More "Earth" Boards

In the end, 3 boards, each containing 8 squares, were built to get a sense of the resulting color during the tests described here. The last square of the 3rd board was rather dark, and - given blue pigment was added incrementally to each - I was confident each of the shades could be replicated. Then the day came when I actually wanted to produce a specific color, in order to patch some of the earth already fixed on the layout. I chose the most similar shade on the test boards, which meant a concentration of 4.2% blue ultramarine pigment in the standard base formula I've been using. However applying it in just one coat resulted in a color more similar to a 2.8% square.

Given things were totally different from what I was expecting, 2 more test boards were built. This time however, the increment percent of blue ultramarine pigment was increased from around 0.3% (in the first 3 boards) to an average of 0.7%. Still 2 coats were used. The sifting would be done until the new specs of formula barely get "wet". However this time the upper portion of each square was left soaked on purpose. It turns out that these soaked spots actually will dry sensibly lighter than the rest of sifted formula, which has an important contribution to the outcome.

  Thinking about what could have gone wrong in the first boards, it might have been some already prepared base formula, which could have been done without the electronic weighing scale. Also, given the final percentage of the last test patch went up to 10% ultramarine blue, I've also considered how this percentage is calculated. So far in the test boards done, formula was already mixed for the previously completed square. This had a known percentage of blue pigment, which has been gradually increased. In my calculations for the next square, the percent of added pigment will only be computed given the total weight of the existing formula. Adding the blue pigment's weight to the equation wasn't necessary since the difference in orders was quite important (30mg for 15g). However given that bigger and bigger percentages were being used - now reaching 10% - this had to be revisited. Simply think of having 10 grams of the formula already available; we need to get the final percentage to 10% blue pigment. The simplest way would be to consider adding blue pigment in an amount equivalent to 10% of the formula's weight. In our example - 1 gram blue pigment. However the resulting formula now weighs 11 grams. Hence the pigment's correct percentage is 9.09% - becoming quite far off from what we had in mind.

  So doing a little math, we'll first consider C the existing's formula weight, X the weight of blue pigment that needs to be added to get to the desired p percentage of blue ultramarine pigment. The equation thus becomes X / (C + X) = p. This becomes 1 / (C/X + 1) = 1/p. Which eventually becomes X = C / (1/p - 1).

  Turning to our example, the needed weight of blue pigment to be added to reach a percentage of 10% is in fact C/9, which amounts to 1.11g. Computing the resulting percentage yields 1.11g / (10 + 1.11g) which comes really close to 10 (9,999%).