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