http://www.gamutstrings.com/calculators/calculator.htm

I play a 16.5 inch viola with 38cm vibrating string length.

I want to tune it E4 A3 D3 G2 or B4 E4 A3 D3, but I dont know which tension in Kg to choose in the calculator to get the right diameter.

Help me please!

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Or perhaps the OP did not make any typos but is in fact considering tunings that could be used in his Arab Andalusian style - I am speculating here because I am not familiar with that music genre. In any event the A3 does not appear to be possible in gut at the moment.

@Cotton: B4/E4/A3 can be plain gut also...

Yes, in fact what you need is a table that lets you change frequency. The tension isn't the problem.

Tension is proportional to

D x f squared x l squared x d squared

where D = density, d = diameter

Which is convenient - it just means f and d are inversely proportional for constant tension.

In other words, go to the default table, put in 38cm, leave the tensions as they are, and then calculate reduced diameters proportional to the raises in frequency.

Thanks to Pythagoras, we know that up a fifth is 3/2, so the strings need to be 2/3 the diameter for the same tension, and up a fourth is 4/3, so the strings need to be 3/4 and so on. You can do others by interpolation - exact calculations won't be necessary. The small changes in tension that will result from small inaccuracy are of no importance.

Like Cotton says, EMail Gamut if you are in doubt.

This sort of scordatura tuning is not uncommon in some genres of folk fiddling, and that today would be with synthetic or steel strings. I've done it myself in the past by tuning a violin synthetic G up a tone so give the added resonance of a 4th interval drone with the open D. Scordatura was also used on the violin in the Baroque era (with gut strings) – see Biber's “Mystery” sonatas for outstanding examples in which the scordatura sonatas would not be properly playable in normal tuning.

For example, a string tuned to A4 (440hz) typically comes in tension ranges from around 5kg (light) to 6.5kg (heavy).

While a viola C3 (131hz) more typically comes in a tension range from 3.5kg to 5.0kg.

A G2 open tuning is 98hz. That is well below any published viola string I've seen, so one cannot be sure what tension range will give a reasonable sounding and responsive string that is not excessively thick.

For D3 A3 E4 B4 you can use the calculator for C3 G3 D4 A4 with the default tensions, then reduce the diameter it gives in proportion to the frequencies.

Example:

D3 is a step up from C3. C3 frequency of 130.81hz and a default tension of 5kg gives a 2.2mm diameter. D3 is 146.83hz. So its diameter would be:

(130.81/146.83) x 2.2 = 1.96mm

Notice how thick a pure gut C3 or D3 string needs to be. That's why these usually come with some metallic sheathing so the diameter can be decreased.

https://www.cs.helsinki.fi/u/wikla/mus/Calcs/wwwscalc.html

Also you may find that Gamut doesn't do pure gut as thick as some others. If they don't have what you need (and do email them), check with the various European suppliers: Aquila (has a US distributor), Toro, Pure Corde (these are made in Morocco I believe), or George Stoppani's Pure Gut project. Do your own research as to quality, I have had good product from Gamut, Stoppani, and Aquila but do not know the others firsthand.

1. Trial and error. Choose a tension in the middle of the range. Try the resulting string. Decide if you want it tenser or less tense.

2. Look at page 17 of this: -

http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1032&context=musicstudent

(except that's for violins, so you'll either have to Google for a viola version or extrapolate)

3. Buy a micrometer and measure your current string diameters, unless it says what they are on the packet. Calculate their tension. Calculate new diameters for new frequencies at same tension.