Designing a strong mortise and tenon joint follows well-established rules of proportion. Here’s the manual process:

1. Rule of Thirds for Thickness: The tenon's thickness should be approximately one-third the thickness of the stock it is cut from. This ensures the tenon is strong, and the mortise walls are thick enough to resist splitting.
2. Determine Tenon Length: The length of the tenon depends on the width of the mortised piece. A good rule is that the tenon length should be about five times its thickness. For blind mortises, leave at least 1/4 inch of material at the bottom of the mortise.
3. Establish Tenon Width: The tenon's width should not exceed five times its thickness to avoid a weak tenon. The tenon should also not remove more than half the width of the tenoned piece to maintain its structural integrity.
4. Define Shoulders: The shoulders are the surfaces that bear against the mortised piece. They should be cut cleanly and square to ensure a tight, gap-free joint.

• Mortise: A recess or hole (typically rectangular) cut into a piece of wood to receive a tenon.
• Tenon: The projecting tongue on the end of a piece of wood, cut to fit into a corresponding mortise.
• Shoulder: The part of the tenoned piece that abuts the face of the mortised piece, providing a stopping point and adding rigidity.
• Cheeks: The two wide faces of a tenon.
• Through Mortise: A mortise that passes completely through the material.
• Blind Mortise: A mortise that does not pass all the way through the material, concealing the end of the tenon.

"A common mistake is to make the tenon a perfect, friction-fit into the mortise. You actually want a tiny bit of clearance, just a few thousandths of an inch, to allow for glue. A joint that is too tight will squeegee all the glue out when you assemble it, resulting in a weak, starved glue line. The strength comes from the mechanical lock and a well-distributed layer of glue. Also, slightly beveling the end of the tenon (a chamfer) makes it much easier to guide into the mortise without catching." - Fine Woodworking Magazine Editor

Table Leg to Apron: A classic example. The apron (the horizontal piece under the tabletop) has a tenon on the end that fits into a mortise on the table leg. This joint has to resist the racking forces on the table.
If the leg is 1.5" thick, the tenon on the 3/4" thick apron would be `0.75" / 3 = 0.25"` thick.

Frame and Panel Door: The rails (horizontal pieces) of a cabinet door have tenons that fit into mortises cut into the stiles (vertical pieces).
For a 1.5" wide stile, a 1" wide tenon would be appropriate, leaving sufficient material on either side of the mortise for strength.

• Making the Tenon Too Thin: A tenon less than 1/4 the stock thickness can be weak and prone to breaking.
• Misaligned Shoulders: If the tenon shoulders are not cut perfectly square, the joint will not close properly, resulting in gaps.
• Mortise Walls Too Thin: If the tenon is too thick relative to the mortise stock, the walls of the mortise can easily split under stress.
• Loose Fit: A sloppy fit provides no mechanical strength and relies entirely on glue, which is not ideal for a structural joint.

• Furniture Construction: The primary joint for building tables, chairs, beds, and cabinet frames.
• Doors and Windows: Used to construct the frames (stiles and rails) of traditional wood doors and window sashes.
• Timber Framing: Large-scale mortise and tenon joints are the foundation of traditional post-and-beam construction.
• Pergolas and Arbors: Joining posts and beams in outdoor structures for immense strength and durability.