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Methods with mirror or horizontal symmetry are still quite rare. These involve pairs of bells ringing mirror blue lines of each other in perfect unison. To achieve this you must use horizontally symmetric place notations, which in Minor are limited to 16, 34, 1256, X. The plain course of Kidderminster Minor sweeps up all 48 of the mirror rows on six bells - the mirror pairs being bells 16, 25, 34 - and to produce a 720 you simply need to visit all 15 of the different possible pairings.<div><br></div><div>I think principles make the most elegant mirror methods as treble-dominated methods require the treble and tenor to both be hunt bells, which seems a bit bizarre. </div><div><br></div><div>On eight bells there are 384 mirror rows, the permitted place notations being 18, 36, 1458, 1278, 3456, 123678, X. The nicest of these are 18, 36, X, but it seems you can only attain 192 rows using just these place notations, in much the same way that you can only ring 60 rows in Doubles without using single changes. </div><div><br></div><div>I produced what is probably the simplest principle to use these three place notations, and called it Horizontal Major: <span style="background-color:rgb(245,245,245);color:rgb(51,51,51);font-family:"Open Sans","Helvetica Neue",Helvetica,Arial,sans-serif;font-size:12px;white-space:nowrap"><a href="https://complib.org/method/40781?accessKey=97137a1a6d26fa1ec7463700b5fb092c394d4856" target="_blank">https://complib.org/method/40781?accessKey=97137a1a6d26fa1ec7463700b5fb092c394d4856</a></span></div><div><br></div><div>Compositeur generated some more principles, one of which I selected and called Mirror Major as it has palindromic symmetry in addition to the horizontal symmetry (this is what distinguishes horizontal from mirror symmetry): <span style="background-color:rgb(245,245,245);color:rgb(51,51,51);font-family:"Open Sans","Helvetica Neue",Helvetica,Arial,sans-serif;font-size:12px;white-space:nowrap"><a href="https://complib.org/method/40782?accessKey=4bd6cbe546d92baa55773f4afc982889190a13df" target="_blank">https://complib.org/method/40782?accessKey=4bd6cbe546d92baa55773f4afc982889190a13df</a></span></div><div><br></div><div>Building on Horizontal Major, I added two 1458 PNs in each lead to produce a principle containing all 384 mirror rows, and called it Double Factorial Major: <span style="background-color:rgb(245,245,245);color:rgb(51,51,51);font-family:"Open Sans","Helvetica Neue",Helvetica,Arial,sans-serif;font-size:12px;white-space:nowrap"><a href="https://complib.org/method/40789?accessKey=32b4c8bd2455dfb1bab2ea528d4d5a021a9284f7" target="_blank">https://complib.org/method/40789?accessKey=32b4c8bd2455dfb1bab2ea528d4d5a021a9284f7</a></span></div><div>Again this does not have palindromic symmetry, but I don't think that's a shortcoming as the whole point of the exercise was to appreciate alternative types of symmetry.</div><div><br></div><div>Bobs-only 720s of Kidderminster Minor are possible because an odd number of courses is required for an extent, meaning that you simply need to add normal q-sets of 3 bobs. This is an incredibly rare feature. Fortunately a bobs-only 40320 of Double Factorial Major is possible for the same reason, an extent being 105 courses. I set to work producing one and the result is below. I imagine this might be the first bobs-only extent of Major (using 'normal' bobs that affect three bells, at any rate) and probably the first where adding q-sets of 3 bobs was the sole technique for constructing the extent. Frustratingly there were two remaining courses I couldn't q-set in and it took me a few more days to shuffle things round and finish the composition. It was very interesting having to deal with courses that were defined by pairings of bells rather than courses defined by a unique course head. <br></div><div><br></div><div>40320 Double Factorial Major</div><div><font face="monospace" size="4">1 2 3 4 5 6 7 8 12345678</font></div><div><font face="monospace" size="4">--------------------------------</font></div><div><font face="monospace" size="4"> 2 - 3 72145386</font></div><div><font face="monospace" size="4">3 - 82145367</font></div><div><font face="monospace" size="4"> 2 - 82751463</font></div><div><font face="monospace" size="4"> - 3 38751462</font></div><div><font face="monospace" size="4"> 2 - 82351764</font></div><div><font face="monospace" size="4"> 2 82513764</font></div><div><font face="monospace" size="4"> - 3 48513762</font></div><div><font face="monospace" size="4"> 2 - - 82451367</font></div><div><font face="monospace" size="4"> - 3 - 3 62145378</font></div><div><font face="monospace" size="4"> - - 12356478</font></div><div><font face="monospace" size="4"> 2 12456873</font></div><div><font face="monospace" size="4"> - - - 31264578</font></div><div><font face="monospace" size="4"> 2 3 51364278</font></div><div><font face="monospace" size="4"> - - 31245678</font></div><div><font face="monospace" size="4">--------------------------------</font></div><div><font face="arial, sans-serif">3-part.</font></div><div><font face="arial, sans-serif">Bob = 34</font></div><div><font face="arial, sans-serif">Double Factorial Major: </font><span style="color:rgb(51,51,51);font-family:"Open Sans","Helvetica Neue",Helvetica,Arial,sans-serif;font-size:12px">18.36-36-18-36-36-36-18-36-36-36-1458.36.18.36.18-18.36.18.36.18-18.36.18.36.18-18.36.18.36.18-1458-36</span></div><div><font face="arial, sans-serif">Link to composition: </font><span style="background-color:rgb(245,245,245);color:rgb(51,51,51);font-family:"Open Sans","Helvetica Neue",Helvetica,Arial,sans-serif;font-size:12px;white-space:nowrap"><a href="https://complib.org/composition/76406">https://complib.org/composition/76406</a></span></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">The method is named after the double factorial number sequence in mathematics, which is what determines how many mirror rows there are for each number of bells.</font></div><div><font face="arial, sans-serif">This is like the normal factorial sequence but uses only odd or even numbers. So 6!! is 2 x 4 x 6 = 48, which is why there are 48 mirror rows on six bells. 7!! is 1 x 3 x 5 x 7 = 105, which is why an extent of Double Factorial Major has 105 courses, and why extents for other even-bell methods that contain all mirror rows in their plain courses always require an odd number of courses in their extents.</font></div></div>