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.

中文翻译：

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音乐音高量化作为特征值问题

如何从连续的声音中产生离散的音调和和弦？使用音调音乐的量子认知模型，我们证明了在连续音高转置下，傅立叶空间中的关联Schrödinger方程是不变的。但是，在和弦换位的情况下这种对称性被破坏，需要离散的环状基团作为换位对称性。我们的研究将量子力学与音乐联系起来，并且与音乐理论和Hermann von Helmholtz的开创性见解相一致。