  The purpose of this paper is to quantify the size of the Lebesgue constants
$(L_m)_{m=1}^{\infty}$ associated with the thresholding greedy algorithm in
terms of a new generation of parameters that modulate accurately some features
of a general basis. This fine-tuning of constants allows us to provide an
answer to the question raised by Temlyakov in 2011 to find a natural sequence
of greedy-type parameters for arbitrary bases in Banach (or quasi-Banach)
spaces which combined linearly with the sequence of unconditionality parameters
$(k_m)_{m=1}^{\infty}$ determines the growth of $(L_m)_{m=1}^{\infty}$.
Multiple theoretical applications and computational examples complement our
study.
