Envelope time curve - midi mapping

After looking at the transfer function of the Little Phatty, I decided to redo the mapping for my XM8. In the same way as with the level function, I'd like to make it dynamically changeable, to be able to dial in the perfect transfer function per envelope.

The LP has an envelope time (per the manual) of 1ms to 10s (10.000ms).

The midi transfer function, with f(x) in ms, is thus

f(x) = 10^(x/(127/4)), for x=0 to 127

To get a changeable function, I looked at my previous work on envelope curves. I have a general function with built in steepness.

A function starting at 0,0 and ending at 1,1 with variable steepness can be written as

f(x) = a * 10^(steepness * x) - a

where

a = 1/((10^steepness) - 1)

Extending this to a case where we want a highest value for input and output, we get

f(x) = Y_max * ( a * 10^(steepness * (x/X_max)) - a)

where again

a = 1/((10^steepness) - 1)

Unlike the LP function, this starts at 0, since I use a 0-indexed time where 0 is the shortest possible time, around 1ms. If we want a different starting point, we get

f(x) = Y_min + (Y_max - Y_min) * (a * 10^(steepness * (x/X_max)) - a)