In a dielectric-barrier discharge between diametrically opposite sides of a narrow tube, discharge filaments stabilize at regular intervals along the tube's length. Three types of periodic patterns are observed, as is a disordered state in which filaments fire at apparently random positions and times. Time-resolved current measurements indicate that for each spatial pattern, a particular number of discharge stages occur during the voltage half-cycle. A preliminary model of the pattern-formation dynamics is described, motivating further work on time-resolved imaging and investigations of surface charge distributions.