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Should the 56K plate and cathode resistors be changed to a different value to better suit a 12AU7 over a 12AX7Back up. What are you driving??
Most self-biased output tubes may use a 500K grid resistor.
Most fix-biased output tubes should use a 100K grid resistor (we often cheat).
To the cathodyne, the two output stage grid resistors are in series for audio signals. Therefore a cathodyne "sees" 1Meg for self-bias, 200K (or a cheat higher) for fixed-bias.
We also know the maximum drive voltage required. Generally a Peak per-grid signal equal to grid-cathode bias is needed for "full" power. And for the usual power tubes it is related to the power tube screen voltage. We may need 25V peak at lower voltage (Princeton) or 50V at higher voltage (Bassman). Again, the two loads are in series, so we need twice this: 50V or 100V.
As Buttery says, in guitar we want "more!'. However we probably do not want "twice too much". that tends to give grid-block. Anyway for 6V6 6L6 6550 EL34 and friends it is hard to make "twice too much" (they were designed that way).
So "significantly more than" 50V-100V peak.
Those are peak voltages. We will want the "RMS" voltage. Multiply by 0.7. 35Vrms to 70Vrms.
In the last century, pocket-protected engineers compiled data for "best" operation for various tubes. Their jobs depended on tubes.
I've collected the 12AU7 and 12AX7 amplifier tables.

First pick a supply voltage. In most gitar amps you can have at least 300V, so use the "Ebb=300V" column. We already know the load, their "Rs", is either 1Meg or 200K. Look for Rs values nearest these numbers and follow over to the Ebb=300V column.
12AU7 driving 2*500K. 6th line down shows Rs=1.0Meg.
Look at the 300V "Eo" column. This is the RMS voltage possible. 54Vrms. This exceeds the 35Vrms needed for lower-volt output stages, OK.
Going back to the left, using Rs=1.0Meg, "Rp" is given as 0.51Meg or 510K. For "Split-Load" Cathodyne, we split this: two 255K resistors. (This is just ballpark: use two 220K or two 270K.)
The table gives "Rk" cathode _bias_ resistor, it should be 23,000 or 23K to get these results.
The "Gain" is not directly useful... Cathodyne voltage gain is always a little less than "1". Actually if amplifier gain is 12 then Cathodyne gain is about 11/12 or 0.9.
Do it again with 12AX7. 1Meg total load, best is Eo=46; however for "0.51meg" load there is a condition where Eo is 47. This may be round-off slop, or the "0.51Meg" condition may be a teensy bit better. Let's use it. Rp should be 0.24Meg or 240K for one load, so two 120K for split-load. Cathode bias should be 1800. Gain is "64", but really 63/64 or 0.98.
The 12AU7 can make 54Vrms, the 12AX7 can make a bit more than 47Vrms. The AU seems "better", but if we need "significantly more than 35Vrms" then either one is fine. If all our other tubes are AX, it may be best to stick with it.
We have design for 47Vrms-54Vrms in 1Meg, but hi-voltage power stages may need as much as 70Vrms. How can we get there?
The chart shows Ebb 90V, 190V, and 300V. Scanning across, "Eo" increases sorta in proportion to Ebb: 90V gives 19V and 300V gives 54V. We may assume that a higher Ebb will give higher Eo in rough proportion. Outputs that need 70Vrms total grid drive will usually be eating 450V. With a little filtering, we can ask for 400V to feed the cathodyne. Assume Eo rises like 400/300 or a factor of 1.33. 50Vrms at 300V would be 66Vrms with 400V supply. Close. Probably works. Must be bench-raced to know.
All of that is for two 500K loads such as used on self-bias power tubes.
Now try two 100K grid loads such as for fix-biased power tubes.
Total grid-grid load is 200K. We look for "Rs=200K" and take Rs=0.24Meg as the nearest suggestion. 12AU7 does best with Rp=100K at 41Vrms; for split-load this is two 50K resistors. 12AX7 does best with Rp=100K at 40Vrms; also two 50K resistors. While you should use different cathode-bias (12AU7=4K4, 12AX7=900), otherwise they are equivalent.
The 40-41Vrms covers lower-volted power stages but falls shy for the higher-volt power stages, even bumped to 400V. Since driver supply voltage can usually be near power stage voltage, it may seem we should be able to hit our mark. However HIGH-voltage power stages must usually run high negative bias to keep from frying at idle. The driver must swing the full bias voltage. So the high-voltage power stages can be very tough to slam with a cathodyne. We often have to re-think as we approach 5F6A turf.