String Tension Calculations

I've been listening to Matt Schofield a lot - really a lot. The last time I went to the gym, I looped 2 minutes of one of his solos for 30 minutes, trying to get it to magically seep into my brain. It made it into my brain in that I can now hear it, but I still can't play it. I can sort of hear that he's using heavier strings, and it does turn out that he's using 11's. I've always been curious what that would be like, but the extra tension on my fingers held me back. Then it occurred to me that using 11's on a short scale guitar might be the same as using 10's on a standard 25.5" scale strat. A little bit of math later (see below), and I now know this: Let $g$ denote the string gauge (e.g. 10, 11, 12), and $S$ the scale length in inches. In order to have the same tension on the E string as a strat using 10's, the scale length would have to be \[ \frac{10}{g} 25.5 \] This is intuitively appealing, but takes some justification. The disappointing part is that, if $g = 11$, then the resulting scale is \( S = \frac{10}{11} 25.5 = 23.18 \) which is well below the other readily available reasonable scale length of 24.5. The other thing that guys who use big strings do is to tune down a half step. So we can ask a different question: Given a string gauge (e.g. 11), if we tune down 1/2 step, what scale length would we need in order to match the tension of the high E of a strat with 10's and standard tuning? The answer is that the scale length would be given by \[ S = 2^{\frac{1}{12}} \cdot \frac{10}{g} \cdot 25.5 \] The extra factor comes in because of the detuning, since the frequency of an E is $2^{\frac{1}{12}}$ times that of the Eb one step lower. Plugging in 11, we get \[ S = 24.56 \] Aha! So if you put 11's on a 24.5" scale guitar, and tune down 1/2 step, the high E will have the same string tension as on a regular strat strung with 10's. This leads one inextricably to this conclusion: the standard "gibson" scale length was selected in order to allow strat players who are used to 10's to comfortably use 11's when tuned down a half step! I'll post the formulae another time. They follow from two relationships. The first gives the string tension as a function of the unit weight of the string, scale length, and the frequency to which the string is tuned. The second is the observation that if an unwound string of gauge (i.e. diameter) $g$ has unit weight $w$, then a similar string of gauge $g'$ will have unit weight scaled up by $ {\left( \frac{g^\prime}{g} \right) }^2 $