Lexicographic Orderings of Modes and Morphisms

The context of this paper is the theory of modes of non-degenerate well-formed scales (generalized diatonic or pentatonic scales), within the framework of algebraic combinatorics of words, specifically musical modes encoded as members of the monoid of words in \(A^{*}\) over a two-letter alphabet A, and the monoid of Sturmian morphisms that act on \(A^{*}\). The paper relates lexicographic orderings of words modes of (non-degenerate) well-formed scales (especially the canonical examples, the diatonic modes) and lexicographic orderings of the special Sturmian morphisms associated with the modes, to the musical scale and circle-of-fifths orderings. These lexicographic orderings are related to Zarlino’s 1571 re-ordering of Glarean’s 1547 listing of six authentic diatonic modes.