Author 
What is Ian Fritz Chaotica math formula? (chaos discussion) 
Tronketz 
Well first question, would it be ok to release a free audio plugin that recreates the sound of Chaotica? If not, maybe just for private/personal use?
Anyway, I can't even do that without knowing the math behind it.
It looks like Ian Fritz website is down, although not sure if it had the needed info there in the first place. Also, not sure if I could've translate that info into C++, heh.
Any help is appreciated! thanks.
I just spent some hours with Lorenz Attractor and it was enlightening. I made a philosophical discovery: maybe the fundamental nature of chaos is based on how quickly or slowly it locks onto a multiple or division of its own frequency. As the speed of changing increases (the time which it remains locked onto a certain harmonic), you get a more noisy signal, but it is never true noise. This gives a digital quantity like sound unless it is fast enough to sound like noise. On the other end of the spectrum, you can generate stable tones when the chaos never changes the frequency it is locking on to. 

A21.12.12 
By no means am I able to help you with your actual request, however have you tried Archive.org? You might be able to access a cache of his website that way. 

egrad 
Tronketz wrote:  Well first question, would it be ok to release a free audio plugin that recreates the sound of Chaotica? 
You should get in touch with Ian Fritz directly. He's a member here thus it is easily possible to send him a PM or email. 

ear ear 

Tronketz 
Thanks for the replies! PMing Fritz now. Also, I checked archive.org, it ends up saying the page could not be found in Way Back Machine.
https://web.archive.org/web/*/http://my.xfinity.com/~ijfritz/ 

ear ear 

Tronketz 
nothing! I couldn't find it! thanks 

oberdada 
As a matter of fact, I've been in touch with him with exactly the same request. My comprehension of electronics is too limited to translate circuit diagrams into equations, and I suspect Chaotica with all its knobs and cv inputs would make a rather complicated formula with lots of parameters. However, it seems to be based on a jerk system of the form
x''' + Ax'' + Kg(x') + Kx = 0.
I don't know what the function g does nor what are the constants A and K.
Tronketz wrote: 
I made a philosophical discovery: maybe the fundamental nature of chaos is based on how quickly or slowly it locks onto a multiple or division of its own frequency. As the speed of changing increases (the time which it remains locked onto a certain harmonic), you get a more noisy signal, but it is never true noise. This gives a digital quantity like sound unless it is fast enough to sound like noise. On the other end of the spectrum, you can generate stable tones when the chaos never changes the frequency it is locking on to. 
I think you're on to something here. In fact, chaotic attractors use to have lots of unstable periodic orbits within them, meaning that if the initial condition is exactly on one of those unstable periodic orbits it will stay there, but in practice the tiniest noise will perturb the orbit away from that period. But again and again, the chaotic trajectory will come arbitrarily close to those periodic orbits, which will turn out as short moments of quasiperiodicity. This is also used in one of the first discovered methods of chaos control. That's also an explanation of why chaotic systems are often susceptible to synchronisation if you feed a periodic signal into it. I've written a bit about that here. 

Tronketz 
Can we continue this thread as a philosophical and technical discussion of chaos?
Oerdada, what do you think about chaos generated by interacting sinewave oscillators rather than a feedback equation? You say, to control chaos, one should feed a periodic signal into its parameters, but why not skip the middleman and only have periodic signals?
Here is an example of this: https://youtu.be/ah8ZGdGbFLM
I really need to implement oversampling next time, there's a bit of behavior due to sampling limit in one part of the video, oops! 

goom 
Tronketz  That's was very cool!
How did the stereo effect happen? Is that a function of the program? 

Tronketz 
Stereo effect: Chaosfly (the synth in the video) uses two chaotically interacting oscillators. I send the output of one to left channel, the output of the other to the right channel. If you're asking about specific stereo behaviors/effects such as when the stereo image seems to jump back and forth (along with the visual image), those are emergent behaviors of chaotic interactions. Actually, it's the first time that has happened, as I was trying out a simple addition to the feedback path. Now that you mention it, it would be cool to explore and exaggerate these stereoeffectemergent behaviors. But that is the beauty of chaos generators. You hear something. You want more of it? You figure out what to weak and you get more of that. Lo and behold there's now something else you hear that you want to exaggerate. The process continues ad infinitum. 

goom 
Very nice! I need to listen to it again sometime using my main speakers, instead of my computer (cheap!) speakers. lol
Thanks for the explanation... 

oberdada 
Nice video, would you mind sharing the equations?
Tronketz wrote: 
Oerdada, what do you think about chaos generated by interacting sinewave oscillators rather than a feedback equation? You say, to control chaos, one should feed a periodic signal into its parameters, but why not skip the middleman and only have periodic signals?

Sure, that's interesting. There's an old paper by Dan Slater in Computer Music Journal that describes a chaotic system with two complexvalued oscillators doing mutual FM, so there certainly are examples of this.
But if you think about it, wouldn't you still end up with a "feedback equation" (more precisely, an iterated function if you do it in discrete time or an ODE if you put it in terms of continuous time) if you have interacting oscillators? The only way I can think of not to make it a feedback system would be to have a masterslave setup. 

Tronketz 
oberdada wrote:  But if you think about it, wouldn't you still end up with a "feedback equation"  I don't know, you're the expert! I just thought an arbitrary feedback equation would be more difficult to tune than a stable sine wave generator based system, but I don't know enough about classical chaos to know if that's true.
I have a question that will lead to more question. What is the difference between feedback and chaos? If you generate a sinewave with this setup:
This is feedback... but can you call it "stable" chaos? Or is that not a thing? You can go from stable amplitude to decaying amplitude to infinitely increasing amplitude (so it becomes a square wave), all the while the frequency changes, so it's not clean or stable by any means. You do this by changing the frequency relation between the two filters and the amount of signal going back in. 

frijitz 
Tronketz wrote:  Can we continue this thread as a philosophical and technical discussion of chaos?
Oerdada, what do you think about chaos generated by interacting sinewave oscillators rather than a feedback equation? You say, to control chaos, one should feed a periodic signal into its parameters, but why not skip the middleman and only have periodic signals?
Here is an example of this: https://youtu.be/ah8ZGdGbFLM
I really need to implement oversampling next time, there's a bit of behavior due to sampling limit in one part of the video, oops! 
Pretty interesting behavior! Could you give more details about what exactly the system is? Two oscillators FMing each other? Then some kind of clamping? Any other nonlinearities?
Ian 

frijitz 
Tronketz wrote:  Well first question, would it be ok to release a free audio plugin that recreates the sound of Chaotica? If not, maybe just for private/personal use?
Anyway, I can't even do that without knowing the math behind it.
It looks like Ian Fritz website is down, although not sure if it had the needed info there in the first place. Also, not sure if I could've translate that info into C++, heh.
Any help is appreciated! thanks.
I just spent some hours with Lorenz Attractor and it was enlightening. I made a philosophical discovery: maybe the fundamental nature of chaos is based on how quickly or slowly it locks onto a multiple or division of its own frequency. As the speed of changing increases (the time which it remains locked onto a certain harmonic), you get a more noisy signal, but it is never true noise. This gives a digital quantity like sound unless it is fast enough to sound like noise. On the other end of the spectrum, you can generate stable tones when the chaos never changes the frequency it is locking on to. 
Chaotica is an extension of the Jerkster, which in turn came from a paper I found in some journal. The modifications consist of adding extra nonlinearities and feedback paths. I just did this experimentally, trying out different configurations until I got some things I liked. I never worked out the math. It should be easy enough to do in a general sense, without numerical values for all the parameters. It's still a thirdorder system, but the math would probably be easiest if expressed as three coupled equations.
Ian 

oberdada 
Tronketz wrote:  I have a question that will lead to more question. What is the difference between feedback and chaos? 
You could ask, Are there feedback systems that are not chaotic? Yes, there are. Any recursive linear filter would be an example.
What about chaotic systems without feedback? In discrete time, you have a map or iterated function x[n+1] = f(x[n]), which is obviously a feedback system. In continuous time, the equation becomes dx/dt = f(x) where x is a vector, so the speed and direction of change is a function of the state variable. Again, it uses a kind of feedback. In that sense, there is no chaotic system without feedback. The function f(x) also needs to be nonlinear for chaos, and still that's not enough.
Tronketz wrote:  This is feedback... but can you call it "stable" chaos? Or is that not a thing? You can go from stable amplitude to decaying amplitude to infinitely increasing amplitude (so it becomes a square wave), all the while the frequency changes, so it's not clean or stable by any means. You do this by changing the frequency relation between the two filters and the amount of signal going back in. 
Stability is indeed an important concept in dynamic systems, but perhaps in another sense. I think what you are describing is what happens to the system as you change some parameter. That's what they do in those bifurcation plots that you have probably seen.
Then there is another sense of stability. If you start the system from two slightly different initial conditions, will the orbits tend to join over time or will they diverge? The divergence of such orbits is what lead Lorenz to his discovery of chaos. 

oberdada 
frijitz wrote:  It's still a thirdorder system, but the math would probably be easiest if expressed as three coupled equations.

Yes, and at least for simulation purposes I would prefer to write it as three coupled equations. I don't even know how to simulate a jerk system directly. 

Tronketz 
frijitz wrote:  Pretty interesting behavior! Could you give more details about what exactly the system is? 
Hah. I wouldn't have thought a veteran Chaostronaut such as yourself would find my results to be curious! Important note: The video shows my latest experiment by putting a filter on the phase incrementor which creates many new interesting behaviors. But my following explanation will be for the most basic "Chaosfly" setup.
Phase modulation is not really a thing in the analog world, but there is still a lot of fun the have without phase modulation. Also, the behavior is very different if frequency is allowed to go through 0, or if you put a min/max value on frequency.
~ .5 amplitude modulation into frequency inputs using 1 pole hp/lp of some frequency and some resonance.
~ .5 amplitude modulation into frequency inputs using 1 pole hp/lp of some frequency, only lp has resonance.
There are many key behaviors with various settings. One weird thing is that adding resonance actually REMOVES superoscillations in some instances, as if the filter resonance cancels out the oscillations of the sinewave oscillators.
Here's a behavior where the modulation seems to "bounce off the walls". I'm not sure what creates the phantom wall, I think it's just the wraparound nature of the sin/cos function.
https://www.youtube.com/watch?v=mtQeXhCrjJs
The next step to having more fun is adding clippers to various parts of the signal path, or inputting an outside oscillator (such as a sawtooth) into the feedback path, or more filters on the modulation inputs. It is also helpful to have a master frequency that the main oscillators filters are based on, with options for frequency offset. 

elmegil 
So your plots are simply X osc vs Y osc?
Are there direct frequency/phase controls on each oscillator in addition to the feedback amount attenuators?
Have you noticed particular classes of behavior between HP/LP as a bandpass and HP/LP as a notch? 

frijitz 
Tronketz wrote:  frijitz wrote:  Pretty interesting behavior! Could you give more details about what exactly the system is? 
Important note: The video shows my latest experiment by putting a filter on the phase incrementor which creates many new interesting behaviors. But my following explanation will be for the most basic "Chaosfly" setup. 
Pretty amazing stuff! So you have one input driving the phase directly and another driving it via the integral of the FM input?
I just patched up a supersimplified variant using a pair of TZFM VCOs and a thirdorder filter all connected in a ring. Definitely lots of chaotic regions!
Ian 

frijitz 
oberdada wrote:  frijitz wrote:  It's still a thirdorder system, but the math would probably be easiest if expressed as three coupled equations.

Yes, and at least for simulation purposes I would prefer to write it as three coupled equations. I don't even know how to simulate a jerk system directly. 
OK! Today I worked out the equations for the oneeye / mild setup. I need to think about what range of parameters should be used. I hope you can figure out how to implement the diode (kink) nonlinearity. It can just be zero below the breakpoint and linear above. I'm glad you like the coupled equation approach, because I'm failing miserably at putting it in thirdorder form.
Ian 

Tronketz 
elmegil wrote:  Are there direct frequency/phase controls on each oscillator in addition to the feedback amount attenuators?  Yes, usually you'd start the oscillators in tune with each other.
elmegil wrote:  Have you noticed particular classes of behavior between HP/LP as a bandpass and HP/LP as a notch?  Yes, that is more advanced exploration that doesn't easily yield interesting results more than the basic setup.
frijitz wrote:  Pretty amazing stuff! So you have one input driving the phase directly and another driving it via the integral of the FM input?  Oh jeez, my terminology skills are lacking. "Integral of the FM"? 

Tronketz 
Here are some possible results with Ian Fritz' simple mockup of my Chaosfly.
3 pole LP/HP with no resonance, oscillators and filters are arbitrarily tuned
Chaos is beautiful, just gotta have the right camera. 

Tronketz 
Here is some more unique chaotic formulas turned into sound and image.


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