Symmetries and dynamics of musical scales
Alpar Sevgen · The Journal of the Acoustical Society of America · 2001
Configurational potential energy, as well as complexity, may be used as a measure for musical scales. To show this, complexities C(n) and energies E(n) of equally tempered scales with the interval structure n={n1,n2,...,nM} for N=12 semitones and M=7 notes are compared. Complexity is the sum of sharps and flats in a scale. Energy E(n) is due to an assumed interaction energy nα between the notes. It is found that the symmetries of C(n) and E(n) are the same: rotations and reflection (i.e., cyclical permutations and reverse ordering, respectively, of the components of n), and note↔no-note transformation (e.g., major↔pentatonic). But beyond that, the complexity and energy models are apparently unrelated. Symmetries explain the groupings of scales but not the differences in complexity or energy of the groups. There is, however, a remarkable agreement between the C(n) and E(n) tables for a range of values of α. (As an example, the harmonic oscillator value α=2 is employed.) That these tables display an almost identical pattern in the ordering of 38 groups comprising 462 scales of 12-tuples implies that energies E(n) can also be used as a measure for musical scales.