Exponental/Linear Modulation of Phase, + new circuit idea

[Sean Costello]

Hi all:

I have been performing some experiments in Csound, using several cascaded
allpass stages (i.e. phase shift stages), with the allpass coefficient being
modulated at an audio rate. Some of the things I have found:

- This is a VERY nice way of getting FM effects. Using a sine oscillator as a
carrier, and another as a modulator, I could get results that were very close to
classical FM sounds, with the amount of harmonic content rising with the volume
of the modulator.

- Using the mirror opcode (an opcode that implements "foldover" of a waveform,
in a manner similar to the middle section of the Serge Wave Multiplier) on the
modulating signal allows for more dramatic FM effects, similar to the
through-zero FM on Yamaha synths. I implemented it out of necessity, as the
allpass stages will "blow up" if the allpass coefficient (g) is outside of the
range -1<g<1.

- Running the modulating signal through an exponential function - i.e. amod2 =
exp(amod) - does not result in a shifting pitch center with increased modulator
amplitude. This is in contrast to the FMing of an exponentially-controlled
oscillator, where the pitch center rises with increased modulator amplitude.

- Using the output of the allpass stages as the modulating signal results in
some very nice feedback FM. As the modulation increases, a sine input is
transformed into a sawtooth-sounding wave, before breaking into chaotic
subharmonics (yummy!).

- Any signal can be subjected to FM in this way, not just oscillator signals.
The results aren't always pretty, but I'm sure this would be useful in many
cases, especially for processing unpitched percussion instruments with sweeping
oscillator frequencies.

Anyway, I think that this would be a nice DIY module. The core of it would be a
12-stage phase shifter, based on either OTA stages (maybe the LM13600 stages
that Tom G. has on his website) or a variant of the ARP Quadra ladder allpass
filter.  The modulation input would have the provision of going through a Wave
Multiplier-type circuit, although I am not sure how this would be done - diode
breakpoints, maybe? The modulation input would be trimmed out in such a manner
so that the modulating signal would "fold over" at the voltages that would
otherwise represent maximum and minimum phase shift. This seems like a nice way
of getting through-zero FM effects, without constructing a dedicated
through-zero FM oscillator. Any signal could FM any other signal - you would not
be restricted to oscillator signals. Plus, you'd have a nice phaser to work
with, as well. :)

Does anyone have the circuitry where they could test this idea out? A
well-outfitted Serge owner, with a Wave Multiplier and a Voltage-Controlled
Phaser, should be able to see if this works. I am interested to see if this
works in the analog realm, with phasers that might have exponential input
stages. I don't think this would make a difference, but I am not sure.  I had
originally thought that the circuit would need an exponential convertor with the
provision for linear FM, but this may not be necessary, if exponential FM of
phase does not result in a changing pitch center.

Any and all feedback welcome.

Sean Costello