Golden Tonnetz

Abstract: Musical concepts have been represented by geometry with tones. For example, in the chromatic circle, the twelve tones are represented by twelve points on a circle, and in Tonnetz, the relationships among harmonies are represented by a triangular lattice. Recently, we have shown that several arrangements of tones on the regular icosahedron can be associated with chromatic scales, whole-tone scales, major tones, and minor tones through the golden ratio. Here, we investigate another type of connection between music and the golden ratio. We show that there exists an arrangement of 7 tones on a golden triangle that can represent a given major/minor scale and its tonic, dominant, and subdominant chords by golden triangles. By applying this finding, we propose "golden Tonnetz" which represents all the major/minor scales and triads by the golden triangles or gnomons and also represents relative, parallel, and leading-tone exchange transformations in Neo-Riemannian theory by transformations among the golden triangles and gnomons.