# [r-t] Mark's latest definition and mutating fixed bells

Don Morrison dfm at ringing.org
Mon Aug 11 04:32:59 UTC 2008

```At the risk of making someone scream in horror at the perversity of my
imagination....

On Sun, Aug 10, 2008 at 7:09 PM, Mark Davies <mark at snowtiger.net> wrote:
> 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. In a two-stage peal, the structure of the composition should make clear
> which changes are treated at which stage.
>
> 4. For the purposes of proof in (1) and (2) above, methods at any lower
> stage may be considered rung at a higher stage by: (a) including leading or
> covering bells; (b) ringing two or more methods at a lower stage in
> parallel; or (c) both of the above.

In 4 that statement "methods at any lower stage may be considered rung
at the higher stage" is deliberately "may" rather "must", right? Or is
it "must be one of a, b or c"? I'm not sure I'm completely following
that clause. And even with that clarified I'm not sure I'd completely
understand what it means with regard to clause 1.

The case I'm worried about is the following perverse one:

Say I ring triples. And only triples. But sometimes I'm ringing it
with a cover. And sometimes I'm ringing it with a continuously leading
bell instead. This is way outside what the current Decisions allow,
but appears to be within the scope of what Mark proposes, and seems
perfectly plausible to me (feel free to scream in horror now). It even
sounds like fun to me.

It's not clear how I test for truth in this case, at least not with
Mark's definition. It may be that definition says how, but I haven't
been able to quite work it out for sure.

In the triples case one obvious way out would be to prove it simply as
major. But I'm not convinced that's what Mark's definition says to do,
though it might.

What if it's not triples, but covered/led minor? Does that mean it's
got to be the extent of triples, or can I have seven extents of minor,
some with a continuously leading bell and some with a continuously
lying bell, but not true if viewed as triples?

What if we mix doubles, sometimes covered and sometimes led, with minor?

I think the "sets" definition of truth provides a clear and
unambiguous answer in any case involving these swapped leading/lying
bells, at any stage or stages. Whether or not that is an answer that
corresponds with our intuition I haven't figured out, though. I'm
hoping someone else may be able to help clarify that.

And I'm not even close to figuring out what it all means in terms of
the other candidate definitions of truth that have been suggested.

Also, if I ring triples this way do I have to ring a 5,040? Or is this
a case, like variable cover triples in the current Decisions (which I
think Mark would view as major) where a 5,000 is sufficient?

--
Don Morrison <dfm at ringing.org>
"May I remind you that I am in command here! Only an idiot would attempt
such a thing. I will do it myself."  -- _The Island at the Top of the World_

```