Modelling the una-corda effect in pianos
Pablo Miranda Valiente, Giacomo Squicciarini, David Thompson, David Owen Norris, Cesar Hernandez · Journal of Physics Conference Series · 2024
Abstract Most notes of a piano are fitted with two or three strings that, in normal playing conditions, are simultaneously struck by the hammer. In grand pianos, the una corda pedal offers alternative musical effects, notably a softer and duller tonal quality. This pedal’s operation is linked to a lever system that displaces the action to one side causing the hammer to hit only one-out-of-two or two-out-of-three strings, thereby altering the sound. Meanwhile, the other strings of the same note undergo sympathetic vibration, owing to their structural connection through the bridge. This paper introduces a dynamic model that replicates the effects of the una corda pedal. It consists of a state-space scheme, serving as a framework to couple the dynamic behaviour of the various components. Stiff-string models describe the strings’ vibration in three dimensions while a reduced modal model captures the dynamics of the soundboard. Results show how the string vibration and the force transmitted to the soundboard are affected by applying hammer excitation to individual or multiple strings. When all strings are struck, the transverse vibration shows beating. This effect is audible and evidenced in the spectrograms, where the amplitude of partials oscillates over time. In the una corda case, the force exerted on the bridge by the passive string increases initially with time, and its contribution to the overall transmitted force is smaller. However, there is no beating, the decay is more even, and the spectrograms do not show irregularities over time.