#48: Crossing the Parallel

Dear Stolf: I’ve heard something about what’s called Cross and Parallel Cousins, but not much. Can you elucidate?  …from Harvey, in Bedford Falls

48.1 Dear Harvey: Yes indeed. A major part of genealogy consists of understanding concepts and the terminology used to describe them. We have been primarily concerned with the English language and English terminology…and by extension, the kinship concepts that apply to a typical “Western” culture. Other cultures and languages will have the same concepts but different terminology. Back in #11  for example, we considered the Hispanic term for your parent’s 1st cousin…”second uncle” instead of “1st cousin once removed.” But there also exist in other kinship systems concepts not in use in our kinship system, which will then have no natural terms in English…and such is the case with Cross Cousins and Parallel Cousins, also sometimes called Ortho-Cousins.

48.2  Now these terms are strictly anthropological terms, since they classify 1st cousins in a way in which we do not…but they do describe a concept found in many societies, past and present. As shown in Chart 161, a Parallel Cousin (PC) is the child of your parent’s same-sex sibling, which is to say, your father’s brother or your mother’s sister. A Cross Cousin (CC) is the child of your parent’s opposite-sex sibling…your father’s sister or your mother’s brother.

48.3  Since we have nothing like this in the West, this particular “grouping” of 1st cousins will seem odd to us. The father’s side versus mother’s side classification of cousins seems natural enough, altho there really isn’t any significance to a cousin of yours being one or the other, apart from who has what last name. Another natural way for us to group cousins is by “boy cousins” and “girl cousins,” regardless of which side. But Parallel/Cross seems like an arbitrary distinction…altho as we shall see, there is a reason why it’s important to some cultures. But notice you can have PCs and CCs on both sides of your family, and they can be other either sex.

48.4  And one point should be understood: the PC/CC distinction is not really dividing 1st cousins into 2 different types…rather, it refers to 2 different relationships…and depending on the culture, there may or may not be a “generic” term that covers all the offspring of all your parents’ siblings, what we would simply call “1st cousins.” For example, mothers and fathers can be grouped together as “parents”…brothers and sisters are “siblings”…but we ourselves have no collective word for uncles and aunts. Thus in some kinship systems, PCs and CCs are what they are, and are not considered “types” or “subdivisions” of anything else.

48.5  Add to this the fact that in some systems, the alternative kinship grouping doesn’t stop with dividing up the 1st cousins…PCs may in fact be referred to by the same word that refers to siblings! In other words, one word will mean, from our point of view, “siblings and PCs” and another word will mean “CCs.”  Thus, what we might translate as “siblings” will mean “siblings and PCs,” while “cousins” will refer to CCs. And the specific nature of this type of grouping hints at the reason for it.

48.6  And that reason is to designate who you may or may not marry. With very rare exceptions (think Cleopatra and the Ptolemies), brother-sister marriages have been universally taboo. Half-siblings are often included in this prohibition, since many systems are what  is called “unilineal.” This means primary or sole kinship relationships are reckoned thru just one line, usually the father’s. By contrast, our system is “bilineal,” as both the father’s and mother’s sides of our families are of equal significance.

48.7  In a unilineal, patriarchal system, as many are, siblings are those individuals who share the same father, regardless of the mothers, and half-siblings are thus equivalent to full siblings. And the question is, since you always know who the mother is, but you may not know who the father is, how do you know your 1st cousin isn’t actually your half-sibling…or for the purposes of marriage, simply your sibling, and thus excluded?

48.8  And that in a nutshell was what was wrong with marrying Parallel Cousins. In Chart 163, can Abe marry his 1st cousin Zoë? Their fathers are brothers…so maybe yes, maybe no! Is it possible that Abe’s father is also Zoë’s father? That Abe’s father had relations with his brother’s wife? Or in Chart 164, mothers are sisters, same problem. Could Abe’s dad have fooled around with his wife’s sister? Mind you, I’m not judging, simply reporting. And it’s significant to remember that these were overwhelmingly matches arranged by the parents…the couple may have never met! To a suspicious culture, the PC grouping makes perfect sense.

48.9  Thus the taboo against marrying your PC…the chance that it might in fact be your half-sibling. But couldn’t Abe’s Cross Cousin also be his half-sibling? Of course, but it was deemed less likely, and since that would rule out all cousin marriages…and they were preferred for any number of reasons…the risk was worth taking. But as we see in Chart 165a, for that to happen, Abe’s father would have to have incestuous relations with his own sister, Zoë’s mother…or alternately, as in Chart 165b, with his brother-in-law’s wife. Either were considered less likely to happen than Abe’s father getting together with his wife’s sister or his brother’s wife.

48.10  And you know what they say: Familiarity breeds attempt! Maybe our more mobile, disconnected society has some benefits after all! Next week, the 5 biggest mistakes you can make in understanding kinship.

Copyright © 2011 Mark John Astolfi, All Rights Reserved

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#47: The Truth About Fractions

47.1  How much mathematics do you need to do genealogy? It’s up to you: a lot, a little, or none at all. You can figure out who your 3G Grandfather and 5th Cousin Once Removed are with any “cyphering” whatsoever. When it comes to deciding which of them is more closely related to you…or which of your “numbered” cousins is as closely related to you as those 2 are to themselves…well, English majors can do it, but a little math makes it a lot easier.

47.2  I hope I can explain it to you “without tears”…altho feel free to cry if you must…let it out, it’s good for you 😉 😉  But the fundamental thing you have to know about genealogical math is that it deals only with “powers of 2.” That’s the fancy term for multiplying 2 by 2, over and over. This is because we have 2 biological parents, no more, no less.

47.3  “2 to the first power” is just 2…”2 to the second power” is 2 x 2 = 4….”to the third” is 2 x 2 x 2 = 8…and so forth. Thus the fraction of genes one relative shares with another will be some combination of ½, 1/4, 1/8, 1/16, 1/32, 1/64, etc. Nobody is related to anybody by any other “even” fractions…no 1/6, 1/10, 1/12, 1/20, etc….nor by any “odd” fractions, 1/3, 1/5, 1/9, 1/15, etc. You can come close, but only as close as the “powers of 2” can approximate. For example, you cannot be 1/3 related…but you can be 11/32 or .343, which is close to 1/3 or .333.

47.4  Now you are related to your sibling by ½…which is to say, half your genes you share, half you don’t. We will see why this is so in a moment. But given the progression of fractional powers of 2, and the progression of numbered cousins, the table on the far left of Chart 158 would seem reasonable, wouldn’t it? But it is in fact incorrect…the center table is gives the true Coefficients of Relationship or CR’s. And on the right, the “gaps” are filled by the appropriate relatives.

47.5 Why is your 1st Cousin only 1/8 related to you, and not 1/4, which seems like it would be so “obvious”? I admit this puzzled me too at first, but it makes sense once you flesh it out. The short answer is: you are ½ related to your father, he is ½ related to his brother (your uncle), and his brother is ½ related to his son (your 1st cousin)…½ x ½ x ½  = 1/8.

47.6  But to see how it works in detail,  a good analogy would be to imagine decks of playing cards. And since we are dealing with powers of 2 only, our decks will have 64 cards, not 52…say there’s a 5th suit, stars, and the ace is missing. BTW, such decks exist, you can buy one here if you’re feeling adventurous.

47.7  Your father is a deck of 64 with blue backs, your mother with red backs. Since everyone has to have 64 cards, you will choose randomly 32 from your father and 32 from your mother. Genealogically, this is illustrated on the left side of Chart 159. Clearly, all the blue cards you have, your father also has…that’s where you got them from! So half of your deck of 64 matches, front and back, a card in your father’s deck…half of them don’t, since they have red backs. The CR between you and your father is thus ½…and by the same reasoning, it’s ½ with your mother as well.

47.8  Your CR with your brother is also ½, but in a different way. You took 32 cards at random from your father’s deck and he did the same. Odds are, 16 of your blue cards will match 16 of your brother’s blue cards, and 16 will be different. If you were to actually try this at home, it could come out 15 to 17, 12 to 20, or even 8 to 26…but since we have thousands of genes, and you’re picking strictly by chance, it will come out very close to 50/50, matches to non-matches. You and your brother will mark your blue cards that don’t match with an X, and leave the ones that do match unmarked. The same thing applies to the cards you both got from your mother’s deck.

47.9  As you can see on the right side of Chart 159, you and your brother share 16 blue cards and 16 red cards, for a total of 32…out of 64…so your CR is again ½. You might say that your relationship to your father is a “vertical” ½, and to your brother a “horizontal” ½, at least according to the way we diagrammed it. The important point is, the ½ comes about in 2 different ways, and that has a bearing on how you’re related to your 1st Cousin.

47.10   Now in the same way that we compared the “decks” of you and your brother, we can compare yours with your 1st Cousin’s. In Chart 160, your father and his brother (your uncle) are in the middle, blue & red, and each marry a different woman. Right off the bat, half of your cards will be different from half of your 1C’s cards…you got 16 green backs from your father and 16 orange backs from your mother, he got 16 grays and yellows from his. The only relationship you can have with your 1C is thru the 16 blues you each got, and the 16 reds. Let’s look at the blues.

47.11 When you picked 16 of your father’s blue cards, picking randomly of course, you got 8 blue X’s and 8 blue non-X’s…picking from his father, your 1C did the same. Now since the blue X’s don’t match to start with, those you got can’t match those your 1C got…so they are out. The only blue cards that can possibly match are those you picked from the non-X’s your father and his father shared. There are 8 of those cards each…and just as before, half will match and half won’t…so you and your 1C share 4 of the blue non-X’s, and 4 of them you don’t. Remember, that’s 4 out of 64 total cards, so thru the blue side of your fathers, your CR is 4/64 or 1/16. But of course the same holds true with your fathers’ red cards, so that’s another 1/16…and 1/16 + 1/16 = 1/8.

47.12  I know, it’s still complicated…but it is what it is…thank you, sexual reproduction!!  Maybe that’s why cloning has so much appeal 😉 😉 And if you’re feeling really adventurous, you can try it all again with half-siblings and half-1st Cousins…but when you count up the squares that match, you’ll ultimately find that the whole thing checks.

47.13   And it’s interesting to see how little you and your first cousin actually are related…it helps to explain why 1st Cousins marriages have been so common down thru history, and still are in many parts of the world today. Plus legal in half the states in the US…you don’t have to like it, but there you go…next time, more letters from the bulging mail-bag…take care…

Copyright © 2011 Mark John Astolfi, All Rights Reserved

#46: More of David, by Half

46.1  Last week were were flexing our genealogical muscles with a chart of descendants of the Biblical King David…and I left you with the question of how Jeush (O) and Abijah (P) are related, given that they have the same father, their mothers are both half-1st cousins and 2nd cousins, and the father is both half-1st cousin and 2nd cousin to one of the mothers, but just half-1st cousin to the other…whew!

46.2  And this seems like a reasonable time to review the genealogical concept of “half-relations.”

46.3  Suppose Abe and Zoë say that their grandfathers were brothers. That would make then 2nd cousins, the offspring of 1st cousins, as shown in Chart 152a. Now this is correct as far as it goes, as long as we bear in mind that it is a short-hand for Chart 152b…which is to say, we take them at their word, and assume by “brothers” they mean “full brothers.” And as we see in Chart 152c, the grandfathers could be half-brothers, so that their children would be half-1st cousins, and Abe and Zoë half-2nd cousins.

46.4  Now all the half-siblings I’ve ever met called their half-siblings their “siblings”…brothers or sisters. Further discussion night reveal that they have “different mothers,” in which case you could say: Oh, so you’re half-siblings…to which they might reply: Yeah, technically, but we don’t think of it that way. This illustrates the difference between genealogical terminology and “everyday” terminology…thus for example you might introduce your 1st cousin once removed as your cousin, or your uncle, or your father’s cousin, or your father’s 1st cousin…but probably not as your 1st cousin once removed. But every family has different customs and habits…which may vary between individuals within the same family.

46.5  The significance of Chart 152 is this: it’s somewhat sloppy to say that the closest common ancestor of 2nd cousins is a great grandparent…if they are full 2nd cousins, they would have 2 common great grandparents…they would share both a great grandfather and a great grandmother, as in 152b. If not, then we have 152c, where the grandfathers are half-brothers, the parents half-1st cousins, and Abe and Zoë half-2nd cousins. Here their closest common ancestor is literally one person, not 2…but in general, we assume we are dealing with full relations unless told otherwise, for simplicity’s sake.

46.6  And what does it matter, really? Well, in everyday life, it may matter a lot, a little, or not at all. But in genealogy and kinship it is very significant…full siblings share all their ancestors…one does not have a relative in the past that the other doesn’t…both in the mother’s line and the father’s line. For half-siblings, there are 3 lines of descent, not 2, and they only share one of them. This certainly could be crucial from a genetic or medical standpoint…less so from a matrimonial standpoint today, but that could still be relevant depending on your religion, personal beliefs, or the laws where you live.

46.7  But where it gets really interesting is with the “clue” I gave you last week for O and P’s relationship: “Enhanced Half-Siblings.” (Sorry, but there is no “everyday” term for this!)  Most of the time, half-brothers with the same father will have mothers who are unrelated. But the mothers could be related, as in Chart 153 where they are sisters….making  Bob and Cal 1st cousins as well as half-brothers. Now you might object: How can they be 1st cousins if they’re “already something else,” in this case, half-brothers?

46.8  Well, as was hinted at in the 3 diagrams of Chart 152, it’s because from a genealogical standpoint, full siblings are “Double Half-Siblings.” It’s thought of this way: you can have the same mothers (half-siblings), the same fathers (half-siblings), or both (full siblings.) Another way to put this would be that full siblings are half-siblings on one side, and half-siblings on the other side…while with “actual” half-siblings, they are half-siblings only on one side, and not related on the other. In the case of Bob and Cal, they are related on both sides…half-brothers on their father’s side…and 1st cousins on their mothers’ side. (And if you’re thinking that technically full siblings could be considered a form of Enhanced Half-Siblings…well, you’re right, but please don’t think that way! It’s bad enough that they’re Double Half-Siblings, you know?)

46.9  The Coefficient of Relationship for full sibs is ½…for half-sibs is 1/4…and not surprisingly, for Bob and Cal’s type of Enhanced Half-Sibs, it’s 3/8, which is half-way between 4/8 (½) and 2/8 (1/4). Of course, the 2 parents can be related in other ways besides siblings…half-sibs, 1st cousins, half-1st cousins, 2nd cousins, both half-sibs and 1st cousins (!!)…the possibilities are endless.

46.10  Now we’re pretty much ready to tackle the relationship between O and P, except for one hitch: unlike “normal” Enhanced Half-Siblings, the 2 sides of the family are related to each otherwhich is to say, the common parent on one side and the 2 parents on the other side are in fact related, over and above the 2 parents on one side being related to each other…duh!…that’s why O and P are such a tangle! So let’s approach it from a simplified case, then apply what we learn to O and P.

46.11  But before the “simplified case,” we need the “super-simplified case,” that of Double First Cousins. And here we can state, as per Chart 154, the DOUBLING RULE:

A’s father is brother of Z’s father  = 1st cousins
A’s mother is sister of Z’s mother = 1st cousins

So A and B are 1st cousins “on both sides of the family”…hence Double 1st Cousins, with a CR of 1/8 + 1/8 = 1/4. or the equivalent of half-brothers.

46.12  Now let’s look at the “Super-Enhanced” Half-Brothers…in Chart 155, we have taken “Normal” Enhanced Half-Brothers Bob and Cal from Chart 153, and made both sides of their families related…in this case their father A is 1st cousin to their mothers Y and Z.

46.13  What this does is add a Double Cousin relationship to Bob and Cal, and to make this relationship clearer to see, in Chart 156 I have moved A over to the other side…and as I have done in the past, his “background” is a different color to indicate that this part of the chart has been duplicated.

Now if you apply the DOUBLING RULE, you will see that Bob and Cal are indeed Double 2nd Cousins, on top of everything else…

Bob’s father is 1st cousin of Cal’s mother  = 2nd cousins
Bob’s mother is 1st cousin of Cal’s father  = 2nd cousins

This is tricky, since the 2 individuals I’ve underlined above are the same person…but this makes no difference to the Double 2nd Cousin relationship. Taking the extreme example of Cleopatra and her sister Berenice (she had 5 siblings, altho whether full or half or some of each is unknown, so we’ll assume full)…their parents where brother and sister…so Cleo and Berry were both sisters and Double 1st cousins. Thus Bob and Cal are half-brothers, 1st cousins, and Double 2nd cousins, for a CR of 1/4 + 1/8 + 1/32 + 1/32 = 7/16, which is just slightly less related than if they were full brothers = 8/16 or ½.

46.14  And now at last we are prepared to look at O and P…and what I have done is to recast their relationship in the form of the Super-Enhanced Bob and Cal diagram in Chart 156,  so that we may compare and contrast, as they say. To start with, we see in Chart 157 that O and P are half-brothers on their father N’s side, with a CR of 1/4. The relationship between their mothers is a complicated one (see the chart after 46.1)…2nd cousins on one side, half-1st cousins on the other…so we will instead use their CR of 3/32…how convenient! The offspring of 2 related individuals have one quarter of the CR of their parents, so thru their mothers O and P have a CR of 3/128…O and P are 3rd cousins (1/128) and half-2nd cousins (1/64 or 2/128)… 1/128 + 2/128 = 3/128…whoo hoo, it checks!

46.15  Finally, applying the DOUBLING RULE, we see that while O and P are related in 2 ways, they are not the same ways, since their father is related to their mothers in different ways…

                         P’s father is 3/32 to O’s mother  = 3/128  (remember, 1/4th)
P’s mother is 1/16 to O’s father  = 1/64  (ditto…)

So taken all together, O and P have a CR of 1/4 (thru father) + 3/128 (thru mothers) + 1/64 + 3/128 (the irregular double cousin relationship, since their fathers and mothers are related)…for a grand total of 40/128…somewhere between half and full siblings, but closer to half…and all I can say is…Hallelujah and Amen!  Next week, why Cousins are 1/8 and not 1/4….see yez…

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Copyright © 2011 Mark John Astolfi, All Rights Reserved