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Basic color theory
The base color is the predominate color that makes the color you are going to apply. If it is "Ash" the base color will generally be green. Green is opposite red on the color wheel and each will "neutralize" the other. Blue is opposite orange and violet is opposite yellow.
The level (the number) is the level of lightness or darkness. #1 is the darkest and #10 is the lightest. Some manufactures go to level 11 and it usually has the word pale in the name.
The underlying pigment is the color your hair will "lift" to. Levels of lift are determined by the volume of developer.
Here is a chart with an excellent explanation:
Everything except the desired is up to your colorist. In a color consult, I use swatches to determine their natural level, I have them pick the color and level of what they want. Then I determine the formula and write it down on the client card before I mix the color. It is up to the colorist to know how color theory works within the product line they are using.