Dennis sent me a message summarising how the screen diodes work to improve the sound:
In a normal Triode, Tetrode or Pentode amplifying valve it is well established that variation in DC plate voltage does not cause a change in through current.

The Control Grid (Grid #1) is most usually used to control electron/current flow through the device.

In the case of a Tetrode or Pentode the function of the Screen Grid is to regulate DC current flow through the device to a predetermined ratio between change in Control Grid voltage to change in plate Current for all input signal levels. This is effected by installing the Screen Grid between the Control Grid and Plate but considerably closer to the Cathode than the Plate. Triodes do not have a Screen Grid.

This physical relationship between the elements in the valve enables the Screen Grid to increase the attraction of electrons to it, thereby increasing electron flow, and to shield the Control Grid from electrons reflected back from the Plate.

Thus the Cathode and Control Grid, not being able to discern the difference between Screen Grid and Plate, always see a "Triode" - but once electrons reach the Screen Grid, because it is made of a wire coil with gaps between the small diameter individual wires (similar to a picket fence) most of the electrons just pass through to continue on to the Plate for collection.

Managing the interaction between the Control Grid (Grid #1) and the Screen Grid (Grid #2) results in improved efficiency compared to a Triode - eg more gain, more output voltage and more power output for any given DC supply voltage.

But in so doing the Screen Grid consumes power, which is lost as heat but also, and most importantly, diverts some of the available AC power output away from the load.

By adding a correctly polarised diode into the line feed to the Screen Grid in the manner described, the DC supply relationship to the power valve circuit remains unchanged but the AC current in the valve is prevented from accessing the Screen Grid as an output terminal, hereby offering a shorter current path. Consequently all the available current is directed to the Plate, as in a Triode.

Since the AC current through the device is the source of output power, the result is improved performance.

Thus it might be said that the addition of the diode causes the valve to behave similar (but not identical) to a Triode but with the increased efficiency of a Tetrode or Pentode.

The sound produced is more similar to the bright, clean sound of Tetrode/Pentode than Triode.

In the case of connecting a Tetrode or Pentode as a Triode, by strapping the Screen Grid to the Plate, the valve behaves as a Triode. Inserting the Diode as described has a similar effect on the electron/current flow but the sound will still remain the "triode" sound - albeit it cleaner and brighter.