First look up Barkhausen criteria, then read the following. For ring oscillators gain will be greater than unity for only those waves whose period matches the gate delay. Only one such wave exists since the gate delay is a fixed number.

Okay, so I was wrong. You can get multiple harmonics if you use a long enough chain of inverters. I will simplify the paper mentioned in a sibling comment. In a long chain at any given point of time certain (variable number of) pairs of inverters drop in and out of the circuit changing the total propagation delay, giving rise to multiple harmonics. You'll have to read the paper for details.