On the Musical Opportunities of Cylindrical Hexagonal Lattices: Mapping Flat Isomorphisms Onto Nanotube Structures

It is possible to position equal-tempered discrete notes on a flat hexagonal grid in such a way as to allow musical constructs (chords, intervals, melodies, etc.) to take on the same shape regardless of the tonic. This is known as a musical isomorphism, and it has been shown to have advantages in composition, performance, and learning. Considering the utility and interest of such layouts, an extension into 3D interactions was sought, focussing on cylindrical hexagonal lattices which have been extensively studied in the context of carbon nanotubes. In this paper, we explore the notation of this class of cylindrical hexagonal lattices and develop a process for mapping a flat hexagonal isomorphism onto such a lattice. This mapping references and draws upon previous explorations of the helical and cyclical nature of western musical harmony, but is not limited to 12-tone equal tempered scales.