[r-t] Queens and Tittums

[Ted Steele]

Richard Pullin wrote:
> May I remind and inform that Queens and Tittums are transpositions of 
> each other.
> You can see that by looking at the below transpositions:
>  
> 12345678
> 13572468
> 15263748
> 12345678
>  
> or:
>  
> 12345678
> 15263748
> 13572468
> 12345678
>  
> Either Tittums came about because of this, otherwise it's plain but 
> amazing coincidence. 
>  
> From Richard Pullin.
>

It is a tiny point in comparison to most of the others made about this, 
and may indeed have been covered elsewhere but in case not it may be 
worth observing that like the transpositions for Tittums and Queen's on 
higher numbers the major group also contains the reversal of the three 
main changes; or at least the changes with the interior  bells reversed. 
It is useful for three part compositions no doubt but perhaps slightly 
misleading to show the group as having just three members; in fact there 
are six.

_12345678
_14736258
13572468
17654328
15263748
16427538
12345678

Would group theorists regard this as being two separate groups or simply 
as one group in which not all members transpose to produce all of the 
others?

The Plain Bob group of lead ends at the royal stage (shown below) is 
another such instance that I have in mind and I have wondered whether 
there is any theoretical reason why such cases arise.

Two lead ends give only three lead courses but all of the others give 
the full nine; it seems odd that the other six don't form their own 
sub-group.

_1234567890
_1860492735
1795038264
1234567890

or

_1234567890_
1795038264
1860492735
1234567890

but

_1234567890_
1573920486
1907856342
1860492735
1426385079
1352749608
1795038264
1089674523
1648203957
1234567890

Looking at Tittums and Queen's on higher numbers we see, obviously that 
more rows are introduced.

_1234567890_
1357924680 Queen's
1594837260  "Back" Tittums
1987654320 "Back" rounds
1864297530 "Back" Queen's
1627384950 Tittums
1234567890 Rounds

and

_1234567890ET_
13579E24680T Queen's
15926037E48T
1963E852074T
16E50493827T"Back" Tittums
1E098765432T "Back" rounds
108642E9753T "Back" Queen's
184E7306295T
1470258E369T
172839405E6T Tittums
1234567890ET

By contrast

_123456_
135246 Queen's
154326 Back rounds
142536 Tittums
123456

_1234
_1324 Tittums and Queen's and Back rounds (on the internal bells only, 
as above)
1234

Of course the "back" changes are only partial reversals since 1 and "n" 
remain the same. However what I am wondering is whether or not these 
extra rows, ie the reversals and, especially the unnamed rows are 
regarded as having any particular musical qualities. It is possible to 
see the queens and tittums sequences in these rows but with more 
intermediate bells splitting them up, (as has been observed elsewhere) 
and I guess that this may give rise to some interesting music but it 
becomes increasingly subjective. The original question put by Richard 
Pullin appears to be whether Tittums was originally recognised because 
it was easily found as a transposition from rounds of the perhaps 
marginally more obvious Queen's. I suspect that rather more than this 
was probably understood. The musicality of the members of the larger 
groups suggests to me that there is no coincidence here, rather the 
basic rule of taking alternate bells from the rounds row and then each 
third bell and each fourth etc. is a rule which by its nature produces 
some pleasing sounds. The musicality is inherent within the rule and 
therefore the groups that it produces will be musical to many people. 
Queen's and Tittums are just the fundamental changes in these groups and 
the most obviously musical. That they are fundamental is suggested by 
the example of minimus where the three named changes are identical.

My knowledge of group theory is virtually non-existent and much of the 
above may be nonsense (and will certainly be glaringly obvious to many) 
but perhaps there may be some answers to the questions I have posed.

Ted