用黄金三角形构建音乐理论新模型，统一表示大小调与和弦关系
Golden Tonnetz
- 基于黄金三角形设计7音排列，映射大小调及其核心和弦
- 通过黄金三角形与胚体的变换，实现调性转换的几何表达
- 适合音乐数学、理论作曲及跨学科艺术研究者
在半音圈中，12个音高以圆周上的12个点表示；在Tonnetz中，和声关系由三角格网表示。近期我们发现，正二十面体上若干音高的排列可通过黄金比例关联至半音阶、全音阶、大调与小调。本文进一步探究音乐与黄金比例的另一种联系：证明存在一种7音在黄金三角形上的排列方式，可表示特定大小调及其主音、属音、下属音和弦。基于此，提出“黄金Tonnetz”——用黄金三角形或胚体表示所有大小调与三和弦，并以三角形间变换表示新里曼理论中的相对、平行及导音交换等调性转换。
原文摘要 · Abstract (English)
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
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