[r-t] Extension of TDMMs

Philip Saddleton pabs at cantab.net
Sun Aug 15 10:39:00 UTC 2004

Richard Smith <richard at ex-parrot.com> wrote at 01:55:23 on Sun, 15 Aug 
>An example of the more unusual 4n + 6 (n != 4 mod 7) is
>Northumberland S with 2ABCD/2EF.  As the exceptional case
>occurs with 7(4m + 3) working bells, this would suggest a
>lead head order of +/- 7.  In fact, on N bells, the lead
>head order is -(N+6)/4.  I've not encountered extensions
>which didn't have a fixed lead end before.  Are they common?

I think that the lead head order will always be a linear function of N 
(with the exception of the sporadic extensions). Extending by two 
stages, the inserted section can cause a bell to stay in the same place 
(perhaps after more than one insertion, e.g. Newgate front work), or to 
move the same distance as the treble. In general this causes a shift of 
0 or +/-1 place bell. The lead head order will then be kN/2 + c. In most 
cases it seems that k is even, as the same shift occurs in each half of 
the lead, but there are cases (e.g. Single Oxford) where k is odd. 
Extending by four stages allows the possibility of +/-1 place bell so 
the lead head order can be kN/4 + c.

>This means that exceptional leads occur whenever (N+6)/4 and
>N-1 are not coprime, or equivalently when n+3 and 4n+5 are
>not coprime.  I can certainly prove that when n=4 (mod 7)
>they are not coprime; however, I cannot prove that they are
>coprime otherwise.  Any ideas?

Put n=m-3, then we have m and 4m-7 not coprime => m=0 (mod 7).

>> >     1  2n + 6 (n != 1, n != 22) ??
>Oops. I think n != 22 was mean to read N != 22 (i.e. number
>of bells != 22).

Good - this was worrying me slightly, as I think the modulus ought to be 


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