The mu (μ) of a valve is its amplification factor, sometimes also referred to as 'M factor'.
For various types of triode the μ value could be anything between 2 and 200 but most types have μ values between, say, 5 (eg. PX4) and 100 (eg. 12AX7).
The μ value depends on grid/anode spacing and is remains pretty constant for a given (triode-type) valve for all normal conditions of use.
Tetrodes and pentodes do not really have definable μ values, except when operated as triodes. Attempts to measure μ in pentodes tend to yield very high values (eg. 1000s) which vary widely according to the method of measurement. Nevertheless, occasionally one sees μ values quoted for certain tetrode and pentode types. More often quoted, however, is the inner amplification factor (μ g1-g2) which relates to the grid/screen spacing but not to the anode. Typical values for mu g1-g2 lie between 5 and 10 for the majority of common valve types. As with triode μ values, μ g1-g2 remains pretty constant for a given valve under all conditions of use.
However, the slope of the grid characteristic, is usually called the mutual conductance gm.
Unlike μ, which is a ratio and therefore a pure number whose value is independent of the system of units in which it is measured, gm is a conductance so its fundamental unit in the SI system is the inverse of resistance, ie. 1/ohms, sometimes called MHOS. Since the internal resistances of valves typically run into 100s or 1000s of ohms, the gm values are numerically small and are commonly expressed in micromhos or millimhos, where 1 millimho = 1 ma/v.
Whereas μ (or μ g1-g2) values are sensibly independent of valve operating conditions, the gm value of a (good) valve is approximately proportional to the square root of the cathode current at which the value is measured. Thus, gm is not constant for a particular type of valve and is meaningful only when quoted in association with the operating conditions under which it is measured. Indeed, the whole principle of AVC (automatic volume control) is based on deliberately altering valve gm values over ranges of 100 : 1 or more merely by modest automatic variation of negative grid bias.
Just to add to the confusion, valves designed for AVC are commonly referred to as vari-mu types, not vari-slope or vari-gm. More confusion.
Because gm is regarded as a measure of valve 'goodness', manufacturers tend to quote maximum values of gm achieved when measured under high-current test conditions. In normal usage the effective gm value will be rather less and the valve type with the highest quoted gm may not in practice be the best.
For most small amplifying valves operating at economical levels of cathode current, a gm value of around 1 ma/v is plenty. Loudspeaker valves, operating at much higher current and handling large peaks, may require gm values up to 10 ma/v but such valves have big cathodes and their high operating currents boost the gm value anyway, so you do not have to be very clever to achieve 10 ma/v.
However, the advent of television (and also radar) in the late 1930s created a requirement for very high gm values at relatively low cathode currents. The requirement is due to the very wide instantaneous bandwidth of radar and TV signals. Receivers therefore need video amplifiers giving at least as much gain as a sensitive radio receiver but covering up to 1000 times the signal bandwidth (MHz instead of kHz).
Amplifier gain is not the main problem. Indeed, a valve oscillator is essentially just a valve amplifier in which the gain has been increased to infinity (finite signal output for zero signal input) by using a small amount of positive feedback.
A much more serious limitation is gain-bandwidth product, which needs to be very high indeed if you need a gain of 10,000 (40 dB) over a 1 MHz instantaneous bandwidth. Once this problem was realised it did not take the theoreticians long to work out that (other things being equal) the gain-bandwidth product achievable in a valve amplifier was directly proportional to the gm value of the valves used. One solution might be to build video amplifiers using loudspeaker valves at high currents in order to boost gm but high current unfortunately meant unacceptably high levels of internal valve noise (hiss) which tended to drown the weak signal and fill the TV or radar screen with 'snow'. There was thus an urgent need for high-gain, high-bandwidth valves giving high gm values but operating at low cathode current in order to minimise hiss.
Other technical considerations meant that pentodes were superior to screen-grid tetrode types, so the later 1930s saw a race to develop low-noise, high-gm pentodes, initially for TV but, as the War Clouds gathered, government funds became available to improve TV types to meet the even greater challenges of radar. The Mazda ACS2/Pen was a fairly early product of this race. It was a great improvement on its screen-grid predecessor (AC/S2) but was in turn soon overtaken by classic video pentodes such as the EF50 which not only had higher gm and better gain-bandwidth product but was more compact, easier to make, and less power-hungry.
Later devices such as the EF80 and EF184 followed the pattern set by the landmark EF50. The all glass valve superseeded all earlier types for high gain-bandwidth devices.