In the western tonal music tradition, when we listen to a melody, our minds generate a system of hierarchical relations among the musical pitches that we hear. This system of relations is known as tonality. The developing of the pitch relations can be visualized as the generating of a tonal space. We propose a spatial analogue for this tonal space, which we call a Spiral Array. In this geometric representation, spatial proximity designates perceptually close relations among the represented musical entities. As a melody unfolds, the accumulating musical information can be charted in the Spiral Array as a walk through this tonal space. Over time, the hierarchical relations that determine the perceived relative stability among the pitches become more settled, and a key (designated by the most stable pitch) is established. We demonstrate that this establishing of key can be derived from the walk within the Spiral Array.