> a grounded cathode where a cap and a resistor are going parallel to ground. What does the value of the cap affect?
Well, what does the cap do?
If you hard-ground the cathode (and get grid bias some other way), the gain is the stage is a function of the dynamic cathode resistance. This is just like plate resistance, only smaller (in fact Mu times smaller). The stage gain is total plate load (plate resistor, plate resistance, following stage input) divided by the dynamic cathode resistance. For historical reasons they talk of Conductance instead of resistance, same thing upside down. Transfer Conductance (transconductance) of a typical small triode is, say, 1,250 microMho or 0.00125 Mho. The inverse of that, 1/0.00125, is 800 Ohms.
Now avoid the complication of grid bias by using a cathode resistor to jack-up the cathode, so a simple ground-biased grid gives the desired negative grid-cathode voltage. A typical value for a small triode might be 1,000 ohms. But if you do that, the cathode bias resistor adds to the dynamic cathode resistance. It acts like a larger cathode resistance, a lower Gm, a lower gain.
We avoid that with a capacitor across the cathode bias resistor. And as with many capacitor problems, you start by checking the two extremes: DC (cap does nothing) and high frequency (cap is near-short). Using a 12AU7, grounded cathode or with a cap large enough to act like a short, the gain is about 16. 12AU7 with a cathode resistor and no capacitor (or capacitor too small for the frequency you are working at) has a gain of about 7. So you get a shelf-curve. Gain is 7 from DC to some low frequency, rises to and stays at 16 above some high frequency.
At a glance, the impedance that the capacitor has to fight to give good bypassing is about 800-1,000 ohms, so we know that it has to be at least 10uFd for semi-hi-fi response.
Actually you should size for the Parallel combination of tube cathode resistance and bias resistance, which is more like 440 ohms. 22uFd will bypass down to 17Hz. The response will not be a full 3dB down at that point: the shelf-curve is usually only 4dB to 10dB high. Still, 22uFd would be a good low-price bypass for a small triode in a simple system. In many-stage systems you can't have a lot of -1dB at ~20Hz droops: they add-up to a many-dB droop at 40Hz where it hurts. So you may find 47uFd or 100uFd on small triodes. At today's prices, 470uFd is no dent on the budget.
Also such sizes are invariably electrolytic, which distorts, but distorts less if you super-size it. Another reason to use caps much bigger than "just enough for 20Hz".
OTOH, guitar amps often use skimpy caps. They are made cheap, also they want to shed some deep bass to keep the open-back speaker from flapping too much.
The other side is: a cathode resistor without a cap gives a higher overload point. Not a lot more output, but less distortion at the same high output. Local feedback. Some Hi-Fi amps are full of stages without cathode caps. A guitar amp may want the deepest bass string tones a little weak and also less distorted than the midrange tones, so a skimpy cathode cap gives the right tone for that special instrument.
