The first examples of triangular AF algebras to be studied were the refinement algebra and the standard algebra. Both are analytic algebras with the property that the cocycle can be taken to be constant on the matrix units of the algebra. The latter property, called local constancy, is quite special and is still ill-understood. In the present paper we examine the class of nest algebras [Tscr ] in AF C*-algebras which share the distinctive features of the refinement algebra:

(1) [Tscr ] is a nest algebra in which the nest generates the diagonal,

(2) [Tscr ] admits a locally constant cocycle.