Signal Period of each stage in a ring oscillator
In any oscillator, the period or frequency of the signal at all points in the feedback loop will be identical (unless the oscillator is using some multiplication or division technique to create a higher or lower frequency from a simpler oscillator). The signal may have a completely different phase, polarity, or waveform, but the period will be the same throughout.

In the particular case of a ring oscillator, consider this argument: you could tap the outputs of all three gates, producing three outputs instead of one, and then (ignoring any fanout/loading effects) the circuit would be completely symmetrical and therefore each stage is doing the same thing at different times.

If there were some signal in some design of oscillator that was at a different frequency (that's not in a deliberate simple ratio with it) it would either be completely irrelevant or making the oscillator more unstable (causing frequency modulation). At least, that's true as far as I know; I'd be interested to hear of examples to the contrary.