[r-t] Queens and Tittums

[Richard Smith]

Richard Pullin wrote:

> May I remind and inform that Queens and Tittums are transpositions of each
> other. [...]
> Either Tittums came about because of this, otherwise it's plain but amazing
> coincidence.

I not sure tittums came about because of this per se, or
that it's simply a coincidence.  Rather, the things that
make certain changes (rounds, tittums, back rounds, etc.)
sound particularly pleasant, are precisely the things that
result in them being transpositions of each other.

Why is queens pleasing to listen to?  Because it runs down
the scale of odd bells, then through the even bells.  I.e.
it takes every second note from the scale.  What about
tittums?  On eight, 15263748, it takes every fourth note
from a scale.  If you take every second note from something
that is already taking every second note, it's equivalent to
taking every fourth note.

But that description is slightly disingenious: Queens
doesn't take precisely every second note as it jumps from 7
to 2; likewise every other jump in tittums doesn't fit.  It
also doesn't explain why there isn't (on eight bells) a
third example where every sixth bell is chosen.

A slightly more accurate would be to consider them as seven
bell changes, rung with tenor covering.  Now rounds is
1234567; queens is 1357246 -- which is precisely every
second bell (if you wrap round to the beginning of the row).
Repeating -- i.e. taking every second bell from queens --
gives 1526374.  And once more goes back to rounds, 1234567.
Why?  Because queens is every second bell from rounds;
tittums is every fourth (2x2) and the next step would be
every eighth bell (2x2x2), but taking every eighth bell and
evrey bbell is the same thing.

On ten bells, every fourth bell isn't tittums -- every fifth
bell is; neither does taking every eighth bell bring rounds
again -- we need every tenth bell for that.  This means we
get some extra rows on ten bells:

  123456789(0)  Rounds
  135792468(0)  Queens
  159483726(0)  'Reverse tittums'
  198765432(0)  'Reverse rounds'
  186429753(0)  'Reverse queens'
  162738495(0)  Tittums
  123456789(0)  Rounds

Queens is every second bell; 'reverse tittums', every
fourth;  'reverse rounds', every eight -- which is
equivalent to counting backwards through the sequence.
Then 'reverse queens' is every second bell counting
backwards'; tittums, every fourth bell counting backwards
(or every fifth bell counting forwards); and taking every
eight bell backwards is the same as taking every bell
forwards and gives rounds again.

RAS