Hsü K J, Hsü A
Eidgenössische Technische Hochschule, Zürich, Switzerland.
Proc Natl Acad Sci U S A. 1991 Apr 15;88(8):3507-9. doi: 10.1073/pnas.88.8.3507.
Suggestions have been made that computer musicians should attempt to compose fractal music, and questions have been raised whether there is such a thing as fractal music. Voss and Clark observed that music is scaling, or 1/f noise, as analyzed on the basis of the amplitude (loudness) of the audio signals; they failed to find a fractal distribution of acoustic frequencies (music notes) in music. Analyzing Bach's and Mozart's compositions, we have shown that the incidence of the frequency intervals, or of the changes of acoustic frequency, has a fractal geometry. Fractal phenomena are characterized by scale-independency. The purpose of this investigation is to demonstrate the self-similarity of music and to explore its implications.
有人建议计算机音乐家应尝试创作分形音乐，也有人提出是否存在分形音乐这样的东西。沃斯和克拉克观察到，根据音频信号的振幅（响度）分析，音乐是标度不变的，即1/f噪声；他们未能在音乐中找到声频（音符）的分形分布。通过分析巴赫和莫扎特的作品，我们已经表明，频率间隔或声频变化的发生率具有分形几何特征。分形现象的特点是尺度独立性。本研究的目的是证明音乐的自相似性并探讨其意义。
