“High gain” vacuum tubes?

Crudely speaking gain (µ) is due to the geometry of the control grid wires and cathode and anode - the ratio of influence between the anode and grid voltages on the electric fields around the grid determines the µ value. Since grid spacing can be a fraction of a mm and the cathode to grid distance is again a mm or less, even a few microns difference in placement can affect µ, so you have to very tightly control grid wire thickness, pitch of winding of grid wire, and the position of cathode within the grid is crucial. Variation of these parameters within one valve (accidental or deliberate) can give a variable-µ characteristic too.

When moving to tetrodes/pentodes the screen grid allows the anode influence to be reduced markedly, raising the µ but adding another potential source of variability dependent on the efficiency of the screening.

Alas most original valve datasheets are lacking in any variation data such as min and max µ specifications, which is a shame as this data was definitely available to the manufacturers - you also rarely see drift-over-time or expected lifetime specifications, despite these being well known non-idealities of the technology. I think you were expected to measure every valve and do your own gain-matching if appropriate.
 
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Mu should be pretty constant from sample to sample in modern triodes, what is not is Rp and transconductance, the math generally works out so that significant variations in both complement each other and mu doesn't vary much. There are exceptions in the case of high transconductance types where depending on operating point the mu may vary over a significant range. (Not talking deliberately variable mu types here)

For tubes like the 12AX7A I have not seen much variation in mu from sample to sample or manufacturer to manufacturer. External circuit design will have a much larger effect on the gain realized than the specific tube you stick in the socket. (plate resistor and load resistance)