Optimization of playing parameters of a trumpet under uncertainty

Abstract

[EN] The sound of a trumpet depends on various parameters such as the mouth pressure and the dynamic properties of the lips of the player. The main objective of this project is to explicitly identify the regions of the parameter space (i.e., the playing conditions) to play in tune for a given range of notes. In this work, a description of the sound generation mechanism in the trumpet is first provided, where the lips act as a basic mechanical oscillator with a single degree of freedom. This oscillator is non-linearly coupled to the air column of the instrument by an airflow equation. All of these components combine to form a model consisting of three ordinary differential equations (ODEs) that must be solved simultaneously. By solving these equations, it becomes possible to obtain the pressure profile for various sounds generated by modifying some of the governing parameters in the acoustics of the trumpet. With the pressure profile it is possible to obtain information regarding the nature of the oscillation, like its frequency and periodicity. Afterwards, a space of parameters is defined for study. Specifically, two parameters are deeply analyzed: the pressure in the mouth of the trumpet player and the frequency of their lips. By varying these parameters, a wide range of different notes can be produced, even without changing the configuration of the trumpet (i.e., without any of the pistons being touched). The main part of the project lies in the construction of frequency maps within this space of parameters, and identifying different regimes of playability. To accomplish this, an initial step was performed using a clustering tool to group all the obtained playing frequencies within a certain set of parameters. The clusters were based on the similarity between the playing frequencies and the expected harmonic series in a trumpet with the specified configuration. After that, a machine learning tool called SVM was used to classify the different regimes by establishing boundaries between them. The regimes classified have been 3: the sound oscillations from the soundless ones, the periodic from the quasi-periodic and the oscillations in tune with respect to the different harmonics from the oscillations out of tune. Each classification has been performed for every frequency cluster. A criteria has also been defined in order to build each type of boundary. The ultimate goal of this project would be to perform an optimization to try to reach more regions where the sound is in tune, in a higher dimensional space of parameters and with a more sophisticated acoustic model, in order to eventually find the geometrical parameters of the trumpet that will lead to the most optimal design in terms of the easiness to play in tune. During the course of this project, unexpected problems and challenges have arisen and needed to be solved, so the optimization of the parameters has not been achieved. However, a glimpse of the optimization process is explained in the last section, where the Kriging method is used to obtain a map of all the space by making an interpolation with calculated samples.

[ES] Univ. Arizona, bajo la supervisión de Sami Missoum

El sonido de una trompeta depende de varios parámetros, como la presión de la boca y las propiedades dinámicas de los labios del intérprete. El objetivo principal de este proyecto es identificar explícitamente las regiones del espacio de parámetros (es decir, las condiciones de ejecución) para tocar afinadas en un rango determinado de notas. En este trabajo, primero se proporciona una descripción del mecanismo de generación de sonido en la trompeta, donde los labios actúan como un oscilador mecánico básico con un solo grado de libertad. Este oscilador está acoplado de forma no lineal a la columna de aire del instrumento mediante una ecuación de flujo de aire. Todos estos componentes se combinan para formar un modelo que consta de tres ecuaciones diferenciales ordinarias (EDO) que deben resolverse simultáneamente. Al resolver estas ecuaciones, es posible obtener el perfil de presión para varios sonidos generados modificando algunos de los parámetros que rigen la acústica de la trompeta. Con el perfil de presión es posible obtener información sobre la naturaleza de la oscilación, como su frecuencia y periodicidad. Posteriormente se define un espacio de parámetros para su estudio. En concreto, se analizan en profundidad dos parámetros: la presión en la boca del trompetista y la frecuencia de sus labios. Al variar estos parámetros, se puede producir una amplia gama de notas diferentes, incluso sin cambiar la configuración de la trompeta (es decir, sin tocar ninguno de los pistones).