用几何配置重构音乐和声网络，揭示调性关系新结构
Configurations, Tessellations and Tone Networks
- 将音程网络建模为12点12线的几何配置，每线3点、每点3线
- 发现其符合特定类型配置{12_3}，具有对称性和双重表示特性
- 为作曲与和声分析提供新视觉化工具，适合音乐理论研究者
调性网络通常以三角形网格铺满平面来表示，也可视为一个三度二分图，包含12个大三和弦顶点与12个小三和弦顶点。我们证明该莱维图可在ℝ²中几何实现为12个点与12条线的配置，满足每条线上有三个点、每个点处有三条线相交，即达布尔布斯基-冯-施特恩尼克型D222的{12_3}配置。这一调性网络几何结构及其各类推广，可作为音乐创作与分析的新基础。
原文摘要 · Abstract (English)
The tonnetz, which is commonly represented as a tessellation of the plane by a triangular network of tones, can also be represented as a bipartite graph of degree three with twelve vertices denoting major triads and twelve vertices denoting minor triads. We show that this Levi graph can be realized geometrically as a system of twelve points and twelve lines in $\mathbb R^2$ with the property that three points lie on each line and three lines pass through each point, in a configuration $\{12_3\}$ of Daublebsky von Sterneck type D222. This tonnetz configuration, alongside various generalizations thereof, can be used as a new basis for the composition and analysis of music.
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