The harmonic balance method (HBM) was originally developed for finding periodic solutions of electronical and mechanical systems under a periodic force, but has been adapted to self-sustained musical instruments. Unlike time-domain methods, this frequency-domain method does not capture transients and so is not adapted for sound synthesis. However, its independence of time makes it very useful for studying any periodic solution, whether stable or unstable, without care of particular initial conditions in time. A computer program for solving general problems involving nonlinearly coupled exciter and resonator, HARMBAL, has been developed based on the HBM. The method as well as convergence improvements and continuation facilities are thoroughly presented and discussed in the present paper. Applications of the method are demonstrated, especially on problems with severe difficulties of convergence: the Helmholtz motion (square signals) of single-reed instruments when no losses are taken into account, the reed being modeled as a simple spring.

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Abstract.A digital clarinet played by a human and timed by a metronome was used to record two playing control parameters, the breath control and the reed displacement, for 20 repeated performances. The regular behaviour of the parameters was extracted by averaging and the fluctuation was quantified by the standard deviation. It was concluded that the movement of the parameters seem to follow rules. When removing the fluctuations of the parameters by averaging over the repetitions, the result sounded less expressive, although it still seemed to be played by a human. The variation in timbre during the play, in particular within a note's duration, was observed and then fixed while the natural temporal envelope was kept. The result seemed unnatural, indicating that the variation of timbre is important for the naturalness.

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