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Here a a couple more illustrations and examples and precedents for my approach to reproducing specular and non-specular whites.
Adams, in creating his Zone System, also created a new vocabulary for tonality based on the appearance of content in scene and print. His system in practice is a process of finding the "Zone 1 - shadow with first detail" and "Zone 9 - solid white" in the scene, then reproducing those same areas in the print with tonal values slightly above solid black and slightly darker than the white paper base of the print, respectively. The "Zones" are tonal values in the print and scene, not f/stop differences (as some erroneously assume today). The "maximum" black the print paper can produce is reserved for reproducing the area with no detail visible by eye, such as the mouth of a cave. The maximum white on the print, the bare paper, is reserved for specular refections on the Zone 9 solid white objects.
Originally focused on print values Adams did not assign a Zone number for specular highlights, stating in the 1968 edition his book "The Negative" I learned from that the white paper base should be reserved for the specular highlights. Thus in a photo of a white ball the solid surface of the ball on the print wouldn't be reproduced by the white paper base, but by a slightly tone of gray silver, with only the specular reflection the sun created on the ball reproduced with the white of the paper. Later, because of confusion on the part of practitioners of his System, Adams added a "Zone 10 to define the specular white which creates 3D shape in solid "Zone 9" whites.
So on a print "white" is always, out of necessity, reproduced one tone darker that the lighting seen by eye. The only way to create a specular highlight brighter than the paper and more similar to what is actually seen by eye to define 3D shape in a solid white object would be to poke a hole in the print and put a light behind it where you wanted specular highlights. That in essence is what you are doing when you expose images so only the specular Zone 10 whites wind up at 255 on screen and printed with the plain paper base.
If you've ever tried to reproduce a white object like a ball on a white background you'll understand the dilemma and the need to have separate definitions specular and non-specular white tones. In the illustrations below the rim-lit circle outline on the ball and the catchlight at 2 o'clock have 255 values (Zone 10 - specular white). The tone of the ball and background change from (Zone 9 - solid white) to (Zone 8 - shaded white).
The "craft" aspect of photography is the ability produce any of those looks at will to meet the artistic goals of the shot by virtue of understanding both the perceptual dynamics and how to precisely control every step of the reproduction process from capture to final JPG. So while you might like the look of a nuked "hot white" Zone 10 background in some photos there will be situations where a Zone 9 or Zone 8 background tone may be more effective in making the foreground content contrast and seem brighter. If you want to become a well rounded photographic craftsman you'll experiment and learn how to create all the looks above with your lighting.
Here's a photographic example. The rationale for using a "hair light" or accent from behind is to separate the subject from the background, as in the illustration of the balls above. To do that the rim lit parts of the foreground must contrast from the background tone. That requires several decisions about "Zone Placement" of the white objects on the photograph when setting exposure.
The first decision is whether or not detail is desired where the rim lighting hits white objects in the foreground. In ZS parlance whether to exposed the rim lit parts as Zone 10 - specular white or Zone 9 - solid white. Since I find blown highlight areas distracting in my portraits I choose to keep everything the lights hit below clipping, except on reflective objects like the catchlights in the eyes, jewelry, buttons, eye glass frames and other similar objects..
When setting my lights I turn the foreground key and fill and back rim lighting on at a level near that I want to wind up at. That's important because spill from all sources affects the amount and character of the fill in the shadows of the image. In the example below I didn't turn on the background light when setting the foreground because I wanted to see how much, if any, change to the appearance of the foreground lighting occurred due to spilled light from the background when added.
I use a target on a stand with a white towel so I can see the 3D wrap and interplay of the rim / key / fill on the foreground in ways a flat target will not show. As others have pointed out in previous threads flat targets create glare. The other objects besides the towel are not used by me to set exposure and ratio so the fact there is some glare off them is irrelevant for this step in my workflow .
I also don't aim for any specific numerical values on the target when setting my lights. I just adjust my brightest light, in this case the rim lighting, so the areas it hit in the image the camera records are below clipping in the camera's clipping warning in the highlights, and have visible separation of shadow detail in the playback on the camera. Yes, the RAW has more detail than seen by eye in the playback, but often the final product in my workflow is a JPG file not much bigger than the one the camera uses for it's playback. So in terms of my entire workflow the camera clipping warning is a very good predictor of how the highlights will be reproduced at the end of it. I allow for processing induced compression of highlights during the post processing steps at capture.
There's no clipping indicator for the shadows in the camera so more recently I've added a black towel to the exposure target.
As the annotations on that shot state the shadow detail in an image is controlled with fill, both the direct fill source and whatever spill is bouncing of walls and ceiling in the shooting environment. The highlights are controlled via the accent and key lights. For me setting fill is pretty much a no brainer because I want to record detail n the darkest objects in the foreground and usually also in the background when using dark ones. Looking at the image of the black towel I raise the fill until I can see the black-on-black folds of the towel.
Setting the hair light is also pretty much a no-brainer for me (YMMV) because I have a very simple criteria for setting it — keeping the area it hits on the white towel below clipping in the playback on the camera. I use that "under clipping" criteria in part because its the only one I can check continually while shooting. The can only control a process to the degree you can measure it. I might be able to measure exposure to the nearest 1/10th stop with an incident meter, but if the subject moves closer to the key light that meter reading will no longer be valid. By setting the exposure of the highlights per the clipping warning I'll know from the change in the warning if that type of problem has occurred and fix it by move the subject, avoiding clipped highlights.
The decision on how to expose the rim lit areas is a subjective one. If the subject is a hot babe in bikini with body glistening with baby oil then the equally "hot / hard" look of specular rim lighting would be appropriate. But for my conventional portraits I like a softer look created by keeping detail in the rim lit areas of skin and clothing. Thus I choose to expose the rim lighting on the target as "Zone 9 - solid" white, not "Zone 10 - specular".
The first decision to render the rim light at Zone 9 has downstream implications in the workflow on adjustment of the frontal "key" lighting and background level. Again YMMV here as to goals, but the reason I use rim lighting is to define shape via separation with the background. So if I peg my rim light at Zone 9 - around 250, out of necessity I need to reproduce the background darker, around Zone 8, if I want to see the effect of the rim lighting. That the same dilemma Adams recognized and addressed when creating the system of zones to explain this situations more clearly. So after setting the foreground lighting around the baseline criteria of keeping detail in the rim lit parts of the foreground and filling for detail in the darkest areas I adjust my background lighting not based on making 255, but based on retaining the ambience of the rim lighting...
The reaction to tonal ranges in photographs is both adaptive and comparative. In person we perceive white as a range of tone. For example if your foreground subject was holding a piece of the white seamless paper next to their face would you want to expose it for 255 white? Probably not because that would also blow the highlights in their skin. To get correct exposure on the skin you'd need to reproduce the white paper in their hand at its accurate 250 value. Then you just need to decide when setting the background lighting on the same paper whether you want the paper in hand to look brighter, darker, or seem to disappear. That's an artistic decision. Being able to do all three at will? That requires craftsmanship and learning to control the reproduction medium. The two are not mutually exclusive.
Here's the photo above with a 255 border...
Here's the photo above with about 30 seconds of vignetting with the eraser tool...
Here the subject I was setting the lights to shoot with that target. I use the target on a stand to set my light so my subject doesn't need to stand around while I futz with the setting them. That allows me to focus 100% of my attention on them when shooting and makes for a much more relaxed atmosphere...
If you measure the background with the eye dropper tool you'll find it is a 254, one unit below clipping. I didn't get that level of consistency with my lighting shot-to-shot because it's not that consistent. What I did instead is what I learned to do running printing plants for years. When I want a consistent background shot-to-shot at a specific value I add a duplicate layer, fill it with that value, in this case 254,254,254 then knock out the foreground with a mask. It allows me to retain a full range of detail on the subject without the flare you'll get from nuking a background above clipping and get the background value I want.
When I want it 255 I can make it 255 because I know how to control the process, I just choose not to so my foreground white values are perceived brighter. What I'd suggest to the OP and others learning lighting is to learn to control the process in the same way and you'll find at the end of that experimentation you'll have more creative control over what the final results look like.