Samuele Antonini

Hamiltonian Cycles in the Topological Dual of the Tonnetz

The Hamiltonian cycles in the topological dual of the Tonnetz (i.e. the successions of triads connected only through PLR-transformations which visit every minor and major triad only once) will be introduced, enumerated on, studied, and classified both from a theoretical and analytical point of view.

Designing Non-constructability Tasks in a Dynamic Geometry Environment

This chapter highlights specific design features of tasks proposed in a Dynamic Geometry Environment (DGE) that can foster the production of indirect argumentations and proof by contradiction. We introduce the notion of open construction problem and describe the design of two types of problems, analysing their potential a priori, with the goal of elaborating on the potentials of designing problems in a DGE with respect to fostering processes of indirect argumentation. Specifically, we aim at showing how particular open construction problems, that we refer to as non-constructability problems, are expected to make indirect argumentations emerge.

On the Group of Transformations of Classical Types of Seventh Chords

This paper presents a generalization of the well-known neo-Riemannian group PLR to the classical five types of seventh chord (dominant, minor, half-diminished, major, diminished) considered as tetrachords with a marked root and proving that it is isomorphic to the abstract group \(S_5 \ltimes \mathbb {Z}_{12}^4\). This group includes as subgroups the PLR group and several other groups already appeared in the literature.