  We review the theory and practice of determining what parts of a data set are
ultrametric. It is assumed that the data set, to begin with, is endowed with a
metric, and we include discussion of how this can be brought about if a
dissimilarity, only, holds. The basis for part of the metric-endowed data set
being ultrametric is to consider triplets of the observables (vectors). We
develop a novel consensus of hierarchical clusterings. We do this in order to
have a framework (including visualization and supporting interpretation) for
the parts of the data that are determined to be ultrametric. Furthermore a
major objective is to determine locally ultrametric relationships as opposed to
non-local ultrametric relationships. As part of this work, we also study a
particular property of our ultrametricity coefficient, namely, it being a
function of the difference of angles of the base angles of the isosceles
triangle. This work is completed by a review of related work, on consensus
hierarchies, and of a major new application, namely quantifying and
interpreting the emotional content of narrative.
