Fig 1.
Dissonance function D (blue) as a function of logarithmic pitch difference Δp over one octave, used here as the interaction between neighboring tones. For comparison, a more typical interaction energy ∝(1 − cos(Δθ)) is shown in red.
Fig 2.
Right: Histograms of pitches p over one octave on the simulated lattice, as temperature is quenched at τQ = 3276t0 from above Tc at T = 40D0 to near Tc ≈ 27D0, and then towards T = 0. The 12-fold octave division used in Western music emerges near T = Tc illustrated by the 12 equally spaced black lines. Left: Simulated lattice of tones at the same simulation steps shown at right. Domains of nearly constant pitch emerge at low temperature, separated by discrete domain walls.
Fig 3.
(a) Magnitude of order parameters |Mk| vs. T for a slow temperature sweep up, showing k = 1 − 20. Prominent values of k are indicated in legend. (b) Same as (a) but for a faster quench down. (c) Example of determination of one value of τ as the area of the shaded region under the curve of the magnitude of the autocorrelation R7(t) of the M7 order parameter over time until R7 first crosses 0.1. (d) Example of determination of one value of ξ by performing an exponential fit to the correlation function C7(r). Due the finite system and simulation time, both Rk and Ck have small imaginary parts that we neglect.
Fig 4.
(a) τ vs. ε extracted from ten simulation runs with the same parameters on a log-log scale. Power law fit is shown in red, yielding zν = 1.39±0.03 and τ0 = (1.1±0.1) × 10−2t0. (b) Same as (a) but showing ξ vs. ε from 15 runs. At each ε, C7(r) was extracted from each run and averaged together before fitting to find ξ. Power law fits are shown in red, yielding ν = 0.45±0.04 and ξ0 = 0.25±0.05, with ν varied in the fit (solid line) or ξ0 = 0.101±0.005 with fixed ν = 0.66 (dashed line).
Fig 5.
Comparison to KZM predictions.
(a) Final simulation state at T = D0, following a quench at three τQ. Color represents pitch p at each lattice site. As τQ increases, larger domain size, lower concentration of defects, and larger can be seen by eye. (b) ξ vs. ε for the three simulation runs shown in (a), extracted by fitting to C7(r). (c) Values of
(blue points) extracted from simulation runs with different τQ, compared to the prediction of the KZM (red).
Fig 6.
Arrangement of pitches in the quenched state.
(a) The Tonnetz, with pitch indices labeling nodes and edges representing consonant intervals. Edges connecting nodes across the graph are indicated by repeated nodes shown in gray. (b) A 2D slice from the simulated 3D lattice at T = 5 D0 following a quench at τQ = 4368t0. Each pixel represents a lattice site, colored according to pitch. (i) shows a domain wall with an interval of p4 or p5, (ii) shows a minor triad vortex, (iii) encircles another triad vortex and two tetrad vortices.
Fig 7.
Vortex strings in the quenched state.
(a) Network of vortex strings in the quenched state of a simulation run with τQ = 1638t0. Color gradient is applied along the x-axis to improve clarity. (b) Zoom in of a small region of (a), showing vortex strings in red. Pitches surrounding the vortex at each string segment are indicated by four colored dots around the string segment. Vortices corresponding to major or minor triads are labeled by the root pitch index and M or m respectively. Inset shows a portion of the Tonnetz spanning the pitches appearing in (b).