Spatial and structural features of the quarto quintic circle: Cartesian Matrix

Abstract

The author of the paper investigates the problem of correlation between the properties of a 12-element even-tempered musical system, its planar model, and the proper dodecagon. Within the framework of the planar model, the constructive role of the connectivity diagonal {12/5} is revealed and a formula for correlating the fifths and chromatic representations of the system is proposed. For the first time, the possibility of translating the planar model of the circle of fifths into a three-dimensional model of a regular polyhedron (dodecahedron) is pointed. Based on the analysis of the properties of the P-R-I-RI complex in music and similar spatially symmetric complexes in other spheres of scientific knowledge, a generalized unification of their definition as a Cartesian Matrix is proposed. The constant "resonance-node" elements of the system are identified. A number of tasks for further research are defined.

Keywords

Cartesian matrix, circle of fifths, dodecagon, connectivity diagonal, music, geometry, continuity