Abstract

We report an approach to obtaining complex networks with diverse topology, here called syntonets, taking into account the consonances and dissonances between notes as defined by scale temperaments. Though the fundamental frequency is usually considered, in real-world sounds several additional frequencies (partials) accompany the respective fundamental, influencing both timber and consonance between simultaneous notes. We use a method based on Helmholtz’s consonance approach to quantify the consonances and dissonances between each of the pairs of notes in a given temperament. We adopt two distinct partials structures: (i) harmonic; and (ii) shifted, obtained by taking the harmonic components to a given power β, which is henceforth called the anharmonicity index. The latter type of sounds is more realistic in the sense that they reflect non-linearities implied by real-world instruments. When these consonances/dissonances are estimated along several octaves, respective syntonets can be obtained, in which nodes and weighted edge represent notes, and consonance/dissonance, respectively. The obtained results are organized into two main groups, those related to network science and musical theory. Regarding the former group, we have that the syntonets can provide, for varying values of β, a wide range of topologies spanning the space comprised between traditional models. Indeed, it is suggested here that syntony may provide a kind of universal complex network model. The musical interpretations of the results include the confirmation of the more regular consonance pattern of the equal temperament, obtained at the expense of a wider range of consonances such as that in the meantone temperament. We also have that scales derived for shifted partials tend to have a wider range of consonances/dissonances, depending on the temperament and anharmonicity strength.

Graphical abstract

中文翻译：

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句法：建立一个受和谐启发的复杂网络通用模型

摘要

我们报告了一种获得具有多种拓扑结构的复杂网络的方法，这里将其称为音调，同时考虑到音阶气质所定义的音符之间的共鸣和不谐调。尽管通常会考虑基音频率，但在实际的声音中，几个附加频率（部分）会与各自的基音一起出现，从而影响木材和同时出现的音符之间的谐音。我们使用一种基于Helmholtz谐音方法的方法来量化给定气质中每对音符之间的谐音和不谐音。我们采用两个不同的部分结构：（i）谐波；（ii）通过将谐波分量取给定功率β来获得，此后称为非谐指数。后一种声音在某种程度上更真实，因为它们反映了真实乐器所暗示的非线性。当沿着几个八度音阶估计这些谐音/不谐音时，可以获得相应的音调，其中节点和加权边缘分别表示音符和谐音/不谐音。获得的结果分为两个主要组，分别是网络科学和音乐理论。关于前一组，我们知道，对于不同的β值，这些音符可以提供，涵盖了传统模型之间空间的广泛拓扑。确实，在此建议，谐振可以提供一种通用的复杂网络模型。结果的音乐诠释包括确认等律性的更规则的和声模式，以更宽范围的和声为代价获得音调，例如在意向性气质中。我们还发现，根据气质和非谐音强度，为偏音分量导出的音阶往往具有更广泛的谐音/不谐音范围。

图形概要