[r-t] Proofs, etc

[Richard Smith]

Robin Woolley wrote:

> When Royal is considered, the maximum false lead set size is 72 and, as can
> be seen by inspection , but also shown algebraically, we have the situation
> where the in - course tenors together members of falseness groups B & D
> always appear together. The Exercise has been fortunate that a more radical
> revision of the FCH groups has not been necessary as we go to stages higher
> than major.

This isn't the only difference though.  For Major, there are
19 in-course tenors-together falseness groups.  For Royal,
there are 21, and for Maximus and higher, there are 22.

Going from Major to Royal, the in-course L, P and U groups
split in two:

  26543 L1     54326 P1     34256 U1
  36245 L2     64352 P1     35426 U1
  42563 L2     56342 P2     42356 U1
               64523 P2     43652 U1
                            52436 U1
                            63254 U1
                            35264 U2
                            42635 U2

This split is retained in Max and at higher stages.  (A
similar splitting occurs with the out-of-course parts of K,
N and a.  More noticeably, moving from Major to Royal
separates the in-course part of the Major falseness groups
from the out-of-course parts.  This means that, for example,
Cambridge Maximus has B falseness out-of-course and D
falseness in-course (usually written B2 D1).)

The remaining difference is that, as Robin notes, the
in-course B and D groups merge into a single falseness group
for Royal, separating again for Max and at higher stages.


[...]

> Once again, I can't see how any of the other treatments proposed deal with
> incidence. For another treatment, why not look at, say RW77/923 and
> RW80/259?

The incidence of falsenes between false courses, and the
list of false leads are just two ways of looking at exactly
the same problem.  If you are looking at compositions that
are entirely in whole courses, then thinking in terms of
false courses makes obvious sense.  As you move to
compositions that are nearly in whole courses (Middleton's
is a good example), then think about the incidence of course
falseness is a good way of making use of your existing
understand of false courses.

But once you get away from compositions that are
course-based, I personally don't find the incidence of
course falseness particularly helpful, and prefer to work in
terms of false lead heads.  Certainly when I've written
computer programs to search for compositions, I've usually
written them in terms of false lead heads rather than
the incidence of course falseness.

Of course, which is the "better" approach is inevitably
subjective as the two approaches achieve exactly the same
thing.


> We can't start every e-mail with a definition
> of plain hunt, but it is justified to expect an explanation of M11 or
> 'geoetrics' which may only be familiar to those researching a particular
> area. (Is the latter peculiar to Bangor?)

I think "geoetric" was a typo for "geometric".


Graham John wrote:

> Robin wrote:
>
> > Falseness groups aren't groups either - they're not closed for one
> > thing - but they're still referred to as that.
>
> Excuse my ignorance, but why aren't they closed? All 720 possible
> courseheads fall into the 28 falseness groups
> (ABCDEFGHIKLMNOPRSTUabcdefXYZ), each having its own distinct transposition
> relationship.

The 28 falseness groups considered as a wholeis indeed
closed, but any one (other than A) considered in isolation
is not.

Imagine one of the B false course heads -- say, 13245678.
Now transpose this by another B false course head -- perhaps
12365478.  We end up with 13265478, which is an F false
course head.  This means that the B falseness "group" is not
closed and so is not, mathematically, a group.

Richard