Place Notation and Blue Lines - Differentials

Stephen also talked about differential methods, so here's another example to look at.

We shall start by drawing out the Minor method with place notation -4-3.2.5,234. There are a couple of things to notice here; firstly the use of a dash instead of a X to show all bells swapping, secondly the comma, which indicates that the method has palindromic symmetry, and thirdly the fact that some of the place notations have been abbreviated. Isn't that typical in ringing? You're just getting the hang of it, then it all changes...

So, again, we've got six working bells and the first change is X, so, from rounds, all three pairs will swap over:

Next is place notation 4, so the bell in 4ths place stays where it is:

It's clear that the bells in 5ths and 6ths places have to swap over, but what about the bells in 1sts, 2nds and 3rds? When place notations have been abbreviated like this, it means that, wherever there's a odd number of bells left, the ones leading or lying stay in the same place. So here, the bell leading stays in the same place, while the bells in 2nds and 3rds swap. The place notation could equally well have been written as 14.

Then we get another X, so everyone swaps:

The next change has place notation 3, so a place is made in 3rds:

The bells in 1sts and 2nds places swap, but one of the remaining three bells must stay in the same place. The one to choose is the one that is at the back, so this place notation is the same as 36. The bells in 4ths and 5ths places swap over.

The next place notation is just shown as 2, so the bell in 2nds place is fixed.

That forces the bell leading to stay in the same place, while the rest of the bells swap (i.e. a 12 place notation):

The place notation at the half lead is given as 5, so the bell in 5ths place stays there:

The bell at the back is stuck there, and the other two pairs swap over (giving a place notation of 56):

Now we've reached the comma, so we do what we've just done, but in reverse order, so the next change has place notation 2 (or, as we saw above, 12):

Then we work back through the rest; 36, X, 14, X:

Now the final change is the one after the comma, given as 234:

The bell leading is trapped there, while the bells in 5ths and 6ths swap, giving the completed grid as:

This is the first lead of the method. We can see that the treble is back in the lead, so we know the method is again a Hunter. This time it got all the way up to the back, though. Here is the grid with the path of each bell coloured:

So, we want to draw a blue line for the method. Let's start 2nds place bell again. Following the orange line brings us to 4ths place at the end of the first lead, so then we follow the line for 4ths place bell, shown in blue. But that ends up back in 2nds place at the end of the next lead, so its blue line is only 2 leads long (which is 24 changes).

But is it rounds? We can see that the 4th is back in 4ths place, as its blue line is also two leads long.

Let's look at the 3rd. The green line in the grid takes it to 5ths place bell, then, following 5ths place bell (in purple), to 6ths place bell and then (in black) back to 3rds place bell. So these three bells are all on a blue line which is three leads long (i.e. 36 changes).

Note that all the blue lines have palindromic symmetry, because the place notation had palindromic symmetry. That is, they have reflective symmetry from top to bottom. For instance, 3rds place bell and 6ths place bell are reverses of each other.

So how long is the plain course? We've seen the blue lines of every bell are either 2 or 3 leads in length. So bells 2 and 4 come into their home places at every even numbered lead end. But 3, 5 and 6 come back into their original positions every 3rd lead. So, after 2 leads, 2 and 4 are back where they started, but 3, 5 and 6 aren't. After 3 leads, 3, 5 and 6 are back, but 2 and 4 have set off for another circuit of their blue lines.

We will get rounds after a number of leads that divides by both 2 and 3. The first time this happens is after 6 leads, which is 72 changes. So this is cunning way of ringing a method where the blue line that you have to learn is shorter than the plain course. In fact, even if you learned both blue lines, it would still be less than the length of the plain course.

In case you wanted to know, this method is Christ Church Differential Bob Minor. Or you could think of it as Thelwall Bob Minor with a Single at every lead.

Now, one for you to do. It's a Major method this time; can you work out the length of a plain course of the method with place notation -1-3456,2 ?

The answer is here.