Discrete-scale invariance deviates from power law

Continuing this post, power laws are a burr in my butt today. From this article:

"Many self-similar systems are scale invariant only in discrete steps. A blood vessel tends to branch into two smaller vessels, a fluid vortex into two or three smaller vortices, and the Sierpinski triangle is self-similar only by powers of two. These systems preserve relative proportions upon rescaling from one step to the next, but not upon arbitrary rescaling. This property is termed discrete-scale invariance or discrete renormalizability. It is a weaker condition than the continuous scale invariance underlying the Pareto distribution. Whereas strict scale invariance implies a power law and vice versa, discrete-scale invariance allows log-periodic modulations in the frequencies of observations that deviate from a pure power law such as Eq.(1). Such modulations are indeed observed in bronchial tube diameter, vortex ens-trophy, and financial asset prices."