I recently finished Dorothy Sayers’ The Nine Tailors, a mystery novel in which bells feature prominently. The setting is a provincial English town with a formidable church and an equally formidable set of bells. These bells are rung in the change ringing tradition, so that each bell (in this case, eight bells) is rung by a separate person pulling on its rope.
Early on in the book, the local Rector organizes a performance of the bells lasting nine hours. That is no small feat considering there is only one ringer per bell (so each person is constantly at work) and they start their task at midnight! The method they accomplish is called Kent Treble Bob Majors. Let’s figure out what exactly they were performing for nine hours.
First of all, we need to know the basic rules of change ringing. The bells are rung in any of the different possible orderings, but for a performance of the bells to be “true,” an ordering of the bells cannot be repeated. In addition, for each order permutation (called a row or change), each bell is played exactly once and never struck simultaneously with another bell. Also, each bell can only move one position within the row with each new row. The bells are numbered, the smallest bell starting as 1, for ease in vocal and written transmission.
So, one possible permutation of a row for eight bells would be.
1 2 3 4 5 6 7 8
This would be equivalent to a descending scale, since the sets of bells are arranged as complete diatonic scales or subsets of them.
But we could have any possible ordering available for performance. 2 3 4 8 7 6 1 5 or 8 1 7 5 6 4 2 3 or whatever your heart desires. With eight bells, there are 40,320 different possibilities for the rows! So how do you organize all the ringers to perform different permutations without pauses, mistakes, or repeats? Well, basically by following simple rules of reordering your bell within the row.
When I looked up Plain Bob Major, I found this grid. My first reaction was something like “holy %$*.” It’s not really as bad as it looks, though.
We can see at the beginning—the top left column—that we start with the sequence of bells from smallest to largest 1 2 3 4 5 6 7 8. We also see lines running through the rows in black, red, and blue. The black line is following bell 1, the red 2, and the blue 8. The 1 bell is the lead, which means it follows its own set of rules while the rest of the bells follow a separate set. Following the black line, you’ll see that the 1 bell moves down to the right, and then turns around and comes back to the left, over and over.
The other bells follow another set of rules. Those bells go back and forth across the grid just like the lead bell, with exceptions. For example, when a bell is in the second position behind the lead bell in first position, the non-lead bell plays in the second position two times in a row. Look at the 8 bell with the blue line in these rows below. After playing in the lead position twice, it begins to move to the right, but plays in the second position for two rows.
This hiccup in the normal pattern creates a conundrum, though. If the second position bell stays in the same spot, another bell that was supposed to have moved in that spot is stuck. In the example above, the 7 bell ordinarily should have moved into second position (see it moving to the left one row at a time?), but it can’t since the 8 bell is there. What to do? The solution is that the other bells form pairs and swap places for two rows, violating the normal rule of moving back and forth along the grid. The bells in the third and fourth positions swap, as do the bells in fifth and sixth positions and seventh and eighth positions (the black lines trace the swapping). In this way, many more rows can be generated without danger of repeating a row already played. This bell swapping and disruption to the normal back-and-forth pattern occurs every time the 1 bell rings in the first position.
And that is the Plain Bob Major method. Tune in next time to read about the even more complex Kent Bob Major that takes nine hours!