The 1∕fα noise displayed by the fluctuation of the nth unfolded eigenvalue, where n plays the role of a discrete time, was recently characterized for the classical Gaussian ensembles of N×N random matrices. It is investigated here for the β-Hermite ensemble by wavelet analysis of Monte Carlo simulated series both as a function of β and of N. When β decreases from 1 to 0, for a given and large enough N, the evolution from a 1∕f noise at β=1 Gaussian orthogonal ensemble (GOE) to a 1∕f2 noise at β=0 Gaussian diagonal ensemble (GDE) is heterogeneous with a ∼1∕f2 noise at the finest scales and a ∼1∕f noise at the coarsest ones.