Sparks produce truly wideband noise; I didn't think it went much into the GHz, but it would seem so!
Still reading, looks pertinent:
http://www.ece.unm.edu/summa/notes/SwN/SwN28.pdfThere will be some effect from the electrode shape (corona priming the gap, self capacitance of the immediate structures -- or really, the characteristic impedance of them as the spark wave propagates over them), but once you have the energy at whatever frequency you want, you just tune it to select a frequency. Spark gap transmitters have always been notoriously noisy, both in repeat rate (except for a few CW methods, which were probably still very hissy, but could at least accommodate intelligible voice) and bandwidth (without careful filtering between generator and antenna, expect bandwidth as a sizable percentage of center frequency -- not to mention harmonics!).
Tim