We propose a local and general dependence quantifier between two random variables $X$ and $Y$, which we call Local Lift Dependence Scale, that does not assume any form of dependence (e.g., linear) between $X$ and $Y$, and is defined for a large class of random variables, singular and absolutely continuous w.r.t Lebesgue measure. We argue that this dependence scale is more general and suitable to study variable dependence than other specific local dependence quantifiers and global dependence coefficients, as the Mutual Information. An outline of how this dependence scale may be useful in branches of applied probability and topics for future research are presented.