We study the simultaneous boundary desensitizing controllability of vibrating elastic strings of the same length and different wave velocities. The problem consists in finding a boundary control for several wave equations such that the solutions are insensitive in some region of the strings with respect to small perturbations of the initial data. The term simultaneous is understood in the sense that the same control function is used to control all the strings. We show that the simultaneous approximate desensitizing controllability holds true in enough large time if and only if the wave velocities of the strings are pairwise incommensurable. The proof is based on a weighted observability inequality for the adjoint system. The inequality also allows us to identify spaces of simultaneously desensitizing controllable initial states depending on Diophantine approximation properties of the ratios of the wave velocities.