Characterizing the clarinet tone: Measurements of Lyapunov exponents, correlation dimension, and unsteadiness

The clarinet tone is produced by a self-sustained oscillation involving nonlinearity between the flow through the reed and the mechanical response of the reed, and acoustic coupling via the air column response. Regimes of oscillation include periodic, biperiodic, and other quasi-periodic signals, yet even a nominally periodic tone has small, but perceptually and musically important, deviations that are obscured in a conventional power spectrum. Such deviations may be due to the nonlinear dynamics underlying sound production or to perturbations in the performer's control of the instrument via changes in lip embouchure, blowing pressure, and vocal tract configuration. Techniques based upon experimental nonlinear dynamics and short-time signal processing are applied to the acoustic signal measured within the clarinet mouthpiece to assess the role of these additional deviations on sound production. These include the Lyapunov exponent, correlation dimension, and a normalized period-synchronous energy variance, termed unsteadiness. Normal tones and multiphonics are indistinguishable with respect to their Lyapunov exponents. The largest exponent is small and positive, indicating a small amount of information loss each cycle. The information in clarinet tones diminishes at rates ranging from 10 to 60 bits/s. Unsteadiness accounts for the variations in correlation dimension for normal tones but not for multiphonics. These measures may be useful in the study of more subtle aspects of tone production in wind instruments.