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<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">I don’t see how the outer automorphism would help, because the numbers of rows involved are different.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Another weird example of “clean joining” comes in April Day Doubles, or indeed April Day Minor. Calling a Grandsire Single brings up the *<b>same</b>* 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). *<b>These</b>* blocks get
cleanly joined by the calls. Perhaps doubles variations experts can provide more insight.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Incidentally, I recently discovered how to make a physical manifestation of the outer automorphism of S_6 (I think the first such):<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><a href="https://www.youtube.com/watch?v=T19oalEPp70">https://www.youtube.com/watch?v=T19oalEPp70</a><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">although this does not answer your question!<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
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<p class="MsoNormal"><b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">From:</span></b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> ringing-theory <ringing-theory-bounces@bellringers.org>
<b>On Behalf Of </b>Richard Pullin<br>
<b>Sent:</b> 01 September 2026 16:09<br>
<b>To:</b> ringing-theory@bellringers.org<br>
<b>Subject:</b> [r-t] Link between Stedman sixes and Kent fours<o:p></o:p></span></p>
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<p class="MsoNormal"><o:p> </o:p></p>
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<p class="MsoNormal" style="mso-line-height-alt:.75pt"><span style="font-size:1.0pt;color:white">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<o:p></o:p></span></p>
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<p class="MsoNormal">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.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">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.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">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.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">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. <o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">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.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
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