The use of higher-order functions, as a method of abstraction and re-use in EC encodings, has been the subject of relatively little research. In this paper we introduce and give motivation for the ideas of higher-order functions, and describe their general advantages in EC encodings. We implement grammars using higher-order ideas for two problem domains, music and 3D architectural design, and use these grammars in the grammatical evolution paradigm. We demonstrate four advantages of higher-order functions (patterning of phenotypes, non-entropic mutations, compression of genotypes, and natural expression of artistic knowledge) which lead to beneficial results on our problems.