Evidence of nonuniformity in the rate of seismicity and volcanicity has been sought on a variety of timescales ranging from ~12.4 hours (tidal) to 103-104 years (climatic), but the results are mixed. Here, we propose a simple conceptual model for the influence of periodic processes on the frequency of geophysical ``failure events'' such as earthquakes and volcanic eruptions. In our model a failure event occurs at a ``failure time'' tF = tI + tR which is controlled by an ``initiation event'' at the ``initiation time'' tI and by the ``response time'' of the system tR. We treat each of the initiation time, the response time, and the failure time as random variables. In physical terms, we define the initiation time to be the time at which a ``load function'' exceeds a ``strength function,'' and we imagine that the response time tR corresponds to a physical process such as crack propagation or the movement of magma. Assuming that the magnitude and frequency of the periodic process are known, we calculate the statistical distribution of the initiation times on the assumption that the load and strength functions are otherwise linear in time. This allows the distribution of the failure times to be calculated if the distribution of the response times is known also. The quantitative predictions of this simple theory are compared with some examples of observed periodicity in seismic and volcanic activity at tidal and annual timescales.