Ionian theorem
Thomas E. Noll · Journal of Mathematics and Music · 2009
Through the application of algebraic combinatorics on words to the study of diatonic modes, the paper characterizes the ascending authentic Ionian mode among the others in terms of divider incidence. This property characterizes positive standard words among their conjugates with respect to plain adjointness. The plain adjoint of the Ionian step-interval pattern aaba|aab is the Ionian fifth-up/fourth-down folding pattern yx|yxyxy. Three different kinds of mathematical objects are associated with one another: affine automorphisms of ℤ n , trajectories {0,.., n}→ℤ×ℤ traversing lifts of the graphs of affine automorphisms to fundamental domains of ℤ n ×ℤ n within ℤ×ℤ, and well-formed words encoding the zigzag patterns of the trajectories in terms of binary words. Inversion of affine automorphisms is thereby associated with an extension of the duality of Christoffel words to their conjugates. Plain adjointness for positive standard words has a transformational expression in terms of the monoid-anti-automorphism of the special standard monoid. Positive standard words are images of the word ab under special standard morphisms f∈⟨ G, D ⟩. Plain adjointness corresponds to the reversal of the order of the occurrences of the generators G and D within the representation of f. The factorization of f(ab) into the two factors f(a) and f(b) coincides with the authentic division.