[r-t] Link between Stedman sixes and Kent fours

[Robert Bennett]

As regards April Day, I think that it can be looked at as a bob after the
lead head, and a three-lead splice before it:
If the three lead splices are left out, we have Old Hudibras variation
(3rds after the lead).

There are probably lots of examples of clean joining out there.
The magic blocks of Stedman Triples bobs only compositions are 5 blocks
which contain the equivalent of 10 bob courses.
*2314567*
3241657   omit
*3426175*
4362157
*3461275*
4316257
*4132675*
1423765  omit
*4127356*
1472365
*1743256*
7134265
*1732456*
7123465
*7214356*
2741365
*7243156*
2734165
*2371456*
3217465
*2314756*
3241765
*3427156*
4372165
*3471256*
4317265
*4132756*
1423576 omit
*4125367*
1452376
*1543267*
5134276
*1532467*
5123476
*5214367*
2541376
*5243167*
2534176
*2351467*
3215476
*2314567*

heads of the other blocks
1354726
5374612
7364251
6324175

The three sixes between the first two omits contain the same changes (but
in a different order) as the first three sixes left out of the last block
6324175.
The extras and omits device in Stedman Triples is another way, but it
doesn't allow bobs-only peals I think.


The same sort of idea is used in a 360 of Bob Minor called WHW x3 where the
H-->W lead is rung backwards;
or a 2520 of Bob Triples called W3H, WM3H x5

Robert Bennett

On Wed, Sep 2, 2026 at 8:49 PM Alexander E Holroyd <holroyd at uw.edu> wrote:

> It’s an interesting analogy.  My inclination is to think that there is no
> particularly deep connection between the two scenarios, although obviously
> I don’t know for certain.  A six itself forms a group (or a coset of it),
> and quick and slow sixes are two different paths through it’s elements, and
> the same applies to fours, but you doubtless know all this.
>
>
>
> I don’t see how the outer automorphism would help, because the numbers of
> rows involved are different.
>
>
>
> Another weird example of “clean joining” comes in April Day Doubles, or
> indeed April Day Minor.  Calling a Grandsire Single brings up the **same**
> lead head as a plain, so one way to think about it is that the natural
> block consists of the internal rows of the lead (starting and ending with
> the treble in 2nds) together with the two rows one away on each side (the
> lead end before and the lead head after).  **These** blocks get cleanly
> joined by the calls.  Perhaps doubles variations experts can provide more
> insight.
>
>
>
> Incidentally, I recently discovered how to make a physical manifestation
> of the outer automorphism of S_6 (I think the first such):
>
> https://www.youtube.com/watch?v=T19oalEPp70
>
> although this does not answer your question!
>
>
>
>
>
>
>
> *From:* ringing-theory <ringing-theory-bounces at bellringers.org> *On
> Behalf Of *Richard Pullin
> *Sent:* 01 September 2026 16:09
> *To:* ringing-theory at bellringers.org
> *Subject:* [r-t] Link between Stedman sixes and Kent fours
>
>
>
> A plain course of Stedman Doubles is made up of alternating quick and slow
> sixes, the resulting plain course being the alternating group 'A5'. Extents
> in whole courses are therefore very easy to come by. A plain course of
> Kidderminster
>
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> A plain course of Stedman Doubles is made up of alternating quick and slow
> sixes, the resulting plain course being the alternating group 'A5'. Extents
> in whole courses are therefore very easy to come by.
>
>
>
> A plain course of Kidderminster Minor is made up of alternating Kent and
> Oxford fours (i.e: the first four rows of Kent or Oxford Treble Bob). The
> resulting plain course of 48 rows is the 'mirror group' as ringers call it.
> Again, extents can be easily constructed out of whole courses.
>
>
>
> A plain course of Erin Doubles - slow sixes only - is half of the
> alternating group, with the rows not forming any kind of set. There are no
> possible 120s in whole courses. A plain course of Forward Minor - Kent
> fours only - is half of the mirror group, with the rows not forming any
> kind of set. 720s are possible but not trivial to construct and certainly
> none exist in whole courses.
>
>
>
> Two consecutive slow sixes can be replaced by two consecutive quick sixes
> and v.v. The substitution cleanly joins two blocks together without
> introducing any new rows (e.g: Artistic Triples). The exact same scenario
> is true when substituting two consecutive Kent fours with two Oxford fours
> or v.v (e.g: the Worcester Variation.) I struggle to think of other
> examples where two consecutive blocks of changes can be inverted to cleanly
> join two round blocks together.
>
>
>
> Does all this demonstrate a group theoretical link going on, perhaps
> something to do with the outer automorphism of S6 as described in Brian
> Price's paper? Or is it just coincidence? Probably it's something far more
> mundane and obvious which I'm just overlooking.
>
>
>
>
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