Ringing Campanologically New methods for Old Church Bells

Ringing Campanologically
New methods for Old Church Bells
Alexander Holroyd
Microsoft Research
Frank Ruskey
University of Victoria
Aaron Williams*
Carleton University
Campanology
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Christ Church Cathedral (photo by YVRJerry)
click to play
campanology - the
study of bells; their
casting, tuning and
ringing
change ringing - the
art of ringing a set of
tuned bells in a series
of mathematical
patterns called “rows”
or “changes”
Change Ringing
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each bell has its
own ringer
each bell is played
once during a row
rows are rung very
quickly (around two
seconds for 8 bells)
after ringing a row,
there is a brief
pause before the
next row is rung
(photo by Britannica)
Rows
treble
tenor
●
8
●
...
1
2
12345678
rounds
13572468
Queens
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Kings
15263748
Tittums
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the treble is the
highest-pitched
and is numbered 1
the tenor is the
lowest-pitched
some rows have
special names
in mathematics
rows are called
permutations
Extents
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if every possible row
is rung once without
repeating any row,
then it is an extent
123
132
213
231
312
321
●
}
45000
40000
35000
30000
25000
an extent on three bells
● each of the six possible
rows appears once
●
the number of rows
can be very large, so
patterns are required
20000
15000
10000
5000
0
4 bells
5 bells
6 bells
7 bells
8 bells
Number of Rows in an Extent
The first extent on 7 bells was played in Norwich in 1715.
The first extent on 8 bells was rung in Loughborough 1963.
The 40,320 rows were played in 18 hours.
Question: What order did they play the rows?
Bells
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bells rotate 360o and ring
up and down stroke
the tenor bell can weigh
up to 4000 kg!
Ringing Restriction:
Bells rung near the end of
a row cannot ring near
the start of the next row
18765432
21345678
bell 2 cannot
re-ring in time
(animation by cathedral.org)
Swaps
●
12345678
21345687
1 and 2
swap
7 and 8
swap
bells move at most
one position to the left
●
●
a swap is when two
consecutive bells
switch places from
one row to the next
swaps ensure the
next row satisfies the
ringing restriction
can swaps be used in
extents? yes! ringers
have focused on swap
patterns for centuries
Plain Bob
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a popular order for
ringing is Plain Bob
blue line diagrams
highlight the bell
position in each row
a ringer memorizes
her order and also
listens to a caller for
variations
Plain Bob Major
(8 bells)
Plain Bob Minor
(6 bells)
(image by Matthew S. Fry)
Many different orders are known such this Bristol Surprise.
Almost all orders are based on swaps.
Question: Can extents be made without using swaps?
New Idea: Shifts
75312468
53124687
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7 is shifted into the last position
75312468
53124678
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7 is shifted into the second-last position
bells move at most
one position to the left
shifts produce different
soundscapes than swaps
●
a shift is when the
first bell in a row
moves into the last or
second-last position
shifts ensure the next
row satisfies the
ringing restriction
can shifts be used in
extents? yes! new
patterns by Holroyd,
Ruskey, and Williams
Holroyd's Shift Extent
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●
major
4213
2134
1342
3421
4231
2314
3142
1432
4312
3124
1243
2413
successive rows are
written clockwise
around the circle
(each row's last symbol is redundant
and is omitted in the circle)
●
4321
3214
2143
1423
shift extents can be
visualized by circles
of numbers
4123
1234
2341
3412
4132
1324
3241
2413
this shift extent
uses the maximum
possible number of
long shifts
Ruskey-Williams Shift Extent
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major
4321
3214
2143
1423
4213
2134
1342
3412
4132
1324
3241
2413
4312
3124
1243
2413
4123
1234
2341
3421
4231
2314
3142
1432
●
this shift extent
uses the maximum
possible number of
short shifts
both shift extents
can be created
efficiently by simple
patterns
the sequences for
the caller can also
be created easily
Applications
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these shift extents have
applications other than
bell-ringing
the stacker-crane
problem tries to deliver
multiple objects along
the shortest route
the new shift extents
give the most efficient
exhaustive solutions
(they minimize the number of ordered pairs
that change between successive permutations)