Proceedings of the 2010 Conference on New Interfaces for Musical Expression (NIME 2010), Sydney, Australia

Designing Custom-made Metallophone∗with Concurrent Eigenanalysis
Nobuyuki Umetani

Jun Mitani

Takeo Igarashi

The University of Tokyo
JST ERATO
7-3-1 Hongo, Bunkyo-ku,
Tokyo, 113-0033 Japan

University of Tsukuba
JST ERATO
Tsukuba Ibaraki
305-8573 JapanThe University of Tokyo
JST ERATO
7-3-1 Hongo, Bunkyo-ku,
Tokyo, 113-0033 Japanumetani@ui.is.s.utokyo.ac.jp

mitani@cs.tsukuba.ac.jp

takeo@acm.org

ABSTRACT

acoustic musical instruments cannot have individual designs
like electric guitars.

We introduce an interactive interface for the custom designof metallophones. The shape of each plate must be determined in the design process so that the metallophone willproduce the proper tone when struck with a mallet. Unfortunately, the relationship between plate shape and tone iscomplex, which makes it difficult to design plates with arbitrary shapes. Our system addresses this problem by runninga concurrent numerical eigenanalysis during interactive geometry editing. It continuously presents a predicted tone tothe user with both visual and audio feedback, thus making
it possible to design a plate with any desired shape and tone.
We developed this system to demonstrate the effectivenessof integrating real-time finite element method analysis into
geometric editing to facilitate the design of custom-mademusical instruments. An informal study demonstrated theability of technically unsophisticated user to apply the system to complex metallophone design.

Figure 1: A custom designed metallophone in the
shape of a fish.
We propose a system for designing original acoustic musical instruments with computational assistance. We utilize
the finite element method (FEM) to predict tone. Although
the standard FEM runs on a fixed initial geometry, our idea
is to continuously update the simulation results in responseto the designer’s shape modifications. Real-time feedback
during editing provides the guiding principles for creating
a better overall design. It assists the designer in obtaining
satisfactory design results without the many trial-and-error
iterations that would be necessary with an offline simulation. We believe that responsive feedback can also be usefulfor educational purposes.
The specific object that illustrates our concept is the metallophone. We define a metallophone as a musical instrument that produces sounds via the vibration of metal plates
struck by a mallet. Metallophone plates are usually rectangular because this practical shape makes it possible toanalytically predict the instrument’s tone [5]. Designing a
metallophone with an arbitrary shape is difficult becauseof the complex relationship between shape and tone. We
believe that our approach is well suited to designing a metallophone of desired artistic shape and tone because thiskind of highly constrained modeling naturally requires atight integration of design and analysis.
This paper is organized as follows. We first outline previous work in this area. In the next section, we describe theuser interface and implementation details of the prototype
system. We then address system performance and user experience. Finally, we discuss the limitations of the current
system and the potential for future work.

Keywords
Modeling - Modeling Interfaces, Modeling - Geometric Modeling, Modeling - CAD, Methods and Applications - Education, Real-time FEM

1. INTRODUCTION
Each acoustic musical instrument has its own typical shapeand appearance. Although the exterior may show subtle differences, the fundamental shape cannot be very different (e.g., metallophone plates are always rectangular).
Although these shapes have become sophisticated throughyears of refinement, the appearance of musical instruments
can be repetitive and characterless. Conveying character,or a message, through appearance is a major challenge forthe makers of acoustic musical instruments. Because theshapes of acoustic musical instruments and their tones are
inseparable, making designing them much more complex,∗The supplemental video can be seen at http://www.
youtube.com/watch?v=7TfMgwuDHS4

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage and that copies
bear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.
NIME2010, June 15-18, 2010, Sydney, Australia
Copyright 2010, Copyright remains with the author(s).

2. RELATED WORKFEM analysis of acoustic musical instrument is an established field of research. There have been a number of studies

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Proceedings of the 2010 Conference on New Interfaces for Musical Expression (NIME 2010), Sydney, Australia

on the use of FEM eigenanalysis for predicting the tones ofvarious musical instruments, such as piano [9], guitar [2],and xylophone [3]. Shoofs et al. [15] used FEM eigenanalysis to determine the optimal shape of carillon bells. However, eigenanalysis have previously been used exclusively as
an offline evaluation tool and has not been directly integrated into geometry editing as in our system
Automatic optimization methods can be used to design
objects that satisfy physical constraints. For example, Smithet al. [16] applied an optimization approach to the designof truss structures. However, automatic methods present
many practical difficulties, such as explicitly specifying constraints and parameter spaces that are too large. The interactive approach offers the advantage of allowing users
to use their own preferences and judgment during the design process while considering less tangible factors such as
aesthetics.
Various techniques have been suggested for making physical simulations interactive. One approach is to use precomputations. James and Pai [6] performed an interactive
physical simulation of deformable objects by precomputing
the matrices of reference boundary value problems. They
also proposed modal analyses of dynamic elastic models forreal-time deformation simulations [7]. Another approachis to use approximations for nonlinear elasticity to achieve
large deformations quickly and stably [12]. These methods
require user input as an external force in a continuously
running simulation. In our system, the user directly modifies the definition of the target problem, and the system
reruns the simulation with the new setting.
Our goal is to aid the design process by using physicalsimulations. Our system shares some concepts with firstorder analysis [13], which emphasizes the importance of using simulation in the early stages of a design. Masry and
Lipson [10] develop a sketch-based three-dimensional (3D)
modeling interface capable of FEM analysis that has a similar objective to our own. However, first-order analysis is notvery different from conventional computer-aided design andcomputer-aided engineering systems, in that the analysis isperformed after some of the design processes have alreadyoccurred. In contrast, in our own system the analysis isperformed during the design process. Several systems havebeen proposed for designing physical objects such as stuffedanimals with the aid of interactive simulations [11]. However, the focus of those systems is mainly the user interface,
and they involve relatively simple simulation procedures.

2D view(Shape design)

3D view(Eigenmode)

Scale / frequency

Figure 2: Metallophone design software. The left
window is used for the original 2D design, whereas
the right window shows the analyzed eigenmode asa 3D graphic. The tone is updated in real time withboth audio and visual feedback for the user.
curves, such as Bézier curves or NURBS, permit the creation of more diverse shapes; this is left to future development.

add
point

drag
point

drag
arc

Figure 3: The modeling operations of our software:adding and deleting a point, dragging an arc and apoint.
The system continuously predicts the tone that the plate
will produce when struck with a mallet and presents it tothe user during the editing process. Visual feedback is given
as a numerical value (Figure 2, bottom) and audio feedback
is provided in the form of a sine wave, with the speaker
emitting the predicted tone. As the user drags the cornervertex, the tone from the speaker and the numerical valueon the screen gradually change. Audio feedback allows theuser to continuously monitor the tone while editing, andvisual feedback is useful for verifying the exact value. Low
tones tend to be produced by large plates (and vice versa),
and the user can observe and learn this phenomenon during interactive editing, which facilitates the overall design
process.
The system also predicts the geometric deformation of
the plate during vibration and visually presents it to theuser as a 3D graphic (Figure 2, right). The actual deformation is too small to be visible, so the system exaggerates
it in the visualization. This helps the user decide where
the plate should be attached to the base, because the plateshould be attached at a point of minimal deformation. It isalso useful for visually evaluating the quality of the vibrations. Vibrations perpendicular to the plate are desirable
because they are excited when the mallet strikes the facevertically, whereas a plane vibration is not likely to occur
during a performance, and its tone is not dominant (Figure4). Thus, the user should strive for a shape that produces
perpendicular vibrations.
The system only supports the design of individual plategeometry, and physical construction must be done manually(Figure 5). The user may export the plate geometry to a
DXF file and then send it to a wire-electrical discharge machine for the actual cutting of the metal plate. The metal
plate can also be cut manually with a band saw. Someerrors always occur during this process, and the finished

3. USER INTERFACE
This section describes our metallophone design systemfrom the user’s point of view. The system basically operates
as a two-dimensional (2D) modeling program in which theuser interactively edits the shape of a metallophone plateusing direct manipulation (Figure 3). FEM eigenfrequency
analysis runs concurrently with the user manipulation andpresents the predicted tone to the user with visual and audio
feedback. The system also provides 3D graphics of the geometric deformation of the plate during vibration. In thisway, the user, guided by artistic inspiration, can explore
various plate shapes while continuously verifying that thedesired tone is produced.
Figure 3 shows a set of modeling operations provided byour system. A metallophone plate is represented as a closed
area surrounded by straight lines and arcs, connected bycorner vertices. The user drags the corner vertices to move
them, and drags the arc to change its radius. The user canalso add and delete corner vertices. Our current implementation does not support plates with holes. More complex

27

Proceedings of the 2010 Conference on New Interfaces for Musical Expression (NIME 2010), Sydney, AustraliaGoodsimple to implement and yield sufficiently accurate results.
The use of thick plate elements will be a subject of future
research.

Bad

2 layers of tetrahedral elements

Figure 4: Perpendicular vibration mode is desirable(left), whereas in plane vibration mode is undesirable (right)

Actualcreation
4mm
150mm

metal plate will not produce exactly the same tone as thesimulated plate. It is therefore necessary to adjust the tone
by rasping the plate. Our system is also useful in this regard, because it is able to predict how the tone will change
when particular edges are rasped. Finally, the user attaches
the plate to a wooden board at the point suggested by thevibration shape analysis.

Figure 6: The mesh used in FEM discretization. We
use double-layered tetrahedral elements extruded
from a triangular 2D mesh.

4.2 Mesh Deformation
To continuously simulate a tone during a user’s shapeediting, the system adjusts the mesh to fit the shape. To
adjust the mesh to fit a user-specified shape, the system
first places the nodes on the boundary and then applies
smoothing to the nodes inside the boundary. We use Laplacian smoothing if the mesh distortion is small (Figure 7).
When the change in the boundary is large and the mesh is
distorted, we employ Delaunay smoothing to suppress themesh distortion.

spectrum analyzer

AdjustDesignCutband saw

Arrange & Fix
Figure 7: The smoothing of mesh using Laplaciansmoothing

Figure 5: The process of producing a metallophone:first, cutting the metal plate; second, adjusting thetone by rasping; finally, attaching the plate to awooden board.

4.3 Eigenmode analysis of the metallophones
We assume that the metallophone is floating in a gravityfree space without any external forces or fixed boundaries(which makes the stiffness matrix singular). Generalized
eigenanalysis with a singular stiffness matrix has been investigated in the field of aerospace engineering [4, 8]. A typical method of obtaining an eigenvalue and its eigenmodeconsists of first shifting the stiffness matrix with the mass
matrix, then calculating the distribution of the eigenvalues
using the Lanczos or Jacobi method to obtain the lowest
nonzero eigenvalue, and finally calculating the eigenmodeby shifted inverse iteration [14]. However, the extensive
calculation of the distribution of all of the eigenvalues is
unnecessary for our purpose. Hence, we compute only thesmallest nonzero eigenvalue, which enables real-time performance. We also take advantage of the problem setting.
First, we already know that the kernel of the coefficient matrix consists of translations and rotations. Second, we canreuse the solution from the previous configuration as a goodinitial guess for the inverse iteration.
Here is a detailed derivation of the algorithm we use. The
eigenvalue problem is formulated using the FEM discretization as

4. IMPLEMENTATION
4.1 FEM model for eigenmode analysis
In our current implementation, a metallophone plate ismodeled as a thin 3D linear solid plate extruded from a 2D
shape designed by the user. We discretize the solid into
tetrahedral elements by first triangulating the 2D shape,
then extruding each triangle to obtain a 3D prism, and finally dividing each prism into tetrahedra. We then apply
FEM eigenvalue analysis to compute its tone, or eigenfrequency. The deformation caused by the vibration is computed as the eigenmode corresponding to the eigenvalue.
Note that the simulation parameters must be calibrated foreach specific material. We use aluminum plate with a thickness of 4mm. The simulation accuracy is very sensitive tothe mesh density, because we use linear tetrahedral elements
to represent a bend in a 3D plate. To alleviate this problem,
known as “shear-locking” [17], we must prevent excessive
distortions of the tetrahedral elements. We divide the plate
into two layers, each filled with tetrahedral elements (Figure 6), such that the longest edge is always shorter than
two times the shortest edge. Thick plate elements mightwork very well here in place of tetrahedral elements, but wehave chosen to use tetrahedral elements because they areM̄ü + Ku

= 0

(1)

where u is the nodal displacement vector, M̄ is the lumped
mass matrix, and K is the positive semi-definite stiffness28

Proceedings of the 2010 Conference on New Interfaces for Musical Expression (NIME 2010), Sydney, Australia

matrix. By splitting the displacement u into the product ofa spatially varying amplitude φ and a harmonic oscillation
with angular velocity ω, u(x, t) = φ(x)eiωt , and substituting this into Eq.(1), we obtain
Kφ = λM̄φ

the task freely and without time limitations and was provided instructions when required. We then used an electoricdischarge machine to manufacture an actual metallophone
based on the artist’s design. Figure 8 shows the shapes
designed to correspond to the musical scale notes from C(523 Hz) to B (987 Hz). After the design of each piaces, wetrimed and assembled the metallophone (Figure 1)(2)√

where the eigenfrequency f = ω/(2π) = λ/(2π). Our
goal is to calculate the smallest nonzero eigenvalue λ andits corresponding eigenvector φ.
Cholesky factorization is employed to represent the lumped
mass matrix as M̄ = LLT , and both sides of Eq.(2) are multiplied on the left by L−1 to obtain the standard eigenvalue
problemAψ = λψ−1−T

Designed
shape

E

(3)

G

A

B

T

F

Figure 8: Metallophone shapes designed by anartist. The upper row shows the designed 2D
shapes, whereas the lower row shows their analyzed
eigenmodes in 3D.
The top three rows in Table 2 show that the frequencies
of most of the pieces conformed well to one another for thetarget, the simulation, and the actual metallophone. To further improve the quality, we manually adjusted the tonesof the actual metallophone pieces by trimming their edges(except for the piece corresponding to F). As mentioned in
Section 3, our metallophone design software was also usefulfor this adjustment process, because it was able to predict
the tone changes caused by edge trimming. The bottom row
in Table 2 lists the frequencies of the actual metallophone
after manual adjustment. Although inharmonic overtones
make the timbre unclear, we found that the sounds produced by an actual metallophone were of acceptable quality
for a hobbyist.

5. RESULTS
5.1 System Performance
Table 1 summearizes the performance of our system. Because the performance depends on user operations, we measured the average number of frames per second during thecontinuous editing of shapes. In our current implementation, the number of elements is proportional to the area ofthe plate, because we have limited the edge length to avoid
element distortions, as described in Section 4. This causesa slight slowdown when the shape is large, but we consider
it acceptable.
#Tetrahedra
2196
3192
4524

D

Analyzedeigenmode

where A = L KL
and ψ = L φ. We solve this byinverse iteration. The standard iteration procedure is modified by adding a step that removes all zero eigenvectors of
A from the current solution. Because of our problem setting, we already know that the zero eigenvectors of K are
φi0 (i = 1 . . . 6), the translations along the three coordinate
axes and the rotations around them. We then apply modified Gram-Schmidt orthogonalization to LT φi0 (i = 1 . . . 6)
to obtain the orthonormal basis vectors ψ i0 (i = 1 . . . 6)that span the kernel of A, and define a projection P thatmaps a vector v toPthe complement
space of the kernel of
A by P(v) = v −ψ i0 ψ i0 · v . In each step of the iteration, we apply this projection to the solution vector andnormalize it. We add a small positive number ε to the diagonals of A to improve the numerical conditions. Once theshifted eigenvalue λ01 and its corresponding eigenvector ψ 1
of A are computed, we finally obtain the smallest nonzero
eigenvalue λ1 = λ01 − ε and the eigenvector φ1 = L−T ψ 1 .
Reusing of the solution from the previous configuration significantly improves the convergence of the inverse iterations.
The processing returns to the main thread in each iteration
step to avoid freezing to user input.

Size (mm)100 × 30
150 × 30
200 × 30

C

FPS
10.7
4.2
3.3

Scale
Targeted
Simulated
Measured
Adjusted

C
523
525
506
523

D
587
588
604
587

E
659
661
621
659

F
698
699
698
698

G
783
786
787
783

A
880
880
860
881

B
987
989
993
987

Table 2: Target, simulated, measured, and adjusted
frequencies (Hz) of the metallophone for each notein the scale, illustrating the accuracy of our analysis.

Frequency (Hz)
1931
860
494

We interviewed the artist afterward to obtain subjective
feedback. The artist reported that the design took roughly
5 hours to complete, with most of the time being devoted
to the C and D pieces. This was mainly because these lower
tones required larger areas than the others, which greatly
slowed the response of the analysis. One of the difficulties the artist encountered during the design process wasneeding to maintain an overall balance in shape among thepieces while keeping their tones true to the intended tone.
This was a major design constraint. Another difficulty wasthat sometimes a small modification of the shape resulted
in a large change tone, necessitating high responsivenessof the analysis. Finally, the artist noted that the relationship between the shape and tone of the metallophone wasstill difficult to understand, even after participating in thisstudy. Nevertheless, the artist at least learned that larger

Table 1: The performance of our system during interactive mnipulation, tested with a 2.5GHz CPU
and 2.0GB RAM. The first column shows the sizeof the rectangular metallophone plates, the second
shows the number of tetrahedral elements, the thirdshows the frame per second, the fourth shows thefrequencies (Hz).

5.2 User Experience
We asked a professional artist to design a metallophoneusing our software. The artist was allowed to work on

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Proceedings of the 2010 Conference on New Interfaces for Musical Expression (NIME 2010), Sydney, Australia

pieces tend to produce lower tones, having previously believed the opposite to be true.

6. LIMITATIONS AND FUTURE WORK
Our current system only computes the smallest nonzero
eigenvalue and its eigenmode. This is sufficient for predicting the basic tone, but the computing higher order eigenvalues and eigenmodes is necessary to more accurately predictthe resulting timbre. We plan to address this problem inthe future. Another interesting possibility is to design a special metallophone plate that produces different tones whenstruck with a mallet in different positions.
The present implementation uses volumetric tetrahedral
elements to model the metallophone plate. However, modeling with locking-free thick plate elements [1] might be more
efficient for this purpose. The simulation for an overtone iscomputationally more demanding, and thick plate elements
might be necessary in this case. One advantage of volumetric elements is that they are able to support truly 3D form
factors, such as those in a sculpture; this is another possible
direction for future research.
Interactive eigenanalysis could potentially be extended tothe design of other musical instruments. We are planningto develop a system for designing a wind instrument, suchas an ocarina. In this case, we would need to perform eigenanalysis on a sound wave equation. Finally, we believe thatinteractive shape design with concurrent FEM analysis will
be useful for more general design problems, and we plan totest it in other domains.

7. ACKNOWLEDGMENTS
This project was partially supported by 2008 IPA Exploratory Software Project (Project Manager: Ikuo Takeuchi),Information-Technology Promotion Agency, Japan. We would
like to thank Yuki Tsujita for designing the fish shaped metallophone, Greg Saul for his help in the construction of themetallophone, JST ERATO project members for valuablesuggestions and help, and anonymous reviewers for theirconstructive feedback.

8. ADDITIONAL AUTHORSKenshi Takayama (The University of Tokyo, JST ERATO, email: kenshi@ui.is.s.tokyo.ac.jp)

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