Multistability of saxophone oscillation regimes and its influence on sound production

Abstract

The lowest fingerings of the saxophone can lead to several different regimes, depending on the musician’s control and the characteristics of the instrument. This is explored in this paper through a physical model of saxophone. The harmonic balance method shows that for many combinations of musician control parameters, several regimes are stable. Time-domain synthesis is used to show how different regimes can be selected through initial conditions and the initial evolution (rising time) of the blowing pressure, which is explained by studying the attraction basin of each stable regime. These considerations are then applied to study how the produced regimes are affected by properties of the resonator. The inharmonicity between the first two resonances is varied in order to find the value leading to the best suppression of unwanted overblowing. Overlooking multistability in this description can lead to biased conclusions. Results for all the lowest fingerings show that a slightly positive inharmonicity, close to that measured on a saxophone, leads to first register oscillations for the greatest range of control parameters. A perfect harmonicity (integer ratio between the first two resonances) decreases first register production, which adds nuance to one of Benade’s guidelines for understanding sound production. Thus, this study provides some a posteriori insight into empirical design choices relative to the saxophone.

The bichromator as the quintessence of a saxophone and the question of harmonicity

“…HM stands for Helmholtz Motion. beyond the extinction threshold of the Helmholtz motion [4][5][6]. The branch of the Helmholtz motion is connected to the branch of the second regime (at the octave) whose direct threshold is…”

Nuancing Stability for Self-Sustained Musical Instruments: Transient Durations and Basins of Attraction

“…In the long term, the goal is to apply them on complete self-sustained musical instrument models. Such models are presented in [1,2] (reed instrument models with a modal description of the resonator).…”

“…µ is the control parameter in this study. In the context of wind musical instruments, it can be interpreted as the blowing pressure that causes sound emergence trough linear instability [1].…”

“…Self-sustained musical instruments sometimes have multistable behavior, that is to say that multiple locally stable solutions can coexist for a given set of parameters [1,2]. As a result, it may be both hard to obtain the desired sound for the musician and to predict the stationary regime of the system for the scientist.…”

Parameter identification of a physical model of brass instruments by constrained continuation

“…Some techniques such as numerical continuation are particularly relevant in order to compute periodic solution branches of wind instrument models [11]. Such a technique has been applied to woodwind instruments [12] and brass instruments [13,14], and recently to the objective comparison of trumpets, through the extraction of descriptors calculated from the solution branches obtained by numerical continuation [15].…”