The circumplex: A slightly stronger than ordinal approach

For some proximity matrices, multidimensional scaling yields a roughly circular configuration of the stimuli. Being not symmetric, a row-conditional matrix is not fit for such an analysis. However, suppose its proximities are all different within rows. Calling {{x,y},{x,z}} a conjoint pair of unordered pairs of stimuli, let {x,y}→{x,z} mean that row x shows a stronger proximity for {x,y} than for {x,z}. We have a cyclic permutation π of the set of stimuli characterize a subset of the conjoint pairs. If the arcs {x,y}→{x,z} between the pairs thus characterized are in a specific sense monotone with π, the matrix determines π uniquely, and is, in that sense, a circumplex with π as underlying cycle. In the strongest of the 3 circumplexes thus obtained, → has circular paths. We give examples of analyses of, in particular, conditional proximities by these concepts, and implications for the analysis of presumably circumplical proximities. Circumplexes whose underlying permutation is multi-cyclic are touched.