  In order theory, a rank function measures the vertical "level" of a poset
element. It is an integer-valued function on a poset which increments with the
covering relation, and is only available on a graded poset. Defining a vertical
measure to an arbitrary finite poset can be accomplished by extending a rank
function to be interval-valued. This establishes an order homomorphism from a
base poset to a poset over real intervals, and a standard (canonical) specific
interval rank function is available as an extreme case. Various ordering
relations are available over intervals, and we begin in this paper by
considering conjugate orders which "partition" the space of pairwise
comparisons of order elements. For us, these elements are real intervals, and
we consider the weak and subset interval orders as (near) conjugates. It is
also natural to ask about interval rank functions applied reflexively on
whatever poset of intervals we have chosen, and thereby a general iterative
strategy for interval ranks. We explore the convergence properties of standard
and conjugate interval ranks, and conclude with a discussion of the
experimental mathematics needed to support this work.
