This is an interesting thread. The pyramid has been a mystery to me for a long time. I think I understand it better after Lynn's explanation.
From Nenand's post you'll see that there is a science of sharpening. I scanned that thesis and it looks excellent. It may answer amny of my questions. It's my style to try to understand things, so when I got into honing I started researching and looking for an understanding of things, as well as experimenting. I still have a lot to learn, but I have learned some principles, and I'll discuss them below. When something is my theory or a guess, I'll say so.
There's a lot of black magic discussed here. from a scientific point of view Jeff comes closest and I agree with him on just about everything. He has a strong curiosity and a obviously a good basic understanding.
First of all, let's talk about striations or scratch lines. They're caused by the grits in an abrasive, which produce furrows that are usually at an agle to the edge. Since hey're on the bevel of the edge, where they meet the edge they produce a series of teeth (microserrations), which get smaller with finer grit. You always have them. A basic principal is that a sharper edge cuts with less effort. So, theoretically the edge should improve as you use finer abrasive, because the edge gets sharper. In other words, you want the finest edge you can get without making it so fine that it doesn't hold up and you need to hone all the time. Sharpness vs. durability is the traditional tradeoff in sharpening. All the usual principles DO aopply, but we have the luxury of being able to create very sharp edges because they will stand up to a beard.
It's been suggested that the teeth produced by a 4K hone are necessary for best for shaving. It seems to be based on the barber's manual discussing teeth in the edge. A barber's finishing hone is around 10K, so the teeth we're talking about are even finer than those produced by an 8K hone. Lynn tells us he finishes with a coticule, which he estimates at 12K. So, the edges he gets are even finer than traditional barber's edges. Frankly, I don't understand how a blade with coarse teeth could shave smoother than one with fine teeth. There's a lso the issue of comfort. The striations on the bevel will come int contact with the skin when you shave. Larger scratches make a rougher surface with more friction.
It's been suggested that the best edge is made up of large teeth with small ones between them. Assuming you hone with the finer hone without eliminating the 4K scratches entirley, you're simply dividing the ridge between 4K scratches into multiple ridges (say 3 if your'e usig a 12K stone). That gives you groups of 12K teeth separated by 4K scratches. Is that better than all 12K teeth?
That brings me to the pyramid. It's a good approach when you're first starting out because you have no idea what's going on, and if you can follow these steps and keep testing you'll eventually get where you want to be. I agree with Jeff that reverting to a coarser grit has to set you back, even if it's a little. When I hone, I don't go back to a coarser grit. I'll start and edge with a 1200 withe spine edges taped, if I need to remove a lot of material or create a new edge. Otherwise I'll start on the 4K. I come off the 4K when I pass the thumbnail test or when the edge looks nice an even. The 8k finishes the edge to look even under a microscope and to pass the thumb test. When I get no more improvement with the 8K I move on to the coticule. I stop the coticule when I get no more edge improvement. I don't know how many strokes I do on each abrasive. It's different with each razor, anyway.