To solve for \( g \) in the equation
\[ 4 - \frac{1}{12}g - 2 = \frac{3}{2}g + 1 - \frac{5}{6}g, \]
we will first simplify both sides of the equation.
Starting on the left-hand side:
\[ 4 - 2 = 2, \]
so the left-hand side simplifies to:
\[ 2 - \frac{1}{12}g. \]
Now, let's simplify the right-hand side. We need a common denominator to combine \( \frac{3}{2}g \) and \( -\frac{5}{6}g \). The common denominator for 2 and 6 is 6.
Rewriting \( \frac{3}{2}g \) in terms of sixths gives us:
\[ \frac{3}{2}g = \frac{9}{6}g. \]
Now we can combine the terms:
\[ \frac{9}{6}g - \frac{5}{6}g = \frac{4}{6}g = \frac{2}{3}g. \]
This allows us to replace the right-hand side:
\[ \frac{2}{3}g + 1. \]
So now, we can rewrite the equation:
\[ 2 - \frac{1}{12}g = \frac{2}{3}g + 1. \]
Next, we can get rid of the fractions by multiplying the entire equation by 12 (the least common multiple of the denominators).
\[ 12 \cdot \left( 2 - \frac{1}{12}g \right) = 12 \cdot \left( \frac{2}{3}g + 1 \right). \]
This results in:
\[ 24 - g = 8g + 12. \]
Now, let's move all terms involving \( g \) to one side and the constant terms to the other side:
\[ 24 - 12 = 8g + g, \]
which simplifies to:
\[ 12 = 9g. \]
Now we can solve for \( g \):
\[ g = \frac{12}{9} = \frac{4}{3}. \]
Thus, the solution is
\[ \boxed{\frac{4}{3}}. \]