tri 2 saw modualtion

[Martin Czech]

> If you look at the harmonic spectra of the waveform you'll see for the
> triangle case that every even harmonic is nulled out, and as you
> change the symmetry, those "nulls" will slide up in frequency over the
> harmonics and move farther apart, up until the wave is a full sawtooth
> shape where there are no more nulls.  A Bessel function describes the
> amount of each harmonic.
> 
> (This is a guess from the harmonic spectra of a PWM square wave -- I
> haven't done the actual math.  The same mechanism applies though.)
> 
Good idea, maybe it is really the bessel function set that rules.
Or maybe a similar function.

> You should be able to hear these nulls sliding around, it should be a
> very pretty vocal sound, something like a phase shifter.  Especially
> with a few of them playing simulataneously each with a sine LFO.
> 
Hmmm, I'll dig up that programm and do the simulation again. 
perhaps 16 bit this time and listening harder ;->.

>    Audio modulation: Hmm, this is the same as using the linear FM-Input with
>    a strange pre-warped input , isn't it? Will it be that much different?
>    I doubt.
> 
> No, it would be closer to audio modulation of a PWM square wave.  But
> warmer. 
> 
True, if you integrate the PWM you get this kind of tri2saw.  Is this
perhaps a way to solve the problem? Simple integration won't do, the
time "constant" has to be time variable, a more and more faster for the
quicker slope, and a slower for the falling.  This could be done with
one ota and some computation circuit for the slope.  I doubt that this
would work, since it is difficult to achieve constant amplitude. But
maybe this is not so important.

But back again to linear "real" FM: If you read my proposal and look
what i wanted to do it turns out that I proposed some strange
warped kind of feedback in the oscillator:  pulse output
feed thru some multiplier (this is the switch) and some function
approximation (diode network) and then into the exponential input of the
osc. again.  Hey, writing this it comes to my mind: This is exactly the
method you would use with a "FM" synth (DX7 etc.). Feedback of
operators or stacking operators with the same parameters is used to
obtain saw-like waveforms.

Now I'm really convinced that it must be besselish, or some kind of
Bessel variation. Wow!


(Note: this is not possible with analog synths, since they do "real"
linear FM, whereas Yamaha choose to use Phase-Modulation instead, if
you look at the formulas they use. The algorithms for FM and PM are
very much the same, I stumbled across this when  I did a small
demo-prg, I simply swaped two programm lines and got FM instead of PM!
But from perception and continous math FM and PM are not the same at
all.
 

 F=d/dt P or int{F dt} = P

PM means constant phase change, FM means decreasing phase change with
higher modulation frequency (integration).  Or the other way: FM means
constant frequency change, PM means increasing frequency change with
rising modulation frequency (differenciation). Try this at home:
compare analog linear FM with dx7, both  very slow modulation (<1 Hz).
And then speed up modulation.  The dx7 won't give large vibrato if
Fmod<10Hz.

I don't believe the we really can get hands on Phase (except sync), but
above formulas show that PM can be obtained by using a differentiator
before feeding into the linear FM input. Ok, these diff. circuits can only be
approximated.  Would this be a possible solution ?)

> probably aren't any especially interesting sounds at the
> last 1% of the range.

Hmmm, it could well be that a lot of higher harmonics jingling occours only
at the last 0.1%, the brightness. I think the last few % may be similar to
filtering a saw, it is already  very saw-like.

m.c.