Thomas A. Alspaugh
Chart/Diagram Generator

 ⌖ ◉

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Chart: Ltcas: Diagram: Landmarks: Text: Changes: more
 On: ☼% Place notation: Name: Error message for name field Error message for moving bells field Tenor Behind / Covers: 0 1 2 Error message for covers input Error message for place notation field Error message for bob masks field · pv · Error message for single masks field ◉ ◉ ◉ bells · treble hunts · 2nds hunts · Plain Hunt · treble bobs · has prefix more rows/lead × leads/course = rows/course (ratio ) coursing(s): coursing(s): more Help for inferring the Single method from which a Double is derived:

Table of Work

Table of Ltcas

Progress through Snowdon Diagrams 1998: #
Progress through Snowdon Diagrams 2011: p
methods/principles.

This page uses a Javascript program to analyze and display methods and principles. Please enable Javascript in order to use it.

(You do not need Java enabled to use this page, and to protect your computer you should keep Java disabled in your browser.)

This page has been tested on recent versions of Firefox, Safari, and Opera.

The help for this page is under construction.

# ⌖ How to Avoid Reading the Rest of This Help

1. Click the for the method or principle you are interested in. For example, for Plain Bob Doubles (5 bells), find the row headed "Plain Bob:" and click the
button in that row.
2. Look on in amazement as a mass of fascinating information appears:
3. Click any of the buttons that display the progress of the method/principle:
• a
• a or a (Plain Course)
• a (Segment) (for Stedman and similar)
• a or if supported
• a or if supported

If you are clicking the button to see a row at a time, the and buttons will be enabled whenever a bob or single respectively can be called.

4. For most methods/principles, there is a table of work showing the work done by each bell, organized by where the bell was at the previous lead end backstroke.

Why use this program?

1. See which strokes are back and which are hand.
2. Refer to rows by their ordinal number.
3. See a diagram with each bell in a different color.
4. Step through the method/principle at your own pace; back up and look again at any point.
5. Add bobs and singles wherever they are allowed; if you are stepping through a row at a time, add them at the stroke where they should be called. Back up later and make a different call, or no call.
6. Compare two methods side by side, or two sequences of the same method.
7. Start from rows other than rounds.
8. Select and copy from a text chart to work with in a text editor.

# Why?

## Why should you bother to use this?

The methods and diagrams and their bobs and singles are primarily derived from Snowdon's Diagrams, 1998 edition.

# ⌖ Chart or Diagram or Text?

The chart, diagram, and text show roughly the same information but in different forms. Each is visible if its checkbox is checked; you can show any combination of them but not none of them.

## ⌖ Chart

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

The chart lists the bells in each successive row, in the order in which they strike. Each row is preceded by its ordinal [1], [2], [3], etc. except for the initial row 0 of rounds. Handstrokes have a white background and backstrokes have a gray background. Lead end backstrokes are distinguished by a darker gray background.

Each lead end backstroke row is followed by the work number (② ③ ④ etc.) identifying the work done in the lead to come. The work number is the same as your bell's place at the lead end backstroke.

Each row at which a call was made is followed by a B for Bob or an S for Single. The row two strokes later at which the call takes effect is followed by a ! (exclamation point).

Rounds and other rows in recognized patterns are marked as in the table below.

Row markers
Rounds
Back Rounds
Queens
Tittums
Whittingtons

To see all the markers, click on Plain Bob and look at a plain course (four leads, (Plain Course)). Other methods with this somewhat uncommon property are:

## ⌖ Diagram

The diagram shows the path of each bell as it moves from place to place. As in the chart, backstrokes have a gray background and lead end backstrokes have a darker gray background. Each bell is indicated by a line of a different color (see table at right); the colors are chosen so that their names rhyme or chime with the bell number.

To the left of row [0.] are your course bell, your bell, and your after bell, shown as a bell character and a bell color.

To the right of row [0.] and every later lead end row is your bell's work, shown as the place number for that work, circled.

The name of the method is echoed above the diagram.

## ⌖ Landmarks

The landmarks list things an attentive ringer of your bell might notice and benefit from at each row.

The top row of the list gives your course bell, your bell, and your after bell as numbers and colors.

## ⌖ Lead-treble-course-after

The Lead-treble-course-after (Ltca) display lists which bells you are over as the method/principle unrolls, not in terms of bell numbers but abstractly in terms of the treble, your course bell, your after bell, and when you Lead.

Knowing which way to look seems to be an important component of ropesight, and in learning Plain Bob an important step for me was learning the rhythm in which a ringer looks toward the treble. The Ltca sequence can be very helpful in this regard.

The Table of Ltcas presents this information horizontally.

## ⌖ Text

The text gives virtually the same information as the chart, but in plain text form so you can copy and paste it easily to work with in a text editor or other program.

## ⌖ Changes

The changes show the place notation change that produced each row.

## ⌖ A second group of displays for comparison

Click the button to start a second group of displays (chart, diagram, and/or text) for comparison, and the button to delete the first group of displays when you are done.

Open a new window in your browser and start a second session of this page, then arrange the windows side by side so you can compare the results.

(An earlier version of this page showed two or more on the same page, but as the page was expanded to show more information about each method/principle, it became difficult to arrange it helpfully. Possibly a future version will display more than one on the same page again.)

# ⌖ Method/Principle Preset Buttons

 Singles 3 Minimus 4 Doubles 5 Minor 6 Triples 7 Major 8 Caters 9 Royal 10 Cinques 11 Maximus 12

Preset buttons are provided for an assortment of methods and principles. The methods and principles were selected to cover the ones ringers often start out with, and to include some in which the treble does not hunt (Bastow and Stedman), more than one bell hunts (Grandsire and Antelope), and the coursing order is not the order in which bells come to the front (Stedman). They are shown in two groups:

# ⌖ You Can Customize the Method/Principle

You can also customize the method/principle by entering parameters of your choice for:

When you make your customizations, the input fields and the button will be highlighted. Press your computer's Return, Enter, or Tab key if this doesn't happen spontaneously. Then click the button, to work with your custom settings.

To enter a mask for a bob or single, you must first check the the corresponding checkbox to enable it.

## ⌖ Place notation

Place notation is a counterintuitive (from a mathematical point of view) but compact way of writing the changes between one row to the next in terms of which bells do not move. You can see examples by clicking any method/principle button.

For example, the place notation for Plain Hunt on 5 is 5.1.5.1.5.1.5.1.5.1. Each change is separated by a period; this place notation shows a sequence of 10 changes.

1. Place notation 5

5 means the bell in 5ths place does not move (and all other bells do move); thus the bells in leads and 2nds exchange places, and the bells in 3rds and 4ths change places. If there were more than 5 bells involved, the bells above 5ths place would exchange places too.

(You might wonder if the bell in leads can exchange places with the bell at the end of the row. That is ruled out; it's too difficult. The bell in leads can only exchange places with the bell in 2nds; and the bell at the back in Nths can only exchange places with the bell in (N-1)ths.)

From rounds 12345, place notation 5 would produce 21435. You may want to work this out with pencil and paper if it is new to you.

2. Place notation 1

The second change 1 meands the bell in leads does not move (and all others do); thus the bells in 2nds and 3rds exchange places, as do the bells in 4ths and 5ths.

From 21435, place notation 1 would produce 24153.

3. The third change 5 is the same as the first change, but now it is operating on a different row, 24153. As before, it indicates that the bell in 5ths stays put while each other bell exchanges with another (the bells in leads and 2nds, and the bells in 3rds and 4ths). From 24153 the change 5 produces 42513.

To see how the rest of this list of changes works out, click Plain Hunt then either or (repeatedly) . The two simple changes 5 and 1 produce a variety of rows (sequences of bells).

Other features of place notation are illustrated by Plain Hunt . Its place notation is x.14.x.14.x.14.x.14.

1. Place notation x

x means every bell swaps places with another. In this case, the bells in leads and 2nds exchange places, as do the bells in 3rds and 4ths.

From rounds 1234 place notation x produces 2143.

2. Place notation 14

The second change 14 illustrates the fact that more than one bell can stay in its same place. Here the bells in leads and 4ths stay there, and every other bell exchanges places with another (in this case, the bells in 2nds and 3rds).

From 2143, place notation 14 produces 2413.

To see how the rest of this list of changes works out, click Plain Hunt then either or (repeatedly) .

In summary:

• Each digit means that bell does not move during that change.
• x is shorthand for 214365…. It can only be used for an even number of bells.
• Successive changes are separated by . (a period).

You may have already realized that it is possible to write place notation that makes no sense. An example is the change 2, which specifies that the bell in 2nds stays put while every other bell exchanges places with another. But that can't work; who does the bell in leads exchange with?

Bell
number
Bell
char.
CCCBR
char.
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
6 6 6
7 7 7
8 8 8
9 9 9
10 a 0 (zero)
11 b E
12 c T
13 d A
14 e B
15 f C
16 g D
17 h F
18 i G
19 j H
20 k J
21 l K
22 m L
23 n N
24 o P
25 p Q
26 q R
27 r S
28 s U
29 t V
30 u W
31 v X
32 w Y
33 x Z
34 y
35 z

If there are 9 or more bells, the higher-numbered bells are represented here by lower-case letters, as is the usual computer science practice (see table at right). Up to "z" or 35 is accepted, but only up to "g" or 16 will be displayed in a diagram.

The CCCBR bell letters (see table at right) are accepted as well, and if the 90ET radio button is checked, they will be displayed instead of lower-case letters. This encoding uses 0 for 10, E for eleven, and T for twelve, then continues alphabetically from A for 13, skipping I and O to avoid confusion with 1 and 0, and E and T because they were already used, ending with Z for 33.

Stedman is traditionally organized into sixes which are out of phase with its leads. These are indicated in the place notation by segments delimited with \$s. Stedman's bobs and singles are applied to segments rather than leads. For methods and principles organized in segments (at this writing Stedman is the only one I have found), the simulation software offers the possibility of displaying a segment () at a time in addition to a row, lead, or plain course at a time; and of displaying a bob segment or single segment rather than a bob lead or single lead.

## ⌖ Place notation abbreviations and alternatives

This page accepts the following place notation abbreviations and alternatives.

X or - for x
The common alternatives X or - are accepted in place of x.
No dots around xs
The common abbreviation in which dots are elided on either side of x (or X or -) is accepted.
Repetition
(P){N}, where P is unabbreviated place notation and N is a positive integer, expands to P.P.···.P, with N occurrences of P separated by dots. This notation is modelled on a common computer science usage.
Palindromic half-end/lead-end
C.D.···.Q.R:S/T, where C, D, … , Q, R, S, and T are changes, expands to C.D.···.Q.R.S.R.Q.···.D.C.T, with the initial sequence C.D.···.Q.R reflected after S in reverse sequence R.Q.···.D.C and terminated by T. S is the half end change and T is the lead end change.

The CCCBR method libraries present place notation where possible as the initial sequence, the half end change, and the lead end change, though not in this textual form; the same information is presented tabularly. The downloadable XML form of these libraries denotes this using a comma only, as C.D.···.Q.R.S,T. The ringingmethods.co.uk site presents this comma-only form directly.

The ☼ button expands any abbreviations in the place notation to show the fully spelled out place notation.

# ⌖ Displaying Rows, Leads, and Segments of the Method or Principle

You have several options.

## ⌖ The button

Clicking the button starts over with the current method at the initial backstroke in rounds.

## ⌖ The button

Clicking the button shows another row.

If the method or principle supports bobs or singles, and a call could be made, the and/or button(s) will un-grey so you can click.

## ⌖ The button

Clicking the button shows a lead of the method/principle.

If the display was partway through a lead, clicking this button shows the rest of it. If you had clicked or , the lead will show the call; otherwise, it will be a plain lead.

## ⌖ The button

If the method/principle is divided into segments shorter than a lead, then the button will be enabled. Clicking it shows a plain segment of the method/principle.

Stedman is the only common method/principle for which this applies.

## ⌖ The (Plain Course) button

Clicking the button clears all visible rows and shows a plain course of the method/principle. A plain course is a sequence of plain leads that brings the bells back into rounds.

The number of courses required is one of the inferred characteristics, displayed as leads/course.

## ⌖ The button

If you step through the rows one at a time using the button, and the method supports bobs, the button will be enabled from time to time. It is disabled except when a Bob call could be made. When is is enabled, you can click it to make the call.

## ⌖ The or button

If the method/principle supports bobs, then the button will be enabled. Clicking it shows a lead of the method/principle with a bob called.

If the method/principle is divided into segments, then the button will be labelled instead. Stedman is the only common method/principle for which this applies.

## ⌖ The button

If you step through the rows one at a time using the button, and the method supports singles, the button will be enabled from time to time. It is disabled except when a Single call could be made at the next row. When is is enabled, you can click it to make the call.

## ⌖ The or button

If the method/principle supports singles, then the button will be enabled. Clicking it shows a lead of the method/principle with a single called.

If the method/principle is divided into segments, then the button will be labelled instead. Stedman is the only common method/principle for which this applies.

## ⌖ The button

If a call has been made on the current row, the button (no call) is enabled. Clicking it removes the call.

## ⌖ The pv, , and buttons

These buttons back up one row (), lead (), and segment () respectively. Each is enabled whenever its action can occur.

## ⌖ The second set of buttons

A second set of the , , , , , , , , and buttons float under the displays, so you can follow a long sequence more conveniently. They produce the same effects as the first set.

# ⌖ Playing Rows, Leads, and Segments:

The button plays the last row in the display, in the pitches of the bells of the Miami tower.

To hear every displayed row, check the checkbox beside it.

To stop the rows being played, unclick the checkbox; playback should stop immediately.

For some reason, two strokes are consistently missing from the sounds for Plain Hunt on Three when a complete lead is played (the 1 in row 4 and the 2 in row 6). Other numbers of Plain Hunt work properly.

# ⌖ Calls

Many methods support calls that are used to ensure the band can ring all possible permutations. The most common calls are bobs and singles.

## ⌖ What is a call?

Each method/principle produces a finite number of different bell rows; the number of different rows is typically equal to the length of a lead times the number of coursing bells. Thus for example Plain Bob Minimus, with three coursing bells and a lead length of 8 produces 24 distinct rows. There are

$4!$ (pronounced four factorial) = 4×3×2 = 24

different rows of four bells, so that's all of them. However, Plain Bob Doubles, with 4 coursing bells and a lead length of 10, produces only 40 different rows out of the

5! = 5×4×3×2 = 120

different rows of five bells, leaving 80 rows that are not reachable in Plain Bob Doubles. An example is the row 12453.

To reach this other rows, ringers have introduced calls such as Bobs and Singles.

## ⌖ Bob

Exceptions:

1. Antelope
2. Canterbury
3. Stedman
4. Stedman Slow Work
5. Union

The common feature of bobs for most methods/principles I've looked at (and I've looked at a few) is that they do not change how many pairs of bells swap places (as a single does) but instead changes which pairs swap places. Most of those I've examined move the swap over one place either in or out, so that a bell that would have swapped keeps its place, the bell it would have swapped with swaps with the bell on the other side, and that bell which would have kept its place swaps instead.

There are exceptions; see the list at right of some methods/principles whose bobs reduce by one the pairs swapping places.

Each bob is specified here as a list of masks that operate on the changes for a plain lead or segment.

A mask for the bob is applied to the changes at the end of the current lead, or (if the mask begins with a < as for Stedman Triples and higher) to the changes at the beginning of the next lead.

To see a bob, click the button or the / button. These buttons are disabled except when a bob is possible.

## ⌖ Single

The common feature of singles for the methods/principles I've looked at (and I've looked at a few) is that they reduce by one the number of pairs of bells that swap places, making one of those pairs keep their places instead.

Each single is specified here as a list of masks that operate on the changes for a plain lead or segment.

A mask for the single is applied to the changes at the end of the current lead, or (if the mask begins with a < as for Stedman Triples or higher) to the changes at the beginning of the next lead.

To see a single, click the button or the / button. These buttons are disabled except when a single is possible.

## ⌖ Mask notation for bobs and singles

Bobs and singles are specified here using a mask notation related to place notation. A bob or single is specified by a list of one or more masks, separated by periods. Each mask is an operator on a change, that produces a related change from it. Each mask is either:

• A string beginning with `=` and continuing with the place notation that is to replace the masked change;
• A sequence of strings each consisting of `+` or `-` followed by one or more bell characters, with `+BC…` indicating that B, C, etc. are to be added to the change's place notation and `-BC…` indicating that B, C, etc. are to be elided from the change's place notation; or
• The identity mask `*` indicating that the change is to remain the same.

The character # may be used in a =, +, or - list to represent the highest-numbered bell abstractly, as for example in Grandsire where the masks for all even numbers of bells are the same, as are the masks for all odd numbers of bells.

If the list begins with `<` then the masks are to be applied starting from the beginning of the change list; otherwise, the masks are to be applied ending at the end of the change list.

The majority of mask lists consist of a single mask applied at the end of the change list, as for example both the bob and the single of Plain Bob Minor.

Some mask lists affect more than one change, as for example:

1. Grandsire in which singles affect two changes; and
2. Antelope in which both bobs and singles affect three changes.

A very few mask lists are applied at the beginning of the next lead or segment, as for example both the bob and single of Stedman Triples.

Stedman Doubles sixes
Which Odd (1st, 3rd, …) Even (2nd, 4th, …)
Plain 5.3.1.3.1.3 5.1.3.1.3.1
Single 5.3.1.345.1.3 5.1.3.145.3.1

Why a mask notation for bobs and singles rather than just place notation? Blame it on Stedman Doubles, in which a single may be called for either kind of six. In both cases 45 is added to the fourth change of the six, but in one case that fourth change goes from 3 to 345 and in the other it goes from 1 to 145. Is that two kinds of single (145 and 345), making Stedman the only method/principle with a second kind of single, or one kind that depends on the context in which it is applied? I chose the second approach (after trying and discarding the first one).

Masks also support abstraction in the actions for bobs and singles so that some commonalitites across methods/principles and numbers of bells can be made evident; the identity operator on changes (mask *) which is convenient for Erin singles and Grandsire bobs; and the attribute of place deltas, the change in the number of placing bells caused by a call.

# ⌖ Initial Row

By default, the method or principle begins at row 0 from rounds. You can if you wish have it begin from an arbitrary row number and/or row, by entering them in the Starting at text inputs.

• The row number must be non-negative.
• The row must be for the same number of bells that the method/principle is for.
• The initial row will be displayed and treated as a lead end, so you should choose the row number and row accordingly.

# ⌖ Transformations:, , , etc.

3.1.3.1.3.123

Plain Bob Singles

Each of these takes the current method or principle and transforms it into a related method or principle. They are presented in sequence from simplest to most complex. We use Plain Bob Singles (right) as an example.

The first two transformations do not in themselves produce another standard method or principle, but are combined to make more complex transformations. When applied one after the other (in either sequence) the result is the same as Reverse.

The remaining three transformations produce recognized methods or principles with standard names. If the original method's name is X, then the result of applying the Backward, Reverse, or Double transformation is the method named Backward X, Reverse X, or Double X respectively. If X contains Plain or Single, that part of the name is elided in the name of the transformed method.

If a tranform button is grayed out, that transformation is either not possible or would result in the same method or principle; hover the cursor over the button to find out which.

Because inversion is an operation that can also be applied to masks, the button inverts the bob and single mask lists along with the place notation. The other transformations do not apply to masks, so their transformation buttons reuse the original bob and single masks lists (if any) if the following conditions hold:

1. The mask lists are one mask long and are applied to the lead end change, and
2. The lead end change of the transformed method is the same as that of the original method.

Otherwise, the transformed method is not provided with a bob or single.

## ⌖ The transformation

Plain Bob Singles, swapped

The button swaps the first and second halves of the method or principle.

You'll notice that although the place notation is the original place notation with its halves swapped, the paths of the bells are not the original paths with their halves swapped.

For some methods, the standard Double transformation does not produce an interesting result, for example with Canterbury Place for which it produces Plain Hunt. The R/S Double transformation (Reverse over Single) takes the places

The swap-halves transformation is its own inverse, which means that swap-halves undoes itself.

The swap-halves transformation has no effect when applied to some methods, such as Plain Hunt: it produces the same method. The button is disabled for such methods.

## ⌖ The transformation

The button inverts each change from top to bottom.

In terms of the place notation, with $M$ moving bells (3 in the example of Singles), each number in the place notation is subtracted from $\left(M+1\right)$. If there are several numbers for a single change, as in the last change of Plain Bob Singles, then we would sort them back into sequence.

The invert transformation is its own inverse, which means that invert undoes itself.

The invert transformation has no effect when applied to some methods, such as Plain Hunt: it produces the same method. The button is disabled for such methods.

## ⌖ The transformation

The button transforms a method to or from its Backward counterpart.

The Backward counterpart of a method is the method in reverse, from end to beginning. The changes are the same ones as the method's but they are listed beginning with the last change and continuing to the first.

The Backward transformation is its own inverse, which means that Backward undoes itself.

## ⌖ The transformation

The button transforms a method to or from its Reverse counterpart. It is the combination of and , in either sequence.

A Reverse method is produced by taking the original method, inverting every change top-to-bottom (as for ), and swapping the second half of the change list with the first half (as for ). The two transformations may be performed in either sequence; the result is the same.

The Reverse transformation is its own inverse, which means that Reverse undoes itself.

The Reverse transformation has no effect when applied to some methods, such as Plain Hunt: it produces the same method. The button is disabled for such methods.

## ⌖ The transformation

The button transforms a method to its Double counterpart.

The button is enabled for every method that begins from rounds and in which the treble Plain Hunts.

The Double counterpart of a method is produced from the method and its Reverse counterpart. For each change, the part of the Reverse method change that lies under the treble's place is joined with the part of the original method change that lies at or over the treble's place.

In terms of place notation, for each change, if the treble is in place P before the change, then all numbers in the place notation for that change of the Reverse method that are less than P are combined with all numbers in the place notation for that change of the original method that are greater than or equal to P.

The Double transformation is not its own inverse. In fact, since the Double transformation discards some information about the original method, it has no inverse transformation. For example, Snowden (1998) lists Double Norwich Court Bob (chart 79), place notation x.14.x.36.x.58.x:18/18, but the Single method Norwich Court Bob is nowhere to be found. 96 possible place notations for the hypothetical Norwich Court Bob would be transformed into Double Norwich Court Bob:

1. x.14.x.16.x.18.x:18/18
2. x.14.x.36.x.38.x:38/38
3. x.14.x.36.x.58.x:58/58
4. x.14.x.36.x.58.x:78/78
5. And every other combination of these choices.

The Double transformation has no effect when applied to some methods, such as Plain Hunt: it produces the same method. The button is disabled for such methods.

The standard Double transformation produces no interesting result when applied to some methods, such as Canterbury Place for which it produces Plain Hunt. A variant in which the Reverse changes at or over the treble's place are combined with the Single changes under its place is often used for them. The button (Reverse over Single Double) produces this transformation.

# ⌖ Course and After Bells

These depend on which bell is Your bell. When the page loads, your bell defaults to . You can enter whichever bell you like for your bell, and then press the return or enter key which triggers recalculation of your course bell and after bell. If you enter a number other than the number of one of the bells, you'll get instead.

Your course bell is the bell doing the work that precedes yours in the coursing order. If your bell isn't a working bell (for example if you are trebling in most methods), then you are not in the coursing order and you have no course bell.

Your after bell is the bell doing the work that follows yours in the coursing order. If your bell isn't a working bell (for example if you are trebling in most methods), then you are not in the coursing order and you have no after bell.

Each plain lead moves each bell to the next work in the coursing order, so your course and after bells do not change though the work each of the bells is doing changes.

However, a call can send bells to do work that was not the next for them in the coursing order. When that happens, your course and/or after bells may change, whether because one or both of them has been sent to different work or because your bell has.

Each colored dot shows the color of that bell in the diagram.

# ⌖ The Coursing Diagram

The coursing diagram shows the coursing order in circular form. The large numbers are the places in the coursing order. The smaller pairs of numbers in the inner ring indicate where the bell in each passes the treble on the way to the next place.

Originally I hypothesized that the Javascript software could infer the work for each place, but now I believe that is not possible, primarily because the names used for the work in various methods/principles show little regularity and indeed vary geographically and over time. The Javascript software instead infers what I call the landmarks, such as where your bell passes the treble, dodges, and similar well-defined phenomena. Instead, I have gone through and manually assigned work names for some of the methods and principles on some numbers of bells, as an experiment.

If the method or principle is one for which I have manually set up names for each place's work, based primarily on Adams Ringing Circles (2000), then each place's work is shown in the ring of arrows, with the place circled just before it, but keep in mind these are my abbreviations for what Adams says, divided as necessary to give each place some work, and may not match what you are used to. Otherwise, the place numbers are shown in the ring of arrows.

# ⌖ Inferred Characteristics

Various facts can be inferred about each method/principle, some directly and others indirectly.

Where numbers are listed for Hunting, Coursing, and Coming to front, they represent not bells but places at the lead end backstroke.

## ⌖How many bells are moving

The tenor behind / covers, if any, are not moving.

## ⌖Which bells hunt only

In Plain Hunt, all the moving bells hunt, of course. In many methods only the treble hunts, but in Grandsire two bells hunt, the treble and the bell in 2nds at the lead end.

Note: in Grandsire the other bell that hunts changes with each Single call, but the fact that the bell in 2nds at the lead end backstroke hunts does not change.

## ⌖Which bells course, and the coursing order

These bells cover (in the mathematical sense) all the working places, from the front to the back, and a lead of the method/principle sends each one to a different place than it started in.

They are listed in the coursing order, an abbreviated representation of the permutor. The coursing order begins with the lowest-numbered place that is permuted to a different place, continues with the place the bell in that place is sent to, continues with the place the bell in that place is sent to, and so on until stopping when the first-listed place would be reached again.

For many (but not all) methods and principles, the coursing order is evens up, odds down: 2 4 … 5 3.

## ⌖Which bells cover

If you specified a tenor behind or two or more covers, the bells performing that function are listed.

## ⌖The order in which bells come to the front

This is related to the coursing order but usually not the same.

## ⌖ Permutor

The permutor of the method/principle is a vector that shows where each bell is sent by a plain lead of the method/principle.

As an example, look at the permutor calculated for Plain Bob Minimus (4). It is [1423].

1. The bell in leads (1) before the plain lead begins is in leads again when the plain lead ends.
2. The bell in 2nds before is in 4ths when the lead ends.
3. The bell in 3rds before is in 2nds when the lead ends.
4. The bell in 4ths before is in 3rds when the lead ends.

You can confirm this by comparing successive lead end rows separated by plain leads.

Successive lead end backstrokes
for Plain Bob Minimus
Row
num.
Row Leads
to
leads
2nds
to
4ths
3rds
to
2nds
4ths
to
3rds
0. 1234 1 2 3 4
8. 1342 1 3 4 2
16. 1423 1 4 2 3
24. 1234 1 2 3 4
1. The bells begin from rounds, 1234.
2. The first plain lead ends with 1342. The bell that had been in 2nds (the 2) has been sent to 4ths.
3. The second plain lead ends with 1423. The bell that had been in 2nds (the 3) has been sent to 4ths.
4. The third plain lead ends with rounds, 1234. The bell that had been in 2nds (the 4) has been sent to 4ths.

## ⌖ The lead length and number of leads in a plain course

$n$ $n!$
1 1
2 2
3 6
4 24
5 120
6 720
7 5,040
8 40,320

Each lead is the same number of rows long. The number of leads listed is how many plain leads are required to take the bells from rounds to rounds.

## ⌖The number of permutations of the moving bells

The number of permutations of $M$ moving bells is $M!$ (pronounced M factorial); its value is $M×\left(M-1\right)×\dots ×3×2×1$. The factorial of 1 through 8 is given in the table at right. It increases very rapidly, faster than any polynomial.

The inferred characteristics include the number of permutations of the moving bells and the ratio between the number of permutations and the number of rows in a plain course.

## ⌖ Method/Principle, treble hunts, 2nds hunts, Plain Hunt, treble bobs, has prefix

These indicators show:

• Whether the pattern is a method or a principle. In a principle, all moving bells follow the same path, whereas in a method, some of the moving bells (called the working bells) follow the same path, but the treble and perhaps another bell Plain Hunt or treble bob. Each working bell starts at a different point in the path, but they all follow the same path.

Plain Hunt, Antelope, Erin, and Stedman are examples of principles. Most other change ringing patterns are methods.

• Exceptions:

Whether the treble hunts. This is true for many methods, especially those new ringers typically start on, but not for all; some exceptions are listed in the table at right. It is not the case for any principle other than Plain Hunt, since all bells follow the same path in a principle and if the treble hunts then all the bells would have to hunt.
• Whether the bell in 2nds hunts, as in Grandsire.
• Whether all the bells hunt.
• Whether the treble Treble Bobs. An example is the method Kent Treble Bob, in which the treble dodges in n-(n+1) for every odd place n, both on the way out and on the way back in.
• Whether the method or principle is organized as a prefix and segments, like Stedman.

Each indicator is normal text if true, or grayed and struck through if false.

## ⌖ Palindromic place notation

Exceptions:

A method/principle's place notation is palindromic if its changes are symmetric about the half end change in the middle, with the exception of the. lead end change at the end. For example, C.D.···.Q.R.S.R.Q.···.D.C.T is symmetrical about half end change S, except for lead end change T; all the changes preceding S are reflected following it, in mirror-image fashion. See the discussion of the palindromic abbreviated form.

Many but not all methods/principles are palindromic in this sense; see the list at right for some examples of non-palindromic methods and principles. If the method or principle is palindromic, its place notation is presented in palindromic form.

## ⌖ Expanded place notations

The expanded place notations present the method or principle's place notation with all abbreviations expanded.

If the method/principle is divided into segments (as is Stedman), the place notation for each non-prefix segment is presented in sequence, separated by \$s. Otherwise a single place notation is presented.

## ⌖ Bob or Single place notations

For methods and principles that support bobs or singles, these are the place notation(s) for a lead in which a call is made.

If the method/principle is defined with two or more segments, like Stedman, then the place notation for a call in each segment is presented, separated by semicolons. Otherwise a single place notation is presented for bobs (if defined) and singles (again if defined).

## ⌖ Bob or Single permutor and their coursing order(s)

For methods and principles that support bobs or singles, these are the permutor(s) and the coursing order(s) for a lead in which a call is made.

Unlike plain leads, which always have one coursing order, bob or single leads may have two or more disjoint coursing orders. For example, in Plain Bob Minor, a lead for which a single is called has two coursing orders: one for 2nds and 3rds, which exchange places, and another for 4ths, 5ths, and 6ths, which cycle among themselves.

Stedman is one of the oldest methods or principles and was created before they were well understood. Thus it has some distinctive features not found in any other method or principle. One of these is that it has two plain segments in each lead (called sixes), each with a different permutor. A single produces different permutors (and thus coursers) when applied to the first kind of six than to the second kind of six.

## ⌖ Follow string for plain course

The follow string lists the sequence of bells that the point-of-view bell follows (rings over) during a plain course, with "L" interspersed for each blow in leads. If the method or principle has more than one lead in its plain course, the divisions between them are marked with dots ().

The follow string can be quite long, and for anything above 5 or 6 bells is too long to fit in the table. By default only the part that can fit is displayed. If you wish to see the whole string, click it; if you wish to hide the overflow again, click it again.

## ⌖ Inferring the source of a Double method

A Double method is created from an original method by a specific transformation in which the parts of the original method lying above the treble are combined with the parts of the Reverse of the original method lying below the treble.

To aid in inferring the Single method from which a Double method was created, for appropriate methods (those in which the treble hunts) this page displays:

• Over the treble: the method's place notation at or above the treble;
• The treble: the treble's place in the method;
• Under the treble: the method's place notation under the treble; and
• Under reversed: the place notation under the treble reversed.

The place notation for the Single method must include all of the places over the treble and all the under-reversed places. You will have to guess the other places, as the Double transformation discards some information about the Single method.

These methods are present in Snowdon only in Double form; inferred Single precursors are given here, confirmed by reference to other definitions if available.

1. Dunkirk Bob Minor.
2. Norwich Court Bob Major.
3. Single Oxford (Single Oxford Bob) Major.

# ⌖ The Table of Work

The table of work summarizes what each working bell does as it moves through the method/principle.

The From column lists the places a bell can find itself in at a lead end backstroke. Each row summarizes what happens for a bell in that place. The rows are listed in coursing order.

The Pass the Treble columns name the places in which such a bell passes the treble, first going in and then going out. The places in which such a bell rings over the treble are emphasized.

The Work column names the work such a bell performs. The names of the work in different methods appear to be arbitrary enough that this is set on a method-by-method basis, which so far I've only set up for a few common methods/principles. For all others, except for bells that hunt, treble bob, or hunt to a place and back (which can be automatically identified), the work is blank.

The To column gives the place such a bell finds itself in at the end of the lead, assuming of course no call was made.

As long as no calls are made, each working bell cycles through the table row by row.

If a call is made, some working bells will find themselves in another place at the lead end backstroke. Such a bell then continues on the row whose From place is that place.

# ⌖ The Table of Ltcas

The table of Ltcas lists who a bell follows as it moves through the method/principle, not in terms of bell numbers but abstractly in terms of

• when the bell is leading (L),
• when the bell is following the treble (t),
• when the bell is following its course bell (c),
• when the bell is following its after bell (a), and
• when the bell is following any other bell (.).

The From column lists the places a bell can find itself in at a lead end backstroke. Each row summarizes what happens for a bell in that place. The rows are listed in coursing order.

The Call column lists the calls that can affect a bell starting from that place. Only calls that affect the bell's Ltcas or To place are listed.

The Follows column lists the bells followed by a bell starting from that place, in terms of Leads, treble, course, and after. The bells starting from some places, namely the treble in methods and in some methods like Grandsire another hunting bell, have no course or after bells and so neither c nor a appear in the list of bells they follow.

If the bell is affected by a call, its Ltca list is presented for the call too.

The To gives the place the bell finds itself in at the end of the lead.

I learned when to look for the treble in Plain Bob this way.

# ⌖ The Error Signal

The software that runs this page is larded with runtime checks for unexpected errors and internal inconsistency. The error signal indicates when any of these checks find a problem.

# ⌖ Why?

## ⌖ Why did I bother to create this?

There are already programs and tables and charts out there — why bother to write my own?

• The ones I found weren't answering all my questions. Granted, as a new ringer my questions might not have been particularly insightful ones, and for that matter it is taking me quite a while to figure out what all my questions are, but still, they were my questions and I wanted to know their answers. Two of my early questions were Which are the handstrokes and which are the backstrokes? and What (ordinal) number is that row?, questions whose answers I suppose knowledgeable ringers already know or don't need to know; but I kept getting mixed up about the strokes and I wanted to be able to know which row I was looking at (the 5th one, or the 125th one, or …) so I could refer back to it later. So I started writing programs that addressed the issues that were puzzling me. You'll note that this page produces charts and diagrams in which backstrokes have distinctive backgrounds, and in which every row is numbered.
• I did a good bit of writing out sequences of rows by hand, of course, but I was troubled by that too. First, I was worried that I would make careless errors and not notice; second, I was worried that I would unknowingly make convenient errors that fitted how I thought things should be turning out. Writing a program to do it for me eliminated the second concern and reduced the first one.
• I wanted to explore my understanding of methods and principles. In my work in software research I'm used to examining data, constructing a theory from the data, and exploring the consequences by writing software that works with the data, so it was natural and easy to take the same approach to understanding this.
• When I moved to Miami and suddenly found myself in charge of a tower and a band, I found I needed charts and diagrams first to make sure I knew what I was trying to teach, and second to post in the tower so the ringers could refer to them instead of asking me everything. We needed to know very basic things: which way does my bell start in Plain Hunt? Who should I be over when I start to hunt in? What does it mean that the 3 is my course bell? I found myself drawing up diagrams by hand, which was enjoyable but tedious at the same time, and of course I worried about making careless errors. It soon became obvious that it would take less time to write and debug a program to make charts than to make all the charts by hand and check that they were correct. Then I became interested in JavaScript and SVG for other reasons, and this seemed like a good opportunity to learn them while producing something useful.
• I wanted diagrams in which I could trace all the bells, not just the treble and the blue line.
• I was baffled by bobs and singles, and none of the programs and charts I found seemed to be answering the questions I knew I wanted answered but couldn't state clearly. So I explored them through creating and evolving software that expressed bobs and singles and applied them.
• I couldn't make sense out of the rule for generating a Double method, so I wrote software to do it and explored the Double transformation by implementing and debugging the software. This revealed something completely unexpected — there are two Double transformations in use, not one:
1. single-over-the-treble plus reverse-under-the-treble, the most commonly used one and the one I found explanations of, but also
2. its dual the reverse-over-the-treble plus single-under-the-treble transformation used to produce Double Canterbury Place Minimus and possibly others.
I learned to thoroughly understand Reverse and Backward along the way too.
• And what about Stedman? It didn't fit my mental model and looked like it was an exception to just about every explanation I found. Of course much of that was due to my lack of understanding. I certainly understand it a lot better now.

And why not just use the CCCBR method libraries to define all the methods and principles?

• Well … when I started I didn't know they existed.
• Plus I wanted to understand the Snowdon Diagrams book and the similar diagrams I saw elsewhere.
• Finally, the method libraries don't seem to say anything about bobs and singles.

## ⌖ Why should you bother to use this?

• You might have some of the same questions I did, in which case you'll find this software helpful.
• You might not realize all the questions you want answered, in which case this software may help you find or formulate some new ones.
• You might want charts and diagrams that are more flexible than the ones you find elsewhere. Here you can put in bobs and singles wherever you like, go through a method/principle row by row, explore place notation you come up with on your own, compare two methods/principles side by side or different sequences of the same one, start from an arbitrary row, and more.
• You might not want only to know what (what chart, what diagram, what permutor, what calls, …) but also why. I give explanations for everything, at least to the limit of my understanding. If you are code-minded, you can look at the code, too, and see what's happening in detail and specifically.
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