Research ArticlePHYSICS

The structure of musical harmony as an ordered phase of sound: A statistical mechanics approach to music theory

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Science Advances  17 May 2019:
Vol. 5, no. 5, eaav8490
DOI: 10.1126/sciadv.aav8490
  • Fig. 1 Quantifying dissonance.

    (A) Pure-tone dissonance curves dx) at different critical widths wc. (B) Dissonance function Dx) for tones with a sawtooth timbre and a fixed wc = 0.03. Just-tuned commonly used intervals are indicated by red dotted lines. Green dots show x = n/12, for integers n. (C) Fourier components dk of the periodic dissonance function Dpx) corresponding to Dx) shown in (B).

  • Fig. 2 Mean field results.

    (A) Solutions P(x) to Eq. 4 for a range of T and wc = 0.03. (B) Modulus ∣pk∣ of order parameters versus temperature T. The dashed line indicates the predicted Tc1 = − d12/2.

  • Fig. 3 Comparison of peak positions of P(x) versus T from the mean field model with ET or JI tuning.

    Two values are shown for the tritone (the pitch near 600 cents) in JI, corresponding to an augmented fourth or diminished fifth. The mean field result shows equal peaks at both positions.

  • Fig. 4 Phase diagram showing significant order parameters ∣pk∣ versus T and wc.

    dpk∣/dT is plotted with color indicated by the color bar, superimposing all values of k. Regions of a single color indicate phases with one or few dominant pk; white regions indicate phases with many significant pk. The lines show −dk/2 versus wc for values of k corresponding to the color bar. wc corresponding to pitches from C4 (middle C) to C8 are indicated.

  • Fig. 5 Tone lattice simulation results.

    (A) Histograms of pitches x mod 1 in a metastable configuration of the tone lattice following a quench to temperature T. At T = 6, no ordering is observed. At T = 5,4, and 3.5, ordering is observed, with the octave divided into 5, 7, and 12, respectively. (B) Pitches on the tone lattice at T = 3.5. Pitch domains are labeled with pitch indices 0 to 11. Major or minor triads are marked with triangles. Junctions of more than three pitch domains are marked with circles. (C) The Tonnetz (edges shown in blue) with connections between neighboring pitch domains at T = 3.5 shown in red.

  • Fig. 6 Comparison of the pitch class distribution on the tone lattice and in a selected piece of music.

    (A) Histogram of pitch classes 12x mod 12 rounded to the nearest pitch class on the T = 3.5 tone lattice and arranged by ascending fifths. (B) Histogram of pitch classes appearing in Bach’s Prelude and Fugue in D major, BWV850, arranged by ascending fifths. The shaded area corresponds to the notes of a diatonic scale.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/5/eaav8490/DC1

    Fig. S1. Mean field solutions versus T at selected wc and phase diagrams for different timbres and sweep directions.

    Fig. S2. Comparison of F versus T for different timbres and octave periodicities.

    Fig. S3. Dissonance functions Dx) for different root pitches (and hence different wc) in legend, calculated as described in Methods for the lattice simulations.

    Fig. S4. Pitch histograms and spectra from the tone lattice simulation.

    Fig. S5. Distribution of intervals (pitch differences Δx = xixj) Cx, r) as a function of distance r between sites i and j.

    Fig. S6. Metastable or disordered configurations of the tone lattice following a quench to T = 4, T = 5, and T = 6.

    Movie S1. Major diatonic scale on the quenched tone lattice.

    Movie S2. Major and minor triads on the quenched tone lattice.

    Description of MATLAB scripts.

  • Supplementary Materials

    The PDF file includes:

    • Fig. S1. Mean field solutions versus T at selected wc and phase diagrams for different timbres and sweep directions.
    • Fig. S2. Comparison of F versus T for different timbres and octave periodicities.
    • Fig. S3. Dissonance functions Dx) for different root pitches (and hence different wc) in legend, calculated as described in Methods for the lattice simulations.
    • Fig. S4. Pitch histograms and spectra from the tone lattice simulation.
    • Fig. S5. Distribution of intervals (pitch differences Δx = xixj) Cx, r) as a function of distance r between sites i and j.
    • Fig. S6. Metastable or disordered configurations of the tone lattice following a quench to T = 4, T = 5, and T = 6.
    • Legends for movies S1 and S2
    • Description of MATLAB scripts

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    Other Supplementary Material for this manuscript includes the following:

    Files in this Data Supplement:

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