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1. Introduction =============== The evolutionary relationship between a set of *k* homologous sequences of *N* nucleotides can be represented by a *k*-leaved bifurcating tree where each leaf node represents a known sequence and each internal node represents an ancestral sequence (which is almost always unknown). The phylogeny of the *k* sequences can be inferred by using maximum-likelihood methods, which rely on models of nucleotide substitution to infer the most likely tree. Popular phylogenetic methods, like those implemented in PHYLIP ([@b10-ebo-01-62]), PAUP\* ([@b41-ebo-01-62]), and Tree-Puzzle ([@b33-ebo-01-62]), use models of nucleotide substitution that assume the evolutionary process is stationary, homogeneous, and reversible. Although a detailed mathematical description of stationarity, homogeneity and reversibility can be found in [@b1-ebo-01-62], we will give a brief description of these terms in the context of molecular phylogenetics. Stationarity implies that the marginal probabilities of the four nucleotides remain constant over all the nodes of a given tree. Homogeneity implies that the instantaneous rate matrix (described in eg, [@b24-ebo-01-62], [@b23-ebo-01-62]) is constant over an edge (local homogeneity) or constant over the entire tree (global homogeneity). Reversibility implies that the probability of sampling nucleotide *i* from the stationary distribution and going to nucleotide *j* is the same as the probability of sampling nucleotide *j* from the stationary distribution and going to nucleotide *i*, where *i*, *j* = {*A*,*C*,*G*,*T*} ([@b6-ebo-01-62]). Reversibility, therefore, implies that the process is stationary and permits us to ignore the direction of evolution. The assumptions of stationarity, homogeneity and reversibility are often violated by the data, resulting in an elevated probability of incorrect phylogenetic results (for examples of the complexity of the problem, see [@b18-ebo-01-62]; [@b21-ebo-01-62]). In a landmark article, [@b3-ebo-01-62] considered a general Markov model for unrooted trees, the assumptions being that the process relating each pair of nodes in the tree is Markovian and the sites are independent and identically distributed. Their model does not make the assumption of stationarity, homogeneity (local or global) or reversibility, so it is more general than the non-stationary but locally homogeneous models considered by [@b45-ebo-01-62] and [@b46-ebo-01-62]. [@b3-ebo-01-62] considered a *k*-taxa tree with 2*k*−3 *Q*-matrices, one for each edge of the tree. Each *Q*-matrix represents the joint probability distribution of nucleotides at the two ends of the associated edge and is a 4 × 4 matrix. Since the sequences at internal nodes are not known, we can only observe the 4*^k^* different combinations of nucleotides at the leaf nodes. These combinations together represent the joint probability distribution of the nucleotides at leaf nodes and can be written as a function of the *Q*-matrices. Thus, the likelihood of the observed sequences is a function of the set of *Q*-matrices and can be maximised by determining the maximum-likelihood estimates of the *Q*-matrices. The algorithm for obtaining the maximum-likelihood estimates was suggested by [@b3-ebo-01-62] but has not received the attention that other aspects of their paper have, especially calculation of LogDet distance ([@b26-ebo-01-62]; [@b39-ebo-01-62]), probably because the large number of parameters was assumed to make the interpretation difficult. We revisit [@b3-ebo-01-62]