[r-t] Semi-interesting observation

[Robin Woolley]

Let, as by convention, H = 13527486 and J = 12436587. Also, let R = 13456782
and S = 12346587. (H & J were discussed a year or so ago and I adopt the
convention of the algebra operating from right to left, e.g. HJ means 'H
operating on J' - I'm too old to change without the danger of making simple
slips in notation.

Since the regular lead heads are all powers of H: (H^j, j=0,..,7), then the
single at the lead end when the tenor is becoming 5ths place bell can be
written as SJH^2.

A little work with a pencil shows R is generated by having the single at
3rds which in this notation is SJH^3 followed by five plain leads (if a
group A method is being used) thus:

R = H^5.SJH^3.SJH^2.

The semi-interesting observation is that if any of the cycles of rounds are
required, then

R^n = H^5.SJH^3n.SJH^2 for appropriate choice of n.

It is the 3n term which I find semi-interesting.

This generates a seven part composition with the cycles as the part ends
and, as has been pointed out by PABS and others, this part end occurs
somewhere in a (full) plain course - note the H^5 and H^2 at the ends of the
expression.

Examples:
i) For Group A, the first call is at the end of the second lead and the
cycle occurs five leads after the second single;
ii) For Group B, the first call is at the first lead end, there being six
leads after the single to obtain the part end;
iii) For Group F, 1st call at end of 5th lead, two more leads after the 2nd
single gives the part end.

John David asks for a use for these courses. A 7 part a.t.w. peal, possibly?

Best wishes
Robin.