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At the risk of asking a stupid question, what's the difference in technique between the two chisel types?
Not a stupid question at all! I had no idea myself until and old joiner patiently showed me how!
You cut out the whole mortice with a mortice chisel of same width, a big mallet, no other tools.
Mark up the sides with mortice gauge, top and bottom if it's through, mark ends with a square
You hold it vertical at all times, no levering at all except a little pull back to loosen it from each cut, and also to clear the bottom of a blind mortice.
You bash it in anywhere but not at the ends, these you do last.
Loosen it out of first shallow cut and then repeat by paring off the vertical face of the cut and going deeper, repeat etc.
When close to the end turn around and work your way back, cutting through the chippings you've already made and going deeper with each cut, paring down the face of the previous one.
It's a fast and furious process and good for body building!
When halfway through turn it over and work from the other side, if it's a through mortice.
Otherwise work to a depth mark, felt tip mark or sticky tape.
Finish by cleaning the faces at the ends of the mortice with a last paring cut.
Clean out the bottom of a blind mortice by levering against the side with the curve of the bevel against it, and lever out any remaining chippings. Most of them will have just fallen out as you go.
Sharpen a little and often through the process.
Do it flat on a bench top un-clamped and not in a vice - there's a lot of heavy bashing and the clamps will mark the sides.
Better still do it between your legs sitting astride the workpiece over two saw stools, or even a purpose made morticing stool.
It's much the same process as you would apply with a hand powered morticing machine, just a series of powerful vertical paring cuts, close together, down the face of the previous one.