Effective length of horns

Abstract

Some authors writing about brass musical instruments have used the term ’’effective length,’’ usually meaning the length of a cylindrical tube having the same resonance frequencies as a given horn, but possibly with different end conditions. In this paper, alternative definitions of effective length are considered, and one definition is chosen and generalized to all frequencies, not just discrete resonance frequencies. Within the framework of lossless plane-wave horn theory, a nonlinear first-order differential equation is derived that yields effective length as a function of frequency and horn contour. Effective length has been calculated for some horn contours resembling French horns and trumpets. The solutions are qualitatively consistent with the experience of instrument makers and players, and with the effective lengths of actual instruments, determined from measured resonance frequencies. Subject Classification: 85.60; 75.40.

Subglottal resonances of American English speaking children

“…In Fredberg and Moore (1978), high frequencies were found to penetrate deeper into the tracheo-bronchial system than low frequencies. Similar frequency-dependent penetrations are well documented in the acoustics of horns (Benade, 1990;Pyle, 1975), particularly relevant due to the similarities between horns and the subglottal system (Jackson et al, 1978;Van den Berg, 1960). This frequency-dependent penetration depth is analogous to the effective acoustic length of the subglottal system or tube length of the quarter-wavelength resonator.…”

“…The height-related correction was empirically derived and modeled to account for the observed decrease in Sg3/Sg2 ratio. However, derivation of a correction from the physical principles of horn acoustics (Benade, 1990;Pyle, 1975) may provide additional insight into the subglottal acoustics of both children and adults. This should be addressed in future research.…”

Comparing measured and calculated frequencies of a demonstration trumpet

“…2,3 The measured frequencies f bell in Table I 1 clearly show this effect. For frequencies less than its own resonant frequency, the effective length of the mouthpiece will increase with frequency. 2,6,7 The measured data in Table I show that, with the exception of the n ¼ 1 mode, the presence of the mouthpiece raises the frequencies f mp . The fractional change decreases with increasing frequency, indicating that the effective length of the mouthpiece increases with frequency.…”

Experimenting with brass musical instruments

“…The reason for this effect is that a mouthpiece, which is essentially a cup connected to a tapered tube called the backbore, behaves, when closed by the player's lips, as a Helmholtz resonator [9]. Since a Helmholtz resonator has its own resonant frequency, the mouthpiece will have an effect on the frequencies of a cylindrical tube to which it is attached; the effect will be greatest at the resonant frequency and less at frequencies above and below [10,11].…”

“…It is considered permissible to use a Bessel-horn to model a real instrument if accounting for the presence of the mouthpiece [9]. Setting (4) equal to the expression f n = (2n − 1)c/4L e for the frequencies of a cylindrical tube closed at one end and solving for the equivalent length of cylindrical tubing gives…”