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In this paper we study the possibility to $q$-generalize the Poisson bracket(where $q = (q_{i, j})$ is a matrix of formal variables) based on the idea tointroduce a collection of `$q$-Planck constants' and to motivate thecorresponding construction by the semiclassical approximation of quantumtheory. This leads to a definition of a $q$-analogue of the Weyl quantizationmap from which we extract the first semiclassical correction and extend it upto a certain collection of bracketed expressions on multiple arguments.Different bracketings are linked to each other in a more complicated way thanis suggested by the $q$-Jacobi identity, and one terms this collection a\emph{distortion} if the parameters in its construction are adjusted in such away that the Hochschild cocyclicity and the $q$-Jacobi identity are fulfilledon the $q$-twisted antisymmetrizations of the brackets. We show that thedistortions are parametrized by symmetric $d \times d$ matrices over therational functions in $\lbrace q_{i, j} \rbrace$ (where $d \times d$ is thesize of the matrix $q$), define the \emph{bracketing (epoch\'e) algebra} as ageneralization of the commutation relations for the $q$-Planck constants, andsuggest to lift the $q$-Weyl quantization to a truncation of a Rees ring.

Distortion of the Poisson Bracket by the Noncommutative Planck Constants