[r-t] Proposed definition of a peal

[Richard Smith]

Don Morrison wrote:

> 7) A piece of change ringing, if all of one stage S, is called true is
>   there is a non-negative integer N such that each of the rows in the
>   extent at stage S occurs at least N times and no more than (N + 1)
>   times.
>
> 8) A piece of change ringing, if of multiple stages, is called true as
>   follows. All the stage fragments contained in the piece of change
>   ringing that are of the same stage with the same non-changing
>   bells, are grouped together, and tested for truth as for a single
>   stage. If all such groupings are true, then the overall piece of
>   change ringing is called true.

Iain Anderson replied:

> I think we may need to tighten this definition up a little 
> bit for things like Spliced Bristol Maximus and Stedman 
> Cinques, which I am interpreting to be a piece of change 
> ringing of multiple stages. Clearly it isn't sufficient to 
> have the group of Bristol rows true separately from the 
> group of Stedman rows.  In this case we really need to 
> define the Stedman as a maximus method and have the cover 
> bell as part of the method ...!

I agree with Iain, but I think this is easy to fix in Don's 
definitions.  Let's take Don's definition 7 and add two 
further definitions:

> 7) A piece of change ringing, if all of one stage S, is
>    called true if there is a non-negative integer N such
>    that each of the rows in the extent at stage S occurs
>    at least N times and no more than (N + 1) times.

  7a) A piece of change ringing, all of one stage S, is
      additionally called complete if there is a positive
      integer N such that each of the rows in the extent at
      stage S occurs exactly N times.

  7b) A piece of change ringing that is not complete is
      called incomplete.

And now let's alter 8 to read:

> 8) A piece of change ringing, if of multiple stages, is
>    called true as follows. All the stage fragments
>    contained in the piece of change ringing that are of
>    the same stage with the same non-changing bells, are
>    grouped together, and tested for truth as for a single
>    stage. If all such groupings are true, and at
>    most one is incomplete, then the overall piece of
>    change ringing is called true.

(All I have changed is "if all such groupings are true" to 
"if all such groupings are true and complete", which is 
arguably tautologous, but probably clearer than simply 
"complete".)

Now we've solved Iain's objection.  If you ring Cinques and 
Max, you have two options.  Either you call it variable 
cover (in which case truth is defined as if it were wholly 
Maximus), or you call if mixed Cinques and Max.  However, if 
you call it Cinques and Max, you have to include a whole 
extent of at least one.  And no-one is going to do that.

This definition of truth is slightly more liberal than the 
current one, but considerably less lenient than Don's 
because I'm allowing at most one incomplete block in a peal. 
This still opens the door to new things like 5016s of minor, 
5039s of triples, and ordinary-length peals of minor and 
triples.

Perhaps this is closer to what we want?

RAS