Chaotic dynamics of woodwind multiphonics
Douglas H. Keefe, Bernice Laden · The Journal of the Acoustical Society of America · 1989
Experimental evidence for chaos in multiphonic tones derives from measured power spectra, and more directly from the measured correlation dimension of the reconstructed phase space of the dynamical system. The frequencies fl,m of the line spectral components of a reed-driven woodwind multiphonic fit a biperiodic spectrum of low- to mid-playing levels; i.e., there exist base frequencies f1 < f2 such that fl,m = lf1 + mf2, for non-negative integers l and m. For an alto saxophone multiphonic, these base frequencies are phase locked, namely, their ratio is equal to a ratio of small integers. Using the same nominal saxophone fingering, this measured ratio equals 8 : 5 or 7 : 4 (within 0.1%) depending upon player's adjustment of embouchure and playing level. A broadband spectrum is present in the saxophone multiphonic spectra at all but the lowest levels, which exceeds instrumentation noise and window leakage associated with signal processing. A phase-locked biperiodic spectrum superposed on a broadband background spectrum is characteristic of low-dimensional chaotic attractors in other fluid mechanical systems. At the highest playing levels, period doubling of the f1 component occurs, the broadband level significantly increases, and phase locking is not observed. The dimension of the attractor is measured by embedding the time series in a higher space. The correlation dimension D (Grassberger and Procaccia, 1983) is measured by embedding a single measured time series in a higher-dimensional space, so to reconstruct the phase space of the dynamical system. For the saxophone multiphonic at the lowest and highest playing levels, D = 2.85 ± 0.06 and 2.84 ± 0.12, respectively. These fractal dimensions suggest that multiphonics are strange attractors. Data for other woodwinds will be presented.