"The frequency of these oscillations seems to be around 5 Hz, drifting up, and seems to be a fundamental parameter of the flame since it appears to be relatively constant in all observations."
It appears that some (optical) flame detectors work by obtaining the frequency of this flicker: 
"This flame scanner also monitors the rate of combustion by analysing the flicker frequency, or the fingerprint, of the flame... Since the flame is always burning back to the fuel source, the flame is always in motion. This motion allows the intensity of the flame to vary across a flame flicker frequency spectrum."
From that link, it appears that the "flicker frequency" would be related to the exact fuel source and possible fuel/air mixture ratio, as the "flame is always burning back to the fuel source".
Some other information about flame frequency-flicker: 
"Under normal gravity conditions, the flames have a well defined oscillation frequency which is inversely proportional to the square root of the burner diameter, D, and to a good approximation can be written as f » 1.5/D½, with D given in meters."
So, it would indeed seem that the flicker frequency is related to the particulars of the candle, its wick, etc.
Very interesting article!