Permutation of the Bells

I came across this picture I took while visiting the bell tower of the Old Post Office in Washington, D.C. a couple of years ago.  It describes how the bells are occasionally rung in a “full peal,” consisting of all possible permutations of their tones.  With 6 bells, you get 6! = 6*5*4*3*2*1 = 720 permutations, which could be rung in under an hour, but with 7 bells, you get 7! = 7*720 = 5040 patterns to ring!

I thought maybe this would involve listing the set of permutations in increasing numerical order:

123456
123465
123546
123564
123645…

But In the example they showed, it seems that the pattern is to first swap neighbors in 3 groups:

12 34 56
21 43 65

And then to keep the first and last positions in place but swap neighbors in the 2 groups inside:

2 14 36 5
2 41 63 5

I can’t tell if that’s all there is to the method or not, and I wonder if there’s a proof that the method produces all permutations.  A quick test with 4 bells shows that you don’t get all 24 permutations before it repeats.

1234
2143
2413
4231
4321
3412
3142
1324
1234