Double Grandsire is an awful method. It has four blows at the back, and four blows at the front. Because normal Grandsire Minor (which isn't much better) has four leads in the plain course, Double Grandsire splits these up and only has two leads in the plain course. Double Grandsire Doubles should be much more rewarding.

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Here is the plain course of Double Grandsire Minor:

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123456     126543

213546     216453

231456     261543

324165     625134

342615     652314

436251     563241

463521     536421

436512     563412

463152     536142

641325     351624

614235     315264

162453     132546

126543     123456

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To make matters worse, each lead is asymmetrical to the path of the treble, so it is not possible to get a true 720 with normal Grandsire Minor bobs and singles alone. You need to have these same calls reflected at the half lead too, to turn the B-Blocks into something symmetrical. (1440s are possible with calls only at the lead end.)

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In February 2013 there was a discussion about Double Grandsire Minor on the change ringers mailing list, and while experimenting with the method I managed to come up with a bobs-only 720 (i.e: with bobs at the half leads as well as the lead ends.)

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The main reason this is possible is that certain blocks can be cut clean in half without any extra material being added in the process. In most other methods, when we add an extra q-set to cut a block into two halves this will add an extra course, so we still have the same parity of blocks as we did beforehand. But in Double Grandsire Minor this is not so, and we can therefore switch the parity of the number of blocks without needing to use singles.

(The fact that the method's plain course is exactly half a plain course of normal Grandsire Minor is analogous to this.)

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Note that being able to attain the out-of-course leads without singles was never a problem in Grandsire Minor or Double Grandsire - as is a problem in Bob Minor - because the Grandsire bobs change the nature of the lead ends from in-course to out-of-course and back. This was not what I was referring to in the previous paragraph: the only apparent problem was getting the right parity of blocks with bobs alone.

 

Here is my bobs-only extent, with each full lead being represented by any combination of BP, the first being at the half lead, and the second being the lead end:

 

720 Double Grandsire Minor

RBP

 

BB  34625

BP  56243

BB  62354

BB  23465

PP  25643 *

BB  56324

BB  63452

BB  34265

BB  42536

BP  65324 *

BB  53462

BB  34256

BB  42635

BB  26543

BP  35462 *

PB  42356

BB  23645

BB  36524

BB  65432

BB  54263

PB  23546

BB  35624

BB  56432

BB  64253

BB  42365

PB  35426

BB  54632

BB  46253

BB  62345

BB  23564

PP  24653 +

BB  46325

BB  63542

BB  35264

BB  52436

BP   64325 +

BB  43562

BB  35246

BB  52634

BB  26453

BP  34562 +

PB  52346

BB  23654

BP  46532

BB  65243

BB  52364

PB  34526

BB  45632

BP  26354

BB  63425

BP  54236 +

BB  42653

BB  26345

BB  63524

BP  45236 *

BB  52643

BP  36425

BB  64532

BB  45263

PB  23456

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The * and + leads are the ones which incorporate important q-sets used for altering the parity of blocks. I composed this on 13th February 2013, and Ander Holroyd very kindly computer proved it for me, as I can't get my own provers to work for a method like this! A more detailed account of how I constructed the 720 is this rather embarrassing post on ringing theory. It is rather nice that this 720 has a few PPs (i.e: an entire lead with no calls), granting this to be a pedantic 720 of Double Grandsire.

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At about the same time I came up with a 720 of normal Grandsire Minor partitioned into four blocks with no singles. This was very exciting, as q-sets have four members, and it seemed that a bobs-only 720 was in my grasp! However, there was no q-set equally split across the four blocks, and it has apparently been computer proved that a bobs-only 720 of Grandsire Minor is impossible. It is still fascinating, though, that an extent can still be made up of four blocks - the same number of elements in the method's bob q-sets.