Computing string tension for strings tuned to a different pitch than their design pitch.

I am of the opinion that for the most uniform bass response, the four strings need to have equal tension. And many string makers agree, though I've not seen that in Kolstein strings nor consistent variations across a set of Kolstein strings. Changing a string from the factory pitch will change the string tension but that is predictable with some math.

Any way frequency

n = (1/2L)sqrt(T/m)

where L is length, T is tension, m is mass per unit length and n is frequency. This formula comes from the Chemical Rubber Company “Handbook of Chemistry and Physics.”
Or rearranged:

T = 4 m n² L².

Same units.

Now each half step is the 12th root of 2 frequency change. 1/2 step is 1.059463, whole step is 1.12246, major third (C to E) is 1.25992 on an untempered scale.

So for a constant length and mass a whole step change in pitch means a tension change of 1.12246 squared or 1.25992 up or down. A third change in pitch means tension change of 1.25992 squared or 1.587. Why that E gets wimpy tuned down to a C only partly compensated by going to a heavy E.

So a 75 pound tension string tuned up a whole step for solo playing becomes a 95 pound tension string. A 54 pound tension gut string tuned up that whole step becomes a 68 pound string with some greater brightness until it breaks. Similarly a solo 75 pound string tuned to orchestra pitch becomes a docile 59.5 pound tension string. Meek and mild and quiet. An 88 pound E tuned down to C gets even quieter at a pitch where the bass is getting even smaller than normal tuning so its a quiet 55 pound tension spring.

Dr. Gerald N. Johnson retired P.E. (electrical engineering)

Corrected December 14, 2012