Chomsky (1959a) presented an algorithm for constructing a finite transducer that is strongly equivalent to a Chomsky-normal-form context-free grammar for all sentences generated by that grammar with up to any specified finite degree of center embedding. This article presents a new solution using a variety of COORDINATE GRAMMAR to assign nonembedding (paratactic) structures strongly equivalent to those assigned by an embedding grammar, which can in turn be directly computed by a finite transducer. It proposes that the bound on center embedding is really a consequence of a bound on alternation between right and left embedding, called here ZIGZAG EMBEDDING. Coordinate grammars can also be used to assign nonembedding structures equivalent to those with up to any specified finite degree of coordinate embedding (the occurrence of a coordinate structure as a member of a coordinate structure of the same type). It concludes that coordinate grammars or the finite transducers strongly equivalent to them are psychologically real, and that the existence of a finite bound on the degree of zigzag and coordinate embedding is a consequence of the increasing size and complexity of such grammars or transducers as the bound increases.