Musical pitch quantization as an eigenvalue problem

Abstract

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.

Keywords:

• scales
• continuum
• discrete
• quantum cognition
• transposition symmetry
• circle of fifths
• cyclic groups

2010 Mathematics Subject Classification:

• 81Q05
• 58D19
• 81R40

Acknowledgments

We thank Reinhard Blutner and Thomas Noll for helpful comments. We are gratefully indebted for the helpful comments of two anonymous referees, and acknowledge the conspicuous service of the editorial board.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This definition can be justified by an experimental finding of Krumhansl and Shepard (1979) using quartertones as probes in a tonal attraction experiment. The attraction rates at quartertones interpolate those of neighboring semitones.

2 This has an interesting physical interpretation: Periodic potentials in the Hamiltonian describe the Bravais lattices of crystals in solid state physics. The Schrödinger equation in configuration space is solved by Bloch waves (Kramers 1935; Kohn 1959) for canonically conjugated lattice sites and wave vectors. In particular, Kramers (1935) proved the emergence of the Lie group $\mathrm{S}\mathrm{L}(2,\mathbb{C})$ as an essential symmetry. This group is also relevant for the canonical transformation of intervals and their respective multiplicities in mathematical musicology (Noll 2007; Clampitt and Noll 2011).

3 With a small caveat: enharmonic equivalence leads to exactly the same sounds but with different names for single notes or for intervals, while octave equivalence leads to different sounds, having an n-octave distance between them, but considered within the same pitch class and hence with the same note names.

4 Condition (43(43) $\tan \frac{q_{\mathrm{\infty }}(a-b)}{2}=\pm \mathrm{\infty }$(43) ) leads accordingly to $\frac{q(a-b)}{2}=\frac{\pi }{2}+p\pi$ that is solved under the same considerations below.

5 A functor is a morphism between categories. A category is constituted by objects and morphisms between them, satisfying associative and identity properties (Mac Lane 1978).