Here is a sketch of topological description of a Tamarkin-Tsygan precalculus. 

Consider the compactified configuration spaces $C_n$ and $D_{1,n}$ of $n$ points on $\mathbb{R}^2$ and $R^2-\{(0,0)}$, respectively. The collection $(C_n,D_{1,n})_n$ form a topological (colored) operad. 

**Claim**: the collection $(H_{-\bullet}(C_n,\mathbb{Q}),H_{-\bullet}(D_{1,n},\mathbb{Q}))_n$ is the operad of Tamarkin-tsygan precalculi. 

It is well-known that $(H_{-\bullet}(C_n,\mathbb{Q}))_n$ is the Gerstenhaber operad. Then the operations $L$ and $i$ are the two classes (of respective degrees $1$ and $0$) in $H_{-\bullet}(D_{1,1},\mathbb{Q})=H_{-\bullet}(S^1,\mathbb{Q})$. 

The two compatibility conditions can be understood as identities in $H^1(D_{1,2},\mathbb{Q})$. 

If you want to get the operator $d$ (i.e. get a calulus rather than a precalculus) you'll have to replace $D_{1,n}$ by its semi-direct product with $S^1$. 

See also Section 11 (especially $\S 11.3$) of [this paper by Kontsevich and Soibelman][1] for a similar point-of-view and its relation to a generalization of Deligne's conjecture.  


  [1]: http://arxiv.org/abs/math/0606241