Basic idea is to provide a scale that is mostly log--like, but can express the extremities $0$ and $\infinity$. How to map this time scale $T = [0,\infty]$ to a control parameter range $C=[0,1]$ in a meaningful way? A mapping that satisfies the interval boundaries is $$c(t) = \frac{t }{a + t}.$$ This maps $c(0)=0$ and $c(\infty)=1$. The inverse is $$t(c) = \frac{a c}{1-c}.$$ The parameter $a$ can be determined by constraining the middle of the scale $t(1/2) = a$. Another important parameter is how fast the dial will move to $0$ and $\infty$. Both are extremes that are part of $T$, but for say $50\%$ of the $T$ range we'd like to have decay rates that change mostly exponential in $c$. To expose the symmetry in $t(c)$, let's introduce a change of variable $d = 2c -1.$ This gives $$t(d) = a\frac{1+d}{1-d}.$$ This scale is logarithmically symmetric around $a$ or $t(d)/a = (t(-d)/a)^{-1}$, meaning that in the middle range it behaves mostly exponential, while tending to $0$ and $\infty$ in the two extremes\footnote{ This hints at the input--scaled variant $\frac{1+ax}{1-ax}$ being a good approximation for $e^x$, which is the case when $a=\frac{1}{2}$.}. The slope of $t(d)$, relative to $a$ is fixed. At $d=0$, the function approximates $a\exp(2d)$. For mapping meaningful parameters, this might be a bit too flat. Successive squaring of $t_0(d) = (1+d)/(1-d)$ can solve this. For $n$ squarings we have $\exp(2nd)$. The curve then becomes $$t_n(d) = a t_0(d)^{2^n}.$$ In practice it seems that a single squaring works good. This gives a reasonably flat log response for $3$ decades, about $70\%$ of the scale, leaving the rest for the extreme range. If necessary, the extremities can be avoided by prescaling $d$. With one squaring, using $0.99$ limits the output range to about $9$ decades. Mapping this to pole radius requires an extra step. Let's use the natural $1/e$ decay to relate decay time $t$ (measured in samples) to pole $p$ as $p^t = 1/e$ or $$p = \exp(- \frac{1}{t}).$$ This approximation needs to be accurate for $t \gg 1$, and extend correctly to $p=0$ at $t=0$. The first degree Taylor expansion is $1 - 1/t$. Modifying this slightly to give $$p' = 1 - \frac{1}{t+1}$$ yields the wanted behavior at $t=0$ without changing the large $t$ behavior too much. For numerical reasons the update equations will use the positive quantity $$q' = \frac{1}{t+1},$$ where $p' = 1 - q'$. Composing the two mappings gives $$q_n(d) = \frac{1}{1+a(\frac{1+d}{1-d})^n}.$$ where $a$ gives the mid--scale time constant in samples. I'm using $n=2$, but some knob twiddling makes me think that maybe $n=1$ is better. What is important is to get the mid--scale value correct. I.e. what is a prototypical note's attack and decay rate? To find the warping % [1] entry://20130321-160005