Normalized Tunable Sigmoid Function

See also:

The Function

As I looked for a way to shape envelope stages, I stumbled across Dino Dini’s beautiful normalized tunable sigmoid function:

$$y = \frac{x - kx}{k - 2k|x| + 1}$$

By beautiful, I don’t mean that the function looks nice (though it looks very nice). I mean that it has three features that make it enormously useful for my purposes:

Here is a graph of the the normalized tunable sigmoid function for inputs from -1 to 1, with each line representing a different curvature, from -0.95 (red) to 0.95 (violet):


The function is normalized in two interesting ways:

Note also that for every value of k:

If I can map some linear signal onto one of those “normal” ranges, I can apply the sigmoid function to shape the signal.

This is how the DHE Modules create curved envelope stages and waveshaper transfer functions.


The normalized tunable sigmoid function has a parameter, k, which tunes the curvature of the function. Each line in the graph represents a different value of k.

Positive k (from green to violet in the graph) gives a J shape over the range [0,1] and an inverted S shape over the range [-1,1]. Negative k (from green to red in the graph) gives a shape that is nicely symmetrical to the positive ks in either range.

And when $$k=0$$, the function is linear: $$y=x$$. The output is the same as the input.

By mapping each module’s CURVE knob to this function parameter k, I give users control over the curvature of envelope stages and waveshaper transfer functions.

Note: For S-shaped curves, the DHE Modules negate the tuning parameter k so that turning the CURVE knob clockwise gives an S curve instead of an inverted S curve.


Originally I sought a function that could produce useful, symmetrical “logarithmic” curves for envelope stages. My initial attraction to this normalized tunable sigmoid function was that in the range [0,1] it produced exactly the kind of J curve I was looking for.

But playing with this curve got me thinking: What if my modules could produce S shaped envelope stages? That would be a distinguishing feature! And this function made S shaped envelope stages easy.

Then I wondered: Might this function work well as a waveshaper transfer function? And just like that, the idea for Swave was born.

Curved Envelope Stages

Stage and Booster Stage use the normalised tunable sigmoid function to produce curved envelope stages.

To produce an envelope stage, each module follows the same general procedure:

  1. Generate a ramp signal that progresses from 0 to 1 over the duration of the stage
  2. Apply a shaping function to shape the ramp signal into a curve
  3. Map the curved ramp signal onto the desired output range

For step 2, each module shapes the ramp signal by applying the normalized tunable sigmoid function.

J-Curved Envelope Stages: Given that the ramp signal is already in the range [0,1], shaping the stage into a J curve is easy: simply call the sigmoid function.

S-Curved Envelope Stages: Booster Stage can produce S-shaped stages. To do this, it applies the shaping function in three steps:

  1. Map the ramp signal from [0,1] to [-1,1]
  2. Apply the sigmoid function
  3. Map the shaped signal back to [0,1] before scaling it to the desired output range


Swave uses the normalized tunable sigmoid function to shape its input signal.

Note: Swave assumes that its input signal is in the range [-5V,5V], the standard range for audio and bipolar CV signals in VCV Rack.

To shape its input signal, Swave:

  1. Maps the signal from [-5V,5V] to the range [0,1] (for J-curved waveshaping) or [-1,1] (for S-curved waveshaping)
  2. Applies the sigmoid function to shape the signal
  3. Maps the shaped signal back to the range [-5V,5V] for output

Other Uses

In addition to shaping signals and generating envelope stages, DHE Modules use the sigmoid function to give some knobs a taper, making the knobs more or less senstive at different parts of their rotation: