[r-t] Decisions / Algorithms for generating the extent

[Philip Saddleton]

edward martin said  on 20/06/2006 17:06:
>> Well, there's another one in Tintinnalogia, though
>> Stedman/Duckworth isn't explicit that it works on arbitary
>> stages.
>>
>> The starting point of the series is the extent on three
>> bells, Original Singles.
>>
>>   123   3
>>   213   1
>>   231   3
>>   321   1
>>   312   3
>>   132   1
>>   ---
>>   123
>>
>> All bells hunt throughout and this is enough to get the
>> whole extent.  If you try this on four bells, though, you
>> only get a third of the extent:
>>
>>   1234  x
>>   2143  14
>>   2413  x
>>   4231  14
>>   4321  x
>>   3412  14
>>   3142  x
>>   1324  14
>>   ----
>>   1234
>>
>> However, we can take this as one lead of a three lead extent
>> by making lead heads and lead ends form the extent on the
>> back three bells:
> 
> But Campanista demonstrates this very point pages 8-10
> and again (5 bells) pages 18 - 29
> and again (6 bells) pages 35 - 52
> He is very explicit & excited about applying Grandsire Doubles to give
> us 'Grandsire Bob Minor (the standard 720 of Bob Minor)  page 114
> 
> mew
> 

These are examples of systems of hunts, the basis of many extents. More 
generally:

- find a block where a subset of the bells occupy each possible 
combination of positions (WHWH)

- find a calling that does not disturb this subset, but cycles the 
remaining bells - this gives an equivalent block for a larger subset (WHWx3)

- repeat as necessary, with a calling that fixes one more bell at each 
step (WHWx3 sH)

Taken to extremes:

362880 Grandsire Caters
23456789   1  3  4
------------------
43628579   -  -  s |  |  |
63847259   -  -  s |  |  |
38765429   -  -  - |  |  |
87532649   -  -  - |A |  |
57284369   -  -  s |  |  |
27456839   -  -  s |  |  |
47623589   -  -  s |  |  |
------------------    |  |
67348259   -  -  s |  |C |
37865429   -  -  s |  |  |
78532649   -  -  - |  |  |
85274369   -  -  - |B |  |
52486739   -  -  - |  |  |E
42653879   -  -  s |  |  |
62347589   -  -  s |  |  |
------------------    |  |
76234       2B        |  |
43625789    2A        |  |
------------------       |
63542        C           |
------------------       |
57263489     A     |     |
63572       4B     |D    |
54263789     A     |     |
------------------       |
35426       2D           |
------------------
25364       3C     |F
42536       2D     |
------------------
24356       2F
------------------
45326        E     |
54236       2F     |G
43256        E     |
------------------
324          G
------------------
Repeat


-- 
Regards
Philip
http://www.saddleton.freeuk.com/