This page follows “Thread Design for Wooden Screws”. The major and minor screw and nut diameters need to be established in order to determine the pitch angles here.
For any particular screw configuration both the tap cutter and the die cutter need to be set at specific angles to cut well. The tap cutter should be angled to match the pitch angle at the minor diameter of the nut. The die cutter should be set between the two pitch angles of the minor and major diameters of the screw so the cutter is ‘aimed’ in the right direction to follow the path of the thread being cut.
A tap cutter set in the body of the die at the correct pitch angle.
The two part die cutter must likewise be mounted in the die at the correct pitch angle.
Below is an example of the method that I use to find these angles. A separate drawing must be made for each screw configuration. To get the best accuracy the drawing should be scaled up. I scaled this drawing up 2-1/2 times to fit on an 8-1/2″ x 11″ sheet of paper.
The pitch for this screw (3 TPI or .333″) is represented by the vertical distance between the two parallel horizontal lines. Using Pi (3.14) and the major and minor diameters the four circumferences are calculated and laid out in vertical lines (A,B,C,D). Angled lines are drawn through where the pitch and circumference lines intersect. These angles are then measured with a protractor. This drawing shows that the recessed face on the top of the tap cutter should be ground at 6 degrees (B) to align the cutting face at that angle when the set screw is tightened against it. This orients the face at 90 degrees to the path of the thread as it is cut from the inside of the nut.
The two piece die cutter needs to be set between the minor and major screw pitch angles; in this case between 6-1/2 degrees (A) and 4-3/4 degrees (C). I would shoot for about 5-3/4 degrees. The main consideration is to leave space behind the cutting edges that is roughly equal on both sides so the sharp edges can slice cleanly without rubbing on the sides of the cut.
Incidentally, this drawing is a good illustration of the fact that a screw is essentially a wedge wrapped around a cylinder. A wedge being one of the most basic tools.