Let’s say your signals are between -10V and +10V.
IN is your (bipolar) input signal (carrier).
CV is your (bipolar) cv (modulator).
Your toolkit consists of op-amps to add and subtract signals, and unipolar linear VCAs which compute V (IN, CV) = IN x CV / 10, but only for unipolar CVs (no restriction on IN though).
You want to use this to build a four-quadrant multiplier that computes IN x CV / 10 even if CV is bipolar. Let’s do a couple of algebraic manipulations…
IN x CV / 10
= IN x (CV / 10 + 1) - IN
= IN x (CV + 10) / 10 - IN
= V (IN, CV + 10) - IN
And that’s it… Shift the CV high enough to ensure that it is unipolar, and compensate by removing the input from the result.
Now let’s get to the tricky bit. Because four-quadrant-multiplication is commutative, you can swap the inputs IN and CV and get a circuit that should (in theory!) output the same thing. Indeed we have V (IN, CV + 10) - IN = V (CV, IN + 10) - CV. At least algebraically…
Blinds’ circuit computes V (CV, IN + 10) - CV, that’s why you see the audio signal going into what you’re believing is the control path of the linear VCA ; and the CV going through two branches.
So why this odd choice, of computing V (CV, IN + 10) - CV instead of V (IN, CV + 10) - IN?
Let’s have a look at V (IN, CV + 10) - IN… In the real world, the two terms in this equation will go through different paths, the first term will go through a 2164 cell and an op-amp before hitting the op-amp that does the subtraction, the other term won’t go through that, so the first term will have some tiny bit of distortion, noise, high frequency roll-off and slew-limiting. Which means that when CV is 0, V (IN, CV + 10) - IN won’t be zero, but will contain some faint garbage (high frequencies that are not completely nulled, higher harmonics from distortion, bonus noise). However, V (CV, IN + 10) - CV will be 0 as expected, first term is zero because the VCA has good offness, the second term is zero.
Since people expect the output to be silent when CV is null, I have chosen the V (CV, IN + 10) - CV variant. This swap is not magical - its downside is that if IN is null, we’ll still hear the high frequencies of the CV bleeding through. But in a typical modular applications, when using Blinds as a VCA, the signal going into IN is rarely silent; and the CV rarely has super hard edges. So it’s less of an issue…