Golden Tonnetz

TL;DR

The paper develops a phi-based geometric framework, the Golden Tonnetz, to represent tonal relationships. It constructs a base seven-tone arrangement on a golden triangle and extends it horizontally and vertically to cover major/minor scales and their triads. The resulting infinite lattice represents all scales and Gregorian modes with golden triangles/gnomons, and maps Neo-Riemannian transformations (relative, parallel, leading-tone) to shape transformations. This approach provides a visually intuitive, topology-based method for analyzing and visualizing tonal structure beyond traditional Tonnetz representations.

Abstract

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

Figures (11)

• Figure 1: a, Golden triangle and gnomon . b, The base figure where the $C$ major scale on the golden triangle. c, Triads represented by golden triangles or gnomons in the base figure. d, Horizontal extension of the base figure. e, Vertical extension of the base figure.
• Figure 2: a, Golden Tonnetz obtained by horizontally and vertically extending the base figure (Fig. \ref{['fig1']}b). b, Major and minor scales represented by golden triangles on the golden Tonnetz. c, Basis of the golden Tonnetz.
• Figure 3: a, Relative, parallel, and Leading-tone exchange transformations by Tonnetz, b, by golden Tonnetz. c, The part of golden Tonnetz used in b.
• Figure 4: a, Gregorian modes (Lydian, Ionian, Mixolydian, Dorian, Aeolian, Phrygian, and Locrian modes) on golden Tonnetz. b, The part of golden Tonnetz used in a.
• Figure 5: a--g, Seven kinds of arrangement of $C$ major scale on the golden triangle where a is shown in Fig. \ref{['fig1']}. Only a is an arrangement consistent with the golden ratio. For example, in the arrangement of b, the triad $CEG$ (tonic) is not represented by a golden triangle or gnomon.