作为特征值问题的音高量化

IF 0.5 2区数学 Q4 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Journal of Mathematics and Music Pub Date : 2020-05-24 DOI:10.1080/17459737.2020.1763488

Peter beim Graben, Maria Mannone

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引用次数: 7

摘要

离散的音高和和弦是如何从连续的声音中出现的?利用调性音乐的量子认知模型，证明了连续音高变换下傅里叶空间中相关的Schrödinger方程是不变的。然而，这种对称在和弦调换的情况下被打破，需要一个离散的循环群作为调换对称。我们的研究将量子力学与音乐联系起来，这与赫尔曼·冯·亥姆霍兹的音乐理论和开创性见解是一致的。

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Musical pitch quantization as an eigenvalue problem

How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum mechanics with music and is consistent with music theory and seminal insights by Hermann von Helmholtz.

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来源期刊

Journal of Mathematics and Music 数学-数学跨学科应用

CiteScore

1.90

自引率

18.20%

发文量

审稿时长

>12 weeks

期刊介绍： Journal of Mathematics and Music aims to advance the use of mathematical modelling and computation in music theory. The Journal focuses on mathematical approaches to musical structures and processes, including mathematical investigations into music-theoretic or compositional issues as well as mathematically motivated analyses of musical works or performances. In consideration of the deep unsolved ontological and epistemological questions concerning knowledge about music, the Journal is open to a broad array of methodologies and topics, particularly those outside of established research fields such as acoustics, sound engineering, auditory perception, linguistics etc.