The maths behind the music

The transformation rule which is the basis for Nick's composition uses the complex function $$f(z) = 0.5(-z + \sqrt{12 - 3 z^2}).$$ Iterating this function starting with a given value $z_0$ gives a sequence of complex numbers: $$z_0, z_1 = f(z_0), z_2 = f(z_1), z_3 = f(z_2), ... .$$ Any complex number z = x+ iy can also be thought of as the pair (x,y) of real numbers, so the formula does indeed transform one pair into another pair, as described above (see Plus article Curious quaternions for an introduction to complex numbers).