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Perrin, R.; Guo, Z.; Eager, D.; Halkon, B. Geometry of an English Church Bell. Encyclopedia. Available online: https://encyclopedia.pub/entry/59110 (accessed on 11 December 2025).
Perrin R, Guo Z, Eager D, Halkon B. Geometry of an English Church Bell. Encyclopedia. Available at: https://encyclopedia.pub/entry/59110. Accessed December 11, 2025.
Perrin, Robert, Zimu Guo, David Eager, Benjamin Halkon. "Geometry of an English Church Bell" Encyclopedia, https://encyclopedia.pub/entry/59110 (accessed December 11, 2025).
Perrin, R., Guo, Z., Eager, D., & Halkon, B. (2025, October 10). Geometry of an English Church Bell. In Encyclopedia. https://encyclopedia.pub/entry/59110
Perrin, Robert, et al. "Geometry of an English Church Bell." Encyclopedia. Web. 10 October, 2025.
Peer Reviewed
Geometry of an English Church Bell
This is a peer-reviewed entry published in Encyclopedia Journal, full entry can be found at 10.3390/encyclopedia5040161
The axisymmetric geometries of both internal and external profiles of church bells, along with the properties of the bell metal, are of vital importance to their acoustics. The role of “golden geometry” in determining the visual characteristics of the outer profile of a modern English church bell is presented. It is shown that a golden rectangle, a golden triangle, and a golden angle are all important factors. Three angular parameters are identified as measures of these attributes of the bell’s “goldenness”. Two of these features are shown to result from an underlying regular pentagon with one side defined by the bell’s mouth. The centre of this pentagon determines the location of the bell’s shoulders. Further minor features are also shown to be determined by golden geometry. The evolution of the three angular parameters over the previous millennium is included, showing a tendency to trend quickly towards the golden values in all the cases. Ultimately, the angular parameters for the English church bell of focus here are used for comparison with other modern European bells studied previously. The bells considered all displayed at least two of the three major golden features to an agreement of better than 1%.
bell profiles divine proportion golden ratio sacred geometry
While also belonging to the musical instrument category, church bells are unique in their purpose and character to other bell types, such as handbells, orchestral bells or “chimes”, bells that are attached to animals (e.g., cows, elephants, and cats), other ornamental bells, and so on. Church bells are rung individually or sequentially in a small number of varying sizes—for example, for marking the hour or announcing a service. A carillon, alternatively, is a musical instrument consisting of a larger number of bells played from a console. While the larger of the bells in a carillon are similar to church bells, the smaller differ significantly. The focus of this contribution is on the large English church bell and, to a lesser extent but for context, several of its modern Western European counterparts.
Church bells and their vibroacoustics have been of interest to physicists at least since Lord Rayleigh’s time [1]. Reports have appeared concerning their acoustical spectra and normal modes, the phenomenon of “warble”, and the psycho-acoustical “strike note” [2][3]. Some success has been achieved in comparing predictions of finite-element models with experimental results for normal modes [4]. Modern founders tune the frequencies of at least the first five (musical) partials in the approximate ratios of 1:2:2.4:3:4. The second, known by founders as the “fundamental”, usually approximates to the “note” of the bell. The third, the “quint”, is a musical minor third above the fundamental and is responsible for the “ethereal” quality of the church bell’s sound. The vital importance of profiles to a bell’s acoustics is obvious but is also well demonstrated by the facts that (1) for a given outer profile, bells are tuned by removing metal from the inner profile in an axially symmetric way, and (2) modern Dutch founders have succeeded in producing bells with a major third, rather than a minor one, by drastically distorting their axisymmetric profiles in the waist region [5].
The geometry of church bells has, historically, mainly been of interest to antiquarians and founders. Much information on the subject has appeared in the relevant literature [6][7], but a large proportion is purely diagrammatic or qualitative in nature and restricted to outer profiles. An exception is the bell historian Elphick [8]. He attempted to parametrise outer profiles using trapezoids, but these are of limited use, as, in most cases, he did not include crowns in his analysis. There have been discussions on how the profiles evolved historically; how straight lines, circular arcs, and elliptical arcs can form parts of some profiles; and how these may influence important partials. However, little has been said about the mathematics underlying the complete profiles. An attempt at this was made by Perrin et al. [9], who produced an analytical formula giving reasonable fits to the outer profiles of modern Taylor church bells but which did not reproduce sharper regions, such as shoulders, sufficiently well.
The approximate axial symmetry of English church bells allows them to be described using cylindrical polar coordinates (r, 𝜃, z), with the z-direction lying on the axis of symmetry. Their geometry allows examination in a single plane for a fixed 𝜃, and any golden features located will be repeated at all the other values of 𝜃. The presence of axial symmetry in bells has important consequences for their normal modes and for the degeneracy structures of their vibration spectra, which have previously been studied for church bells [10] and for gamelan gongs [11]. For axially symmetric systems, it has been shown using group theoretical arguments that the modal functions must vary like either sin(m𝜃) or cos(m𝜃), where m = 0, 1, 2, … Those with m = 0 are singlets—either so-called breathing modes or twisting modes—while all the others are bending modes of some form in degenerate pairs [4].
A basic problem with seeking analytical formulae for axisymmetric bell profiles is that although overall forms are largely similar, details vary between founders, sometimes considerably, having evolved independently over centuries. What seems to be required is a more general approach, seeking to fix major features analytically while leaving room for minor variations. In this paper, we present evidence that “golden geometry” can fulfill this role for the outer profiles of many founders. Inner profiles, although important for acoustics, are not considered herein for reasons explained in Section 2. Research into various aspects of the geometry and its association with the acoustics of bells is still being undertaken, with a recent example being from Whyte et al. [12].

References

  1. Strutt, J.W. On Bells. Phil. Mag. 1890, 29, 1–17.
  2. Rossing, T. Acoustics of Bells; Van Nostrand Reinhold: New York, NY, USA, 1984.
  3. Rossing, T.D. Science of Percussion Instruments; World Scientific: Singapore, 2000.
  4. Perrin, R.; Charnley, T.; de Pont, J. Normal modes of the modern English church bell. J. Sound Vib. 1983, 90, 29–49.
  5. Lehr, A. Profielconstructies van Luid-en Beiaardklokken in het Verleden; National Beiaardmuseum: Asten, The Netherlands, 1991.
  6. Lehr, A. Designing Chimes and Carillons in History. Acust.-Acta Acust. 1997, 83, 320–336.
  7. Elphick, G.P. Sussex Bells and Belfries; Phillimore: London, UK, 1970.
  8. Elphick, G.P. The Craft of the Bell Founder; Phillimore: London, UK, 1988.
  9. Perrin, R.; Charnley, T.; Samson, J.H.; Gottlieb, H.P.W. The Campanoid: An Equation for the Church bell Profile. J. Sound Vib. 1991, 151, 163–167.
  10. Perrin, R.; Charnley, T. Group Theory and the Bell. J. Sound Vib. 1973, 31, 411–418.
  11. Perrin, R.; Elford, D.P.; Chalmers, L.; Swallowe, G.M.; Moore, T.R.; Hamdan, S.; Halkon, B.J. Normal modes of a small gamelan gong. J. Acoust. Soc. Am 2014, 136, 1942–1950.
  12. Whyte, H.; Perrin, R.; Halkon, B.J. A New Analysis of the Normal Modes of a Large English Church Bell. In Proceedings of the Acoustics 2024: Acoustics in the Sun, Broadbeach, Australia, 6–8 November 2024.
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Robert Perrin
, Zimu Guo , David Eager , Benjamin Halkon
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