Nonlinear modeling of the ‘‘mitnahme-effekt’’ in coupled organ pipes

Abstract

Two slightly mistuned organ pipes of the same nominal tone placed on neighboring holes of the same groove of the slider chest can synchronize if simultaneously sounded. The frequency difference between the two pipes disappears (mitnahme-effekt) giving up on a common frequency which is, in general, slightly higher than both natural frequencies of the pipes [Angster et al., ‘‘Coupling between simultaneously sounded organ pipes,’’ AES preprint 3534 (F1-2), Berlin (1993)]. A related phenomenon is observed when the separation between the two natural frequencies is bigger than 1%: this gives up on an increased difference frequency. This paper shows that a numerical simulation, where the two coupled pipes are modeled with a pair of coupled nonlinear oscillators, qualitatively obtains both these results. Furthermore the existence of a parameter range fitting all the experimental data is presented.

Synchronization of organ pipes: experimental observations and modeling

“…He describes the so-called "Mitnahme-Effekt" (loosely the take-along effect), known by organ builders [16,17]. In [19] the phenomenon is interpreted in the frame of synchronization theory.…”

“…Hereafter we will use the terms frequency locking or synchronization to be synonymous with the "Mitnahme-Effekt". In [17], the effect has been analyzed heuristically, while Stanzial has investigated the dependence of the frequency locking on the detuning or frequency difference of two single pipes in [16] by acoustic measurement. He modeled the effect by two coupled oscillators, without giving a physical reason of the coupling.…”

“…In [16], the frequencies ν 1 , ν 2 of the coupled pipes were plotted against the frequency detuning of the uncoupled pipes, ∆f . As explained above, a clearer characterization is achieved when the frequency difference ∆ν is plotted versus ∆f [19,32].…”

“…The results are interpreted within the frame of synchronization theory. We give an explanation of why a model of two oscillators works out so well, as indicated in [16]. Further, the dependence of the frequency spectrum on the distance between two detuned pipes is investigated for a fixed amount of detuning.…”

“…Then a synchronization of the pipes, a frequency locking, occurs [14,15,16,17]. This has a direct importance for the arrangement of organ pipes in a common Orgelwerk.…”