Kulkarni Suman, David Sophia U, Lynn Christopher W, Bassett Dani S
ArXiv. 2023 Sep 5:arXiv:2301.00783v2.
Music has a complex structure that expresses emotion and conveys information. Humans process that information through imperfect cognitive instruments that produce a gestalt, smeared version of reality. How can we quantify the information contained in a piece of music? Further, what is the information inferred by a human, and how does that relate to (and differ from) the true structure of a piece? To tackle these questions quantitatively, we present a framework to study the information conveyed in a musical piece by constructing and analyzing networks formed by notes (nodes) and their transitions (edges). Using this framework, we analyze music composed by J. S. Bach through the lens of network science and information theory. Regarded as one of the greatest composers in the Western music tradition, Bach's work is highly mathematically structured and spans a wide range of compositional forms, such as fugues and choral pieces. Conceptualizing each composition as a network of note transitions, we quantify the information contained in each piece and find that different kinds of compositions can be grouped together according to their information content and network structure. Moreover, we find that the music networks communicate large amounts of information while maintaining small deviations of the inferred network from the true network, suggesting that they are structured for efficient communication of information. We probe the network structures that enable this rapid and efficient communication of information--namely, high heterogeneity and strong clustering. Taken together, our findings shed new light on the information and network properties of Bach's compositions. More generally, our framework serves as a stepping stone for exploring musical complexities, creativity and the structure of information in a range of complex systems.
音乐具有复杂的结构，能够表达情感并传递信息。人类通过不完善的认知工具来处理这些信息，从而产生一种对现实的整体的、模糊的认知。我们如何量化一段音乐中包含的信息呢？此外，人类推断出的信息是什么，它与音乐作品的真实结构有何关系（又有何不同）？为了定量地解决这些问题，我们提出了一个框架，通过构建和分析由音符（节点）及其转换（边）形成的网络来研究音乐作品中传达的信息。利用这个框架，我们从网络科学和信息论的角度分析了J.S.巴赫创作的音乐。巴赫被视为西方音乐传统中最伟大的作曲家之一，他的作品具有高度的数学结构，涵盖了广泛的作曲形式，如赋格曲和合唱作品。将每一部作品概念化为一个音符转换网络，我们量化了每一部作品中包含的信息，发现不同类型的作品可以根据其信息内容和网络结构进行分组。此外，我们发现音乐网络在传达大量信息的同时，推断出的网络与真实网络的偏差很小，这表明它们的结构有利于信息的高效传播。我们探究了能够实现这种快速高效信息传播的网络结构，即高度的异质性和强大的聚类性。综上所述，我们的研究结果为巴赫作品的信息和网络特性提供了新的见解。更一般地说，我们的框架是探索音乐复杂性、创造力以及一系列复杂系统中信息结构的一块垫脚石。
