I think Ken may be suggesting the WHDT is an American joint. Which makes sense, being, it is used mostly in the dropped tie configuration, with common rafters. Ken, do you see that set up on your side of the pond? I have a book, Recording Timber-Framed Buildings: An Illustrated Glossary By N.W. Alcock, M. W. Barley,.... It shows a half dove tail with a wedge. No detail so I am guessing but the wedge looks like it has no taper. This book is a recording of English joints. It is on page F-28. So there may be examples on you side of the pond, Ken just rare.

On one of my own first buildings I used the WHDT on the foot of the king posts (6 trusses 30') In the recient past I noticed one of the wedges was not present! I can look right up through the wedge slot, the joint shows no gaps to speak of. There is no floor loads either. I should jam in a new wedge. These joints were cut incorrectly. My wedge side had no taper, and some of the wedges went in from the top, some bottom, it is a bottom wedge which is missing. In a king post set up, I put the wedge in from the top now. This is the same as going in from the inside as with a dropped tie. I also like the ratios Jack Sobon talks about 1:5. I would think a wedge with even less taper would be better yet. So, on a 10" post the wedge would be something like 2"-2-1/2" in 10". How about putting a nail in the wedge so it can't fall out? This would have to be pulled to tighten later.

Another situation I have come across is where the king post was square ruled into the top of the tie by about 1", this housing prevented the half dove tail from fully contacting its seat before being completely assembled. The joint needed some futher adjustments. I had to extend the housing/reduction on the tie to allow the king to seat onto and than slide over to the half dove tail. A haunched shoulder eliminates this problem on the dropped tie. I now only use the top of the tie as reference in a king post truss. For futher info on that, I used the bottom as ref. so I wouldn't have to reduce at the posts, Trial and error. Tim