Useche Jorge, Hurtado Rafael
Departamento de Física, Universidad Nacional de Colombia, Carrera 45 No. 26-85, Bogotá 111321, Colombia.
Entropy (Basel). 2019 May 25;21(5):532. doi: 10.3390/e21050532.
One of the most relevant features of musical pieces is the selection and utilization of musical elements by composers. For connecting the musical properties of a melodic line as a whole with those of its constituent elements, we propose a representation for musical intervals based on physical quantities and a statistical model based on the minimization of relative entropy. The representation contains information about the size, location in the register, and level of tonal consonance of musical intervals. The statistical model involves expected values of relevant physical quantities that can be adopted as macroscopic constraints with musical meaning. We studied the occurrences of musical intervals in 20 melodic lines from seven masterpieces of Western tonal music. We found that all melodic lines are strictly ordered in terms of the physical quantities of the representation and that the formalism is suitable for approximately reproducing the final selection of musical intervals made by the composers, as well as for describing musical features as the asymmetry in the use of ascending and descending intervals, transposition processes, and the mean dissonance of a melodic line.
音乐作品最相关的特征之一是作曲家对音乐元素的选择和运用。为了将旋律线作为一个整体的音乐属性与其组成元素的属性联系起来，我们提出了一种基于物理量的音程表示法和一种基于相对熵最小化的统计模型。该表示法包含有关音程的大小、在音区中的位置以及音调和谐程度的信息。统计模型涉及相关物理量的期望值，这些期望值可作为具有音乐意义的宏观约束。我们研究了来自西方调性音乐七部杰作的20条旋律线中的音程出现情况。我们发现，所有旋律线在表示法的物理量方面都有严格的顺序，并且该形式体系适合于近似再现作曲家对音程的最终选择，也适合于描述诸如上行和下行音程使用中的不对称、移调过程以及旋律线的平均不和谐度等音乐特征。
