The dual resolution graphs of rational triple point (RTP) singularities can be seen as a generalization of Dynkin diagrams. In this work, we study the triple root systems corresponding to those diagrams. We determine the number of roots for each RTP singularity, and show that for each root we obtain a linear free divisor. Furthermore, we deduce that linear free divisors defined by rational triple quivers with roots in the corresponding triple root systems satisfy the global logarithmic comparison theorem. We also discuss a generalization of these results to the class of rational singularities with almost reduced Artin cycle.