论文标题
音乐中的分形图案
Fractal Patterns in Music
论文作者
论文摘要
如果我们的审美偏好受自然的分形几何形状的影响，则预计规律将以所有的艺术形式出现，包括音乐。尽管已经提出了各种统计工具来分析声音中的时间序列，但关于音乐中最有意义的复杂性衡量，或者如何首先识别构图中的分形模式，尚无共识尚未达成共识。在这里，我们根据各种时间尺度上反复出现的旋律线的自相似性提供了一种新方法。与最近文献中的统计分析相反，所提出的方法不取决于时间浪费的平均，并且是独特的局部情况。分形维度的相应定义基于时间缩放层次结构，并取决于音乐主题的音调轮廓。新概念在Cantor Set和Koch Curve的音乐“演绎”上进行了测试，然后应用于跨越了五个世纪音乐制作的许多精心选择的精巧作品。
If our aesthetic preferences are affected by fractal geometry of nature, scaling regularities would be expected to appear in all art forms, including music. While a variety of statistical tools have been proposed to analyze time series in sound, no consensus has as yet emerged regarding the most meaningful measure of complexity in music, or how to discern fractal patterns in compositions in the first place. Here we offer a new approach based on self-similarity of the melodic lines recurring at various temporal scales. In contrast to the statistical analyses advanced in recent literature, the proposed method does not depend on averaging within time-windows and is distinctively local. The corresponding definition of the fractal dimension is based on the temporal scaling hierarchy and depends on the tonal contours of the musical motifs. The new concepts are tested on musical 'renditions' of the Cantor Set and Koch Curve, and then applied to a number of carefully selected masterful compositions spanning five centuries of music making.
