It still doesn't answer the question of what "scale" we use to determine skin colour.
That's because you're asking the wrong question.
People don't use a "scale" to determine race; it's a set of prototypes. (The other usual term of art in psychological literature is "family resemblance.") Skin colour itself can be semi-scaled. Certainly variations in darkness can be measured, but a proper description would have to take hue, saturation, and value into account -- Asians aren't considered "white" in part because their skin varies too much from the typical Caucasian norm, just not on a dark/light axis.
Someone with (relatively) fair skin but displaying many other traits associated with the "black" racial prototype -- Colin Powell is a good example -- will still be considered "black." Skin colour is a single and very relevant feature for racial determination, but it's not the only one.
If you're looking for single thin bright line to separate racial categories, you won't find it. Which isn't surprising, since people don't typically categorize things that way. It's actually rather hard to find
any natural category that can be broken down by thin bright lines that way, because that's not how people think.
For further details, start with
this Wikipedia article (with the usual caveat that almost all Wikipedia articles are wrong in detail), and then continue with the complete works of the brilliant
Eleanor Rosch. A good article is Rosch, Eleanor. 1977. "Human Categorization." In N. Warren, ed., Studies in Cross-Cultural Psychology. London: Academic, or Rosch, E. 1973. "Natural categories." Cognitive Psychology 4: 328—350.