I think every techie nerd experimentalist has fantasized about making a Tesla coil. So going down the rabbit hole I looked it up on Wikipedia. The first question I have is how do you determine the self resonance of a big coil that is the Tesla coil? Perhaps you can use it as part of an oscillator to see where it wants to oscillate?

 

Drive it with an RF generator and look for the max in voltage across it.

 

I would expect the resonant freq would likely be low given this is likely to be a really huge coil. White noise and a FFT oscilloscope might be able to look for the notches there gonna be more elegant solutions.

For example I don't remember what kind of oscillator this is but I have seen it many times and it works:

​

 

I think every techie nerd experimentalist has fantasized about making a Tesla coil.

Not this one. I fantasized about making usable things, like radio receivers and transmitters with clean, stable RF signal generation. Building Tesla coils are pretty much a modern fantasy.
There are online calculators that estimate the distributed reactance of a typical coil and give a rough Resonant Frequency.

https://www.teslascientific.com/products/coil-resonant-frequency-calculator/

http://www.hvtesla.com/tuning.html

You will most likely find that there are several other points where the voltage peaks (harmonics), but there will always be one dominant peak. This will be located closest to the expected frequency, whilst the other harmonics will be either side. Having a good idea of the expected frequency before you use the oscilloscope method, can be essential if you get a lot of harmonics, as otherwise you could easily end up tuning to the wrong frequency.

 

The common oscillator circuit is used in your Tesla Coil and in my FM transmitter:

 

Actually it is my lame brained idea to measure self resonance once you have that is well on the way to getting there.

 

I am trying to figure out what kind of oscillator this is, so far the closest match is a Dynatron oscillator, any ideas?

It is not a colpitts nor is it a clapp oscillator.