Correlation dimension of woodwind multiphonic tones

A multiphonic is a regime of oscillation of woodwind musical instruments that is perceived as two or more simultaneously sounding pitches. The frequencies fl,m of the line spectral components of a measured woodwind multiphonic tone fit a biperiodic spectrum at low- to mid-playing levels. For the saxophone and clarinet multiphonics investigated, the two basis frequencies of the biperiodic spectrum are phase locked, that is, their ratio is equal to a ratio of small integers. A broadband spectrum is present in multiphonic spectra that exceeds instrumentation noise and window leakage associated with signal processing. The correlation dimension D of P. Grassberger and I. Procaccia [Physica D 9, 189–208 (1983)] is measured by embedding a single measured time series in a higher-dimensional space, so as to reconstruct the phase space of the dynamical system. The time delay used in the dimensional reconstruction is chosen using information theory. For the particular multiphonics analyzed, the correlation dimension ranges from 2.5 to 2.9 for the saxophone and from 1.3 to 2.2 for the clarinet. One clarinet multiphonic shows possible additional dynamical complexity at small length scales in the embedding space, with a correlation dimension of 3.3. These results give quantitative evidence that some, but not all, multiphonic tones possess a strange attractor.