This could be a really cool project! Kind of makes me think of a Bassman or early Marshall in terms of the gain structure. I definitely think this is possible, though your preferences in terms of coupling the cathode follower will dictate how difficult this might be to implement. If you're looking just to lower the output impedance to drive the tone stack and aren't looking for some DC cathode follower compression, an AC coupled cathode follower would be fine and the coupling caps coming to and from the mixing control make this a lot easier. Something like the Echoplex mix control comes to mind for the control, and you actually wouldn't really need a tube cathode follower if the goal is just to lower output impedance, since a MOSFET source follower would lower output impedance while being more linear and more rugged/reliable.
DC coupling makes this a lot more complicated since you'd more than likely be changing the bias point of the cathode follower by changing the mix of pentode vs triode. My suggestions if you want have a DC cathode follower summing the signals would be to implement Merlin's "morph" control or to have some type of active mixer which in turn could be DC coupled to a cathode follower. This page from Merlin's site shows the general topology for the morph control:
https://www.valvewizard.co.uk/ef86pcb.html
There are a number threads here, on the Hoffman amps forum, and on the Telecaster forum showing different values people have tried with 5879's and EF86's, so you wouldn't have to start from scratch figuring out values that work well to make the circuit work.
As far as active mixers go, I haven't personally tried or seen a report describing successfully using this circuit, but I came across this schematic on the Hoffman amps forum and thought it was interesting:
Zero gain mixer.jpg
One option that I think would probably be easiest if you wanted to have it perform similarly to a DC cathode follower and didn't want to use a morph control would be to AC couple the outputs of the pentode and triode stages and use a resistive mixer (either an Echoplex-style mixer or just a dual-ganged pot wired as two volume controls but with lugs 1 and 3 swapped for one gang such that turning one direction makes one side louder while making the other side quieter; the second option would still require mixing resistors off of each gang's wiper). From there you could send the mixed signal into the input of an AC cathode follower whose cathode is DC coupled to the grid of a DC cathode follower. That would provide a lot of current to drive the grid of the DC cathode follower with an added benefit of providing a softer startup condition for the DC cathode follower. Another option would be to use an anode follower, but at the expense of more noise thanks to the very large resistors required.
I don't think there's a clean solution that lets you "have your cake and eat it" strictly coupling the plates of the pentode and triode gain stages to the a DC cathode follower's grid. The two most common pentodes I know of for guitar amps -- EF86 and 5879 -- will probably not idle with similar voltage drop magnitudes across their plate resistors when compared against a 12AX7. Most schematics I've seen describe voltage drops considerably greater across the plate resistors in both cases as compared with a 12AX7 with the same value plate resistor and similar voltage. Additionally, pentodes generally have much higher output impedance than triodes, connecting connecting their plates as a kind of common plate load mixer would almost certainly involve a very heavy load for the pentode. A solution to would be to add resistance between it and the triode's plate, amounting to a kind of resistive mixer, but any kind of resistive mixer DC coupled to the cathode follower's grid would limit the amount of driving current available. That would limit the amount of current available to drive a DC cathode follower's grid current, which in this setup kind of defeats the purpose in my opinion since grid current is what causes the asymmetric wave form compression.
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