Nonlinear Oscillators?

[Martin Czech]

> Hmm...isn't Yamaha FM based on phase modulation, as opposed to strict
> frequency modulation?  If so, are there any analog oscillator designs
> that allow phase to be controlled directly?  Maybe the way to get these
> sounds would be to have an allpass network of several stages, where the
> phase shift frequency of the network is modulated by the output of the
> network.  I always thought that this would be a good way to get a sitar
> sound out of a guitar (actually, to simulate a sitar, probably
> non-feedback phase modulation is called for, as the sitar spectrum has
> tightly clustered harmonic peaks that closely resembles the spectrum
> from non-feedback FM, while feedback FM generates a spectrum closer to a
> sawtooth wave).  Running the output of the allpass network through a
> nonlinear element, then into the modulation input, would also be a
> useful experiment.
> 
Sorry folks, I can not suppress this:

Yes (!) this is important, Yamaha uses a phase modulation algo.  I think,
with analogue osc. you can't do it in the same way.  If they have a
linear FM input, well, it's really FM, and as I have learned, Through
Zero Modulation is also important for deep linear FM, most oscillator
designs don't have this feature (see J. Haible's homepage for an
example). The difference between PM and FM becomes evident if you use a
square wave as modulator: The PM carrier jumps in phase, ie. the signal
is non-steady whereas the FM carrier just changes the "speed" of it's
phase rotator, there is distortion of the carrier waveform, but no
jumps.  So, the PM algo will sound brighter than FM. And another fact
is that PM remains at the same phase modulation depth if the modulator
frequency moves far away from the carrier, with FM the phase modulation
depth will decrease with - i think - -6dB per octave. Simple
experiment: try different ratios with a Yamaha "FM" instrument and then
try this with analogue gear (of course sine wave output !) and you'll
hear how the carrier will get closer to the original sine shape, if the
modulator moves away in the later case.  On the other hand, slow PM and
slow FM cause vibrato, but in the case of FM the vibrato depth remains
constant, no matter how slow it is (of course dc-coupled fm input
required for this !) whereas PM gets inaudible if slow enough.

For the math lovers out there:
                             _
f(t)=d/dt phi(t) or phi(t)= / f(t) dt + c
                           -
ie. frequency is the time derivation of phase, or the other way
arround: phase is the integral of frequency plus some constant (initial
value). Now, this shows how the above mentioned observations can be
explained. Remember, integration is some kind of low pass filter.

And writing this it comes to my mind, that above statement "amalogue PM
is not possible" is obviously not true, the second formula above shows
what to do: The analogue FM input is integrated by the oscillator, to
reverse this operation, we have to use a differentiator (spelling ?) If
you have an analogue osc. with linear FM input and Trough Zero
Modulation, just insert a differentiator before the FM input. Viola!  The
problem is, a *real* differentiator is not possible, you can only
approximate it. And differentiation means: raising the noise floor.  In
generall this all means using filters before the FM input of analogue
gear. You allready have this idea in mind.


> Also, has anyone tried filter FM, with the modulating signal being the
> output of the filter that is run through a nonlinear element (full-wave
> rectifier, clipper, suboctave divider, etc.)?  Both the allpass-FM and
> filter-FM methods would allow any signal to be run through the circuit,
> not just an oscillator.  I wouldn't think that either of these methods
> would produce the octave-jumping effect I am after, but I could be very
> much wrong.  
> 
Well, I know that PM causes octave -sliding- not jumping.  Imagine
carrier and operator with ratio 1:1, it is obvious that a some point of
modulation depth the fundamental is inaudible and the second harmonic is
very strong (Bessel functions).

Another interesting way to go is sync. I mean syncing is a way of PM,
and it is nonlinear and easy to do. This would require a switch (mosfet
etc) that sets the oscillator to some predefined (or even better
voltage controlled) level if triggered.  One could add a window
comparator that enables sync only under some condition , eg. osc level
above 75% (this is called weak sync) or master osc between 10% and
20%.  If the loop is closed (ie. two osc.  sync each other) this could
get pretty chaotic. Sweeping osc. with sync can easily jump into
another octave, or another harmonic, if you use strong sync. I remember
a sound from Reinhold Heil (Splif, "Schwarz auf Weiss" ?), where this
feature is used (Oberheim, Prophet ?). I'm still after this sound.
If you use weaker sync, more "dirt" is produced.

m.c.

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