Change ringing is a musical art form that emerged in England in the 1600s. It involves ringing a set of bells (tower bells or handbells) one at a time to cycle through different possible orderings. Some key terminology helps in discussing change ringing: a round is a sequence of bells in highest-to-lowest order; a plain change is a technique in which one bell is swapped with its neighbour; and a cross-change involves swapping multiple bells in a single move. For our project, we chose to visualize a plain change: a sequence of changes involving a cross-change followed by a plain change, beginning and ending with a round.
Specifically, we represented a plain hunt on six—a pattern on six bells defined as the course followed by a single bell as it moves from first position to last and back again (Merriam-Webster, 2025). Each 4×4 block of colour in the artwork represents one bell being rung. One thing to notice is that the first and last rows are identical: both are rounds. The artwork also makes the plain changes and cross-changes visible, as only neighbouring colours ever swap.
Another feature worth observing is the surface beading. Each line of beads of the same type and colour traces the path of a bell throughout the plain hunt. The over-under pattern of these beaded strands forms a braid—another elegant way of visualizing the structure of a plain hunt on six bells.
Plain hunt on six bells is not a full extent, meaning it does not cycle through all 720 permutations of six elements. A fun historical fact is that a full extent on eight bells has been rung. Polster and Ross (2007) describe one such performance at the Loughborough Bell Foundry in the U.K., which began at 6:52 a.m. on 27 July 1963 and ended at 12:50 a.m. on 28 July after 17 hours and 58 minutes of continuous ringing.
Change ringing is still alive today. It is practiced in English-speaking countries—Canada, England, Australia, and New Zealand—reminding us that historical traditions are continuously reinterpreted and that music and mathematics have long been interdependent.
References
Arthur T. White. (1983). Ringing the changes. Mathematical Proceedings of the Cambridge Philosophical Society, 94(2), 203–215. Cambridge University Press & Assessment
Gresham College. (2021, January 5). The mathematics of bell ringing [Lecture]. https://www.gresham.ac.uk/watch-now/maths-bellringing Gresham College
Jongrsde, et al. (2015). The mathematics of change ringing. (Bachelor’s thesis / project). Leiden University. Leiden University Math Publications
Nelson, A. C. (2020). The mathematics of bell ringing [Colloquium presentation]. https://annacnelson.github.io/assets/pdf/MathematicsOfBellRinging_2020Talk.pdf
Polster, B., & Ross, M. (2007, October 15). Ringing the changes. The Age / QEDcat. https://www.qedcat.com/archive_cleaned/ringing.html
Polster, B., & Ross, M. (2007). Mathematical Impressions: Change Ringing. Simons Foundation.Ringing Systems Web. (n.d.). Plain Bob Minor – Methods – Blueline. https://rsw.me.uk/blueline/methods/view/Plain_Bob_Minor
