The committee’s conclusions are based on the theory that correct relationship of place-positions in the extension with those in the parent is the “first principle” in method extension, which can be ascertained impersonally by the application of a mathematical formula based on certain rules suggested by accepted standard methods of varying types.
The principle adopted is that methods extend in sections, or by front and back subsections singly, and that an extension can only become effective from a cross-subsection place, or from the first places of the subsection. The place positions in the new subsection are relative to the subsection immediately before it and are not related to parts of adjacent subsections or to subsections that are not adjacent. All stages have the same relationship with the parent.
(In treble dominated methods, the treble’s path cuts each interior section into two - that before the treble being known as the front sub-section and that behind the treble as the back subsection. The initial section has a back subsection only and the last section (in the ½ lead) a front subsection only.)
This formula, which is suitable for checking any treble dominated even bell method except Little Bob, has been applied to extensions obtained from the Surprise Major and Royal Methods in Corrigan’s Collections, the C.C. Collection of Minor Methods and the C.C. Collection of Plain Methods. Results show that many of the extensions already published are not mathematically true. On the other hand, out of about 500 methods extended, over 100 Surprise methods have acceptable extensions in one or more stages. This is contrary to what is generally concluded that correct Surprise extensions are few. The formula should prove a generous one in investigating extensions and should satisfy the needs of ambitious bands.
All methods are divided into three classes for extension purposes, viz., “Expanding,” “Static” and “Double.” “Expanding” methods are those where the internal place positions of both front and back subsections expand laterally from the lead, commencing from where the extension becomes effective. In “Static” methods, the internal places of the front subsections remain static, i.e., they are identical with those in the parent, although the interior places of the back subsections laterally expand as in the “Expanding” type.
In all methods, the new front and back subsections, not necessarily from the same secion, are inserted between any two front and back subsections of the parent. A back subsection may be added after the last back subsection. A front subsection may be added after the last front subsection if the pivot place is an interior one or in the penultimate position. The adding of a new subsection after the last front subsection and a new subsection after the last back subsection - although from different sections - has the same effect as adding a complete new section after the last section. Lincolnshire, Yorkshire and Rochester all extend from different front subsections, and Pudsey, Yorkshire and York from different back subsections. Some methods extend in complete sections, e.g., Lincolnshire; in other methods, e.g, Yorkshire, the front and back subsections extend singly.
The new subsection is always an associate of the subsection it follows and in a further extension the new subsection is an associate of the latest new subsection and immediately follows it. This means that if any method extends from, say, the first front subsection and the second back subsection, all successive stages continue from the same subsection. The remaining subsections in the extension are counterparts of the remaining subsections of the parent.
The insertion of a subsection between two existing subsections allows the extension to commence either from a cross subsection place or from the beginning of a subsection. The front extension may be effective from a cross subsection place and the back from the beginning of a subsection or vice versa; or both the front and back from a cross subsection place or from the beginning of a subsection.
The first front cross subsection place is not affected in any extension and in the formula this place is not notated. The reason for the omission is that to extend the first front cross subsection place will result in the next cross subsection place, which may adjoin the treble, being transfered to lead in the extension.
(NOTE.- A cross subsection place “adjoining the treble” is the place made immediately before or immediately after dodging with the treble, or in plain methods “passing the treble.” All the cross subsection places in Cambridge and Double Norwich (Major) adjoin the treble.)
If an extension commences on the cross subsection place, the cross subsection place is related to the place positions in the following subsection. On the other hand, if the extension commences at the beginning of a subsection, the cross subsection place is related to the place positions of the subsection immediately before. Where the last front subsection is reproduced and the extension becomes effective from the beginning of a subsection place, the pivot place is related to the front subsection immediately above, provided the normal pivot position is not affected.
In a “Static” method, where the extension affects a front cross subsection place adjoining the treble, the subsection place either continues to adjoin the treble (as in Lytham Bob) or remains static (as in Northampton).
The pivot place may be in (a) the lead position, (b) the penultimate position (under the treble) or (c) in an interior position. In all stages the lead position and the penultimate position remain the same whether the method is “Expanding,” “Static” or “Double,” but an interior place expands in “Expanding” methods and remains static in “Static” methods.
In Double Methods, the formula is rearranged because the cross subsection places and the places of the front subsections are identical with those of the back except that they are inverted. Methods are restricted to “Expanding” only because in all methods the place positions of the initial section are identical in any stage. While the interior places in front of the treble expand from the lead, the places behind the treble expand from the back. The pivot place is always the reverse of the lead-head place (it would not be a double method otherwise) and is therefore restricted to either the “lead” or penultimate position. The front subsection may extend from the beginning of a subsection or from a cross subsection place in the same way as a single method, provided the back extends similarly but in an inverted position.
In “Little” methods, an extension does not increase the number of sections and the formula is unnecessary. In all successive stages, the places are identical with those of the parent.
The interim report of 1950 stresses in general terms the necessity of reproducing the characteristics of the parent method. The committee has been advised to amplify the recommendations in regard to half-lead (pivot) places, which have now been dealt with in a previous paragraph of this report, and to modify the requirement of rigid conformity of “place pattern.” In view of the considerable number of methods with backward hunting, and others, which, owing to their construction, would eliminate any possibility of the extension being recognised, it is felt that requirements will be met by the application of the formula subject to certain safeguards in subsequent paragraphs, and to the preservation of “lead-head place sequence” which is dealt with in the section of the Report on “Pivot bells.”
In order to reproduce contiguous places made in 1-2, no “Expanding” method or “Double” method may extend from a point prior to the subsection or cross subsection containing such a contiguous place. In “Static” methods this restriction is not necessary. Any other contiguous places will be automatically reproduced in the extension whether the method is “Expanding,” “Static” or “Double.”
It is essential that the number of cross subsection places adjoining the treble, whether before or behind the treble, shall be not less in number in the extension than in the parent. The formula already provides for the back places, but in “Static” methods a front extension may not be effective from a point prior to a front cross subsection place adjoining the treble, otherwise the corresponding places in the extension will no longer adjoin the treble. “Expanding” methods or “Double” methods when extended automatically ensure that this requirement is effected.
The formula which follows is based on the above conclusion.
Every parent method, whether treble bob or plain hunt, with the exception of double methods, has a standard notation, commencing with “A” from the first front subsection, “B” at the following cross section place and continuing with initial letters of the alphabet up to the pivot place. If the pivot place is at the lead, the letter is omitted. In Static methods it is omitted also if the pivot place is penultimate. At the back, terminal letters are used, commencing from the first back subsection and concluding with “Z” at the last cross subsection place.
In Double methods, the front is labelled in the same way as ordinary methods. As the pivot place depends on the lead-head place, it is excluded from the notation. The back is labelled with the same initial letters as the front, except that this operation is inverted.
Examples of the notations in respect of each parent method are shown hereunder. All subsections and cross subsection places in the extensions which correspond with those in the parent bear the same letter for comparison purposes. Where the interior places expand from the lead, each successive stage is indicated by 1, 2 or 3 against the appropriate letter.
Any of the following will comply with the rules (one for the front and one for the back) provided that only the subsections and cross subsection places given in the examples are notated, and that regard is paid to the extension restrictions in Rule 6 which are illustrated at the conclusion of the section of the formula.
|(a) Back subsections|
|Parent - Minor||W||X||Y||Z|
|W||Major (1st stage)||..||W||X||W1||X1||Y1||Z1|
|Royal (2nd stage)||..||W||X||W1||X1||W2||X2||Y2||Z2|
|Maximus (3rd stage)||..||W||X||W1||X1||W2||X2||W3||X3||Y3||Z3|
|X||Major (1st stage)||..||W||X||Y||X1||Y1||Z1|
|Royal (2nd stage)||..||W||X||Y||X1||Y1||X2||Y2||Z2|
|Maximus (3rd stage)||..||W||X||Y||X1||Y1||X2||Y2||X3||Y3||Z3|
|Y||Major (1st stage)||..||W||X||Y||Z||Y1||Z1|
|Royal (2nd stage)||..||W||X||Y||Z||Y1||Z1||Y2||Z2|
|Maximus (3rd stage)||..||W||X||Y||Z||Y1||Z1||Y2||Z2||Y3||Z3|
|Parent - Major||U||V||W||X||Y||Z|
|U||Royal (1st stage)||..||U||V||U1||V1||W1||X1||Y1||Z1|
|Maximus (2nd stage)||..||U||V||U1||V1||U2||V2||W2||X2||Y2||Z2|
|V||Royal (1st stage)||..||U||V||W||V1||W1||X1||Y1||Z1|
|Maximus (2nd stage)||..||U||V||W||V1||W1||V2||W2||X2||Y2||Z2|
|W||Royal (1st stage)||..||U||V||W||X||W1||X1||Y1||Z1|
|Maximus (2nd stage)||..||U||V||W||X||W1||X1||W2||X2||Y2||Z2|
|X||Royal (1st stage)||..||U||V||W||X||Y||X1||Y1||Z1|
|Maximus (2nd stage)||..||U||V||W||X||Y||X1||Y1||X2||Y2||Z2|
|Y||Royal (1st stage)||..||U||V||W||X||Y||Z||Y1||Z1|
|Maximus (2nd stage)||..||U||V||W||X||Y||Z||Y1||Z1||Y2||Z2|
|Parent - Royal||S||T||U||V||W||X||Y||Z|
|(b) Front subsections|
|(i) “Expanding” methods|
|Parent - Minor||A||B||C||D*|
|A||Major (1st stage)||..||A||B||A1||B1||C1||D1|
|Royal (2nd stage)||..||A||B||A1||B1||A2||B2||C2||D2|
|Maximus (3rd stage)||..||A||B||A1||B1||A2||B2||A3||B3||C3||D3|
|B||Major (1st stage)||..||A||B||C||B1||C1||D1|
|Royal (2nd stage)||..||A||B||C||B1||C1||B2||C2||D2|
|Maximus (3rd stage)||..||A||B||C||B1||C1||B2||C2||B3||C3||D3|
|C*||Major (1st stage)||..||A||B||C||D||C1||D1|
|Royal (2nd stage)||..||A||B||C||D||C1||D1||C2||D2|
|Maximus (3rd stage)||..||A||B||C||D||C1||D1||C2||D2||C3||D3|
|Parent - Major||A||B||C||D||E||F*|
|A||Royal (1st stage)||..||A||B||A1||B1||C1||D1||E1||F1|
|Maximus (2nd stage)||..||A||B||A1||B1||A2||B2||C2||D2||E2||F2|
|B||Royal (1st stage)||..||A||B||C||B1||C1||D1||E1||F1|
|Maximus (2nd stage)||..||A||B||C||B1||C1||B2||C2||D2||E2||F2|
|C||Royal (1st stage)||..||A||B||C||D||C1||D1||E1||F1|
|Maximus (2nd stage)||..||A||B||C||D||C1||D1||C2||D2||E2||F2|
|D||Royal (1st stage)||..||A||B||C||D||E||D1||E1||F1|
|Maximus (2nd stage)||..||A||B||C||D||E||D1||E1||D2||E2||F2|
|E*||Royal (1st stage)||..||A||B||C||D||E||F||E1||F1|
|Maximus (2nd stage)||..||A||B||C||D||E||F||E1||F1||E2||F2|
|Parent - Royal||A||B||C||D||E||F||G||H*|
|* Interior or penultimate pivot place only.|
|(ii) Static methods|
|Parent - Minor||A||B||C||D†|
|A||Major (1st stage)||..||A||B||A||B||C||D|
|Royal (2nd stage)||..||A||B||A||B||A||B||C||D|
|Maximus (3rd stage)||..||A||B||A||B||A||B||A||B||C||D|
|B||Major (1st stage)||..||A||B||C||B||C||D|
|Royal (2nd stage)||..||A||B||C||B||C||B||C||D|
|Maximus (3rd stage)||..||A||B||C||B||C||B||C||B||C||D|
|C†||Major (1st stage)||..||A||B||C||D||C||D|
|Royal (2nd stage)||..||A||B||C||D||C||D||C||D|
|Maximus (3rd stage)||..||A||B||C||D||C||D||C||D||C||D|
|Parent - Major||A||B||C||D||E||F†|
|A||Royal (1st stage)||..||A||B||A||B||C||D||E||F|
|Maximus (2nd stage)||..||A||B||A||B||A||B||C||D||E||F|
|B||Royal (1st stage)||..||A||B||C||B||C||D||E||F|
|Maximus (2nd stage)||..||A||B||C||B||C||B||C||D||E||F|
|C||Royal (1st stage)||..||A||B||C||D||C||D||E||F|
|Maximus (2nd stage)||..||A||B||C||D||C||D||C||D||E||F|
|D||Royal (1st stage)||..||A||B||C||D||E||D||E||F|
|Maximus (2nd stage)||..||A||B||C||D||E||D||E||D||E||F|
|E†||Royal (1st stage)||..||A||B||C||D||E||F||E||F|
|Maximus (2nd stage)||..||A||B||C||D||E||F||E||F||E||F|
|Parent - Royal||A||B||C||D||E||F||G||H†|
|† Interior pivot place only.|
|Front and Back Subsections|
|Parent - Minor||A||B||C|
|A||Major (1st stage)||..||A||B||A1||B1||C1|
|Royal (2nd stage)||..||A||B||A1||B1||A2||B2||C2|
|Maximus (3rd stage)||..||A||B||A1||B1||A2||B2||A3||B3||C3|
|B||Major (1st stage)||..||A||B||C||B1||C1|
|Royal (2nd stage)||..||A||B||C||B1||C1||B2||C2|
|Maximus (3rd stage)||..||A||B||C||B1||C1||B2||C2||B3||C3|
|Parent - Major||A||B||C||D||E|
|A||Royal (1st stage)||..||A||B||A1||B1||C1||D1||E1|
|Maximus (2nd stage)||..||A||B||A1||B1||A2||B2||C2||D2||E2|
|B||Royal (1st stage)||..||A||B||C||B1||C1||D1||E1|
|Maximus (2nd stage)||..||A||B||C||B1||C1||B2||C2||D2||E2|
|C||Royal (1st stage)||..||A||B||C||D||C1||D1||E1|
|Maximus (2nd stage)||..||A||B||C||D||C1||D1||C2||D2||E2|
|D||Royal (1st stage)||..||A||B||C||D||E||D1||E1|
|Maximus (2nd stage)||..||A||B||C||D||E||D1||E1||D2||E2|
|Parent - Royal||A||B||C||D||E||F||G|
|Cross subsection||D||A & B|
|Subsection||E||A, B & C|
|Cross subsection||F||A, B, C & D|
|Subsection||G||A, B, C, D & E|
* No restrictionRule 6.- “Static” methods. Cross subsection places adjoining the treble.
Rule 5 (b) not
|D||A & B|
|F||A, B, C & D|
|H||A, B, C, D, E & F|
* No restriction
|1 (i)||A new front subsection may be inserted between any 2 front subsections of the parent. A new back subsection is inserted between the corresponding inverted back subsections.||A new front subsection and a new back subsection (not necessarily forming the same section) may be inserted between any 2 front or back subsections of the parent. A new front subsection may be added after the last front subsection of the parent, if the pivot place is an interior one or penultimate. A back subsection may be added after the last back subsection||A new front subsection and a new back subsection (not necessarily forming the same section) may be inserted between any 2 front or back subsections of the parent. A new front subsection may be added after the last front subsection of the parent, only if the pivot place is an interior one. A new back subsection may be added after the last back subsection.|
|(ii)||Each new subsection is an associate of the subsection it follows.|
|(iii)||In a further extension, the new subsection is an associate of the latest new subsection and immediately follows it.|
|(iv)||Any subsequent subsections of the extension remain counterparts of those in the parent and lower stages.|
|2 (i)||Any extension may take effect from (a) the beginning of a subsection or (b) at any cross subsection place subsequent to the first subsection.|
|(ii)||The front extension may be effective from a different position to that of the back.|
|3 (i)||Where the extension commences at the beginning of a front subsection, the cross subsection place and the places in the subsection above are relative. Where the extension commences at the beginning of the last subsection, the pivot place is relative to the subsection above, subject, however, to the requirement of Rule 10.|
|(ii)||Where the extension commences at a cross subsection place, the cross subsection place and the places in the subsection below are relative.|
|4 (i)||The places of the new front subsection are the same as those of the subsection immediately above with which it is associated, except that the interior places expand by 2 places from the lead.||The places of the new front subsection are identical with those of the subsection immediately above with which it is associated.|
|(ii)||Any subsequent interior subsection and cross subsection places expand 2 places from the lead.||Any subsequent interior subsection and cross subsection places are identical with their counterparts in the parent.|
|(iii)||In each successive stage the interior subsection and cross subsection places expand 2 places further from the lead.||In all subsequent stages, the interior subsection and cross subsection places are identical with their counterparts in the parent.|
|5||A front cross subsection place adjoining the treble may either:-|
(a) continue to adjoin the treble, or
(b) remain static
In (a) any later front cross subsection places which adjoin the treble continue to adjoin the treble, but ordinary front cross subsection places remain static.
In (b) any further front cross subsection places, whether they adjoin the treble or not, remain static.
|6||A front extension may not become effective before:-|
(a) a subsection containing contiguous places in 1-2, or
(b) a subsection which is connected to the following subsection by contiguous places in 1-2.
|Notwithstanding 5 (b) above, a front extension may not become effective from a point prior to any front cross subsection place adjoining the treble, unless the affected place continues to adjoin the treble as required by 5 (a).|
|7 (i)||The interior places of back subsection are “expanding” to coincide with the front subsection, but in an inverted position. The places of the new subsection are the same as those immediately below with which it is associated except that the interior places expand 2 places from the back.||The interior places of the back subsections are “expanding.” The places of the new back subsection are the same as those of the subsection above with which it is associate except that the interior places expand 2 places from the lead.|
|(ii)||Any subsequent interior and cross subsection places expand 2 places further from the back than their counterparts in the parent.||Any subsequent interior and cross subsection places expand 2 places further from the lead than their counterparts in the parent.|
|(iii)||In each successive stage, the interior subsection and cross subsection places expand 2 places more.|
|8||In all stages the place positions prior to the points where the extension becomes effective are identical with those in the parent.|
|9||All lead positions and ultimate places, whether subsection or cross subsection places, are identical with their couterparts in the parent.|
|10 (i)||The pivot place is either at the lead or penultimate position in every stage, to correspond with its position in the parent. It is the reverse of the lead-head place.||The pivot place, if at lead or penultimate position under the treble, remains in the lead or penultimate position in all stages.|
|(ii)||An interior pivot place expands 2 places further from the lead in each successive stage.||An interior pivot place remains static in all stages.|
|11||Every extension bears the standard lead-ends and the method type of the parent, e.g., Surprise, Delight, Exercise, etc., as appropriate. The lead-head “place sequence” is preserved as required by the Report on “Pivot Bells.”|
Application of the foregoing formula and rules will ensure that the treble lead is a correct extension of the parent. It now remains to join the leads of the course together to ensure that the order in which a bell occupies the different positions at the successive lead-heads of the parent is maintained in the extension; e.g. Cambridge Minor - 126.96.36.199.5; Major - 188.8.131.52.4.8.5; etc. The reasons for this and examples of pivot sequences were given in the 1952 report and need not be repeated here, but further investigation has revealed that satisfactory results are obtained by some additional sequences from Minor to Royal, omitting Major and Maximus; from Minor to Maximus, omitting Major and Royal; and from Major to Maximus, omitting Minor and Royal. The full table of these sequences is given below.
|Extensions from Minor||Extensions from|
The Committee is aware that the insistence on correct lead-head sequence may cause apparent anomolies. On the other hand, there is evidence that the waiving of these requirements will cause further anomolies. The matter will be kept under review and any adjustments found to be necessary will be reported to the Council.
The adoption of these recommendations will result in so-called border-line extensions being excluded, as indeed would be the case whatever criteria were adopted. It is considered that such a position is preferable to the spate of worthless extensions which would result from a broadening of these recommendations.
In view of the impossibility of regularising the position of incorrect extensions which have been rung in the past, except by wholesale renaming, the Committee proposes that these recommendations become operative in respect of any new method extension rung after the date of the adoption of this report and that those previously rung to peals may continue to hold the names under which they were originally rung.
Examples of how the parent method should be notated (plain or treble bob hunt methods) when applying the formula
“EXPANDING” or “STATIC”
|PARENT - MINOR||PARENT - MAJOR||PARENT - ROYAL|
|Treble in 1-2||W||Treble in 1-2||U||Treble in 1-2||S|
|Cross section places||X||Cross section places||V||Cross section places||T|
|A||Treble in 3-4||Y||A||Treble in 3-4||W||A||Treble in 3-4||U|
|B||Cross section places||Z||B||Cross section places||X||B||Cross section places||V|
|C||Treble in 5-6||C||Treble in 5-6||Y||C||Treble in 5-6||W|
|D*||Pivot place||D||Cross section places||Z||D||Cross section places||X|
|E||Treble in 7-8||E||Treble in 7-8||Y|
|F*||Pivot place||F||Cross section places||Z|
|* If pivot place at lead, omit this letter in “Expanding” methods. If pivot place at lead or penultimate, omit the letter in “Static” methods. (See Rule 10 (i))||G||Treble in 9-0|
|PARENT - MINOR||PARENT - MAJOR||PARENT - ROYAL|
|Treble in 1-2||C||Treble in 1-2||E||Treble in 1-2||G|
|Cross section places||B||Cross section places||D||Cross section places||F|
|A||Treble in 3-4||A||A||Treble in 3-4||C||A||Treble in 3-4||E|
|B||Cross section places||B||Cross section places||B||B||Cross section places||D|
|C||Treble in 5-6||C||Treble in 5-6||A||C||Treble in 5-6||C|
|Pivot place||D||Cross section places||D||Cross section places||B|
|E||Treble in 7-8||E||Treble in 7-8||A|
|Pivot place||F||Cross section places|
|Pivot place not notated in Double methods. (See Rule 10 (i))||G||Treble in 9-0|
The Ringing World, July 17, 1953, pages 472 to 473, July 24, 1953, page 489, and July 31, 1953, page 504