with 50.49KG or 111lbs
I think the bazouki option was expecting an instrument tuned down an octave

[Allen]

I talked with a mando builder down around Byron who put me onto the way he chooses strings. He likes to have between 9 and 11kg per string. The D'Addario J66 (tenor guitar) string set has string tension around 85lbs, so with that in mind I reversed the equation to have a string tension of 10 kg each and then found strings that pretty closely matched that scenario.

G - .024" woiund
C - .017" plain
E - .035" plain
A - .010" plain

Giving a total tension of 87.25lbs.

I think I will be going with this option.