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.
翻译:如果我们的美学偏好会受到自然的分形几何的影响,那么,所有艺术形式,包括音乐,都会出现比例定态。虽然提出了各种统计工具来分析时间序列是否合理,但对于音乐最有意义的复杂性衡量方法,或者如何辨别成份的分形模式,还没有形成共识。在这里,我们提供了一个基于不同时间尺度反复出现的旋律线的自异性的新办法。与最近文献的统计分析不同,拟议方法并不取决于时间窗口中的平均率,而且具有独特的地方性。对分形维度的相应定义以时间缩放等级为基础,并取决于音乐模型图案的图案轮廓。新概念先在康托尔赛和科奇曲线的音乐“翻调”上进行测试,然后应用到一系列精心挑选的、跨越五个世纪的音乐制作的精细精选主构。