[论文解读] Harmony Explained: Progress Towards A Scientific Theory of Music

Daniel Shawcross Wilkerson|arXiv (Cornell University)|Feb 20, 2012

Music Technology and Sound Studies被引用 3

一句话总结

本文通过将物理声学与计算感知相结合，提出了一种音乐和声的科学理论，通过感知一致性和和声简洁性原则，推导出大调音阶、标准和弦字典以及大调与小调三和弦之间的情感差异。该理论挑战了赫尔曼·冯·赫尔姆霍兹的经典‘协和’理论，表明和声源于结构化的互动，而不仅仅是减少的不协和。

ABSTRACT

Most music theory books are like medieval medical textbooks: they contain unjustified superstition, non-reasoning, and funny symbols glorified by Latin phrases. How does music, in particular harmony, actually work, presented as a real, scientific theory of music? The core to our approach is to consider not only the Physical phenomena of nature but also the Computational phenomena of any machine that must make sense of sound, such as the human brain. In particular we derive the following three fundamental phenomena of music: * the Major Scale, * the Standard Chord Dictionary, and * the difference in feeling between the Major and Minor Triads. While the Major Scale has been independently derived before by others in a similar manner [Helmholtz1863, Birkhoff1933], I believe the derivation of the Standard Chord Dictionary as well as the difference in feeling between the Major and Minor Triads to be original. We show to be incomplete the theory of the heretofore agreed-upon authority on this subject, 19th-century Physicist Hermann Helmholtz [Helmholtz1863]: he says notes are in "concord" because the sound playing them together is "less worse" than that of some other notes. But note that, in this theory, more notes can only penalize, some merely less than others, and so the most harmonious sound should be a single note by itself(!) and harmony would not exist as a phenomenon of music at all. I intend this article to be satisfying to scientists as an original contribution to science and art, yet I also intend it to be approachable by musicians and other curious members of the general public who may have long wondered at the curious properties of tonal music and been frustrated by the lack of satisfying, readable exposition on the subject. Therefore I have written in a deliberately plain and conversational style, avoiding unnecessarily formal language.

研究动机与目标

• 开发一种科学严谨、基于实证的音乐和声理论，以可测试的原则取代模糊的传统解释。
• 通过建模物理声波与感知计算，填补现有音乐理论的根本空白——即缺乏对为何某些和弦与音程听起来和谐的连贯解释。
• 通过展示和声源于结构化的多音互动，而非仅减少不协和，来解决赫尔姆霍兹理论中的逻辑缺陷，该理论暗示单个音符应是最和谐的。
• 通过使用通俗易懂的口语化语言，同时不失科学深度，使理论对科学家和音乐家都易于理解。

提出的方法

• 将人类听觉系统建模为处理声音的计算机器，将感知一致性视为和声的核心原则。
• 通过分析音乐音程的频谱，识别出产生最小感知冲突和最大协和性的和谐比例。
• 通过识别在协和性、音阶连贯性和感知编码效率之间取得平衡的一系列音程，推导出大调音阶。
• 通过将音阶扩展为能最大化和声稳定性和感知独特性的三和和弦，构建标准和弦字典。
• 通过分析其和声频谱及大脑处理中产生的感知张力或解决感，区分大调与小调三和弦。
• 通过展示和声依赖于建设性互动而非仅减少不协和，来否定赫尔姆霍兹的“更不差”不协和模型。

实验结果

研究问题

• RQ1 为何大调音阶在调性音乐中成为基本结构？其构建遵循何种原则？
• RQ2 从感知与物理一致性的角度，什么定义了标准和弦字典？
• RQ3 为何大调与小调三和弦尽管结构相似，却引发不同的情感反应？
• RQ4 若赫尔姆霍兹的模型暗示和声本不应存在，科学的和声理论如何解释和声作为现象的存在？
• RQ5 大脑对声音的计算处理在塑造和声感知中扮演何种角色？

主要发现

• 大调音阶被推导为一种感知上最优的音程序列，其在协和性与音阶连贯性之间取得平衡，源于和声频谱与感知效率。
• 标准和弦字典被证明是将大调音阶扩展为能最大化和声稳定性和感知独特性的三和和弦的自然结果。
• 大调与小调三和弦之间的情感差异源于三度音程的相对位置，该位置改变了和声频谱，从而在大脑处理中产生不同的感知张力或解决感。
• 该理论通过证明和声依赖于多个音符之间的建设性互动，而非仅减少不协和，从而否定了赫尔姆霍兹的‘协和’模型。
• 该模型解释了为何和声作为一种现象存在：它源于结构化的、计算高效的声学处理，而非不协和的缺失。
• 本文为统一的、基于科学的音乐理论奠定了基础，整合了物理学、感知与音乐结构。