We have a deep well using a jet pump with a double drop, using a large older metal air tank. I recently noticed the pump would cycle and run for about 12 seconds every 9 minutes or so. It would do this while no water was being used in the house, and there are no leaks anywhere.

The tank had apparently lost some of its air supply and needed to be refilled. This can happen over time, or during a power outage when water is used and the pump cannot run. Simply have your take re-charged and everything should work normally again.