音乐和弦的几何结构
1Department of Music, Princeton University, Princeton, NJ 08544, USA. dmitri@princeton.edu
概括
音乐和弦是几何轨道折叠中的点,线段显示音符的变化. 作曲家利用这些空间的近对称性来创作音乐,和和不和的和弦表明不同的用途.
科学领域:
- 音乐理论 音乐理论
- 计算音乐学的音乐学.
- 几何音乐理论 几何音乐理论
背景情况:
- 音乐和弦可以被概念化为一个称为orbifold的几何空间内的点.
- 这个空间中的线段代表了和弦之间的转换,说明了音乐音符的映射.
- 这些空间的非欧几里德几何学被作曲家在各种风格中利用.
研究的目的:
- 探索音乐和弦及其转换的几何表示.
- 研究作曲家如何在音乐创作中利用圆形几何学的特性.
- 了解和弦对称与音乐应用之间的关系.
主要方法:
- 代表音乐和弦作为点在一个orbifold空间.
- 分析线段作为和弦之间的映射.
- 在和弦中识别近对称性 (翻译,反射,变换).
- 关联和弦对称与音乐使用.
主要成果:
- 在结构上相似的和弦之间发现了代表音乐转换的短线段.
- 这些短段的存在取决于和弦内部的近对称性.
- 同音和异音的和弦表现出明显的近对称性.
- 这些独特的对称性表明不同的潜在音乐应用.
结论:
- 轨道折叠几何学为理解音乐结构和转变提供了一个框架.
- 弦对称是音乐创作中的一个关键因素,影响了转换的选择.
- 弦的几何性质,特别是它们的对称性,直接决定了它们的音乐实用性和应用.
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