On a 4-dimensional subalgebra of the 12-tone Equal Tempered Tuning

Abstract

An operation of associative, commutative and distributive multiplication on { Euclidean vector space} $\mathbb{E}_4$ is introduced by a skew circulant matrix. The resulting algebra $\mathbb{W}$ over $\mathbb{R}$ is isomorphic to $\mathbb{C} \times \mathbb{C}.$ The related algebraic, geometrical, and topological properties are given.There are subplanes of $\mathbb{W}$ isomorphic to the Gauss and Clifford complex number planes. A topology on $\mathbb{W}$ is given by a norm which is a sum of two norms. A hint how to apply this 4 dimensional algebra over $\mathbb{R}$ to the 12-tone Equally Tempered Tuning algebra is given.