Musical intervals and special linear transformations

This paper presents a transformational approach to musical intervals with particular focus on their constitutive role for well-formed scales. These scales have the property that their binary step-interval pattern is maximally even. Transposition classes of well-formed scales are therefore characterized by two step intervals and their characteristic binary pattern, or, more abstractly, by four numbers: two step intervals and two associated multiplicities. The proposed transformational approach therefore studies group actions (1) on interval pairs, (2) on multiplicity pairs, such that the two intervals with their associated multiplicities form two pairs of canonically conjugated variables, and (3) on binary cycle words. The group is the same in all three cases: the modular group Γ=SL(2, ℤ). For any free commutative interval group G we have a faithful action of Γ on G×G through transvections. This action has a refinement in terms of an action of the braid group B 3 on the product F×F of any non-commutative free group F with itself. The non-commutative interval group considers intervals as pathways rather than sums. The action of the modular group on ℝ4 through canonical transformations is given in terms of a representation of this group through symplectic 4×4-matrices. This left action can be comfortably rewritten in terms of a right action on 2×2-matrices. The submonoid SL(2, ℕ) exemplifies the Stern–Brocot tree and provides a link to the classical theory of well-formed scales. We recapitulate how the processes of approximating a scale generator g through its semi-convergents and of generating smaller and smaller step interval sizes are transformationally interconnected. The action of SL(2, ℤ) on cycle words with directed letter is based on parallel rewriting rules. The maximally even patterns form exactly one orbit of this group action: the orbit generated by the one-letter-word . Looking to the future, the author indicates how the extension of this theory can be musically explored by the technology of Sethares spectra and how new questions arise from an inspection of intrinsic properties of the modular group Γ.