Lead Head Types

Complib has very detailed functionality to search for methods, based on a variety of criteria. One of these criteria concerns the different types of lead head which are possible.

We shall look here at what these various lead head types mean. But first, we need to recap on the definitions of a few terms. More details can be found in the Central Council's Framework for Method Ringing; Fundamentals and Method Classification.

Nomenclature

A Row is a sequence, in which each bell rings once, for instance Rounds or Queens.

A Change is the process of swapping bells around, to get from one Row to the next Row. An example might be that each pair of bells in a Row swaps; this change would transform 123456 to 214365, or 524613 to 256431. Changes are often represented by a code known as Place Notation; see here for more on Place Notation. However, confusingly, the word Change is often used to mean a Row.

Lead End is another term that is used in several different ways (and also spelled in different ways; the CC prefers "leadend").

The formal definition uses Lead End to refer to the penultimate row in the lead. In most common treble-dominated methods, this refers to the handstroke row of the treble's full lead. This is shown as 'B' in the diagram below.

"Lead End" is also commonly used to refer to the row shown as 'C' in the diagram. It can also refer to rows 'B' and 'C' together, or to the change between 'B' and 'C'. This change is shown in red in the diagram, with two lines (for the bells staying in the same place) and two crosses (for those swapping over).

The source of confusion is that it's arguable where each lead starts and ends. Here's a plain course of Plain Bob Doubles, written out in leads, as you often see it. Each column is a lead and each column shows one of the place bells. Each column has 10 changes, but each column has 11 rows. The last row in each column is written out again to start the next column. Hence the ambiguity; each ending is also a beginning.

The row which appears at the head of each column is called the Lead Head. That is, the backstroke of the treble's full lead. This was shown in the first diagram as 'A', but is also row 'C' in that diagram, as that row becomes the head for the next lead.

So, formally, the rows 12345, 13524, 15432 and 14253 here are Lead Heads, while the rows 13254, 15342, 14523 and 12435 are Lead Ends.

However, I think it's probably true to say that if you see "Lead End" used in a context which doesn't also refer to "Lead Head", it is most likely to refer to the backstroke lead ('C') rather than the handstroke one ('B').

In this article, I shall stick to the formal definition, and refer to which is why the Complib search criteria refers to Lead Heads, in the same way that the Complib search criteria above refer to Lead Head (or "leadhead").

Plain Bob Lead Heads

Here are the lead heads in Plain Bob at various stages, ignoring rounds.

Note that at each stage, the working bells appear in the same coursing order at the lead head. To see this, let's look at an example, say 1573624 in Triples. First look at rounds, writing down which bells are in the even places, starting from 2nds place and going in order (so you get 246), then look at the bells in the odd places, going in order down from the back to the bell in 3rds place, getting 753, giving the coursing order 246753, which should look familiar as the order the treble would follow the bells in the next lead of Plain Hunt.

Then we do the same for 1573624:

We get the coursing order 532467. As first sight this looks different, but remember coursing orders are cyclic, as it just indicates relative positions. Starting from the 2, we do indeed get 246753 again. This is an easy way to check that a given row is a Plain Bob Lead Head, without having to remember all the lead heads of Plain Bob.

But it's not just Plain Bob; these lead heads come in a wide variety of methods, not just 2nds place methods. The lead heads may not come in the same order, but they will (apart from some differential methods) all appear in the plain course. The order in which they come is used to classify methods, with the different lead head orders being given codes, each with a different letter (or letter and number). This can be useful, as this code then implies the place bell order of the method and thereby, to some extent, its overall shape. Looking at a unfamiliar method, knowing it has the same lead head code as a method you already know gives you a framework to support your new learning.

For instance, here are the lead heads from a few other methods, given in the order they appear. Each list matches with the Plain Bob list, but may be in a different order; in several cases the reverse order.

I'll resist (or at least, postpone) the temptation to talk more about lead ends, 2nds and nths place methods, group theory and which lead heads will generate a method that contains every lead head in the set. But I can't resist this quotation from Tintinnalogia (1667), talking about the relationship between the lead ends and lead heads in an extent of Old Bob Doubles (Plain Bob with different calls), compared with a plain course of Plain Bob Minimus.

Grandsire Lead Heads

In a plain course of Grandsire, the 2 is also a hunt bell, so every lead head will start 12.

Here are the lead heads for Grandsire at different stages, ignoring rounds:

Structurally, Grandsire at one stage is the same as Plain Bob at the next lower stage, but with two hunt bells. This fact also reflects in the lead heads. Ignoring 1 and 2, then subtracting one from each other bell number gives the same lead heads as for Plain Bob. And a similar technique looking at the coursing also works.

As we saw in Twin Hunt Methods, methods with Grandsire lead heads don't have to have the 2 ringing Plain Hunt in a plain course. They include many of the methods formerly known as Slow Course methods. And (apart from Minimus!) as with Plain Bob lead heads, the lead heads don't have to come in the same order as in Grandsire itself.

Original Lead Heads

Original is the official name for Plain Hunt. It's tempting to think that the lead end is just rounds, but remember that a lead refers to the section of a method which is repeated to produce the plain course, not necessarily to the section until the treble comes back to lead. This is a principle; there is no hunt bell. The repeated section is just two changes in length, so that is one lead. In Minor, for instance, the place notation for one lead is X16.

As you can see, the lead heads for Minor, apart from rounds, are 241635, 462513, 654321, 536142 and 315264. If we look at these rows using the coursing order technique from above, we have to include the treble now. Again we start from 2nds place, go up the evens and down the odds, ending up with the bell in the lead.

From rounds, we get the coursing order 246531. From the lead head 536142, we get 312465, which is a rotation of 246531.

(If we tried this technique to look at the coursing order for a lead end, we end up with the reverse coursing order. For instance:

The lead end 426153 gives a coursing order of 213564, which is the reverse of a rotation of 246531.

Here are the lead heads for Original at different stages, ignoring rounds:

Once again, there is a connection with the Plain Bob lead heads. Ignoring the treble in the Plain Bob lead heads, and subtracting one from each bell number, you get the Original lead heads for the stage below.

All methods with Original lead heads must also be principles. As we have already seen, the lead heads will not always come in the same order as in Original. Here are some examples:

Cyclic Lead Heads

Cyclic lead heads have the treble in the lead, then the working bells are a rotation of rounds. That is, they are in ascending numerical order, but starting from any bell; when you run out of bells, you start again with the 2. So you get:

Note that in Minimus, the cyclic lead heads are exactly the same as the Plain Bob lead heads. You can also get cyclic lead heads in methods with more than one hunt bell; here just the working bells are rotated, for instance Union Triples, below.

Runs of bells like this tend to sound good and if the first lead head of a method is cyclic, then all the subsequent ones will have the same property. So if you construct a method with runs of bells (either forwards or backwards) in the first lead, then you will also get further runs (on different sets of bells) in the other leads. However, as we saw in Palindromic Symmetry, cyclic methods like these do not have normal palindromic symmetry.

Here are some examples. Once again, the lead heads don't always come in the same order as listed in the table above.

These methods were popular in the early years of this century, but the emphasis in more recent years has been for cyclic compositions, rather than cyclic methods. In these, "normal" methods are rung, but then the composition shuffles the bells into the right positions to generate run music. If the first part end of the composition is a cyclic lead head, then any runs produced in the first part on one subset of the bells will give runs in the other parts, on different sets of bells. The ultimate example is perhaps Alan Reading's composition of 12-spliced Major, which gives all 480 of the possible runs of bells at the front or back. The first part end of this composition is 23456781, so, as you can see, this goes to the next level as it also includes the treble in the runs - all 8 bells have been rotated, so this is a variable hunt composition.

Rounds Lead Head

This option on the search list gives methods which just have one lead, ending in rounds. It includes methods that don't have any shorter sections that have been repeated to give the plain course. They are mostly Singles and Minimus methods, with a few Minor ones.

For instance:

Shipping Forecast Singles (which has two courses of wrong hunting within one of normal hunting)
Spirolux Treble Place Minimus (a variation based on St Nicholas)
12 Victoria Street Surprise Minor