A particularly interesting one is
F(x) = a x (1-x)
where a is any number. You choose a value of x, plug it in, get something
out, plug that in for x again, ad infinitum. (a procedure similar to this
generates the Mandelbrot set!) The numbers gravitate to a fixed point x0 =
F(x0). As a increases from zero, the "fixed point" increases as well, until
you get to about a=3. Suddenly, the fixed point disappears. Instead, the
values of x converge on a pair of numbers, x1 and x2, such that F(x1) = x2 and
F(x2) = x1. That is, you start oscillating between x1 and x2 with period 2.
For even larger values of a, this "period doubling" happens again, and you
start oscillating between four values of x. It happens again and again,
faster and faster, 8, 16, 32, 64, 128... at about a=3.5, you hit infinity.
Here, the numbers never repeat, and behave completely chaotically. It looks
like this: a increases to the right, x increases up. Asterisks represent
"stable" points or "orbits".
******
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*** ***
** * ****
* ******
***** ***
*** * ******
** * * ****
* **** ***
* * ****
* ******
After that crude attempt at character graphics, perhaps you'd prefer to look
at a real image of the same thing.
Anyway, the chaos continues for a while, until suddenly a cycle of period 3 forms (!) Each of these three stable points doubles and doubles again in a miniature copy of the whole tree (like the little mandelbrot sets in the main one), giving periods of 3, 6, 12, 24, 48... until we hit infinity again, and chaos ensues. It continues for a while, until each of the three copies triplicates again, giving nine strands. Each of these doubles, giving 18, 36, 72,... by the time we reach a=4, we've generated all integer periods. The resulting figure is much more beautiful than I've drawn it here.
Anyway, what does this have to do with music? Imagine 16, 32, 64, or 128 orchestral instruments playing a fugue. They begin in a single voice, playing a single melody in unison. then, sudddenly they split into two interlinking variations on that theme. Each of these themes split, giving a four-part harmony. This continues, over and over, getting more and more complicated, and more and more dissonant and confusing, until each instrument is playing its own melody, the whole sounding totally chaotic. Then, suddenly, a three-part harmony forms, splits into six, and off we go into chaos again.
Here's another way to do this "bifurcation diagram" in music. The whole idea
is a cycle of repeating numbers, right? Why not write it as a round? All the
instruments begin playing a simple melody. Suddenly, half of them shift to a
point halfway through the melody. Then half of each of these shift to points
1/4 and 3/4 of the way through. Then each of these split... at the end, each
instrument is playing the whole song, but none of them are on the same beat in
the same measure. You know that's going to sound weird. John Costello
observed that "Row Row Row Your Boat" is perfect for the job, being 16 beats
in 4 verses. It goes like this:
Verse 1: All start on "Row, Row..."
Verse 2: Half start on "Row, Row", half on "Merrily, Merrily"
Verse 3: 1/4 on "Row, Row", 1/4 on "Gently", 1/4 on "Merrily", 1/4 on "Life"
Verse 4: You get the idea.
By verse 5, all instruments are playing different parts. A bifurcating
three-part round should follow, but dividing 16 beats and 16 instruments into
three, six, twelve, etc. equal parts might be tough. I suggest switching to
12 instruments, and John suggests converting to 3/4 time, which I don't know
enough music to fully understand.