Role of the Resonator Geometry on the Pressure Spectrum of Reed Conical Instruments

Spectra of musical instruments exhibit formants or anti-formants which are important characteristics of the sounds produced. In the present paper, it is shown that anti-formants exist in the spectrum of the mouthpiece pressure of saxophones. Their frequencies are not far but slightly higher than the natural frequencies of the truncated part of the cone. To determine these frequencies, a first step is the numerical determination of the playing frequency by using a simple oscillation model. An analytical analysis exhibits the role of the inharmonicity due to the cone truncation and the mouthpiece. A second step is the study of the input impedance values at the harmonics of the playing frequency. As a result, the consideration of the playing frequency for each note explains why the anti-formants are wider than those resulting from a Helmholtz motion observed for a bowed string. Finally numerical results for the mouthpiece spectrum are compared to experiments for three saxophones (soprano, alto and baritone). It is shown that when scaled by the length of the missing cone, the anti-formant frequencies in the mouthpiece are very similar for the three instruments. The frequencies given by the model are close to the natural frequencies of the missing cone length, but slightly higher. Finally, the numerical computation shows that anti-formants and formants might be found in the radiated pressure.