Naming and ordering the modes, in light of combinatorics on words
David Clampitt, Thomas E. Noll · Journal of Mathematics and Music · 2018
One of the most telling features of the application of combinatorics on words in mathematical music theory is the extent to which aspects of the history of theory are captured by the word theory model. Well-formed modes, including the usual diatonic modes, are modeled by Christoffel words and their conjugates. A class of Sturmian morphisms produces words representing the Glarean modes, and gives rise to standard factorizations, which in turn model the authentic and plagal divisions of the diatonic modes. The principal discussions are the renaming of the ancient Greek modes in the tenth-century Alia musica, and the reordering of the authentic modes of Glarean's Dodecachordon by Zarlino in 1571. The latter exemplifies a recently demonstrated property in combinatorics on words, the matching of the conjugation order with the lexicographic order for the conjugation class of a given special Sturmian morphism. The two main topics are unified in this presentation.