The word root, and the harmonic metabolism it helps to describe, plays a central role in how musicians think about musical structure. Vaguely implicit in some early writers' descriptions of intervals and scales, it awaited Rameau's discussions (1722/1971) of the fundamental bass to become explicit. Since then, music theorists have sought to explain its perceptual nature and causes. Their theories usually turn on some version of physical weightings, a root being the pitch class more powerfully reinforced than its companion pitch classes. After representative versions of influential explanations are reviewed, and their generic shortcomings are noted, Ernst Terhardt's "virtual pitch" theory is recognized as uniquely reasonable; it embodies a condition of pattern perception rather than physical reinforcement, thereby skirting a principal flaw of past theories. And yet, a troublesome paradox surfaces, regardless of which conceptualization one favors: empirical studies of interval perception have fallen short of confirming the phenomenal reality our concepts describe so confidently. In an attempt to outflank these empirical/phenomenal clashes, a scheme of pitch/time interaction, or "vectoral dynamics", is outlined. Its consistency with the linear perspective of vision is noted, and the model is applied to the opening of a Wolf song and to a painting by Titian.