I think what you are planning is to put a pulley on the spindle or preferably on the first shaft after the tumbler reverse, and another pulley on the input to the gearbox. the finest feed on the BH600 is about 4.7 thou per rev.
To work out the relative sizes of the pulleys you first need to work out the ratio of the gearing that normally connects spindle to input shaft. Luckily on the BH600 I think that is 1:1. Therefore you can easily reduce it by a 1:2 or 1:3 ratio.
Pulleys provide a ratio in proportion to their relative diameters but for this you would probably want to go for toothed belts which will be just as quiet as smooth belts so it is just a case of selecting appropriate tooth counts.

toothed 'timing' pulleys

My lathe is a Engineers ToolRoom BL12 /24, which other than having metric leadscrews, dual dials, and different paintwork, is virtually the same as the BH600.

To produce a finer feed rate, I made a 80T gear to replace the 40T on the input to the Norton box.

From memory, the gears are 1.5 Mod.

This then means that the closing arrangements for the Gear Cover have to be modified.

On my lathe, sadly, to remove the closure fitment from the cover I had to destroy the composite knob.

(Fortunately, a suitable replacement was available,) A new tapping had to made into the Norton gearbox casting, and a fresh hole drilled in the Gear Cover, to match. But it works. With the gear train set to 40T : 127/120T : 80T, and the gearbox set to to E 8, my calculations suggest a feed rate of 0.00225"/rev.

The tumbler gears were noisy on mine, so eventually, they were replaced by nylon ones. If memory is correct, the two cost £41 from Davall Gears!

Obviously, if the gears are meshed too tightly, they will be noisy, not to mention being more prone to wear. Backlash is set by trapping a piece of paper in each mesh, before tightening the fixings. This should give about 0.003" of backlash, after the paper has been removed.

The big compound gear is, I suspect, the source of the ringing. The holes in it SHOULD reduce sound radiation, but I wonder if filling the holes with lead would be a means of deadening the sound further? Because this would increase the inertia, and so possibly reduce sped variation, this may also contribute to noise reduction.

Lubricating the gears with a really thick grease helps to lessen the noise, very slightly. (I was given some specialised gear lubricant which is really gooey!)

Hope that this may be of some help.

Howard

If you are worried about repositioning the stud, it is only a M8 tapping, and a matching hole in the sheet metal cover, large enough to take the fitting.

It might be worth making a knurled metal knob, before hand, just in case, like me, you destroy the composite one.

Howard