I'm trying to formally understand the free Gerstenhaber algebra on a set $X = \{x_1,...,x_n\}$. I've just started working with graded objects and I am not used to the complexity of structures such as Gerstenhaber algebras. I don't really have any references for this, so I thought I'd ask here.
I am familiar with other, simpler free constructions, such as the free associative algebra on $X$. This can be constructed by either taking the free vector space $V(X)$ with basis $X$ and then taking the tensor algebra $T(V(X))$, or alternatively by taking the free monoid $M(X)$ on $X$ and then taking the free vector space $V(M(X))$ with basis $M(X)$.
However, if I try to do something similar with the Gernstenhaber algebra, since I need to have two operations, I would need to consider words that are a combination of multiplication and bracket, so they would look for example like $[[x_1,x_3],x_4x_2]x_1$. Then I might define two products on these words, one being concatenation and the other being bracketing, then take the free vector space on this and take quotients to get the relations needed on the multiplication and bracket. Somehow degree will need to come in to play, but I don't understand how. I'm not sure if I'm close or not, but somehow a clear understanding of what to do is evading me. Can someone explain how to do this formally or provide a good reference?