All conditions for synchronic harmonic oscillations are presented. These are for discrete and continuous undamped linear systems in free vibration. A vibrations literature review indicates that only one or the basic condition for synchronous harmonic free vibrations is known. This is, initial conditions as an eingenvector (or eigenfunction) for the initial displacements, and no generalized velocities for t=0. Moreover, although assumed or expected for continuous systems, this result appears solely or mainly for discrete systems in the literature
