There is an equation to find the position of a fret on a guitar fretboard, given the length of a string is given by
\begin{eqnarray} d = s – \frac{s}{2 ^ {(n / 12)}}, \end{eqnarray} where $d$ is the distance of the fret from the 'nut' (the start of the string), $s$ is the length of the string, and $n$ the fret number (the fret whose width we want to know).
So to get the width, one computes $d(n) - d(n - 1)$.
I need to find the relative width of a given fret as a percentage of the string, regardless of the string's length.
This goes above my math level, could you help me please?
The width of a fret whose number is $n$ is given by
\begin{align} d(n)-d(n-1)&=\left(s-\frac{s}{2^{\frac{n}{12}}}\right)-\left(s-\frac{s}{2^{\frac{n-1}{12}}}\right)\\ &= \frac{s}{2^{\frac{n}{12}}}\left(2^{\frac{1}{12}}-1\right). \end{align}
Thus, the percentage width, relative to the length of the string, is given by
\begin{align} \frac{d(n)-d(n-1)}{s}\times100&=\frac{\left(2^{\frac{1}{12}}-1\right)}{2^{\frac{n}{12}}}\times100. \end{align}