Coordinate systems can be thought of as families of (often real-valued) functions able to specify ``points'' in a space in some way. If the functions are set-valued, then it may still be the case that they are able to ``separate'' points or subsets sufficiently to allow their ``numerical'' characterization. Such characterizations may be useful in applications where it is important to know that a given set of scales (i.e., the functions doing the characterizing) is indeed able to separate points or subsets adequately. In this paper subsets \(P\) of Cartesian products \(A\times X\) of non-empty sets \(A\) and \(X\) are considered, with pairs of maps \(F:A\to J(\downarrow)\) and \(\widehat F:A\to J(\uparrow)\) such that for all \((a,x)\in A\times X\), \((a,x)\in P \Leftrightarrow F(a)\cap \widehat F(x)= \emptyset\) or \(F:A\to J(\downarrow)\), \(G:A \to J(\downarrow)\) such that for all \((a,x)\in A\times X\), \((a,x)\in P \Leftrightarrow F(a) \subset G(x)\). These real-interval representations have been looked at in the past and are important especially when \(A=X\) and the resulting relations represent types of order, e.g., biorders, interval orders and semiorders, for which necessary and sufficient conditions of representability in these ways are considered in considerable detail in this interesting follow up to some well-established theory.