Galois field theoretical approach to the dislocational and disclinational aspects of the musical lattice

Abstract

The pure tone lattice is practically on the parallelogram enclosing 54 = 9 × 6 quanta with sides carrying 9 points at perfect fifth steps and 6 points at major third steps, respectively. Of these quanta the 54th is excluded owing to the limit of human auditory perception capacity. The remaining 53 tones form a pentagon reminiscent of the Burgers circuit materializing the torsion tensor. Each of the triangular meshes so forming the lattice has the character of a two-dimensional simplex. The meshes so mutually connected offer indeed a model for projective connexion that is to entail a field of a torsion-curvature tensor. The entire construction of the musicological manifold is, therefore, imbued with analogues of what we call dislocations and disclinations. Various kinds of commas are so materialized. Consequently owing to the combinatorial wild topology inherent in the projective plane, the musicological scales fixed on this basis are represented on a Moebius band. An absolute algebraical approach leads to a new thesis that a musical scale needs to have the character of a Galois field. With three fundamental elements based on the tristimulus principle, and the prime field of characteristic 2 based on two-valued logic as the coefficient domain, we are led to the scale consisting of 7 = 2 3 − 1 tones, i.e. a heptatonic. Formulae for the addition and multiplication within the scale are Galois-theoretically given. Each such addition relation defines a projective triangle; with their repetitions there is entailed a projective connexion, inevitably. The hepatatonic based on the tristimulus principle is a reflexion of statical standpoint where harmony dominates. A tetrastimulus formulation implies an extension to non-statical instationary feasibilities, where 15 = 2 4 − 1 quanta are distinguished so that a 15-tonic scale is given significance. The statical heptatonic tonality is imbedded in the more general 15-tonic which is responsible for a more atonal musical edifice. Many more important musicological revelations are so afforded, with more dislocational and disclinational effects playing their roles.