Abstract
This paper studies the mathematical basis for a new study of modes of well-formed (WF) scales, and presents a new characterization of special standard Sturmian morphisms.
We introduce WF words, which coincide with the step-interval patterns of modes of well-formed scales. WF words can be represented as conjugates of some Christoffel word (generalized Lydian mode).
To every WF word we may assign a pair of affine automorphisms f w and g w . These assignments induce a pair of involutions over the set of WF words: the plain adjoint and the twisted adjoint. We study the properties of these adjoints; in particular we show how the plain adjoint coincides with duality over the set of Christoffel words and also that the twisted adjoint extends Sturmian involution to the set of WF words. Thomas Noll’s divider incidence result holds, inter alia, that w is special standard if and only if f w (1) = 1.
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Clampitt, D., Domínguez, M., Noll, T. (2009). Plain and Twisted Adjoints of Well-Formed Words. In: Chew, E., Childs, A., Chuan, CH. (eds) Mathematics and Computation in Music. MCM 2009. Communications in Computer and Information Science, vol 38. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02394-1_7
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DOI: https://doi.org/10.1007/978-3-642-02394-1_7
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