9.3 Branching Numbers as Finite Dice
The branching at each Mandelbrot bifurcation is not arbitrary but fixed by local algebra, always a finite whole number equal to the bulb’s period, and assigning uniform probability to each branch produces a natural measure that weights simpler branches more heavily, which is exactly what the ear already knows.
Proto
At every bifurcation point in the Mandelbrot set, the branching is not arbitrary. The number of branches is exactly the period of the bulb the point belongs to, a period-3 bulb sprouts 3-branching lightning bolts, period-5 sprouts 5, and so on.
This means the “choice space” of the Mandelbrot set is not a continuous dimension. It is a tree of finite dice. At each node you roll a die whose number of sides is fixed by the mathematics. The dice get more sided as you go deeper, main bulbs have small periods, and as you zoom into the boundary the periods grow.
Assigning uniform probability 1/n to each branch of an n-branching node gives a natural probability measure on the boundary, the harmonic measure[douady]. This measure weights simpler branches more heavily. The octave gets more probability than the minor seventh. Which is exactly what the ear already knows. The consonance weighting is not imposed, it falls out of the uniform die assumption applied to the tree structure.