Scaling of brasswind instruments

Consider two musically similar brasswind instruments pitched an octave apart, like trumpet and trombone. The steady-state tones of the trumpet, shifted downwards in pitch an octave by playing a recording at half speed, sound very similar to those of the trombone. The bore shape of the trumpet, however, is not a half-size dimensional replica of that of the trombone. The trumpet bell is about half the size of the trombone bell, but the narrower parts of the trumpet are considerably more than half the diameter of the corresponding parts of the trombone. Acoustic impedance measurements show the Q of air-column resonances to be similar in both trumpet and trombone. This is consistent with the dimensions of the instruments, combined with the dependence of viscous and thermal damping on tubing diameter and frequency. (Viscous and thermal damping dominate radiation damping throughout the normal playing range.) These observations suggest that a plausible scaling law for a family of brasswinds would preserve the pitch-shifted spectrum and the Q of the resonances across members of the family. How well (or poorly) brass makers follow this in practice will be shown from measurements of dimensions and acoustic input impedance on a variety of instruments.