Bifurcator docs rework, draft 1.

This isn't a final form, but an interim draft to test both LaTeX and HTML methods of displaying the equation and such.

doc/4-instrument/bifurcator.md

@@ -1,6 +1,6 @@
 # Bifurcator instrument editor
 
-Bifurcator instrument editor consists of these macros:
+the Bifurcator instrument editor consists of these macros:
 
 - **Volume**: volume sequence.
 - **Arpeggio**: pitch sequence.
@@ -10,30 +10,5 @@ Bifurcator instrument editor consists of these macros:
 - **Pitch**: fine pitch.
 - **Load Value**: changes the current output value.
 
-## audio generation description
+not all combinations of Parameter and Load Value generate tones. for a description of how these affect the sound, see the [Bifurcator](../7-systems/bifurcator.md) system page.
 
-Bifurcator uses logistic map iterations for sound generation.
-basically it runs the following function over and over:
-
-```
-r = (1 + (p / 65536)) * 2
-x = (r * x) * (1 - x)
-```
-
-where `x` is the current output value and `p` is the "parameter".
-
-by varying the parameter, the value of x may change drastically, producing a variety of sounds.
-the higher the parameter, the more "chaos" is present, effectively yielding noise.
-
-the default parameter is `47360`, which results in a square wave.
-
-if the parameter is set to 0, there's no sound at all.
-as the parameter approaches 32768, a decaying square wave is produced.
-the square wave stops decaying past 32768 and becomes louder until the parameter hits ~47496 (`r = 1 + sqrt(6)`).
-a second square wave one octave lower then starts appearing, until the parameter reaches ~51443 (`r ≈ 3.56995`). this is where chaos begins.
-anything higher results in a total mess.
-however, at ~59914 (`r = 1 + sqrt(8)`) you can hear a 33% pulse wave.
-
-## the importance of loading the value
-
-you must load a value that isn't 0 in order to get sound. otherwise the function will always output 0.

doc/7-systems/bifurcator.md

@@ -1,6 +1,38 @@
 # Bifurcator
 
-this is a fantasy sound chip which uses a unique sound generation method: logistic map iterations.
+this is a fantasy sound chip which uses a unique sound generation method: [logistic map](https://en.wikipedia.org/wiki/Logistic_map) iterations.
+
+## usage
+
+the core of the Bifurcator is the (iterative) logistic map function:
+
+<!-- LaTeX -->
+$x_{n+1}=\lambda x_n (1-x_n)$
+
+<!-- MD/HTML -->
+_x_<sub>_n_+1</sub> = _λx_<sub>_n_</sub>(1 - _x_<sub>_n_</sub>)
+
+which when iterated across $2\leq\lambda\leq4$ produces this graph:
+
+_{{ 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 }}_
+
+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.
+
+### load value
+
+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.
+
+different load values will cause different "attack" sounds before the iterations stabilize (if they do).
+
+### parameter
+
+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.
+
+for most parameter values below 32768 the output converges to a single value (effectively no sound).
+
+above 32768 the output starts to oscillate. from 32768 to 47496, it bifurcates to oscillate between between 2 values, generating a square wave. from 47497 to 51433, it bifurcates again and becomes a period of 4 values. beyond that, the system rapidly becomes chaotic (noise). there's an "island of stability" starting at around parameter value 59914 where the output oscillates between 3 values, then bifurcates from there.
+
+the default parameter value is 47360.
 
 ## effects