[r-t] Extension of TDMMs

[Richard Smith]

Philip Saddleton wrote:
>
> I was thinking of a higher parent stage - with more methods, more
> sections and more modes available (not to mention more treble paths),
> how many possibilities are there for Major?

I don't have an exact figure to hand.  A back of an envelope
calculation suggests one or two billion treble dodging
major methods (including irregular ones), or about quarter
of a billion regular ones.  That seems like a big enough
sample to produce something.

> >  Freq  Series
> >  ------------------------------------------
> >   136  4n + 6 (n != 1 mod 3)
> >    76  2n + 6 (n != 2 mod 3)
> >     4  4n + 6 (n != 4 mod 7) ??
> >     2  8n + 6 (n != 2 mod 3) ??
> >     1  6, 4n + 8 (n != 2 mod 3)
> >     1  6, 8n + 12 (n != 2 mod 3)
>
> Are these exceptions because they have short courses?

Yes, whenever I have looked (which isn't often).  I hope to
have some better data on this by tomorrow.

An example of 4n + 6 (n != 1 mod 3) is Cambridge S extended
with 2DE over 3FG which has -3 (i.e. 'd' n>6) lead head
order.  When n = 1 mod 3, we have 3(m + 3) working bells
which is divisible by 3.

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?

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?

> >     1  4n + 6 (n != 7 (mod something?)) ??
> mod 11?

Yes.  This occurs when extending an unnamed Delight method
(&5-34.1-5-23-4-3,2) using 1ABCD/5CDEF.  This is another one
like Northumberland -- the lead head order on N bells is
+(N+10)/4.  And again, whilst I can substitute N = 4n + 6 to
verify that these do produce differentials with 11 cycles, I
can't prove that no others produce differentials.  (As this
is the only indefinite extension of this method, it is of
slightly more relevance.)

> >     1  2n + 6 (n != 1, n != 22) ??

Oops. I think n != 22 was mean to read N != 22 (i.e. number
of bells != 22).

> Does n=1 come round? In which case (n != 1 mod 21)?

This is Brentford S extended by 2BC/1EF, and yes n=1 comes
round, so this one is n != 1 mod 7 (because of the above
mistake).  The lead head order is -7.

Richard