acidently deleted this thing once so have to remember the thoughts as I was free specuating rambling
I played with this in One Live Slythern so it is not a new thought.
The thought was that given how the number of ancestors double each generation basically
1stgen = 1 - the child
2ndgen = 2 - the parents
3rdgen = 4 - the grandparents
4thgen = 8 - the great,grandparents
5thgen = 16 - the great,great,grandparents
6thgen = 32 - the great,great,great,grandparents
7thgen = 64 - the great,great,great,great,grandparents
8thgen = 128 - the great,great,great,great,great,grandparents
9thgen = 256 - the great,great,great,great,great,great,grandparents
The whole concept of bloodbinding Grimoire,Books of Shadow etc and other objects to the Bloodline a but iffy the further away from the origonators generation it gets the weaker the link but also the number of people who are decended from that orginator would be rediculus and very few would be the mainline.
With that in mind they obviously had to update the bloodbinding on a fairly regular basis to cut the peopel able to acces such things done to a freasonable level.
After looking at those generation calculations and given the love of the number three and Seven for magicals I think it would end up being renwed on the third generation after the last bloodbind as so counting baxk from the child the last bloodbinding would be the great,grandparent.
eg: The Black Family say Arcturus dis the last bloodbinding so it would go Arcturus--Orion-1gen--Sirius/Regalus-2gen--one of their children was due to do the rebinding.
The nastly thing about this is that the generations of the younger Black bloodlines would end up cut out of the access when the new bloodbind occurs. It also means that if the main line dies after that bloodbind there is no obvious Heir Line except via bith order and sex males going first to continue the name.
I cant find a readable copy of the Black Family Tree but I think Wallburga was the generation after Orions so she and her siblings would normally be the last to have access through Arcturus after the next reset and that would only last until Orion had a grandson but it also meant the next generation of her particular Black Bloodline only had access until that child was born and all of their children would not have access at that point as the magic would solidify back into only the main bloodline having access.
Wallburga solved that by marrying Orion the Heir as any children they have would also have the DNA of her Black Bloodline so her grandson as would be binding her line as well not just the main line.
Plot wise it brings into question the marriage of Orion and Wallburga as you have to wonder if he was spelled or potioned into that marriage as the books made it seem like Orion was just enduring the marriage very not happy with it.
This could also be another reason for the disowning if it was Magical it could remove their bloodline from access in a Magical disowning ritual perhaps. The Bloodbinding might also have inbuilt boobytraps perhaps killing the mainline to get access could automatically remove your bloodline from the matrix given Maguc is mainly about intent.
I also suspect having the mainline die without a viable heir already with access might be a big thing as well aka the family having to gather and let Magic decide who they will bind it to and I doubt anyone plotting to have their bloodline stay secondary and primary Heirs if the main line dies as such a Ritual would probably let Magic see all that and include that knowledge in the Judging again the intnt factor in play.
The HP timeline is interesting on that front as Narcissa got pregnant with Draco after Regalus death and at the time Wallburga and the others thought Sirius disowned so they lost the Heirline and with Alphard also dead and only three daughters to their line the heirship would most likely reverted to the older Black birth bloodline because Wallburga's Black bloodline line was actually third the other brother was older but he had three daughters - so if it had reverted to bith order it meant the next Black Heirs would probably be either a Longbotton or a Crouch as the Weasley Line was disowned.
Huh? That gives Crouch Senior a viable reason to get rid of Sirius by shoving him into Askaban without a trial as it stopped him having children and with that ensuring he had a good chance of inheriting the Black title and vaults for the Crouch bloodline. It might also have been extra incentive for the Longbottom attack to remove the Longbottom's from being in line to inherit the Black line as they were next in line then Crouch if so I suspect Bellatrix would have killed Crouch Jr at the end of the raid to kill both off booth of the senior heirlines ahead of Draco.
So that was my thoughts now I just have to contemplate how everyone else had been plotting because the plot might be hidden but the sudden marriage of Orion and Wallburga had all the supplty of a bombarda in turning the families atention their way and working out the nature of that plot.
OH dam you could easily have Ablus plotting the death of the Potter family to get the cloak as it was a Peverell Heirloom so probably bloodbound to the family so given his obsession with the Deathly Hallows he could easily have been arranging for the family to die out to get the cloak perhaps believing he could bind it to himslef with the Potters dead as he already had the Wand and could use sympathetic Magic to do so. It was only after he borrowed it and tested it that he realised it wouldn't work Tom Riddle was next in line to get the cloak as he was both a Peverell decendent and had the Stone. So plot plot plot on his part followed!
Hell he might have truely started working against Riddle while he was at school first because he was a vicius little boy and got sorted into Slytherin so his best candidate to create a new dark lord which he would eventually need to fight to secure his place in history as the next Merlin by defeating two dark lords.
When Tom set the basilisk on the school and accidently killed Myrtle as he didn't really care about a Ravenclaw Muggleborn dying but he realised he realised Tom could not only speak to snakes but open the chamber so descended from both Slytherin which would have just confirmed his being perfect as his Dark lord candidate.
Then he came back from summer holidays wearing the resurection stone therefore a Gaunt and decended from the Middle Peverell brother and therefore in line to inherit the ring. So Tom became a double target in Albus eyes his next dark lord and the holder of the resurection stone.
He probably didn't get confirmation that the Potter's had the cloak until James brought it to school as there were no new mainline Potter Heirs during his time as Teacher and Headmaster until then so no cloak brought to school until the seventies.
shaz