Principle
It is of interest to give a closer look to the logistic system's subharmonic cascade.
The following movie gives both spectrum and audio translation for values of R which follow the subharmonic cascade :
- period 2, then period 4, then period 8.... until period 64 and the beginning of chaos.
The problem is, what we hear doesn't correspond to what we see : whereas the fundamental seems to go lower and lower, no changes are heard.
The reason for this is simple - if we look at the "period 16" file ( R=2.565 ), the fundamental is 70dB below the higher harmonic, and 40dB below what we perceive to be the fundamental.
Thus the file has to be processed, using what could be called "spectral compression", which is like dynamic compression but in the spectral domain.
The goal here is to reduce the level difference which is between the different harmonics.
For instance :
 |
...becomes... |
 |
"Invisible" or, in our case, "inaudible" spectral rays will become audible.
The compression is computed the following way :
- the spectrum is normalized to values between +1 / -1
- then, in absolute value, NewValue = OldValue ^ (1/CompressionFactor), where CompressionFactor >= 1.
A compression factor of 1 corresponds to no compression.
A compression factor of 4 corresponds to a serious compression.
Application to the subharmonic cascade
This principle is applied to the subharmonic cascade with various compression factors.
Here is the result - the first movie is without compression, all the movies display the uncompressed spectrum.
( 4 Quicktime movies, mpeg4 video, mpeg4 audio at 64kbs mono, base freq 44k1, 120kB each )
Compr Factor = 1 (ref ) |
Compr Factor = 2 |
 |
|
Comp Factor = 4 |
Comp Factor = 8 |
The results are quite convincing for Comp Factor = 4 or 8.
The optimal compression factor for this cascade would be 4. With 8, the chaotic parts just after the cascade are a bit "crushed".
The same kind of result would have be obtained using a conventional filter, but then a different filter for each sample (each R value) would have been necessary to get satisfying results, and we are looking to a process that can be applied automatically, in order to eventually adapt the system to "automated" audio synthesis.