Dual lattice-path transformations and the dynamics of the major and minor exo-modes

Abstract

The article investigates an extension of the theory of well-formed modes and proposes a model of the major and minor modes in harmonic tonality. The established theory of well-formed modes is well adapted to the description of the medieval diatonic modes. Its core is the conversion of the circle-of-fifths encoding into the circle-of-steps encoding of the seven generic scale degrees. This conversion is a linear automorphisms of the cyclic group of order 7 (known from well-formed scale theory). This automorphism can be lifted to three refined levels of description: (1) to the diatonic Regener transformation on the two-dimensional note-interval system and (2) to a conjugacy class of Sturmian morphisms, corresponding to the filling of the authentically (or plagally) divided octave by the modal species of the fifth and the fourth and (3) to associated lattice path transformations on chains of anchored note intervals. In the present paper the place of the authentic division of the octave into a perfect fifth and a perfect fourth is taken by a triadic division of the octave into a major third, a minor third, and a perfect fourth. The three triadic intervals are filled with patterns formed by three different step intervals. In the consequence, all the above-mentioned constructions on two-dimensional note-interval system have to be replaced by analogous constructions which are based on a three-dimensional system of Euler-notes and intervals. The resulting constructions are mathematically inspired by Arnoux and Ito [2001. “Pisot Substitutions and Rauzy Fractals.” Bulletin of the Belgian Mathematical Society Simon Stevin 8 (2): 181–207] and include the lifting of the diatonic Euler-Regener-transformation to substitutions on words in three letters, their geometrical action as lattice-path transformations and the study of their duals. A constitutive innovation is the music-theoretical interpretation of the eigen-values, eigen-vectors and eigen-co-vectors of the Euler-Regener-transformation and its dual. A modified instance of the classical Euler-Oettingen-Riemann tone net can be defined on the Handschin plane, the kernel of the linear eigen-height form. Finally, comparing the image under the dual lattice-path transformations of the dual of the step-interval trihedron anchored at the Euler-note C₄ and of the dual of the step-interval anti-trihedron targeting the Euler-note C₄, a dynamical system arises on a region of the continuous Handschin plane.

Keywords:

• major mode
• minor mode
• Euler-Regener interval system
• Euler-Oettingen-Riemann tone net
• diatonic Euler-Regener transformation
• lattice-path transformation
• dual lattice-path transformation
• eigen-height
• eigen-width
• Handschin plane

Acknowledgments

The author wishes to thank the editors and anonymous reviewers for valuable comments and for their carefulness and patience in the supervision of the revision.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 See also Section 8 for an assessment of the current situation in comparison to Noll and Clampitt (2019).

2 The interval group of the continuous three-dimensional interval system is isomorphic to the additive group ${\mathbb{R}}^3$. Starting from the discrete interval group $G_{\mathcal{E}}$, we obtain it as the tensor product: ${\overline{G}}_{\mathcal{E}}=\mathbb{R}{\otimes }_{\mathbb{Z}}{G}_{\mathcal{E}}$.

3 The term trihedron stands for the sum of the three unit vectors which share a given lattice point as their anchor. The term anti-trihedron stands for the sum of the three unit vectors which are jointly targeting a given lattice point. The dual of a trihedron is consequently the sum of the (right, back and top) unit squares in the unit cube corresponding to the shared anchor point. The dual of an anti-trihedron is the sum of the (left, front and bottom) unit squares in the unit cube corresponding to the shared target point.