We have used ab initio calculations to compute all of the tensor elements of the electric field gradient for each carbon-deuterium bond in the ring of deuterated 3-methyl-indole. Previous analyses have ignored the smaller tensor elements perpendicular to principal component V-zz which is aligned with the C-H-2 bond (local bond z-axis). At each ring position, the smallest element V-xx is in the molecular plane and V-yy is normal to the plane of the ring. The asymmetry parameter eta = (vertical bar V(yy)vertical bar - vertical bar V(xx)vertical bar)/vertical bar V-zz vertical bar ranges from 0.07 at C4 to 0.11 at C2. We used the perpendicular (off-bond) tensor elements, in concert with an improved understanding of the indole ring geometry, to analyze prototype H-2 NMR spectra from welloriented, hydrated peptide/lipid samples. For each of the four tryptophans of membrane-spanning gramicidin A (gA) channels, the inclusion of the perpendicular elements changes the deduced ring tilt by nearly 10 and increases the ring principal order parameter S-zz for overall "wobble" with respect to the membrane normal (molecular z-axis). With the improved analysis, the magnitude of S, for the outermost indole rings of Trp(13) and Trp(15) is indistinguishable from that observed previously for backbone atoms (0.93 +/- 0.03). For the Trp(9) and Trp(11) rings, which are slightly more buried within the membrane, S-zz is slightly lower (0.86 +/- 0.03). The results show that the perpendicular elements are important for the detailed analysis of H-2 NMR spectra from aromatic ring systems.