Introduction (1)
Chaotic systems used as audio oscillators - basic principles

Let's consider two fundamental audio synthesis methods : additive synthesis and substractive synthesis, from a order / disorder point of view.

When doing additive synthesis, one starts from extremely ordered systems, and tries to enrich the basic oscillators by combining them.
When doing substractive synthesis, one starts from extremely noisy ~ disordered systems, and tries to simplify them by filtering.

From now on, we'll call any kind of synthesis that consists in enriching simple systems "positive synthesis", and the reverse process - making noisy systems understandable, "negative synthesis".

From this point of view, amplitude and frequency modulation synthesis methods can also be considered as "positive synthesis".

The same "axial" principle, illustrated waveforms and what is called "phase space display" - représentation dans l'espace des phases, or attractors.
( more details about attractors and how to read them in "methods - reading attractors" )

	sine wave attractor	oboe sample attractor	white noise attractor
	sine wave waveform	oboe sample waveform	white noise waveform
	~ order ~ ----->	<------ ~ balance ~ ------>	<----- ~ disorder ~

NB - the goal here isn't to emulate traditional instrument sounds, but taking music instrument samples as a reference is a safe bet when it comes to audio synthesis, which, ultimately, consists in producing audio samples than can be used in a musical context.

Remark : the "waveform table" synthesis method, used for instance in Korg Trinity / Triton synths, can be considered in light of this order / disorder point of view.
This method is neither positive nor negative : it aims at starting from samples which are already "perceptively acceptable".

Let's have a look at the Lorenz system, a classical example when it comes to study chaos.

	exple 1, attractor	exple 2, attractor
	exple 1, waveform	exple 2, waveform

It's obvious that those Lorenz samples are somewhere close to the balance. Not too bad for an arbitrary 3 variable mathematic system.

Again, this point of view is more accurately explained and detailed in "methods - reading attractors" - but briefly, let's consider again the oboe sample.
Its attractor display can be considered from two different perspectives - or observation scales ( échelles d'observation ).

	global shape ( macro )	detail (micro )

An obvious observation : what makes the Lorenz attractor close to the "perceptively balanced zone", what makes it look like the oboe sample attractor, is the micro aspect.

- First remark

It is well known that these kind of attractors have a fractal structure : the closer one looks, the more details one sees, but they always follow the same pattern.
Though it is this fractal aspect that fascinates most people, it is completely uninteresting when it comes to audio synthesis.

For our purpose, it is enough that the attractor's micro structure is reasonably complex - some would say that it exhibits a reasonable entropy - an entropy close to what could be found studying a real instrument.

- Second remark

It is not our purpose to consider the chaotic systems macro aspects.

"Macro shaping" is a well studied process - amplitude modulation and frequency modulation methods, for instance, are efficient macro shaping methods.
AM & FM can indeed be applied on chaotic samples to "macro shape" them, but it's another story.

Our point here is to study the audio synthesis adaptation of chaotic system outputs, focusing on the micro shape aspect in an audio context - as some would say, the "grain".