Kaleidoscope substitutions and pairwise well-formed modes: major-minor duality transformationally revisited

Abstract

Pairwise well-formed modes are investigated as substitutions on words over a three-letter alphabet. Motivating examples are the major and minor modes, here described in terms of the substitutions a↦ac,b↦ba,c↦cab and a↦ab,b↦ca,c↦bac. Pairwise well-formedness is investigated through the inspection of the two-letter-mergings of a substiution, namely concatenations with letter insertions from the right and letter projections from the left. The appropriate generalizations of the two mentioned examples are the plain and twisted kaleidoscope substitutions. Four of their mergings turn out to be Sturmian morphisms with interesting interdependencies. The diatonic mergings are musically related to the medieval modes, which are fifth-generated. The syntonic mergings are third-generated and provide a common neutral anchor for the major and minor modes. The paper thus provides a new proposal in order to integrate the diatonic side of harmonic tonality with its triadic aspects, which we characterize as its syntonic side.

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