Applied Acoustics 217 (2024) 109850 0003-682X/© 2024 Elsevier Ltd. All rights reserved. Contents lists available at ScienceDirect Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust Tuning of idiophones using shape optimization on finite and boundary element 1D models Christos G. Panagiotopoulos ∗ , Spyros Kouzoupis ∗∗ Department of Music Technology and Acoustics, Hellenic Mediterranean University, GR-74133 Rethymno, Greece A R T I C L E I N F O A B S T R A C T Keywords: Multi–modal design Tuning Model updating Optimization Finite element method Boundary element method We address the multi-modal tuning of idiophones and approach it as a shape optimization design problem. The Finite Element Method (FEM) and the Boundary Element Method (BEM), are adopted to discretize continuous models and lead to eigenvalue problems for the shape optimization of both the idiophone vibrating bars and the resonators. The methods presented here consist of two main components. The first component involves solving the eigenproblem of a discrete dynamical system that approximates a real idiophone bar (beam) or a resonator (acoustic tube), while the second component is the implementation of an appropriate design procedure to achieve the desired frequency ratios of the bar and resonator eigenfrequencies through optimization. For the shape optimization problem, both a gradient algorithm and a population–based algorithm are tested. We introduce the use of analytically computed sensitivity indices, which significantly accelerate the optimization when gradient– based algorithms are used in conjunction with the FEM. For illustrative purposes, the presented methods are applied to 1D formulation however, these can be expanded in a straightforward way to more involved/realistic 2D or 3D models. 1. Introduction It is well-known that the most common boundary conditions im- posed at the ends of a finite, ideal string or an air column yield solutions with frequencies that are integer multiples of the fundamental fre- quency. The first partials are harmonic or form intervals which are con- sidered pleasant (consonant) in most music cultures. This phenomenon has led to the development of many important musical instruments, which produce characteristic timbres by shaping the original harmonic spectra of these sound production mechanisms. On the other hand, instruments that are struck may produce pleasant sounds and even produce easily perceived pitches (tuned idiophones), provided certain construction procedures have been followed. The mu- sical instruments that belong to this category include, but are not lim- ited to, the xylophone, the marimba, the vibraphone, the glockenspiel and to a lesser degree various bells and chimes. We will focus solely on mallet percussion instruments and their transverse bar vibration. Throughout this work, the term “bar” will refer to an idiophone’s key (as is customary in musical literature), despite the fact that we are working with a beam in terms of the structural model. * Corresponding author. ** Principal corresponding author. E-mail addresses: pchr@hmu.gr (C.G. Panagiotopoulos), skouzo@hmu.gr (S. Kouzoupis). URLs: https://www.hmu.gr/en (C.G. Panagiotopoulos), https://www.hmu.gr/en (S. Kouzoupis). From a musical standpoint, only the free-free boundary conditions (as found in instruments such as the marimba, xylophone, vibraphone, glockenspiel, and bell lyra) and the clamped–free boundary conditions (as found in instruments such as the kalimba and tuning fork) are impor- tant and are commonly used in musical instruments among all possible boundary condition combinations. The frequency content of a bar can then be estimated using beam theory assumptions with the appropriate boundary conditions. In practice, to ensure that the bars of an instrument produce well- defined pitches, an empirical process of removing material from the bottom of the bar is followed. This process aims to achieve specific fre- quency ratios for the first three partials: 1:3:6 for the xylophone and 1:4:10 for the marimba [18]. For the instrument’s high register only the first two partials are generally tuned. In commercially made xy- lophones and marimbas, the bar’s dimensions (bar length, width and height of section) for a single note usually vary. The undercut shape also shows great variability among different manufacturers, and even across similar models of the same brand. This is due to the fact that there is no unique profile that results in an accepted tuning, plus to dif- https://doi.org/10.1016/j.apacoust.2024.109850 Received 10 April 2022; Received in revised form 3 December 2023; Accepted 1 January 2024