[sdiy] Continously Variable Symmetry Triangle

[Don Tillman]

   > Date: 08 Jul 2003 00:46:20 -0700
   > From: Don Tillman <don at till.com>
   > 
   > So the control is not linear.  While this nonlinearity is perhaps an
   > annoyance on an LFO, I'll claim that it's a big feature on an audio
   > frequency VCO.  Our sense of hearing doesn't really hear duty cycle as
   > such; we hear the harmonic content changing, and as the duty cycle of
   > the waveform approaches 0% or 100% the harmonic content changes very
   > quickly.  So slowing down that change at the endpoints makes the
   > control more musically functional.  Both for this circuit and a
   > standard PWM square wave.

I should explain that one a little better...

A sawtooth wave contains all harmonics.  A triangle wave contains only
odd harmonics with zero levels for the even harmonics.  When you look
at the Fourier series for a variable duty cycle sawtooth that morphs
from saw to triangle to inverted saw (or a PWM square wave for that
matter) you'll see a classic comb filter with a lot of equally spaced
nodes, and those nodes move up and down together as the duty
cycle changes.

For the triangle case (50% duty cycle) those nodes are at the even
harmonics.  As you change the duty cycle from there those nodes move
up in frequency.  And at the sawtooth extreme, the nodes are at
infinity.

These flying patterns of nodes are what we hear in modulating the duty
cycle of sawtooth or PWM square waves.

The frequency of the first node is inversely proportional to the
distance the duty cycle is from 100%, so as the the duty cycle
approachs 100% (or 0%) the nodes rise in frequency very fast.  So, if
you think of the duty cycle control as tuning these nodes, it makes a
lot of sense to slow down the PWM effect at the extremes.  This is
why I say that the nonlinearity of the variable sawtooth circuit is a
feature. 

If the VCO in question is going to be used as an LFO, you won't be
hearing that flying patterns of nodes effect, and you'll really be
more interested in accurately setting the duty cycle percentage, so
the nonlinearity would be a bad thing in that case.

  -- Don

-- 
Don Tillman
Palo Alto, California
don at till.com
http://www.till.com