[r-t] Proposed definition of a peal

[Iain Anderson]

Mark Davies wrote:

> How about a peal is true if:
> 
> 1. It is rung on one stage, and each change in the extent at 
> this stage is rung either N or N+1 times in the peal, and no 
> more, for N>=0.
> 
> 2. It is rung on two stages, A and B, where |B-A|=1, and each 
> change in the extent on A is rung M times in the peal, and no 
> more, and each change in the extent on B is rung N or N+1 
> times, and no more, for M>0 and N>0.
> 
> 3. Peals of methods at different stages may be rung as a 
> single-stage peal as (1) above by considering the covering 
> bell two be included in the change for methods at the lower stage.
> 
> [...]
> 
> I've added an extra consideration to prevent part-extents 
> being rung at both stages. [...] Which seems reasonable.

I'm not convinced by the last bit.  If you are going to allow part extents
at the higher stage and part extents at the lower stage, why not both
provided they are mutually true?

Also, what about a true 720 that is, say, a 660 of minor and a 60 of
doubles.  Ring three extents of minor, one of these mixed 720s and three
720s of doubles and it fails all three conditions.  I think the intention is
there in the rules, but it just doesn't quite work.  Nor does it work for
Minor and Minimus, or Maximus and Royal, or Royal and Doubles, etc..  Or
peals rung on three stages.

There is a sort of recursive thing going on.  Here's my attempt at a truth
test:

1) Consider the set of rows and take the current stage to be the maximum
stage at which each row is rung.
2) If the number of rows in the set is less than or equal to the extent at
the current stage, the rows must be distinct.
3) Otherwise remove a true extent's worth of rows from the set and re-apply
the truth test on the remaining rows.  (If you can't do this, it's false.)

So for 5040 of Minor and Doubles, you initial treat everything as Minor,
remove a true 720 (including as many genuine Minor rows as possible).
Repeat on the remaining 4320 of Minor and Doubles, then 3600 of Minor and
Doubles, and so on until you only have Doubles rows left and the current
stage becomes 5.

For Maximus and Cinques, you shouldn't get beyond step 2, and we are left
with the current understanding of truth.
It also works for 3 stages.