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frijitz
Joined: May 04, 2007
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 Posted: Thu Apr 01, 2010 10:28 pm Post subject:
 Billiard-Dynamic Pseudo Chaos Generator
Subject description: The next big thing. (Edited) |
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So ... if you feed two Tri waves into your scope and look at the xy display (Lisajou figure) you see what looks like a ball bouncing around a billiard table. This makes sense, because the ball has constant speed in the x and y directions corresponding to the constant positive and negative slopes of the two waves.
In mathematical dynamics there is a class of problems called "billiards" related to billiard balls bouncing around tables (which may have a variety of shapes). The tables may also have objects placed on them (again, with a variety of shapes), so the ball bounces off both the table boundaries and these objects.
These billiard trajectories have unusual and interesting properties. For example, a square table with a square obstacle aligned with the table has some trajectories that are periodic (i.e., retrace themselves repeatedly), or quasiperiodic, and some that are highly irregular. The irregularity is often called "pseudo chaos", meaning that it seems as irregular as in chaotic systems, although the system does not satisfy the usual mathematical criteria for chaos.
Getting back to the two Tri oscillators, it is quite easy to electronically simulate an object on the table by using a bunch of comparators and a means of reversing the direction of the oscillators at any time. A familiar oscillator is the Tri VCO based on comparators and an RS flip-flop, such as in the Electronotes TZFM VCO that a number of folks here have built.
So I set up circuit like this, with trajectories modified when the "ball" is in certain regions. (These are not true physical reflections, but never mind, the idea is the same.) Here are some results:
Figure 1 is a scope shot of a trajectory that is periodic, with many trips around the system before repeating. Figure 2 is a photo of a nearby trajectory that shows the irregular behavior of pseudo chaos. This is seen in the areas that look like bands, where the trajectories densely fill the areas.
Figures 3 and 4 show the frequency spectra of the two trajectories. The pseudo chaotic one has a broad and quite strong background feature, just like seen in spectra of the more familiar chaotic systems. The periodic one has many strong overtones.
Finally, the mp3 has two sections, corresponding to the photos, so you can hear what pseudo chaos sounds like.
I'll try to post some more examples later, if anyone is interested.
Ian
Edit: Computer simulations seem to indicate that the trajectories in this system are periodic, often with many thousands of bounces per period. The irregularities seen above are probably related to noise in the system modulating the path in a discontinuous manner. Still pretty interesting, though!
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Last edited by frijitz on Mon Jul 05, 2010 10:29 pm; edited 2 times in total |
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andrewF
Joined: Dec 29, 2006
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| Posted: Fri Apr 02, 2010 3:00 am Post subject:
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love the scope shots Ian, very beautiful.
Is this circuit leading up to a synth module or experimenting?
If its not too much trouble, a block diagram or schematic would be good to see. |
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RF
Joined: Mar 23, 2007
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| Posted: Fri Apr 02, 2010 5:25 am Post subject:
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Very interesting Ian. Thanks for sharing the 'Billiards' graphics and clips  That's a new concept for me, and I agree with Andrew - It would be great to understand a bit more how this was done.
bruce
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Uncle Krunkus
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| Posted: Fri Apr 02, 2010 5:32 am Post subject:
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I'm thinking a couple of window comparators on two triangle or sawtooth generators, then use boolean operations on the comparator outputs to determine whether one or both waves get inverted. Is that close?
It kind of reminds me of my ReAnimator idea. I'm gonna have to resurrect that project soon. (pun intended) 
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frijitz
Joined: May 04, 2007
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| Posted: Fri Apr 02, 2010 9:56 am Post subject:
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| Uncle Krunkus wrote: | | I'm thinking a couple of window comparators on two triangle or sawtooth generators, then use boolean operations on the comparator outputs to determine whether one or both waves get inverted. Is that close? |
Yes, that's it exactly! 
There are two Tri VCOs, similar to the EN TZ VCO, which I believe is available at Bernie's website. These are based on a CD4013 RS flipflop chip. When the signal reaches either 6V or 0V (the "walls" of the table) R or S reverses the ramp in the usual way. In addition, a pulse on the Clock line will also reverse the wave at any time (upon hitting an "object").
The "objects" are colums or pillars extending out from the walls. Each is defined by a window comparator (presently one of the output stages of an LM3914 bargraph chip) to determine the base of the pillar and another (single) comparator to determine the height. There are two pillars, one extending up from the x axis to an adjustable height y_h. The other pillar extends to the right from the y axis.
When the ball crosses into one of the pillars, one of the oscillators is reversed, depending on how the subsystems are jumpered together. For example, if the pillar extending up from the x axis is entered and the x oscillator is reversed, then it looks like the wave is reflected from the sides of the pillar, unless the pillar is entered from the top, when the reversal is the same but it doesn't actually correspond to a reflection.
If the y oscillator were reversed instead (as in the scope shots) then the wave reflects if it hits the top of the pillar, but bends if it enters the side of the pillar. You can see the pillars in the scope shots as the places where the wave direction changes inside the square.
Yes, I'll try to get some diagrams made up at some point.
My original goal was to make a 3D system, with z reversing when the xy pillar is entered and similar for the other coordinate permutations. (I've never seen any studies of 3D billiards.) But my whiteboard is already full. 
Anyway, the present system is much richer in complexity than I expected, so I think a simpler system might be just as good. I'll post a clip with one of the oscillators being swept so you can see how complex the system is.
Ian |
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Sound
Joined: Jun 06, 2006
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| Posted: Fri Apr 02, 2010 10:04 am Post subject:
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| Very ineresting Ian, waiting for more clips |
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nobody
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| Posted: Fri Apr 02, 2010 10:40 am Post subject:
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Nice, Ian! Will listen to the clip soon (not at a computer with working sound at the mo).
This makes me want to set up some oscilloscope software and play around with this idea. |
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frijitz
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| Posted: Fri Apr 02, 2010 10:52 am Post subject:
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Here are two more mp3s. The first has a very slow sweep of one of the VCOs where you can hear the many patterns the system goes through with parameter changes. I couldn't even count the number of different patterns, but it's probably a hundred or so.
The second clip has the system operating in LFO range. The two outputs are controlling the frequencies of two VCOs. You can hear the effect of the system going around a complicated path, similar to that seen in the previous photos.
Ian
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Uncle Krunkus
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| Posted: Fri Apr 02, 2010 2:48 pm Post subject:
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That second clip sounds very acoustic in the way the voices interact with each other. Almost like two child ghosts trying to sing harmonies.
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TekniK
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| Posted: Sat Apr 03, 2010 9:24 am Post subject:
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| first sound like a fax and second like pac-man lol |
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frijitz
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| Posted: Sat Apr 03, 2010 9:32 am Post subject:
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| Uncle Krunkus wrote: | | That second clip sounds very acoustic in the way the voices interact with each other. Almost like two child ghosts trying to sing harmonies. |
What is interesting to me is the rhythmic patterns it creates. Lots of variety, but more regular that what you get from a true chaos generator.
Ian |
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frijitz
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| Posted: Sat Apr 03, 2010 9:33 am Post subject:
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| TekniK wrote: | | first sound like a fax and second like pac-man lol |
I see. |
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TekniK
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| Posted: Sat Apr 03, 2010 9:59 am Post subject:
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| what is the point, to use it as modulation device or sound source? |
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frijitz
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| Posted: Sat Apr 03, 2010 10:50 am Post subject:
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| TekniK wrote: | | what is the point, to use it as modulation device or sound source? |
Either, whatever you like. I always thought the chaos modules would be most useful in the LFO range, but people have also found them useful in the audio range.
http://www.electro-music.com/forum/topic-17893.html
Ian |
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TekniK
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| Posted: Sat Apr 03, 2010 10:56 am Post subject:
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| keeping things simple and 4 of these on one pcb,yes that would be nice! |
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frijitz
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| Posted: Sat Apr 03, 2010 11:31 am Post subject:
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| TekniK wrote: | | keeping things simple and 4 of these on one pcb,yes that would be nice! |
We'll see. The billiard part might could be simple, since there is such a huge range of patterns. But each circuit needs two VCOs, so eight of those on one board? And whenever something is simple, of course, there are complaints about not enough features.
Ian |
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TekniK
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| Posted: Sat Apr 03, 2010 2:06 pm Post subject:
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| frijitz wrote: | | TekniK wrote: | | keeping things simple and 4 of these on one pcb,yes that would be nice! |
We'll see. The billiard part might could be simple, since there is such a huge range of patterns. But each circuit needs two VCOs, so eight of those on one board? And whenever something is simple, of course, there are complaints about not enough features.
Ian |
Exacly, i did understand it very well,thats why i did mention 'simple'
There are plenty full vco's out there already. |
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frijitz
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| Posted: Sat Apr 03, 2010 2:37 pm Post subject:
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| TekniK wrote: | | Exacly, i did understand it very well,thats why i did mention 'simple' There are plenty full vco's out there already. |
Yeah. Too bad it won't work with external VCOs, but hardly any have the reversing capability. Another possibility would be to use it with external Saws -- I'll be exploring that when I have a chance.
Of course, if it did have internal VCOs it would be a shame not to bring the outputs to the panel, but then I'd want it to be accurate and stable and then somebody would want the standard wave shapers .... and on and on. Maybe a bypass switch to bring the VCOs out directly ... or a switch to remove the cross coupling ... Hmmm ....
Appreciate your input.
Ian |
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bridechamber
Joined: Oct 06, 2007
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| Posted: Sat Apr 03, 2010 9:38 pm Post subject:
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Droool.......
I would think you could skip the waveshapers on the VCO cores. I mean, there's already a nice TZ VCO out there, right?
Just an idea -- separate VCO core boards and 8 Ball board?
I love this idea as a VCO/ tone machine. Sequenced or with a sample-and-hold, that would be loads of fun. |
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Uncle Krunkus
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| Posted: Sun Apr 04, 2010 1:02 am Post subject:
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You could use external VCOs if you used the core idea of my ReAnimator and then just incorporate the billiard concept into the ReAnimator rules. Then you'd be able to hang it off any oscillator and leave all the stability issues, waveform options etc. as external concerns.
Ian, let me know if you want a rundown on how these two ideas might be hybridised. You'd be welcome to take the ReAnimator idea and run with it if that helped get the idea completed.
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frijitz
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| Posted: Sun Apr 04, 2010 9:05 am Post subject:
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| bridechamber wrote: | | I would think you could skip the waveshapers on the VCO cores. |
Definitely! But there would be complaints.
| Quote: | | Just an idea -- separate VCO core boards and 8 Ball board? |
Good idea. Especially if we wanted to go Euro. 
| Quote: | | I love this idea as a VCO/ tone machine. Sequenced or with a sample-and-hold, that would be loads of fun. |
I want to look at using this with the TGTSH -- I have a hunch it would be a good match.
Ian |
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frijitz
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| Posted: Sun Apr 04, 2010 9:08 am Post subject:
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| Uncle Krunkus wrote: | | You could use external VCOs if you used the core idea of my ReAnimator and then just incorporate the billiard concept into the ReAnimator rules. |
Thanks for thinking about this. But I'll have to think more about what you are suggesting. The VCO core has to reverse direction on command, and I don't see how you could do that with an external standard core.
Ian |
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Uncle Krunkus
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| Posted: Sun Apr 04, 2010 2:43 pm Post subject:
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I was thinking about the ReAnimator which can easily "reverse direction on command"
Now that I think a bit more, I realise that the ReAnimator takes a sync pulse off the external oscillator, and therefore would re-start the wave based on that pulse. It would keep the reversed direction until told otherwise, but I can see that it would give different (still interesting) results compared to a dedicated, unsynced, reversable core.
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bridechamber
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| Posted: Mon Apr 05, 2010 9:00 am Post subject:
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| frijitz wrote: | | bridechamber wrote: | | I would think you could skip the waveshapers on the VCO cores. |
Definitely! But there would be complaints.
There always are : )
| Quote: | | Just an idea -- separate VCO core boards and 8 Ball board? |
Good idea. Especially if we wanted to go Euro.
: )
| Quote: | | I love this idea as a VCO/ tone machine. Sequenced or with a sample-and-hold, that would be loads of fun. |
I want to look at using this with the TGTSH -- I have a hunch it would be a good match.
Definitely!
Ian |
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frijitz
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| Posted: Tue Apr 13, 2010 5:02 pm Post subject:
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Here are a couple more demos of the mighty MegaCycler. In the first a pair of TGTSH modules are cross coupled to the generator to produce discrete voltages for defining pitches. At the same time the continuous voltages of the system sweep a couple of figures.
The second demo is similar, except four TGTSHs are used, controlling two continuous signals and two percussive ones.
Ian
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