This paper utilizes the notion of 'effective global regularity' and the intuition stemming from Cooper and McLaren (1996)'s General Exponential Form to develop a family of 'composite' (product and ratio) direct, inverse and mixed demand systems. Apart from having larger regularity regions, the resulting specifications are also of potentially arbitrary rank, which can better approximate non-linear Engel curves. We also make extensive use of duality theory and a numerical inversion estimation method to rectify the endogeneity problem encountered in the estimation of the mixed demand systems. We illustrate the techniques by estimating different types of demand systems for Japanese quarterly meat and fish consumption.