[sdiy] analogue DX-7-style Phase Modulation using quadrature signals ?

[jhaible]

This is just an idea at the moment, and I wonder if this
would work. It's about making DX-7-like Phase Modulation
(often misnamed as Frequency Modulation) with analogue
oscillators. From earlier discussions on synth-diy I know that
FM of a VCO is easy, thru-zero FM is difficult but possible,
but PM - easy in digital - would be most demanding on an
analogue VCO.

But what about using a quadrature signal? Either using a
quadrature VCO, or simply running an ordinary VCO thru
a dome filter such as used in a Frequency Shifter. Then you
have sin(wt) and cos(wt).

Now there will always be factors A(phi) and B(phi) such that
sin(wt + phi) = A * sin(wt) + B * cos(wt).
So we can modulate phi indirectly by modulating A and B
with the right function, can't we?

We could run the outputs of the Dome Filter thru two
multipliers (ring modulators) and sum the two products,
just as a Frequency Shifter would do. But instead
of using a Quadrature Oscillator for the modulation,
we could derive both, A and B, from a CV input.

The difference to ordinary frequency shifting is that
*there* (FS) you always go round the circle (0 ... 360deg),
while this PM scheme would allow both a smaller or a larger
range, performing partial circles depending on the
modulator level  modulator envelope.

I haven't done the maths for this, but probably A and B
would be sin and cos functions of an arbitrary CV input
signal to get the desired result.
If so, all we need is a sin(x) and cos(x) nonlinear network
which works over a certain number of periods.
I think I remember from earlier discussions that there are chips
that do this, and of course such a thing could also be implemented
with diode networks etc.

Two questions:

(1) Will this work at all? Is this a true emulation of PM a la DX-7 ?

(2) How many periods do we need for the nonlinear network?
     (a) How many periods does a DX7 with maximum modulation depth
           cover? And, if this number is too high,
     (b) How many periods is the maximum PM depth in a typical "DX Piano"
           sound on the DX7 ?

I have a feeling that it can't be so easy, but at the moment (early morning,
time to go to bed) I can't see an error. Too tired to go and consult the
Bronstein. Someone tell me if this will work, please.

JH.