Clock Pivot Polishing Using

 Jacot Drum in WW Lathe
 

  
Jacot Drum for working on pivots from 0.35 mm to 1.45 mm diameter.

 
Jacot Drum mounted in the same runner that is used as a carrier for a centring plate when drilling an arbour. 

The same centring runner is used to properly orient Jacot Drum 
  
Jacot Drum mounted in the lathe tailstock with carrier runner and wheel supported in a female centre with a driver plate screwed on a lathe nose and a lathe dog used to drive the wheel. 

With a lathe motor set to a slow speed all is ready to apply burnisher. 
 
To get overall picture completed here is a Pivot File - Burnisher Combination that I use on clock pivots.

A very high quality one with a matching price to go with it.

For filing I do not use any lubricant and for burnishing I use kerosene (the smelly staff) in fairly liberal quantities as it serves to wash the swarf away as well as help burnishing process. 
 

Here is a close up image of a Jacot Drum face. 

If You observe closely You can see that the openings are not half of the diameter of the pivot.

The openings are exposing only 0.05 mm (0.002") of the pivot so that when You burnish the pivot You only can work on the exposed section. 

To illustrate this I have placed a red dot on the top of the image so that it can be readily observed.

  Detalied view of Jacot Drum geometry
 
  Jacot Drum is made of steel and is heat treated to be as hard as glass so that if You do touch it with a file it just runs over it with almost no friction. It is precision ground to enable secure use of a pivot file and burnisher so that the pivot when finished has sharp edges with planes at right angles to each other and most important that it has no taper.

If You use a pivot file then You can not get a pivot tapered as only a tiny section is exposed and You do have a guide for filing in  area ground flat on top of the opening.

Furthermore I believe that the Jacot drums, which expose half of the pivot circumference can be used only as a support for burnishing as they provide an easy means to get pivot tapered.  

The cost of these thingies is rather on the  prohibitive side as it runs in the neighbourhood of some US$ 1200 and more for a Jacot Drum, one centring plate, one Carrier runner, two drill holding runners and one centring runner. Nevertheless they are still substantially lower in price then Rollimats, providing that You already have a WW style lathe.