Change Ringing: The Mathematic Music of Church Bells

When one imagines a classic European cathedral, the mental picture is rarely complete without the musical accompaniment of resonant church bells. Yet many early church bells are incapable of producing conventional music. Enter change ringing, a mathematic art that gave cathedral bells their own unique form of musical expression and inspired a host of devoted followers who continue this hybrid auditory science even today.

The origins of change ringing lie in the construction of church bells for English cathedrals during the Reformation era of the early 17th century. During this time period a relatively novel method of hanging massive brass church bells gained popularity: one that involved mounting the bells on large wooden wheels. Bells hung from these wheels were so precisely balanced that even small children could ring them simply by pulling on an attached rope, despite the fact that the bells themselves ranged in weight from several hundred to several thousand pounds.

The great drawback of this engineering feat is that wheel-hung bells consistently take roughly two full seconds to complete a 360-degree swing, limiting how frequently a given note can be repeated and making it impossible for an array of such bells to play conventional melodic music. Change ringing arose as a form of music that could accommodate the limitations imposed by the massive size of the wheel-hung church bells.

Most wheeled church bells are hung in groups of 3 to 12 bells, arranged in order of tone from the highest, or treble bell, to the lowest, or tenor bell. Rung in order these bells sound out a simple musical scale in a precise rhythm. Change the order the bells are rung, however, and a fascinating array of mathematic---and musical--- permutations becomes possible. These permutations are the basis of change ringing.

Change ringing is guided by a strict set of rules and jargon, one of the earliest versions of which was set out by Fabian Stedman in 1668 in his treatise Tintinnalogia. In simplest terms, an entire set of bells is rung in order from treble to tenor, called a "round," and these rounds are repeated with slight variations, called "changes." According to the rules of change ringing, no bell may move more than one place in the order between rounds. Thus, if in the first round the bells are rung in order (1-2-3-4-5), the third bell could be rung second or fourth in the next round (1-3-2-4-5 or 1-2-4-3-5) but the third bell could not be rung first or fifth (3-1-2-4-5 or 1-2-4-5-3). This "no more than one place" rule is enforced each round, explicitly setting out a progression that is the hallmark of change ringing. As the rounds advance, the mathematic permutations governing the change-ringing exercise grow more numerous and complex. 

A full permutation of changes for a seven-bell array requires 5,040 rounds, without break, which requires roughly three-and-a-half hours to complete. This is the minimum number of rounds necessary for a standard change-ringing exercise, known as a "peal." Any change ringing that includes less than 5,040 rounds is referred to as merely a "touch." The first complete peal is said to have been completed in England in 1715. The first American peal is often attributed to the legendary showman P. T. Barnum, who staged the event at Christ Church Cathedral in Philadelphia in 1850.

Despite its comparative age, change ringing remains a widely pursued musical pastime even today. Though still largely an English phenomenon, change-ringing societies continue the practice in Australia, Canada, New Zealand, the United States, and a host of other countries. This timeless combination of mathematics and music seems never to go out of style.