In this paper, the moments of the Riesz-N$acute{a}$gy-Tak$acute{a}$cs(RNT) distribution over a general interval [a, b] $subset$ [0, 1], are found through the moments of the RNT distribution over the unit interval, [0, 1]. This is done using some special features of the distribution and the fact that [0, 1] is a self-similar set in a dynamical system generated by the RNT distribution. The results are important for the study of the orthogonal polynomials with respect to the RNT distribution over a general interval.