The analytical potential of the hexatonic organization of the consonant triads was demonstrated in a recent article by Richard Cohn (1996). This was accomplished by providing the consonant triad with a status independent of the contexts of diatonic scales, tonal centers, or indeed of triadic roots, a status that is, however, anything but free-floating; rather, a position established within highly constrained spaces carved out by maximally smooth hexatonic cycles, defined by Cohn and rehearsed below. Cohn illustrates the range of application of this framework of ideas with, among other late-Romantic examples, various transformations of the Grail motive from Wagner's Parsifal. In this paper, I hope to thicken the hexatonic analytical description by superimposing on Cohn's framework two complementary interpretations. I suggest that the correlation of all three perspectives addresses how one navigates, in the absence of any tonal center or of any other sort of privileged point of reference. These interpretive complements might be applied to any of Cohn's examples (and could be extended by analogy to the passages involving dominant and half-diminished seventh chords discussed in some of the companion papers to this one), but I will focus on one of the Grail transformations, where the alternative interpretations pay an extra dividend. The passage in question is the chromatic-enharmonic form of 'Grail'