Rank-2 tensors are unable to represent multi-modal diffusion associated with intra-voxel orientational heterogeneity (IVOH), which occurs where axons are incoherently oriented, such as where bundles intersect or diverge. Under this condition, they are oblate or spheroidally shaped, resulting in artefactually low anisotropy, potentially masking reduced axonal density, myelinisation and integrity. Higher rank tensors can represent multi-modal diffusion, and suitable metrics such as generalised anisotropy (GA) and scaled entropy (SE) have been introduced. The effect of tensor rank was studied through simulations, and analysing high angular resolution diffusion imaging (HARDI) data from two volunteers, fit with rank-2, rank-4 and rank-6 tensors. The variation of GA and SE as a function of rank was investigated through difference maps and region of interest (ROI)-based comparisons. Results were correlated with orientation distribution functions (ODF) reconstructed with q-ball, and with colour-maps of the principal and second eigenvectors. Simulations revealed that rank-4 tensors are able to represent multi-modal diffusion, and that increasing rank further has a minor effect on measurements. IVOH was detected in subcortical regions of the corona radiata, along the superior longitudinal fasciculus, in the radiations of the genu of the corpus callosum, in peritrigonal white matter and along the inferior fronto-occipital and longitudinal fascicula. In these regions, elevating tensor rank increased anisotropy. This was also true for the corpus callosum, cingulum and anterior limb of the internal capsule, where increasing tensor rank resulted in patterns that, although mono-modal, were more anisotropic. In these regions the second eigenvector was coherently oriented. As rank-4 tensors have only 15 distinct elements, they can be determined without acquiring a large number of directions. By removing artefactual underestimation of anisotropy, their use may increase the sensitivity to pathological change.