Gauge symmetries of musical and visual forces

Abstract

After reviewing the physicalistic or metaphorical accounts to musical and visual forces by Arnheim and Larson, respectively, which were inspired by the basic tenets of gestalt psychology, I present a novel, naturalistic, mathematical framework, based on symmetry principles and gauge theory. In musicology, this approach has already been applied to the phenomenon of tonal attraction, leading to a deformation of the circle of fifths. The underlying gauge symmetry turns out as the SO(2) Lie group of a musical quantum model. Here, I present an alternative description in terms of Riemannian geometry. Its essential constraint of invariance of the infinitesimal line element leads to a deformation of the circle of fifths into a heart of fifths. In vision, the same approach is applied to Fraser's twisted cord illusion where concentric circles are deformed to squircle objects by means of an optical gauge field induced through a checkerboard background.

GRAPHICAL ABSTRACT

Keywords:

• Music
• vision
• force
• gauge invariance
• Riemannian geometry
• deformation
• quantum cognition

Acknowledgments

I am gratefully indebted to Maria Mannone, Thomas Noll and Reinhard Blutner for fruitful discussions. I also appreciate computational resources provided by BTU Cottbus-Senftenberg through a guest scientist account.

Disclosure statement

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

Notes

1 Confer Costère's ‘law of cardinal gravitation’ (Ellard, 1973, Part II, p. 3).

2 Note the differences between Lerdahl's Table 1 and Costère's ‘cardinal table’ for the C major scale (Ellard, 1973, Part II, p. 28, Figure 8).

3 We here refer to the latest version of the hierarchical model presented in Lerdahl (1996), omitting another fifth level.

4 I used MATLAB's edge detector, invoking edge(Image, ‘log’, 0.03, 1.5) with Image being a jpg file of the Fraser illusion generated from Fraser (1908).

5 Note that the graph is not directed: for each directed edge $(i,k)\in E$, also its directed counterpart $(k,i)\in E$. Moreover, the graph Γ contains also all fundamental loops $(i,i)\in E$, which are omitted for the sake of clarity in Figure 7.

6 Note that we follow the convention for the notation of adjacency matrices in graph theory here. Hence, first index: destination, second index: source.

7 Note that my present exposition slightly deviates from that given in beim Graben (2016) in order to facilitate its generalization to the case of continuous manifolds.

8 Note that we normalize particular physical quantities such as mass m, charge q, light speed c and Planck's quantum of action ℏ to natural units $=1$ here.

9 My propaedeutic exposition is somewhat simplified since one has also to distinguish between co- and contragredient vector and tensor components in non-Euclidean geometry.

10 A diffeomorphism is an invertible continuously differentiable mapping on $\mathcal{M}$ whose inverse is continuously differentiable again.

11 Compare with Kandinsky (1947), Costère (1954), and Ellard (1973) for visual arts and musical harmony, respectively.

12 The corresponding German term ‘Quintenherz’ replaces the undeformed ‘Quintenzirkel’ of canonical musicology.

13 Note that Laplacian edge detection exhibits another gauge symmetry. For an edge is defined as a solution curve $p\left(s\right)$ of the Laplace equation $\mathrm{\Delta }u\left(p\right)=0$ for a scalar field $u(x,y)$, it is invariant under the addition of any harmonic function $h(x,y)$ with $\mathrm{\Delta }h=0$. This is the essential gauge symmetry of classical electrostatics (Jackson & Okun, 2001).

14 Nowadays, mathematical physics has clarified that the rigorous transition from statistical mechanics to thermodynamics is only possible through an algebraic field-theoretic approach (Sewell, 2002).