Part of The Beauty of Physics, Mathematics, and Music (and Maybe Cats).

The new bell had a conspicuous bulge around its waist. It was also taller than a conventional bell. Those changes followed a musical request: move one important partial from a minor third to a major third while keeping several other frequency ratios close to their familiar values.

In the 1980s, researchers at Eindhoven and the Royal Eijsbouts bell foundry approached that request as a structural optimisation problem. They varied the bell’s profile, calculated its vibrations, and searched for a shape with the desired ratios. A cast bell could then be brought to the targets by tuning. [[1]] [[2]]

The result was a major-third bell. Its name describes a particular change in the spectrum. It does not mean that every partial became a harmonic, or that the computer had established the most beautiful possible bell.

The Third Is an Interval, Not the Third Harmonic

A struck bell vibrates in many modes. Each mode has a characteristic displacement pattern and natural frequency; those contributing to the sound produce its partials. Their frequencies do not generally follow the integer sequence of an ideal string.

For a conventional tuned bell, five important partials are called the hum, prime, tierce, quint and nominal. The names refer to their musical relationships. In a useful idealisation with the hum as reference, their ratios are approximately

\[ 1:2:2.4:3:4. \]

The tierce at \(2.4\) times the hum is a minor third above the prime at \(2\). It is not a third harmonic at \(3\) times the hum. Changing it to a just major third would replace \(2.4\) with \(2.5\), leaving the other listed targets in place. These are ideal tuning ratios, not a claim that every ordinary bell achieves them. [[2]]

There is a further distinction between tuning systems. Schoofs’s account lists just ratios for swinging bells and equal-tempered targets for carillon bells. A just minor third has ratio \(6/5\); an equal-tempered minor third has ratio \(2^{3/12}\). Similarly, the just major third \(5/4\) differs from the equal-tempered \(2^{4/12}\). The distinctions are small enough to disappear in casual descriptions and large enough to matter in a tuning problem.

Calling this “harmonisation” would also obscure the objective. Neither the minor-third set above nor its major-third replacement is a sequence of integer multiples of the hum. The project sought selected musical relationships within a richer spectrum.

A Semitone Is a Small Shift With Several Constraints

For an illustrative equal-tempered target calculation, choose a prime at \(440\ \mathrm{Hz}\). The two tierce targets are

\[ f_{\mathrm{minor}}=440\,2^{3/12} \approx523.25\ \mathrm{Hz}, \]\[ f_{\mathrm{major}}=440\,2^{4/12} \approx554.37\ \mathrm{Hz}. \]

The required increase is about \(31.11\ \mathrm{Hz}\), or \(5.95\%\). In cents, an interval measure defined by

\[ C=1200\log_2\left(\frac{f_2}{f_1}\right), \]

the change is exactly \(100\) cents: one equal-tempered semitone.

These numbers specify a model target, not a reported measurement of the Eindhoven bell. The historical design used ratios referenced to the nominal, also called the octave in the engineering accounts. Changing the reference does not change an interval between two partials.

A six per cent shift sounds modest. The difficulty is preserving other targets at the same time. Enlarging or shrinking the entire bell cannot do that: it changes the frequency scale while leaving the dimensionless ratios essentially unchanged under ideal geometric similarity.

To see why, scale every length, including wall thickness, by a factor \(s\), keeping the material the same. In linear elasticity the stiffness matrix scales as \(s\), whereas the mass matrix scales as \(s^3\). Squared frequencies therefore scale as \(s^{-2}\), and frequencies as \(s^{-1}\). Every mode moves together. Retuning one interval requires changing the shape rather than merely the size.

All the Partials Share the Same Metal

The calculation starts with a generalised eigenvalue problem,

\[ K(\mathbf p)\mathbf u_i =\omega_i^2 M(\mathbf p)\mathbf u_i. \]

Here \(\mathbf p\) contains geometric design variables, \(K\) is the stiffness matrix, \(M\) the mass matrix, \(\mathbf u_i\) the displacement pattern of mode \(i\), and \(\omega_i\) its angular frequency. A finite-element model approximates a continuous bell by a finite collection of displacement degrees of freedom.

The bell can be rotationally symmetric in its resting shape while its vibration is not. An oval deformation of a circular cross-section, for example, is not axisymmetric at any instant of the motion. The original work explicitly used an axisymmetric structure with non-axisymmetric deformations. [[1]]

Changing a wall thickness alters both mass and stiffness. It also alters the modes differently because their displacement and strain patterns differ. A region that moves strongly in one mode need not be a region where another mode stores much elastic energy.

For a smooth change of one parameter \(p\), a simple, isolated eigenvalue \(\lambda_i=\omega_i^2\) obeys

\[ \frac{d\lambda_i}{dp} =\frac{\mathbf u_i^{\mathsf T} \left(\frac{dK}{dp}-\lambda_i\frac{dM}{dp}\right) \mathbf u_i} {\mathbf u_i^{\mathsf T}M\mathbf u_i}. \]

This follows by differentiating the eigenvalue equation and using the symmetry of \(K\) and \(M\). It assumes the mode can be followed smoothly; degeneracies require a more careful treatment.

The two terms express the competing effects directly. A geometric change modifies elastic resistance and inertia. Removing metal therefore has no universal frequency effect independent of where it is removed and which mode is considered.

The same parameter enters the corresponding expressions for every mode. The tierce has no private piece of bronze that can be adjusted without consequences elsewhere.

A Planned Search Found Shapes That Trial and Error Had Missed

The Eindhoven team described the profile and wall thickness with smooth curves. In one stage of the search, they chose five radii and two thicknesses as seven adjustable variables. Each was varied five per cent above or below its nominal value. All combinations required

\[ 2^7=128 \]

finite-element analyses. The calculation results and sensitivities were used to build faster regression models for the frequency ratios. Candidate shapes found with those models were subsequently checked with direct finite-element calculations. [[1]]

This is a useful distinction between a model of the mechanics and a model of the model’s output. The finite-element analysis approximates the bell’s elasticity. The regression model approximates how its calculated ratios vary within a chosen region of design space. The second approximation makes a search cheaper; checking its candidates with the first reduces the risk of building a bell around an error in the shortcut.

Schoofs’s later thesis records eight candidate designs in this search. It distinguishes an untuned casting from a subsequent bell tuned to the intended ratios. The resulting profile had the pronounced waist bulge and increased height described at the beginning. [[2]]

Those features were an outcome of the chosen targets, geometric parameterisation and search. They do not establish a theorem that every possible major-third bell must have that particular bulge. A different design space, material or set of constraints could lead elsewhere.

The achievement was concrete enough: a calculation identified workable profiles, casting tested the mechanics, and the foundry completed the tuning.

Correct Frequencies Do Not Complete the Sound

A frequency list tells us where spectral components can occur. It does not tell us how strongly they are excited, how efficiently they radiate, or how long they remain.

For an idealised impulsive strike at a point, the excitation of a mode depends on the mode’s displacement at that point and on the direction of the force. A mode with little displacement in that direction at the strike location receives little energy. The displacement pattern over the rest of the bell then affects how its motion couples to the air.

Damping adds another set of parameters. A schematic contribution from one weakly damped mode has the form

\[ p_i(t)=A_i e^{-t/\tau_i} \cos(2\pi f_i t+\phi_i), \]

where \(p_i\) is its sound-pressure contribution at a specified observation point, \(A_i\) its initial amplitude, \(\tau_i\) its amplitude-decay time and \(\phi_i\) its phase. Two bells can have similar \(f_i\) and very different \(A_i\) and \(\tau_i\). This expression is an explanatory modal model, not a fit to either bell.

The limitations were tangible in the subsequent research. Roozen-Kroon’s 1992 thesis describes disturbing partials that had not been included in the original optimisation, and differences in sonority and decay that motivated a broader design problem. [[3]]

That account is a stronger lesson than a claim that the computer made bells “more consonant”. The chosen frequency objective was achieved sufficiently well to build the instrument. Whether its complete sound is preferred requires a perceptual question, with listeners and musical context, as well as a mechanical calculation.

Optimisation Makes the Musical Assumptions Explicit

An optimiser needs a numerical objective. One possible objective is a weighted sum of squared errors in cents,

\[ J(\mathbf p)=\sum_i w_i \left[ 1200\log_2\left(\frac{f_i(\mathbf p)}{f_{\mathrm{ref}}(\mathbf p)}\right) -C_i^{\mathrm{target}} \right]^2. \]

Here \(w_i\) sets the importance of partial \(i\), \(f_{\mathrm{ref}}\) is the chosen reference partial, and \(C_i^{\mathrm{target}}\) specifies the desired interval. This is an illustrative objective; the historical work also used searches within permitted ratio bands.

The mathematics can tell us which allowed shape lowers this quantity. The musician and designer decide the targets, weights, manufacturing limits and acceptable sound. Adding decay or amplitude targets changes the question being solved.

The bulging bell makes that division unusually visible. A musical interval became a structural constraint, and satisfying it meant reshaping metal shared by every mode. The calculation could say whether the tierce had moved.

Only listeners can say whether the bell is better for it.

Literature checked against the full technical report and relevant thesis chapters on 12 October 2026. Interval, scaling and eigenvalue-sensitivity calculations were checked independently.

References

[1] A. J. G. Schoofs, F. van Asperen, P. J. J. Maas and A. Lehr, Experimental Design Theory and Structural Optimization: Design of a Major-Third Church Bell, report WFW 85.043, Technische Hogeschool Eindhoven (1985), especially section 5. Full report.

[2] A. J. G. Schoofs, Experimental Design and Structural Optimization, doctoral thesis, Technische Universiteit Eindhoven (1987), chapter 6. Full thesis.

[3] P. J. M. Roozen-Kroon, Structural Optimization of Bells, doctoral thesis, Technische Universiteit Eindhoven (1992), especially sections 1.3 and 5.2–5.3. Full thesis.

Changelog

  • 2026-10-12: Research and calculation review. Major-third tuning as a shape problem, the original numerical search, and the limits of optimising frequency ratios.