Music viewed by its entropy content: A novel window for comparative analysis

Polyphonic music files were analyzed using the set of symbols that produced the Minimal Entropy Description, which we call the Fundamental Scale. This allowed us to create a novel space to represent music pieces by developing: (a) a method to adjust a textual description from its original scale of observation to an arbitrarily selected scale, (b) a method to model the structure of any textual description based on the shape of the symbol frequency profiles, and (c) the concept of higher order entropy as the entropy associated with the deviations of a frequency-ranked symbol profile from a perfect Zipfian profile. We call this diversity index the ‘2nd Order Entropy’. Applying these methods to a variety of musical pieces showed how the space of ‘symbolic specific diversity-entropy’ and that of ‘2nd order entropy’ captures characteristics that are unique to each music type, style, composer and genre. Some clustering of these properties around each musical category is shown. These methods allow us to visualize a historic trajectory of academic music across this space, from medieval to contemporary academic music. We show that the description of musical structures using entropy, symbol frequency profiles and specific symbolic diversity allows us to characterize traditional and popular expressions of music. These classification techniques promise to be useful in other disciplines for pattern recognition and machine learning.

本研究采用生成最小熵描述（Minimal Entropy Description）的符号集合对复调音乐文件进行分析，我们将该符号集合命名为基础尺度（Fundamental Scale）。借此，我们构建了一种用于表征音乐作品的全新空间，并在此过程中提出了三项方法与概念：(a) 一种可将文本描述从原始观测尺度转换至任意选定尺度的方法；(b) 一种基于符号频率分布（symbol frequency profiles）的形态特征，对任意文本描述的结构进行建模的方法；(c) 将高阶熵定义为与按频率排序的符号分布与理想齐夫分布（Zipfian distribution）之间偏差相关联的熵的概念。我们将该多样性指数命名为“二阶熵（2nd Order Entropy）”。将这些方法应用于各类音乐作品后，我们发现“符号特异性多样性-熵空间（symbolic specific diversity-entropy）”与“二阶熵空间”能够有效捕捉不同音乐类型、风格、作曲家与流派所独有的特征。实验结果显示，各类音乐范畴的相关属性均呈现出一定的聚类特性。借助这些方法，我们可以在该空间中可视化展示学院派音乐从中世纪到当代的历史发展轨迹。研究表明，通过熵、符号频率分布以及符号特异性多样性对音乐结构进行描述，能够有效刻画传统音乐与流行音乐的表现形式。此类分类技术有望在其他学科的模式识别与机器学习领域得到广泛应用。