Following is an illustration of the output from a rectangular waveform showing how it actually disrupts ringing and harmonics as well as not actually producing even harmonics.
The second primary wave is started out of synch with the original ringing waveform. The third primary wave form is even worse, because it is the opposite polarity of the ringing waveform, thereby canceling any ringing that remains.
Yes, there are more frequencies, but what good are they if the ringing is not continuous? At least according to Rife and Clark.