[sdiy] PM vs FM was: Re: Buchla 295 10-band comb filter topology

[Ian Fritz]

Thanks for the discussion! (Inline)

> On Nov 29, 2022, at 10:12 AM, Donald Tillman <don at till.com> wrote:
> 
> 
>> 
>> On Nov 28, 2022, at 9:42 AM, Ian Fritz via Synth-diy <synth-diy at synth-diy.org> wrote:
>> 
>> OK, I've figured out what the FM/PM disagreement most likely stems from.
>> 
>> On Fig 8 of the patent the DX7 output is given by the expression
>> 
>> Sin{kwt + f(w_m t)}
>> 
>> From fundamental theory, this expression is THE SAME for FM and PM. The difference between the two is in the function f().
> 
> 
> Phase Modulation is:
>  out = sin(kwt + f(w_m t))
> 
> Frequency Modulation would be:
>  out = sin((kw + f(w_m t)) t)

Ooopsy. You have to integrate the oscillating part, not just multiply it by t.
Int(cos wt) = (1/w)sin wt.
(Oh and how many times have I done the same thing!)
It’s the (1/w_m factor that makes the FM waveshape (ideally) invariant over pitch.

> Where:
>  k is the carrier frequency ratio, 1 in the regular case
>  w is the carrier frequency
>  w_m is the modulator frequency
>  f is the modulator's enveloped sine operation
> 
> 
> But what are the implications?
> 
> * If you have FM, you can fake PM by taking the derivative of the modulation input.
> 
> * Likewise if you have PM, you can fake FM by taking the integral of the modulation input.
> 
> * PM naturally goes from sine wave to really bright.  The crisp acoustic-like sounds of the DX-7 are due to PM.  And that was a big deal because nothing like that had been heard from a synth before, 
> 
> * The concept of FM isn't helpful in designing voices.  We don't have a feel for what FM variations sound like.  Explanations like "vibrato, only faster" are useless.

Oh, you should look at the Chowning/Bristol book again.  They go step-by-step teaching how to understand FM.  And as mentioned before they treat the DX7 as an FM synth.  Their DX7 exercises and examples would not work for a PM synth. 

> 
> * The concept of PM, which I described as "warping the waveform along the time axis", is much simpler.  If you know this, everything makes sense.  And it's a lot easier to design voices.
> 
> * With PM you can literally sketch out what the waveform will look like.  Much more difficult with FM.
> 
> * In the simple example case (modulator feeding carrier, same frequency) does the phase of the modulator wrt the carrier make a difference?  PM tells you exactly.  FM is actually deceptive; Bessel functions assume a 90 degree difference between the modulator and carrier, but that's never mentioned.

And a hushed up feature of our analog FM modules is that the phase relation is never nailed down. Actually the system always looks the same on power cycling. So power-on transients or a little bit of the unavoidable sync artifacts may play a role.

> Why am I so insistent about this?  Because the *mechanism* is important.  If you understand the mechanism, you are empowered to use it to do innovative things.

Agreed. This is what the Chowning/Bristol book does — explain the FM mechanism of the DX7, with many examples of DX7 settings as illustrations.

> And the "FM Synthesis" mechanism has a weird history of being hidden.  Not intentionally.  But here we are almost 40 years later and there's still a crazy amount of confusion about the basics.
> 
>  -- Don
> --
> Donald Tillman, Palo Alto, California
> https://www.till.com
> 
Ian