Well, just checking in with my version. I'm using the extra diode, a BC337-25 transistor, a blue LED and 2 toroids of 1mH each, about 30 - 32 turns of #33 wire on each one. I still haven't been able to duplicate tinman's double-peaks but as you can see the collector and base waveforms are pretty similar to Itsu's, but at a higher frequency due to the lower inductance.
As I said above, I've also confirmed that this frequency is the natural ring frequency of the tank formed by L1 and the transistor (with no batteries connected). I looped a couple of turns thru the L1 toroid, connected that to the FG thru a 50 ohm resistor, connected scope across coil-transistor connections, and swept the FG until I got maximum voltage on the scope, then read frequency on the Philips counter.
"The easiest person to fool is yourself" -- Richard Feynman