A manual for tuning a lyre/harp from ancient Iraq (or "Mesopotamia"), dating to the early second millenium BCE, uses a cyclical procedure of tuning pairs of strings (dichords). It is the earliest known example of music theory, predating anything comparable from other cultures by approximately one millenium. This manual enables a lyre/harp player to use a diagnostic method of determining the current mode or tuning of the lyre; it directs the musician to transform the instrument from one mode to another, through a series of tightening or loosening dichords (pairs of open strings). The tuning procedure, which may more accurately be called a modulation procedure, is clearly cyclical. The text employs a complex and precise terminology for strings, dichords and modes. This paper presents a step by step analysis of the tuning procedure as described in this ancient text, which has recently been supplemented by the identification of a new manuscript. In addition, it attempts to outline the basic characteristics of the Mesopotamian tuning system as revealed by the tuning text.