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

TL;DR

This paper introduces a 4-dimensional algebraic structure using skew circulant matrices, explores its properties, and discusses potential applications to 12-tone Equal Tempered Tuning in music theory.

Contribution

It presents a novel 4D algebra over real numbers with specific properties and links it to musical tuning systems, expanding algebraic tools in music theory.

Findings

Algebra is isomorphic to a9 imes a9.

Contains subplanes isomorphic to Gauss and Clifford complex planes.

Provides a topology via a sum of two norms.

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.