[r-t] Spliced Doubles Variations

[Richard Smith]

Alex Tatlow wrote:

> Is anyone able to provide/produce a 120 of Spliced Gold & 
> Melchior Doubles? They're variations - Westminster II with 
> Old Hudibras Singles, and Maltby B with Wallflower Singles 
> respectively. If it isn't possible, a 240 or 360 would 
> also do. Thanks in advance for any replies.

I'm not really sure what it means to splice variations, but 
in this case it seems pretty clear as both methods have the 
same underwork (St Simon's).  Effectively you have four 
different overworks that you want to ring with the common 
underwork:

Westminster II B   m   [145.3.125.3.145]
Maltby B           n   [145.5.123.5.145]
Old Hudibras S     je  [1.5.125.3.1]
Wallflower S       f   [1.345.123.345.1]

The fact that Old Hudibras is asymmetric is a complication, 
and so initially I shall replace the Old Hudibras single 
with a Grandsire single (c: 1.3.123.3.1), which also has 
seconds made at the lead-end and has the same effect on the 
coursing order.

All four overworks have a seconds place lead-end which (for 
symmterical overworks) means they all share three-lead 
splices with seconds place bell fixed, and so the choice of 
overwork is determined solely by the bell making seconds. 
If we want all four overworks to be present, we just assign 
an overwork to each inside bell -- say 2=m, 3=n, 4=c, 5=f -- 
and follow the rules to see what happens.  And what happens 
is that it comes round after 100 changes.  The choice of 
overwork assignments is irrelevant, as a change of 
assignment simply rotates or reflects the touch, so any 
choice of overwork assignments will either bring it round in 
100 changes or in 20.  A 120 is not therefore possible.

Because I chose c (the Grandsire single) to have the same 
effect on the coursing order as je (the Old Hudibras 
single), a 120 in the original choice of methods is also 
impossible: it'll also come round after 100 or 20 changes, 
though it may have already run false due to the Old Hudibras 
single.

To formally proof it's not possible we would have to 
demonstrate that the extent can only be done by using the 
three-lead splices.  We can certainly consider each bell 
making seconds independently, because each row can be 
uniquely mapped to particular bell making seconds 
irrespective of the choice of overwork.

For a given observation bell, we must either have 145 every 
time the treble moves between 2-3, or we must have a 1.  We 
cannot have a mixture with any given observation bell. 
This is because of the particular overworks available.  In 
the available 145 overworks, the 45 places effect all three 
working bells, removing the 45 places would cycle those 
three bells, and by the Q-set rule, we would need to do it 
to all three leads.

>From a simple inspection, we can see that the two overworks 
with a 145 (m and n) have a three-lead splice, as do the 
other two (je and f).  We have therefore proved that a 
fixed-treble 120 is not possible containing just those four 
overworks above St Simon's.

A 240 or longer is obviously trivial.  Take an extent (or 
more) of each, and chop them up (or not) as desired.

RAS