# [PD] Resonant filter using cpole~ czero~

Mathieu Bouchard matju at artengine.ca
Thu Feb 2 17:40:46 CET 2012

```Le 2012-02-02 à 12:50:00, Patrice Colet a écrit :
>> De: "Ed Kelly" <morph_2016 at yahoo.co.uk>
>> Here is my first attempt at making a resonant lowpass filter using
>> cpole~ and czero~ objects.
>
> using biquad~ gives very nice results for resonant filtering, ggee audio
> filters show how it sounds. It should be possible to use vanilla's audio
> math objects only for the biquad mathematical functions instead of an
> external.

[biquad~] is actually a shortcut for a certain set of special cases of
having (up to) two [cpole~] and two [czero~] at once.

It's related to how quadratics like ax²+bx+c can't be factorised into
e(x-f)(x-g) in real space because you need negative square roots to make
it always factorisable, otherwise the squareroot fails when b²-4ac < 0,
which is also the condition for resonance ; yet e(x-f)(x-g) in complex
space also covers things that are impossible in real ax²+bx+c.

So a real ax²+bx+c is something between a pair of [rpole~] and a pair of
[cpole~]. It's equivalent to a pair of [cpole~] in which you force two
numbers to be the same, and two other numbers to be negatives of each
other.

When you have a pair of [rpole~], it's like specifying f and g as real
numbers, and thus being unable to construct any «irreducible» quadratics
(b²<4ac). the numbers that are negatives of each other are the ± square
roots : ± sqrt(4ac-b²)/2a.

(I hope I didn't make a mistake, and if I did, I'd like to know)

______________________________________________________________________
| Mathieu BOUCHARD ----- téléphone : +1.514.383.3801 ----- Montréal, QC
```