Hey I have been reading a lot about chaos equations and the like recently. I found interesting analog chaos computers and have seen some chaos equations that can be fed an input such as an oscillator or audio stream and output a very interesting and complex output.

Anyway I am a little lacking in the differential calculus skills and I am having troubles inputting equations with t) type things into an fexpr~ or expr~. I did see the lorenz attractor example in the help for expr~, but if someone could help me figure out how to get from a differential equation to an expr~ or fexpr~ that calculates it that would be awesome. Or maybe I just need to get some calculus books.

the page below has some examples of the types of equations I am talking about.
http://www.viewsfromscience.com/documents/webpages/chaos.html

Thanks

I would add that you should try to find before sounds made by people before you with these diff equs, and then challenge us with something ... musically not too boring

I mean that if you wanna ask 'us' to 'work' for you, the challenge should be appealing .

Nau

Noizehack, thanks for pointing to that page on chaos equations. Are chaos equations fundamentally different from random number routines? I still don't know.

Anyway, the logistic map equation as illustrated on that page can be implemented in Pd with regular objects. The point is, that viewsfromscience gives the equation as:

t+1) = lambda * t) * (1-X(t))

In dsp engineering this could be implemented as:

y[t] = x[t] + lambda * y[t-1] * (1-y[t-1])

where x is input and y is output, and [t-1] is one sample delay. This is by the way an integration process, not differentiation.

See attached patch for Pd implementation of Logistic Map Simulation as depicted on the viewsfromscience page. It's a very simple routine with intriguing output, I'm still thinking about how to use it in musical context.

Katja

http://www.pdpatchrepo.info/hurleur/chaos.pd.zip

ok thanks for that patch, I was playing around with it and it clarifies a few things. Seeing the equation and then the patch helped me out. So what I need to figure out is how to do the integration part. Which is calculus?

As far as the difference between chaos equations and random number routines. I think sometimes complex chaos equations are used for pseudo random numbers, but like the first quote says on the view from science page chaos systems are deterministic and non-linear.

My idea for implementing chaos equations musically was mostly to use them as controls for creating not quite random patterns that are hopefully interesting. But I also like the way they sound as an alternative to noise or for generating complex waveforms.

Another idea I had was to get ideas of what equations are interesting and somehow use those equations as inspiration for slightly less complex equations which would create less random but somewhat chaotic controls.

oh also what does [click~] do? there is no help for it.

ok, sorry for my previous message.

I was clearly thinking about audio, but for control it makes more sense to look at pictures and figure out the type of behaviour (is it confined in a fixed volume, etc).

Nau

Yes, I was joking about having someone else make a C external for me, unless it is something they are interested in as well. Also looking at the patch that katja made I realized it might be a little simpler than I thought to get the equations into fexpr~

So I just need to study up on my calculus now.

@noizehack said:

what does [click~] do?

[click~] is in cyclone library. On bang, it sends a single- or multiple-sample impulse with value(s) as specified by arguments. Zero elsewhere.

Noizehack, please do post if you have some musical result with chaos generators.

Katja

ok here is something I made using the calculation. It makes some patterns I think are kind of interesting. Also some of them seem repetitive at first, but if you listen for a while you will hear variations. I am kind of surprised how repetitive it is. I would think it would be a little more random sounding.

Sorry I didn't get anything up sooner, I got distracted editing a film for a friend.

http://www.pdpatchrepo.info/hurleur/chaosFilteredNoise.pd

To clear it out. Chaos is neither integration nor diffenrential equations. It is based on iteration (tjeck out wiki.pedia). It is based upon complex numbers and is used to compose visual and auditive fractals.

I've written a paper on it a couple of months ago. I am very sorry, but it's difficult for me to explain it profound in english, since I'm a dane

It is indeed a very interesting thread!

Noizehack, thanks for sharing your patch. It is a clear demo of what the logistic map simulator can do with audio. I have compared with [random] applied to your filtered noise and the character is completely different indeed. What I like about the logistic map:

• combines recurring patterns with impression of randomness
• different levels of complexity can be emulated by setting lambda parameter

What I do not like however, is the dominance of a small region in the high end. This region of emphasis could be transposed of course, but it can not be structurally avoided since it is obviously a characteristic of the system. What do you think of this, Noizehack? There may be a way to use it to our advantage. After all, points of emphasis are often deliberately composed in music.

@JensChr said:

I've written a paper on it a couple of months ago.

Can you share a patch implementing chaos iterators with complex numbers? Or if not, can you share an equation for discrete case so we can try and build a patch for it, like we did with the logistic map?

Katja

Making a patch with chaos theory seems difficult, because complex numbers are construced of both a real number and an imaginary number: c = a + bi, where c is the complex number, a is the real number and bi is the imaginary number. b is a real number that is multiplied with i and i is the square root of -1. For calculation tjeck wiki: http://en.wikipedia.org/wiki/Complex_number ...

It is also difficult for me to describe a discrete case, since I haven't worked with waveshaping by the use of complex numbers, but I know fourier analysis and transformation (PD has objects for these) is based on complex numbers.

Maybe i can fix a patch that calculates complex number, but I don't know what for. It would be fun to make one so I'll try.

I'm sorry that I can't help more, but complex numbers is a complex matter, but I'll think of what i can do for sure!

This is iteration on real numbers:

http://puredata.hurleur.com/sujet-4825-infinity-series

Complex numbers in Pd isn't really that difficult. You just have to know when you're using them, since there isn't a complex type. All you really have to do is treat the real and imaginary part as a two element list and apply the appropriate arithmetic for complex numbers. And, of course, it makes things easier if you just treat real numbers as complex numbers with a 0 imaginary part.

Anyway, if it makes things easier, I've attached a set of complex math abstractions I originally made for designing filters and calculating their frequency response, which requires complex numbers. They currently rely on [cartopol] and [poltocar] from cyclone, but they could easily be made without them.

http://www.pdpatchrepo.info/hurleur/complex_math.mmb.zip

So I added some different ways of scaling the number from a 0 to 1 range to a frequency range. Some of them sound cool, but they don't really change the fact that the high end of the frequency range seems more prevalent than the low end. I came up with the idea based on the 1 volt per octave idea in analogue modulars, if you had a linear frequency response most of the knob range would be high piercing sounds, but with a 1 volt per octave range you get way more range in the bottom end.

I think the reason this patch sounds higher pitched though is just because the higher frequencies of filtered noise seem louder. Maybe using pink noise or having the volume scale down as the pitch scales up would fix this. I didn't have time to try those out yet.

thanks for the iteration info JensChr I will research that some.

http://www.pdpatchrepo.info/hurleur/chaosFilteredNoise2.pd

@noizehack said:

So I added some different ways of scaling

The linear inverse and log scaling do help indeed for the case of filtered noise.

I also considered skipping points of too heavy emphasis. [cyclone/Histo] can make a histogram of the scaled output and it is easy to identify the region.

@JensChr said:

complex numbers is a complex matter

Attached is a patch which does complex integration at control rate. It is similar to what [cpole~] does at signal rate. The impulse response is a resonator/oscillator, depending on the radius setting. It's formula is like a regular (leaky) integrator:

y[n] = x[n] + r * y[n-1]

But input x and output y are implemented as complex numbers.

Now compare with the logistic map equation:

y[n] = x[n] + lambda * y[n-1] * (1-y[n-1])

Here (1-y[n-1]) is the non-linear aspect. A similar non-linear component should make a chaos generator for the complex case. But what exactly? JensChr, it must be in your paper, please help us out.

Katja

http://www.pdpatchrepo.info/hurleur/complex_integrator.pd.zip

There is by the way this chaos library of Pd classes by Ben Bogart and Michael McGonagle. The library is about fractals. An entry point for help is flatspace/readme-fractals.pd.

The actual calculations are done within the c objects, so in order to know how they are done you need to read the through the source code. Using the library would be very instructive as well, of course.

Katja.

I don't see how I missed this. Here is a patch that has the mandelbrot in an expr.
It is an old patch and I have actually changed some things since then you can more recent see videos on my youtube page.

http://sites.google.com/site/billystiltner/Astronauts/mandel3.pd

I have been into chaos and sound for quite sometime.
I like mapping the orbits directly to sound as well as generating notes with the orbits. The patch in the link is the generate notes from orbits kind.

The patch was inspired by Elaine Walker's max patch in her thesis on chaos melody maker. It is a good read if you are interested in fractals and music.
http://www.ziaspace.com/elaine/chaos/

Thanks for sharing your patch and the pointer to Elaine's thesis. Together, they are very informative. I found it impossible to find an interesting fractal 'by hand' though. Everytime when a pattern sounds interesting, it's flying to the stars and reset, being not part of the Mandelbrot set. The logistic map function is much easier to handle, like Elaine mentioned in her thesis.

Katja

http://www.geocities.ws/billy_stiltner/music/pd/manit.pd is an abstraction that accepts as input:

a bang to perform an iteration

a bang to restart orbit

k,l(real and imag of C)

boundary ie. 4 if the escape boundary has a radius of 2

and numits - 0=free running

outputs are:

itnum(resets to 0 on escape or numits reached)

x,y(real and imag of Z)

xesc,yesc(real and imag of Zescape) only when an orbit escapes boundary

a bang when an orbit escapes

a bang when an orbit reaches numits

to put x or y to notes use this expr

[expr (($f1 + 4) / 8 * $f2) + $ f3]

f1 = x or y(make one expr ezch for x and y)
$f2 = scale factor (this stretches or compresses the range of notes)
$f3 = offset(this transposes the notes up or down a step)

output is a note number which can feed a mtof or table lookup for nonstandard tunings.

You could also use the old scaling algorithm.

This implementation uses abs( in the boundary check to get rid of some of the flying off to space notes)

One of these days I should make an example and help file. Curently the only patches I have using this have way too many other patches and tangled messes of wires to submit as an example.