Genetic algorithms (GA’s) are global search mechanisms that have been applied to many disciplines including music composition. Computer system MELEC composes music using evolutionary computation on two levels: the object and the meta. At the object-level, MELEC employs GAs to compose melodic motifs and iteratively refine them through evolving generations. At the meta-level, MELEC forms the overall musical structure by concatenating the generated motifs in an order that depends on the evolutionary process. In other words, the structure of the music is determined by a geneological traversal of the algorithm’s execution sequence. In this implementation, we introduce a new data structure that tracks the execution of the GA, the Genetic Algorithm Traversal Tree, and uses its traversal to define the musical structure. Moreover, we employ a Fibonacci-based fitness function to shape the melodic evolution.