Tuning idiophone bar torsional modes with three-dimensional cutaway geometries

Abstract

The bars of marimbas, vibraphones and similar idiophones are tuned by shaping their “cutaway” or “undercut” geometries. Makers commonly shape bar cutaways to tune three flexural modes of vibration. The first flexural mode is tuned to the fundamental frequency of the bar’s musical note. Two other flexural modes are tuned such that their frequencies become specific multiples of the fundamental. The remaining flexural modes and all other modes are left untuned. Makers have complained of untuned torsional modes polluting bar timbre over specific sections of the keyboard. In wooden marimba bars this problem has proven difficult to predict, filling reject bins with valuable tonewood. This work investigates tuning these torsional modes by varying bar cutaway geometry in three dimensions. No additional mass or heterogeneous materials are employed. Modal frequencies are determined via finite element analysis. Mode shapes are identified algorithmically, enabling analyses to explore the parameter space unsupervised. Geometries are tuned using a gradient-based search method. The approach, designed to efficiently solve underdetermined systems with multiple objectives, is readily applicable to other problems. A selection of tuned example models will be showcased, including rosewood and aluminum bars with common and uncommon tuning ratios.