The codenominator is a function F that extends the Fibonacci sequence to the index set of positive rational numbers. Many known Fibonacci identities carry over to the codenominator. One can express Dyer's outer automorphism of the extended modular group PGL(2, Z) in terms of F. This automorphism can be viewed as an automorphism group of the trivalent tree. The real -covariant modular function Jimm J on the real line  is defined via the codenominator. J relates the Stern-Brocot tree to the Bird tree. Jimm induces an involution of the moduli space of rank-2 pseudolattices and is related to the arithmetic of real quadratic irrationals.