[r-t] Extension of TDMMs

Philip Saddleton pabs at cantab.net
Sun Aug 15 10:59:55 UTC 2004

Richard Smith <richard at ex-parrot.com> wrote at 18:06:35 on Sat, 14 Aug 
>They're all mode-4 AB over mode-4 FG extensions.  The methods in 
>question are
>  &-3-45-5-3.2-4.5,2  Eminent Nick Lawrence S.
>  &-3-1-5-3.4-34.5,2  Ripple D.
>  &-3-1-5-3.4-2.5,2   Droitwich D.
>  &-3-4-2-3.4-34.5,2  Beverley S.
>  &-3-4-2-3.4-2.5,2   Surfleet S.
>I checked Beverley and Surfleet further -- up to 100 bells -- and it 
>does appear to be an indefinite series: 6, 28, 52, 76, 100.  If you 
>ignore the first step, it does form an arithmetic progression.

This does make sense. In each case the repeated sections rotate four 
bells above the treble, and three below, so you need to insert 12 to get 
all of them back in the same place. The lead head is f in each case, but 
b for the parent, which is another indication that it is not part of the 
series. I would imagine it might be possible to contrive a Royal method 
extending by 4 places below the treble (swapping 2 and rotating 5) and 
by 6 places above (rotating 3), where the gap is 16x3x5 = 240.

It is significant that although there are some big gaps in the series, 
the only prime factors are 2 and 3. The interesting ones from a 
practical point of view are those where there are two small ones 
(including a sporadic 6) then a big gap, e.g. 6, 8, 20... or 6, 10, 
24... . Are any of these the only indefinite extension for that method?


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