More Bifurcator doc work.
Oooooh, a pretty picture!
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@ -6,27 +6,24 @@ this is a fantasy sound chip which uses a unique sound generation method: [logis
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the core of the Bifurcator is the (iterative) logistic map function:
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the core of the Bifurcator is the (iterative) logistic map function:
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<!-- LaTeX -->
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_x_<sub>_n_+1</sub> = _rx_<sub>_n_</sub>(1 - _x_<sub>_n_</sub>)
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$x_{n+1}=\lambda x_n (1-x_n)$
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<!-- MD/HTML -->
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which when iterated across 2 ≤ _r_ ≤ 4 produces this graph:
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_x_<sub>_n_+1</sub> = _λx_<sub>_n_</sub>(1 - _x_<sub>_n_</sub>)
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which when iterated across $2\leq\lambda\leq4$ produces this graph:
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_{{ IMAGE HERE – will look sort of like [this](https://en.wikipedia.org/wiki/Logistic_map#/media/File:Logistic_Bifurcation_map_High_Resolution.png) but matching house style }}_
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in Bifurcator the current output (and initial "load value") ranges from 0 to 65535, which maps directly to _x_ in the range 0.0 to 1.0. the parameter ranges from 0 to 65535, which maps directly to _λ_ in the range 2.0 to 4.0.
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in Bifurcator the current output (and initial "load value") ranges from 0 to 65535, which maps directly to _x_ in the range 0.0 to 1.0. the parameter ranges from 0 to 65535, which maps directly to _r_ in the range 2.0 to approximately 4.0.
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### load value
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### load value
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the first value of $x$ is set with the **load value**. it must be set to a non-zero value, otherwise the next iteration would also be zero... and the next one too... and so on, generating silence. default is 1.
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the first value of _x_ is set with the **load value**. it must be set to a non-zero value, otherwise the next iteration would also be zero... and the next one too... and so on, generating silence. default is 1.
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different load values will cause different "attack" sounds before the iterations stabilize (if they do).
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different load values will cause different "attack" sounds before the iterations stabilize (if they do).
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### parameter
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### parameter
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by varying the parameter, the values of $x$ may change drastically, producing a variety of sounds. the higher the parameter, the more "chaos" is present, effectively yielding noise.
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by varying the parameter, the values of _x_ may change drastically, producing a variety of sounds. the higher the parameter, the more "chaos" is present, effectively yielding noise.
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for most parameter values below 32768 the output converges to a single value (effectively no sound).
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for most parameter values below 32768 the output converges to a single value (effectively no sound).
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@ -36,10 +33,10 @@ the default parameter value is 47360.
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## effects
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## effects
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- `10xx`: **set low byte of channel sample state.**
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- `10xx`: **set low byte of load state.**
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- `11xx`: **set high byte of channel sample state.**
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- `11xx`: **set high byte of load state.**
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- `12xx`: **set parameter low byte.**
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- `12xx`: **set low byte of parameter.**
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- `13xx`: **set parameter high byte.**
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- `13xx`: **set high byte of parameter.**
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## info
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## info
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BIN
doc/7-systems/logisticmap.png
Normal file
BIN
doc/7-systems/logisticmap.png
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