Development of a non-destructive bell-tuning technique through optimized structural modifications

Carillons are musical instruments constituted by a set of bells that forms a musical scale encompassing several octaves. For carillons to work in a musically satisfying manner, bell tuners pre-define commonly accepted harmonic relationships between the first five modal frequencies of all bells, typically 0.5:1:1.2:1.5:2 (internal tuning), and a specific target pitch for each bell (external tuning) in order to suit a given musical scale. Currently, bell-tuning is performed by removing metal on the inside of the bell wall. In most cases this process is made empirically, through trial and error, frequently leading to ineffective results. Moreover, this approach can weaken the bell structure and is irreversible, a far from ideal situation in the case of historical carillons. Following our recent work on bar tuning, this paper addresses a non-destructive multimodal tuning method, which consists of attaching suitably designed masses to the bells, in order to comply with the pre-defined set of target frequencies. This is achieved through the combination of dynamical modelling and structural modification techniques with optimization methods, in order to compute the optimal location and magnitude of the tuning masses. First, a full-sized finite element model of a real-life bell is used to compute the modal properties of the original, imperfectly tuned, system. A physical model based on the modal formulation is then built from the modes of the original system, allowing the reduction of the number of equations by several orders of magnitude, compared to the finite element model. This modal formulation is particularly effective for the subsequent iterative optimization process, leading to significant improvements in convergence, as well as computational efficiency. Outcomes from this work attest the effectiveness and robustness of the proposed tuning method and provide encouraging results towards the development of feasible non-destructive methods for bell-tuning.