  Two generalizations of the Minkowski ?(x) function are given. As ?(x) maps
quadratic irrationals to rational numbers, it is shown that both
generalizations send natural classes of pairs of cubic irrational numbers in
the same cubic number field to pairs of rational numbers. It is also shown that
these functions satisfy an analog to the fact that ?(x), while continuous and
increasing, has derivative zero almost everywhere. Both extend earlier work of
Beaver-Garrity on the Farey-Bary map.
