I for one have NEVER measured a sharpening angle. People ask what angle I sharpen at and I just shrug. I freehand and eyeball and I am perfectly happy with how my knives perform. I am not searching after the Holy Grail of Sharpness and stropping to 1/64th of a micron. There is no prize for finding the lowest possible angle for each and every knife or the highest polish for the edge bevel. I sharpen to have a nice edge and touch up if it's feeling like it needs it based on how it performs while cutting food. I sharpen the knives and try thinner angles by eye if I feel it needs to be thinner; if I get chipping, I go back up a bit and try it again. Most of the times, I eyeball it and am happy with how it cuts. Not once have I thought "oh, this knife cuts well, but I want to drop down exactly 2 degrees per side more and see how it does". I don't have hours to spend sharpening knives at various angles and then testing them out to see what the difference is and logging it in a journal. If that's your idea of fun, so be it and enjoy it. For most people here, we are looking for a sharp edge that cuts well. I would rather have a more durable edge than one that it right on the cusp. I would rather lose a hair of performance than have a blade so thin that it sticks into the cutting board every damn time it hits the board and now I gotta worry about not torquing it and chipping it, or worry about slowing down so it hopefully won't stick. I don't care if my bevels are a mirror finish or not or if the blade has the absolute lowest angle that it can take. If it cuts well, I am happy. People insist on mirror polishing the bevels with nanocloth and compounds and have to know the exact angle they are using to be happy. I don't shave with my edges and I don't mind if it's not polished to 1/64 of a micron. If it cuts well, I am happy.
People get very agitated over sharpening/angles/grits/etc and after sharpening and being involved with knives for over 15 years, guess what? It doesn't really matter if it's a degree or two or three off. If the knife cuts well, great. If not, change something and try it out. By the time you are done with the trial and error part of sharpening and have everything tuned up and the exact recipe for the perfect edge, the geometry behind the edge will be changed and you get to start all over. There is no prize for sharpening a knife to it's 100% fullest potential except personal satisfaction, with a lot of frusturation along the way for something that is a moot point honestly. If a knife has an edge a few degrees off it's optimum, most people will not be able to tell. You won't ever know until you go too low and get chipping or edge failure. Then you go back up slightly, thinking it was the edge that failed and not some other variable that you may not truly have control over. And when you get to that perfect point, how do you tell?? Can you really HONESTLY feel the difference between a 10 and 11 degree edge angle on your knife cutting thru food???????? Can you pick up a knife and cut something and say, "oh, thats a nice 11 degree angle on there, cuts better than it would if it was at 12 degrees!" Some people chase the rabbit down the hole and get caught up in the minutia of sharpening and are never happy. Me, I sharpen away until it's sharp and use it and then sharpen it again when it needs it. I enjoy using the knife, not picking apart my sharpening routine.
Hardness and angle plays a part, as does the board you are cutting on, items being cut and the steel and HT itself. A steel with larger carbides or coarser grain may not do as well with a lower angler than a steel with smaller carbides. Softer steel may not support as fine/thin of an edge because the steel is softer and rolls over. Each steel has an optimum heat treating, but how do you know your steel is at that level???? Optimum for one knife and it's purpose is different than another knife/purpose. A camping knife is going to have different criteria than a chefs knife, petty knife or Yanagiba. Do you get several of the same knife and have them tested to see what they are rockwell wise? Most manufacturers have a range of Rockwell Hardnesses and MANY Japanese manufacturers do not test; they use age old recipes and methods; many do it by eye, which can lead to variances. Steel batches can vary, as do the heat treatments and rockwell hardness. Oven have hot and cold spots, so following a HT recipe spot on may still have variables. Or maybe the steel didn't get fully heated in a spot or quenched too quickly in a spot or got overheated in a spot and now there is an area that is different from the rest of the blade? Basing everything off an advertised Rockwell Hardness is pretty sketchy...how do you know it's actually what they say it is??? A range of 1-2 points or more is common; how do you know exactly where yours falls?? I had a Fujiwara Terayasu Nakiri that was advertised to be between 63 and 65 rockwell. Did not feel anywhere near that on the sharpening stones compared to the Goko White #1, felt pretty similar. Others have had the FT and they definitely felt harder on the stones than mine did. I sharpened a Gengetsu and it felt as hard or harder than my Kono Honyaki on the same stones.
There are so many variables to sharpening and people have been discussing/arguing about them for many years. Softer steel generally won't take as low of an edge angle, harder steels usually have the ability to take lower edge angles, even if it is not practical to do so. The grind behind the edge bevel plays a HUGE role in how a knife performs and since many are different, it is very difficult to compare different blades and edge angles because the geometry on the sides of the knife can affect performance. If the knife is thin enough behind the edge, the edge bevel is pretty tiny and won't really matter all that much. If it's really thick behind the edge or very convexed, a low edge angle may hit the blade face. Will the geometry of a small sliver of an edge bevel really matter in the larger scheme of things compared to 2" of steel behind that sliver?
With a steel with larger carbides/grain and softer, the convexed edge lets there be a little more metal behind the edge, so carbide fallout/tearout will be less and the steel will be less prone to issues. I have tried sharpening a German style knife with water stones. Low edge angles and high grits don't work well, but a nicely convexed edge off a belt sander gives it a great edge that performs well, even though it is thicker. So for some knives, yes, the convex edge does improve edge retention. Most people who freehand get slightly convexed edge bevels from their natural wobble. People who use the guides systems can also get slightly convex edges if they do not flatten their stones frequently, too.
Maximum performance in a cutting task is a theoretical goal since there are SO many more variables. Unless you are cutting a particular medium that is 100% consistent all the way through and every time you cut it, it means nothing. Those that use their knives to cut food know that the foods are inconsistent and we look for something that does well in all circumstances, not just one. Hardness is one part of the equation with edge angle, but grain size/structure also comes in to play, too. Hard with large carbides does not always work well with low angles!
Tall, Dark and Swarfy; what you postulate is an interesting idea. It seems that generally, softer steels don't do as well with ultra low edge angles and harder steels have the potential to do better, based on a number of factors. 10-15 degrees seems to be the Japanese range for 60-65 rockwell knives and the softer German stuff at 52-56 rockwell seems to fair well at 15-22 degrees edge angle. It may be an interesting starting point to play around with, but there are more variables to consider than just hardness and angles themselves.
For a math equation for sharpening, stones + knife should = sharp knife. There are just so many other variables to come up with a hard/fast rule, but what you proposed is an interesting starting point and people can tweak it from there.
Jeff, I think a Man Soda is a beer