Greek mathematicians. These tunings allow composers to add to the expressiveness of their music by taking advantage of the greater variety of intervals available and of the richness of the enhanced overtones. In particular, in The Harp of New Albion, contemporary American composer Terry Riley has used a five-limit system of just intonation. An analysis of the structure of Riley's system reveals a deviation from more symmetric systems, such as the system of just intonation of Friedrich Wilhelm Marpurg from 1776. Musical and mathematical reasons for the deviation will be considered.