Group theoretic description of just intonation

Abstract

In this paper we present a group theoretic description of just intonation. All possible musical intervals belonging to 5-limit just intonation can be represented by the mathematical group {2 p 3 q 5 r |p,q,r ∈Z}. Considering only the intervals within one octave, an isomorphism with the 2-dimensional lattice Z 2 can be made. Plotting the intervals on the lattice according to the isomorphism, we can identify certain connected and convex sets of elements that represent the major and minor diatonic scale and the 12-tone chromatic scale. These sets of elements remain convex and the area each set spans remains invariant under lattice transformations. The fact that the major, minor and chromatic scale arise naturally from our representation of 5limit just intonation triggers discussion about the origin of scales and might suggest that those scales have a mathematical origin. With this, we challenge the framework of the 12-tone system. Our representation of 5-limit just intonation is compared with Balzano's `thirds-space ' in which he builds the intervals of the 12-tone scale from major and minor thirds also using group theory. This comparison gives a new insight on the compromises regarding intonation that are made when introducing a 12-tone temperament. 1.

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