Music, Mathematics, and Magnetic Ordering
Year: 1999 Authors: Richard Krantz; Jack Douthett; John Clough
Core claim
Mathematical tools from maximally even set theory unify musical, extra-musical, and magnetic-ordering problems under a shared interval-based framework.
Topics
maximally even sets, circular distributions, Ising model, magnetic ordering, music theory
Domains
combinatorics, discrete mathematics, mathematical physics, interval analysis, music theory, sound structure, visual patterning
Methods
interval counting, recursive definition, periodic boundary conditions, spin configuration analysis
Media
12-note equal-tempered scale, round table seating, one-dimensional lattice, piano keyboard
Paper text
The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.
BRIDGES Mathematical Connections in Art, Music, and Science
Music, Mathematics, and Magnetic Ordering
Richard Krantz Department of Physics Metropolitan State College of Denver Denver, Colorado 80217-3362 krantzr@mscd.edu
Jack Douthett Department of Arts and Science Albuquerque Community College and Technical Vocational Institute Albuquerque, New Mexico 87106
John Clough Department of Music SUNY at Buffalo Amherst, New York 14260
Consider the problem of seating a dichotomous group (say, five scientists and three musicians) about a round table so as best to intermingle the two constituencies. There is an intuitive solution (up to rotation), at least for small numbers. The distribution of the seven months with 31 days among the 12 months of the year appeals to a similar intuition. Many Western musical scales and chords (considered as a selection from the 12 notes of the equal-tempered scale) reflect comparable distributions—called Maximally Even (ME) sets [1]. These musical patterns (and corresponding extra-musical patterns: seating arrangements, etc.) are well defined mathematically in terms of intervals between pairs of selected notes: specific intervals (measured by counting over the full 12-note scale) associate in a particular way with generic intervals (measured by counting only the selected notes). The definition of ME applies recursively to account for three-note chords (triads) drawn from the Western diatonic scale, as well as the seven-note gramas of ancient Indian music [2]. More generally, any set of notes (or extra-musical circular distribution) may be characterized in terms of its degree of evenness (minimal through maximal) through further interval analysis [3].
The physical connection is apparent in the application of the music theory formalism (mentioned above) to the interaction of Ising spins [4] on a one-dimensional lattice. The distribution of white and black keys on a piano can be visualized as a distribution of white and black dots on a line. Each white dot can represent an up spin-1/2 site on the lattice and each black dot can represent a down spin-1/2 site. In the Ising model the up (down) sites interact pairwise. In solid state physics (for large systems, i.e., the “keyboard” gets large) it is mathematically convenient to use periodic boundary conditions in which the “last” spin in the line interacts with the “first” spin and minimizes the energy of interaction of all (or a subset of) spins. By invoking periodic boundary conditions (referred to as “octave equivalence” in music theory) the mathematical theory developed to analyze music (Western as well as Eastern) can be extended to address the problem of interacting one-dimension Ising spins [5].
296 Richard Krantz, Jack Douthett, and John Clough
The confluence of music theory, mathematics, and physics leads to surprising connections among seemingly disjoint disciplines. Further generalization of the mathematics of music theory to the physics of magnetic ordering leads to interesting fractal magnetic phase diagrams [6] which has been discussed previously in the technical literature [7].
- Clough and Douthett. 1991. “Maximally even sets.” J. Music Theory 35: 93-173.
- Clough, Douthett, Ramanathan, and Rowell. 1993. “Early Indian heptatonic scales and recent diatonic theory.” Music Theory Spectrum 15/1: 36-58.
- Block and Douthett. 1994. “Vector products and intervallic Weighting.” J. Music Theory 37: 21-41.
- Simon, The Statistical Mechanics of Lattice Gases, (Princeton U. P., Princeton, 1993).
- Douthett and Krantz. 1996. “Energy extremes and spin configurations for the 1-dimensional antiferromagnetic Ising model with arbitrary-range interaction.” J. Math. Phy. 37: 3334-3353.
- Krantz, Douthett and Doty. 1998. “Maximally even sets and the devil’s-staircase phase diagram for the one-dimensional Ising antiferromagnet with arbitrary-range interaction.” J. Math. Phys. 39: 4675-4682.
- Bak. 1986. “The devil’s staircase.” Phys. Today, Dec. 1986, 39-45.