Department of Science, Math, & Technology

=Classificatory Kinship Systems People speaking one language may have different kinterms for two types of kinsmen (gender-neutral interpretation meant) that some other language has just one kinterm for. If so, scholars writing in that first language might call the second language’s kinterm “classificatory”; meaning they classify those two kintypes as the same kintype. To a certain extent nearly every culture’s kinterm system is partly “classificatory” from the point-of-view of nearly every other culture’s kinterm system. For instance, according to https://www.rd.com/culture/grandma-grandpa-different-languages/ , in Swedish, mother’s mother is mormor but father’s mother is farmor; in English they’re both grandmother. So to a Swedish-speaker English’s kinterm “grandmother” is classificatory. Likewise, In Swedish, father’s father is Farfar but mother’s father is morfar; while English classifies them both as grandfather. In Mandarin, there are different terms for siblings depending on relative birth-order as well as gender. gē ge Older brother dì di Younger brother jiě jie Older sister mèi mei Younger sister Swedish has morbror for mother’s brother and farbror for father’s brother. Mandarin has eight different terms for: Father’s older brother Father’s older sister’s husband Father’s younger brother Father’s younger sister’s husband Mother’s older brother Mother’s older sister’s husband Mother’s younger brother Mother’s younger sister’s husband But in English these are all just Uncle. Some languages have one term for a guy who has the same father (whether or not they have the same mother) and another term for a guy who has the same mother but a different father. In other words, one word means full-brother or agnate half-brother, and the other word means enate or uterine half-brother. Other languages have one term for a guy who has the same mother (whether or not they have the same father) and another word for a guy who has the same father but a different mother. In other words, one word means full-brother or enate (aka uterine) half-brother, and the other word means agnate half-brother. Yet other languages have three terms; meaning full-brother, uterine (i.e. enate) half-brother, and agnate half-brother. At least one language has a term for someone with the same father whose mother was your mother’s sister; or who has the same mother but whose father was your father’s brother. And it’s (or they’re) different from the word(s) for full-siblings, and agnate half-siblings whose mothers weren’t sisters, and uterine half-siblings whose fathers weren’t brothers. And some language (languages?) has(have) terms for same-sex sibling versus opposite-sex sibling, rather than male sibling (brother) versus female sibling (sister) _________________________________

= The Kinds of Classificatory Kinterm Systems that Chiefly Interest Me Here When talking of Prescriptive Marriage Systems along with closely conceptually-related Classificatory Kinship Systems, the above-mentioned anthropologists of the 1960s, and also to some degree at least some of the anthropologists of the 1990s and today, are mostly interested in the following kind of “classificatoriness”: If EGO calls ALTER by a certain kinterm, and ALTER is Male, then EGO calls every brother of ALTER by that same kinterm. If EGO calls ALTER by a certain kinterm, and EGO is Male, then every brother of EGO calls ALTER by that same kinterm. If EGO calls ALTER by a certain kinterm, and ALTER is female, then EGO calls every sister of ALTER by that same kinterm. If EGO calls ALTER by a certain kinterm, and EGO is female, then every sister of EGO calls ALTER by that same kinterm. I will write one’s “actual” relative’s kinterm without quotes, and one’s “classificatory” kinterm with quotes. Your father’s brother is your “father”. So is your father’s “brother” and your “father’s” brother and your “father’s brother”. Your mother’s sister is your “mother”. So is your mother’s “sister” and your “mother’s” sister and your “mother’s sister”. Your brother’s brother is your “brother”. So is your brother’s “brother” and your “brother’s” brother and your “brother’s brother”. Your sister’s sister is your “sister”. So is your sister’s “sister” and your “sister’s” sister and your “sister’s sister”. Your husband’s brother is your “husband”. So is your husband’s “brother” and your “husband’s” brother and your “husband’s brother”. Your wife’s sister is your “wife”. So is your wife’s “sister” and your “wife’s” sister and your “wife’s sister”. Your son’s brother is your “son”. So is your son’s “brother” and your “son’s” brother and your “son’s brother”. Your daughter’s sister is your “daughter”. So is your daughter’s “sister” and your “daughter’s” sister and your “daughter’s sister”. ———————————————————————————————— Among the people who are your “father” are: Your father, your father’s brother, your father’s “brother”, your “father’s” brother, and your “father’s brother”; Your mother’s husband, your mother’s “husband”, your “mother’s” husband, and your “mother’s husband”; Your brother’s father, your brother’s “father”, your “brother’s” father, and your “brother’s father”; Your sister’s father, your sister’s “father”, your “sister’s” father, and your “sister’s father”. Among the people who are your “mother” are: Your mother, your mother’s sister, your mother’s “sister”, your “mother’s” sister, and your “mother’s sister”; Your father’s wife, your father’s “wife”, your “father’s” wife, and your “father’s wife”; Your brother’s mother, your brother’s “mother”, your “brother’s” mother, and your “brother’s mother”; Your sister’s mother, your sister’s “mother”, your “sister’s” mother, and your “sister’s mother”. Among the people who are your “brother”, if they aren’t actually you yourself, are: Your father’s son, your father’s “son”, your “father’s” son, and your “father’s son”; Your mother’s son, your mother’s “son”, your “mother’s” son, and your “mother’s son”; Your brother, your brother’s brother, your brother’s “brother”, your “brother’s” brother, and your “brother’s brother”; Your sister’s brother, your sister’s “brother”, your “sister’s” brother, and your “sister’s brother”; Your wife’s husband, your wife’s “husband”, your “wife’s” husband, and your “wife’s husband”; If you are male, your son’s “father”, your “son’s” father, and your “son’s father”; If you are male, your daughter’s “father”, your “daughter’s” father, and your “daughter’s father”. Among the people who are your “sister”, if they aren’t you yourself, are: Your father’s daughter, your father’s “daughter”, your “father’s” daughter, and your “father’s daughter”; Your mother’s daughter, your mother’s “daughter”, your “mother’s” daughter, and your “mother’s daughter”; Your brother’s sister, your brother’s “sister”, your “brother’s” sister, and your “brother’s sister”; Your sister, your sister’s sister, your sister’s “sister”, your “sister’s” sister, and your “sister’s sister”; Your husband’s wife, your husband’s “wife”, your “husband’s” wife, and your “husband’s wife”; If you are female, your son’s “mother”, your “son’s” mother, and your “son’s mother”; If you are female, your daughter’s “mother”, your “daughter’s” mother, and your “daughter’s mother”. Among the people who are your “husband”, if you are female, are: Your sister’s husband, your sister’s “husband”, your “sister’s” husband, and your “sister’s husband”; Your husband, your husband’s brother, your husband’s “brother”, your “husband’s” brother, and your “husband’s brother”; Your son’s father, your son’s “father”, your “son’s” father, and your “son’s father”; Your daughter’s father, your daughter’s “father”, your “daughter’s” father, and your “daughter’s father”. Among the people who are your “wife”, if you are male, are: Your brother’s wife, your brother’s “wife”, your “brother’s” wife, and your “brother’s wife”; Your wife’s sister, your wife’s “sister”, your “wife’s” sister, and your “wife’s sister”; Your son’s mother, your son’s “mother”, your “son’s” mother, and your “son’s mother”; Your daughter’s mother, your daughter’s “mother”, your “daughter’s” mother, and your “daughter’s mother”. Among the people who are your “son”, are: Your son; If you are male, your brother’s son, your brother’s “son”, your “brother’s” son, and your “brother’s son”; If you are female, your sister’s son, your sister’s “son”, your “sister’s” son, and your “sister’s son”; Your husband’s son, your husband’s “son”, your “husband’s” son, and your “husband’s son”; Your wife’s son, your wife’s “son”, your “wife’s” son, and your “wife’s son”; Your son’s brother, your son’s “brother”, your “son’s” brother, and your “son’s brother”; Your daughter’s brother, your daughter’s “brother”, your “daughter’s” brother, and your “daughter’s brother”. Among the people who are your “daughter”, are: Your daughter; If you are male, your brother’s daughter, your brother’s “daughter”, your “brother’s” daughter, and your “brother’s daughter”; If you are female, your sister’s daughter, your sister’s “daughter”, your “sister’s” daughter, and your “sister’s daughter”; Your husband’s daughter, your husband’s “daughter”, your “husband’s” daughter, and your “husband’s daughter”; Your wife’s daughter, your wife’s “daughter”, your “wife’s” daughter, and your “wife’s daughter”; Your son’s sister, your son’s “sister”, your “son’s” sister, and your “son’s sister”; Your daughter’s sister, your daughter’s “sister”, your “daughter’s” sister, and your “daughter’s sister”.

=Important to Notice! In the type of Classificatory Kinship Systems I’m most interested in for this thread —— often called “FB=F and MZ=M systems”, for reasons I will explain in my next reply —— [size 1.5][b]Parallel Cousins Are “Siblings”[/b][/size] That is, =father’s brother’s sons are “father’s” sons are “brothers” =mother’s sister’s sons are “mother’s” sons are “brothers” =father’s brother’s daughters are “father’s” daughters are “sisters” =mother’s sister’s daughters are “mother’s” daughters are “sisters”. !!= Also note that full brother, agnate half-brother, uterine half-brother, and stepbrother, are all just “brother”. In English, your half-brother’s half-brother stands a good chance of being your stepbrother. Your half-brother’s stepbrother and your stepbrother’s half-brother stand a good chance of being no relation to you. Your stepbrother’s stepbrother is highly probably no relation to you. But in these classificatory systems we’re discussing here, they’re all your “brothers”.

=Notations and Abbreviations To make things shorter and easier to read: F means father or father’s or “father” or “father’s” M means mother or mother’s or “mother” or “mother’s” B means brother or brother’s or “brother” or “brother’s” Z means sister or sister’s or “sister” or “sister’s” H means husband or husband’s or “husband” or “husband’s” W means wife or wife’s or “wife” or “wife’s” S means son or son’s or “son” or “son’s” D means daughter or daughter’s or “daughter” or “daughter’s”. Thus FB=F means father’s brother is classified as “father”, and MZ=M means mother’s sister is classified as “mother”. FBS means father’s brother’s son FBD means father’s brother’s daughter MZS means mother’s sister’s son MZD means mother’s sister’s daughter We have the classifications FBS=B and MZS=B, and FBD=Z and MZD=Z, in these Kinship Systems. ———————————————————————————————— Other notations and terminologies; The person from whom the Kinship is reckoned is called the propositus, and always denoted as EGO. The person who is the target of the kinterm is always denoted as ALTER. m.s. stands for male speaker and signifies that EGO is male. f.s. stands for female speaker and signifies that EGO is female. For instance for m.s. we have SF=B, but for f.s. SF=H. e stands for elder and y stands for younger. For instance a man’s eZDy is his elder sister’s daughter, younger than himself. A woman’s MyBe is her mother’s younger brother, older than herself.

!!= Preferences, Prescriptions, and Proscriptions = Proscriptions of Marriage Some societies have rules against certain marriages. For instance every society prohibits father-daughter marriage, and father-daughter sex, unless both parties believe the father is the only remaining fertile man alive, and the only remaining fertile women in existence are his daughters. Symmetrically, mother-son marriage, and mother-son sex, is strictly prohibited, unless everyone believes the mother is the human race’s only remaining fertile woman, and no fertile man other than her son(s) still exist. Many societies forbid parent-child mating for livestock, too, as well as people. Every society also forbids marriage, and/or sex, between full siblings, unless they are royalty. (Some societies do or once did permit full brother-sister marriages if th couple were royals). Half-sibling mating is allowed in some societies, or historically has been allowed. But some prohibit it. —————————— Some societies prohibit marriages between other pairs of close blood relatives; for instance, uncle-niece or aunt-nephew marriage, or double-first-cousin marriage. Some prohibit marriages between certain affine relatives. For instance, a man may be prohibited from marrying one or some or any of: His father’s wife, widow, or ex-wife; His son’s wife, widow, or ex-wife; His brother’s wife, widow, or ex-wife; The mother or daughter or sister of his wife or late wife or ex-wife The corresponding proscriptions for women also occur. Some societies don’t allow a man to marry his brother’s wife’s sister (his BWZ). Some don’t allow a woman to marry her ZHB. There are proscriptions not based on kinship. Some societies won’t let you marry someone from your own village. Some won’t allow a man to marry a woman from a higher caste. In some societies men never marry women older than they are. Etc. ————————————————

=Preferential Marriage Systems If there is a socially-preferred type of candidate spouse, and there are sanctions (however mild or severe) against marrying anyone else, and/or against failing to marry a preferred-type spouse, that’s called a “preferential marriage system”. For instance, in the Dravidian system, the (male speakers’) word for “older sister” is the same as the word for “mother-in-law”. A man’s preferred bride is his older sister’s daughter (younger than himself). A woman’s preferred groom is her mother’s younger brother (older than herself). But some men don’t have any older sisters. Or none of those older sisters have any daughters. Rarely, that or those daughter(s) are all older than the man. Or maybe they’re all already married. Or otherwise unavailable as his bride. So when modern anthropologists these days write about preferential marriage systems, they usually mention a second, and often a third, preference. For instance, if i recall correctly, there’s some minority society in China in which a man’s preferred bride (that is, the bride society prefers for him) is one of his first-cousins. It might be something like this: His MBD; then his FZD; then his MZD; then any other girl not his first-cousin. His FBD is the only first-cousin he is forbidden to marry (as if she were his sister). Other preferred marriage systems address re-marrying widows of deceased close male relatives, such as brothers and fathers and sons. Many societies have a custom called the levirate. It requires or expects a decedent’s brother to inherit his widow(s), so that she/they become(s) the surviving brother’s wife/wives. If the society has secondary marriage, this might apply to all the widows, or only the primary wife, or only the secondary wives. If the society has secondary marriage, and the surviving brother already has a primary wife, the decedent’s widows may become the living brother’s secondary wives. Some levirate customs are “junior levirate”. Only the younger brothers of the deceased can or must or are expected to marry their dead older brother’s widow(s). Some levirates are “senior levirates”. It’s the deceased’s older brother(s) who are expected to take his widows to wife. In many societies, a man’s heir is his oldest living son by his primary wife; or, by any wife; or by his primary wife if there is such a son, or by some other wife if he has no living sons by his primary wife. In some such societies, the man’s heir inherits all the man’s secondary wives, except for the heir’s own mother. And there are societies in which, if a living father’s married son dies, the father inherits/marries all of his dead son’s widows, or just his late son’s primary wife, or just the son’s secondary wife/wives. All four of these preferences can co-occur and co-exist in a given culture; probably with a widely-known socially-mandated order-of-preference. Maybe, for instance, Son > younger brother > older brother > father There could be up to 24 different orders. I doubt more than 12 of them are attested in real life, but I don’t know which, if any, are in fact attested, nor by whom. Among the Rukuba, who do have secondary marriage, a woman’s oldest daughter’s “preferred” groom, is her mother’s last premarital boyfriend’s oldest son. If mom’s last lover’s oldest son isn’t available, the next preferred groom is any of mom’s old flame’s other sons. If mom’s last hot date before her first wedding has no available sons, the third choice is any man from his patriclan. Rukuba women can’t, or won’t, or prefer not to, marry a man from the same patriclan as a full-sister’s or uterine-half-sister’s husband. So a woman’s second and younger daughters’ preferred grooms are men who are ritual-brothers* of mom’s last fling, but not in the same patriclan as him (nor the same patriclan as any other full-sister’s husband or uterine-half-sister’s husband). *Two Rukuba men are each the other’s ritual-brother, if they were both initiated into manhood at the same place. Ritual-brothers can’t marry each others’ wives. Preferential marriage systems that aren’t prescriptive marriage systems, tend to express preferences that sometimes are impossible to adhere to. Also, people in these societies, more or less frequently, don’t adhere perfectly even when it’s possible to do so. If they don’t, there may be a sanction to pay for marrying a less-preferred spouse when a more-preferred candidate spouse is available. Two examples. In many Middle Eastern and African societies, a man’s “preferred” bride is his FBD, and a woman’s “preferred” groom is her FBS. In some of those societies, the sanction is as follows: If a woman is about to marry someone who is not her FBS, and (one of) her FBS(s) wants to marry her, he (the FBS) can interrupt the wedding, and demand that she marry him instead. Among the Rukuba, only 10%%%%%%%%%%%%%%%% of women marry their “preferred” groom [u]first[/u]. However, 97%%%%%%%%%%%%%%%% of them marry their “preferred” groom [u]eventually[/u]. (Usually second, apparently.). This could be viewed as a sanction.

= Prescriptive Marriage Systems Prescriptive marriage systems can be (but not always are) viewed as a subtype of preferential marriage system. Prescriptive marriage systems are always coupled with an “FB=F and MZ=M” type of classificatory kinship system. Everybody who marries, must marry someone who is related to them in a certain way in the classificatory kinship system. If Miss X can marry Mr Y, then every woman in Miss X’s class —— her classificatory “sisters” —— can marry any man in Mr Y’s class —— his classificatory “brothers”. And Miss X and her “sisters” [i]cannot[/i] marry any other men than Mr Y and his “brothers”. = First-Cousin Marriage Systems The most popular type of Prescriptive Marriage Systems, world-wide, are those in which a man must marry his MBD or a girl so classified, and a woman must marry her FZS or a man so classified. In other “words”, W=MBD and H=FZS. Perhaps the second most popular type —— in any case, the second most written-about —— is marriage to the other kind of cross-cousin. A man must marry his FZD or a woman so classified, and a woman must marry her MBS or a man so classified. In other “words”, W=FZD and H=MBS. The third most written-about, and maybe the third most-popular, is marriage to a double cross-cousin. A man must marry a woman whose father is his mother’s brother (or thus classified) and whose mother is his father’s sister (or so classified). So a woman must marry a man who is both (classified as) her MBS and is (classified as) her FZS. That is, W=MBD and W=FZD and H=FZS and H=MBS. = Second-Cousin Marriage Systems There are also attested prescriptive marriage systems in which a person’s prescribed spouse is (classified as) (one kind of) their second-cousin. Maybe W=FMBSD and H=FFZSS. Or H=MFZDS and W=MMBDD. Or W=FFZDD and H=MMBSS. Or five other possibilities. W={FlM}{FZ|MB}{S|D}D and trace that backwards to pick from H={FlM}{FZ|MB}{S|D}S. (In an FB=F-and-MZ=M classificatory kinship system, there are eight kinds of second-cousin of each sex.)

= Direct Sister Exchange In a prescriptive marriage system such as most interests us here, and such as I’ve described (I hope!) above, it is often the case that a man can=must marry (a woman classified as) his ZHZ, and a woman can=must marry (a man classified as) her BWB. If that’s so, there’s no difference between a man’s WB and his ZH. Iow WB=ZH. They are either the same person; or they are brothers; or they are “brothers”. And there’s no difference between a woman’s HZ and her BW; HZ=BW. They’re the same person, or sisters, or “sisters”. Such a system is said to have “direct sister exchange” in marriage. That’s a somewhat sexist way of putting it; the women are exchanging brothers just as much as the men are exchanging sisters, unless there are other factors in the culture which haven’t been discussed so far. Second in popularity to W=ZHZ H=BWB WB=ZH BW=HZ Systems, with only one kind of brother-in-law and only one kind of sister-in-law, are systems in which: W=ZHZHZ and WB=ZHZH and WBW=ZHZ and H=BWBWB and HZ=BWBW and BW=HZHZ and BWB=HZH and ZH=WBWB. In such a system a man can have two kinds of brother-in-law; wife’s brother, vs sister’s husband. And a woman can have two kinds of sister-in-law; husband’s sister, and brother’s wife. This is not direct sister exchange. B=HZHZH H=BWBWB W=ZHZHZ Z=WBWBW BW=HZHZ HZ=BWBW WB=ZHZH ZH=WBWB BWB=HZH WBW=ZHZ ___________________ I may, or may not, have found a refernce on the Web to a real-life (I think Australian) society where the sibling-and-spouse “circles” contained eight classes (four of men and four of women) instead of six or four. If I correctly understood the author, he was talking about a system in which W=ZHZHZHZ H=BWBWBWB and so WB=ZHZHZH ZH=WBWBWB HZ=BWBWBW BW=HZHZHZ But it’s possible he was talking about a system in which W’s class was ZHZ’s class and H’s class was BWB’s class, and so WB’s class was ZH’s class and HZ’s class was BW’s class, and there was just a particular group of eight individuals who went around the circle twice before landing back with the same person, even though they went through each class twice. That was the same paper in which he pointed out that, if in every marriage the husband was older than the wife, or, in every marriage the wife was older than the husband, then any particular chain of individuals who were each the sibling of one and the spouse of another in the chain, would be more of a “helix” than a “circle”.

Of the real-life Classificatory Kinship and Prescriptive Marriage systems I have read reports of, the one with the most classes had 18 classes of men (and 18 classes of women). The men’s classes were organized into six patrilines of three generational classes each. Each patriline’s classes were in a cycle, so that every man was in the same class as his FFF and as his SSS. So for male speakers the term for “brother” also meant FFF and SSS. And the term for “father” also meant SS, and the term for “son” also meant FF. These patrilines were grouped into two moieties of three patrilines each. Nobody married anyone from their own moiety. Everyone married someone from the other moiety. The three generations in each patriline, would take turns, rotating through a cycle of three, marrying women from one of the three patrilines in the other moiety. It was a “direct sister exchange” system. If a man could marry a woman, then her brother could marry the man’s sister. It is my impression that, as near as I can tell, having on average 17 out of every 18 otherwise eligible members of the opposite sex, be unavailable because of being born into the wrong kinship class, is rather rare in real life. More common fractions might, if I am not mistaken, be 1 out of 2 unavailable, or 2 out of 3 unavailable, or 3 out of 4 unavailable. I have long wanted a system in which a [i]circulum connubium[/i] would contain five (classes of) men and five (classes of) women. The reason is that some anthropologists have proposed a parallel to the Sapir-Whorf hypothesis, that in WB=ZH systems, and by a slightly different mechanism* also WBW=ZHZ systems, people were pre-disposed to see the universe as organized into dichotomies. Folk-taxonomies have (so they hypothesised) each taxon divided into exactly two taxa of the next lower rank. *In WB=ZH systems people are grouped as Us and Them. In WBW=ZHZ systems other men (i.e. men who aren’t in my line) are either (wife-)givers, like my WB, or (wife-)takers like my ZH. I worried that if the sibling-and-spouse circle had only four classes of each sex (men could be like me, or like my WB, or like my ZH, or like my WBWB=ZHZH), if these anthropologists were right, the people might think the whole universe was organized into trichotomies; and I didn’t want that. I thought that, if they were all familiar that, for instance for a man, there’d be men of his own line, men of his WB’s line, men of his ZH’s line, and at least two other lines (namely his WBWB’s line and his ZHZH’s line); then, folk-taxonomies would always, in addition to the (two or) three most prominent classifications, always include a “miscellaneous/other” category, which could itself be further divided whenever anyone wanted to take the trouble to do so. Just recently I invented such a system. It has twenty classes of men and twenty classes of women. I thought if a real-life society could put up with being able to date only 1/18 of the single MOTS one met, it wouldn’t be too much of a stretch to let people court only 1/20 of the otherwise eligible members of the opposite sex.

=The System I’m Using in my Unnamed(-as-yet) Multi-Racial Fantasy ConWorld In this system are five patriclans. Everybody is born into one, and only one, of them —— namely the one of which their father is a member —— and stays in it their whole life. The population is, almost independently, also divided into five pairwise-mutually-exclusive and jointly-exhaustive matriclans. Everyone is born into one and only one matriclan —— their mother’s matriclan —— and stays in it their whole life. The members of each patriclan are divided into four generational groups, which are organized into a cycle. A male speaker’s word for “brother” also covers his FFFF (whom he probably never met) and his SSSS (whom he’ll probably never meet). His son will belong to the same class his FFF belonged to; his SS will belong to the same class his FF belonged to; and his SSS will belong to the same class his father belonged to. Likewise the members of each matriclan are divided into four generations, which also form a circle. From a female speaker’s perspective, her term for “sister” is also her term for MMMM and for DDDD. Her daughter she will call by the same term as her MMM. Her DD she will call by the same term as her MM. And her DDD she will call by the same term as her mother. Society is also divided into four generational groups, which are spouse-and-sibling circles. There are five classes of men, and five classes of women, in each such circle. Each class of men are “brothers” to one class of women in the circle, and “husbands” to one other class of women in the circle. Each class of women are “sisters” to one class of men in the circle, and “wives” to one other class of men in the circle. Each patriclan intersects each “spouse”-and-“sibling” circle in exactly two classes; a class of men, and the class of their “sisters”. Each matriclan also intersects each “spouse”-and-“sibling” circle in one class of women, and the class of their “brothers”. The men in each patriclan take turns by generations, rotating through the other patriclans, as the source of their bride. A man can’t marry anyone from his own patriclan. Nor can he marry anyone from his mother’s patriclan (such as his MBD), like his father did. Nor can he marry from his FM’s patriclan (such as his FMBSD), like his FF did. He can’t even marry someone from his FFM’s patriclan, like his FFF did. Instead he must marry a woman from his FFFM’s patriclan, like his FFFF did. At the same time, the women in each matriclan, rotate, generation-by-generation, through the other four matriclans, as the source of their groom. A woman can’t marry a man from her own matriclan, nor her father’s matriclan, nor her MF’s matriclan, nor her MMF’s matriclan. She must marry a man from her MMMF’s matriclan. If a patriclan and a matriclan have any common members, they are all and only two classes of people, one class of men and one class of women. The men are the women’s “brothers” and the women are the men’s “sisters”. Each patriclan has members in common with four of the matriclans; and each matriclan has members in common with four of the patriclans. For every man, his WBW’s matriclan has no members in common with his own patriclan; and for every woman, her HZH’s patriclan has no members in common with her own matriclan. This is a “third-cousin” marriage system, because a man’s wife must be classifiable as his FFFZDDD and a woman’s husband must be classifiable as her MMMBSSS. But a man’s wife is also classifiable as his FZHZD, and a woman’s husband is also classifiable as her MBWBS. And there are other blood- and affine- relationships between a man and his bride-to-be and vice versa; or at least they look different in our notation. For instance if a man can marry a woman, then his FFFF could have married her FFFFZ. So W=FFFMBSSSD. And his MMMMB could have married her MMMM, so H=MMMFZDDDS. So there’re two different ways to consider this system a “fourth-cousin” system.

!!= Pr[r]o[/r]scription Can Imply Pr[r]e[/r]scription If the Options are Few =The Prescriptive Marriage System and Classificatory Kinship System of Early Adpihi For my first conworld and conculture, Adpihi, my classificatory kinship system was as follows. There are 216 (kinship) classes (or (marriageability) sections, or “skins”); 108 of men, and 108 of women. There are three matriclans, and three patriclans, and three “ropes” or [i]geuns[/i]. Everybody is born into their mother’s matriclan and stays for life; and everybody is born into their father’s patriclan and stays for life. Everyone inherits “rope”-membership from their parent-of-the-opposite-sex. Women belong to their father’s “rope”, while their brothers belong to their mother’s “rope”. A person’s class or “skin” or section is determined by the following six data: Their matriclan; their patriclan; their “rope”; their father’s matriclan; their mother’s patriclan; and their sex. It’s also determined by their and their parents’ matriclans and patriclans and ropes. It turns out that if you know a person’s “rope” and their sex, you can deduce what their same-sex parent’s “rope” must be; and if you know a person’s “rope” and their same-sex parent’s “rope”, you can deduce which sex they must be. For any man, any other man of the same matriclan and patriclan and “rope”, whose father shared his matriclan with your own father and whose mother shared her patriclan with your own mother, must be your “brother”. For any woman, any other woman of the same matriclan and patriclan and “rope”, whose father shared his matriclan with your own father and whose mother shared her patriclan with your own mother, must be your “sister”. For a man and a woman, if they’re in the same matriclan and the same patriclan, and their fathers were in the same matriclan and their mothers were in the same patriclan, and he’s in her mother’s “rope” and she’s in his father’s “rope”, then they must be “brother” and “sister”. ____________________ =Adpihi’s Marriage Proscriptions It is proscribed to marry anyone from your own matriclan or patriclan or rope, or anyone from your father’s matriclan or your mother’s patriclan or your same-sex-parent’s “rope”. It is proscribed to acquire a spouse or a parent-in-law from your own matriclan or patriclan or “rope”. But there’re only three matriclans. So if W and H are from two different matriclans, and WF isn’t from either of them, and likewise HF isn’t from either of them, then WF and HF must be from the same matriclan. And, since there’re only three patriclans, and W and H are from two different ones, and neither WM nor HM can be from either of them, then both WM and HM must be in the only remaining patriclan. And there’s only three ropes; and W (and WF) aren’t in the rope H (and HM) are in; and WM can’t be in the same rope with WF nor with H; and HF can’t be in the same rope with either HM or W; so both WM and HF are in the only remaining rope. So, even though I described this system by proscriptions, it turned out that the only eligible brides for a man are classifiable as his MFMBDSD. .................... There are a few other homonymies in this system. For instance, MF=DS. And, MBWB=WBWF. .......... In this system only 1/108 (less than 1%%%%%%%%) of the otherwise-eligible MOTS one meets won’t be ruled unavailable because of coeponymy. That stricture can be loosened by having more than three of one or more kinds of lineal descent groups — matrilines or patrilines or “ropes”. For instance, if there are 15 of each, slightly over half of the otherwise-eligible MOTS will not be too co-eponymous to marry. !!= But that will no longer be a [u]Prescriptive[/u] marriage system!

I'm looking forward to the group theory!

!!= So where’s the Group Theory? =Where’s the group? In a society with this kind of coupled system, a Classificatory Kinship System coupled with a Prescriptive Marriage System, the people of each sex are divided into (Kinship) classes, aka (marriageability) sections, aka “skins”. The Kinship terms form a (mathematical) group, which acts transitively on the classes in each sex. This group is generated by two of its elements. For instance; any kinship relation between men can be generated by composing some sequence of “is the father of” (and its inverse “is the son of”), and “is the wife’s brother of” (and its inverse “is the sister’s husband of”). For me, it was easier to comprehend this all by realizing that there are four one-to-one kinship relations between classes of men and classes of women. For each class of men, there is one and only one class of women who are their “mothers” (M); one and only one class of women who are their “sisters” (Z); one and only one class of women who are their “wives” (W); and one and only one class of women who are their “daughters” (D). These relations are all invertible. The inverse of M is S. A class of women are the “mothers” of one and only one class of men, namely their “sons” (S). The inverse of Z is B. A class of women are the “sisters” of one and only one class of men, namely their “brothers” (B). The inverse of W is H, and the inverse of D is F. We can compose these and come up with ten simple one-to-one functions relating classes of men to each other. MF “mothers’ fathers” MB “mothers’ brothers” MH “mothers’ husbands”, identical to ZF “sisters’ fathers” ZH “sisters’ husbands” ZS “sisters’ sons” WF “wives’ fathers” WB “wives’ brother’s” WS “wives’ sons”, identical to DB “daughters’ brothers” DH “daughters’ husbands” DS “daughters’ sons” MF and DS are inverses; MB and ZS are inverses; MH=ZF and WS=DB are inverses; ZH and WB are inverses; WF and DH are inverses. Something similar can be seen about creating a group of one-to-one relationships between classes of women. FM “fathers’ mothers” is inverse to SD “sons’ daughters” FZ “fathers’ sisters” is inverse to BD “brothers’ daughters” FW=BM “fathers’ wives” aka “brothers’ mothers” is inverse to HD=SZ “husbands’ daughters” aka “sons’ sisters” BW “brothers’ wives” is inverse to HZ “husbands’ sisters” HM “husbands’ mothers” is inverse to SW “sons’ wives” Since there must be only finitely many classes, the groups are also finite. And they are 2-generated. The permutations on the classes of men can all be generated by; Either of MH=ZF or WS=DB, together with either of ZH or WB; or either of MB or ZS, together with either of ZH or WB; or any of several other pairs*. For instance, MH and MB, or MH and WF. The permutations on the classes of women can be generated by either of BW or HZ, together with one of FW=BM or HD=SZ, or FZ or BD. *Pick any two of the five pairs of inverses {MF, DS}, {MB, ZS}, {MH=ZF, WS=DB}, {ZH, WB}, {WF, DH} Then pick one operator/kinterm from each of those two pairs of inverses. The resulting pair of operators probably is sufficient to generate the whole group. There are exceptions. If you choose one of {MF, DS} with one of {ZH, WB}, that probably won’t generate the group. I believe I can remember that I’ve proven this. But I’d have to take the effort to re-prove it, since I didn’t memorize the proof and have lost it. I hope it’s OK for me to “leave that as an exercise for the reader”? ————— =How I made the system in my multiracial fantasy world In order to generate a system with five patrilines each containing four generations, and five matrilines each containing four generations, and four “circula connubial” (“sibling”-and-“spouse” circles) each consisting of five classes of each sex, I represented the WB operator on the classes of men by the five-cycle (12345). I wanted each of the patrilines of men to be a cycle of generations each of whom married women who were (the sisters of men in, and therefore themselves) members of the other four patrilines. So I decided that the MH=ZF operator on the classes of men should be represented by one of the following four-cycles; (2345) (2354) (2435) (2453) (2534) (2543) But I also wanted the matrilines to consist of cycles of four generations, which took turns marrying women from the other four matrilines. So I needed MB = MH WB to be a four-cycle, and needed WF = WB ZF to be a four-cycle. (For men, if a matriline has four generational classes of men, a matriline of men consists of; Him and his “brothers”; His “mothers’ brothers”; His “sisters’ sons”; and his MBMB who are also his ZSZS.) This works with only three of those six 4-cycles. If MH=ZF is (2354), then MH WB is (2354)(12345)=(1243) (MB) and WB ZF is (12345)(2354)=(1325) (WF). If MH=ZF is (2453), then MH WB is (2453)(12345)=(1254) (MB) and WB ZF is (12345)(2453)=(1435) (WF). [r]And if MH=ZF is (2435), then MH WB is (2435)(12345)=(1253) (MB) and WB ZF is (12345)(2435)=(1425) (WF). But the group this generates has 120 classes per sex, instead of 20; so I won’t use it. For a hint that this is true, check out that WSWB=(2534)(12345) is not 4-cycle. [/r] I picked (2354) at random. ———— =Other things that could have happened It isn’t always true that a kinship class that could be reached by ascending or descending to another generation in the same patriline (i.e. ZF or WS or ZFZF=WSWS), then moving left or right to a WB or ZH or WBWB or ZHZH, could also be reached by first moving left or right along the sibling-and-spouse circle, and then up or down within the patriline. In both of the cases I considered for this system, however, it is possible. Suppose the whole group could be generated by two elements, x and y. <x> is the cyclic subgroup of all powers of x; <y> is the cyclic subgroup of all powers of y. <x><y> is the set of all elements that can be represented as a power of x followed by a power of y. <y><x> is everything that’s a power of y followed by a power of x. If <x><y> = <y><x>, that’s the whole group. I lucked out in this case, because that happens here. So there are only 5*4=20 operators in the group. If that had not been the case, it might have been true that <x><y><x> = <y><x><y> ; in which case that would have been the whole group. But in that case there would have been more than 20 operators in the group acting on the classes of men (or the isomorphic group acting on the classes of women). [r]Actually the union of <x><y><x> with <y><x><y> doesn’t cover the whole subgroup generated by ZF=MH=(2435) and WB=(12345). One more reason not to use it![/r] —————— Is there more group theory you want to know here? Do you have any trouble with it so far? @EN:

=The system in earliest Adpihi @EN: It turns out that in my very first I-made-it-up CKS/PMS, (the one with three patriclans, three matriclans, and three “ropes”), however you pick the two generators x and y, you don’t get <x><y> = <y><x>. No non-identity element has an order other than 6 or 3 or 2. But to act transitively on the classes of men, the group needs 108 members. If x and y both have order 6, you might get 216 expressions out of each of <x><y><x> and <y><x><y>. So each of them could and probably does redundantly cover the whole group. OTOH if x has order 6 but y has order 3, <x><y><x> contains 108=6*3*6 expressions; but it couldn’t cover the whole group, because if one chooses an expression with the identity in it for one or more of the factors, one might come up with more than one way of re-expressing a member of <x> or <y> or maybe <x><y> or <y><x>. And <y><x><y> gives only 54=3*6*3 expressions, half enough — less than half once the redundancies are corrected for. I can give you details if you ask for them.

BTW I just realized my multiracial fantasy world has three humanoid species that will have one of these five-sides ring systems so since I mentioned two possibilities earlier in this thread, I could have each system followed by at least one race. So I’ve decided to do that. I have a feeling the MH=ZF=(2354) and the MH=ZF=(2453) systems might be isomorphic, essentially identical except for re-labeling. But in case they’re different, I’ll have them both anyway. [r]The MH=ZF=(2435) system is different; it has MFM=DSD, while they have MF=DS. As a result it has at least 60 classes (in fact it has 120) per sex. So I won’t use it. [/r]

To be honest I've just been skimming the relationship stuff because it's pretty notation-dense and not really what I'm here for. I have a math background and so your mention of group theory interested me. I think it's neat that you were able to incorporate it into your world-building.

Did you understand my notes about permutation groups and cycles and so on? @EN: I represented, or labeled, the patriclans by the numeral digits from 1 to 5. When I said I’d represent WB by the cycle (12345), I meant: At a certain generation, The guy from patriclan 1 would marry a woman from patriclan 2; Her brother (from patriclan 2) would marry a woman from patriclan 3; Her brother (from patriclan 3) would marry a woman from patriclan 4; Her brother would marry a woman from patriclan 5; And finally, her brother (from patriclan 5) would marry a woman from patriclan 1. When I said I’d represent ZF by (2354), I meant: In patriclan 1, One guy would marry a woman from patriclan 2; His father would’ve married a woman from patriclan 3; His father would’ve married a woman from patriclan 5; His father would’ve married a woman from patriclan 4; And finally his father would’ve married a woman from patriclan 2. Was that clear? Did you already know how to notate permutations by decomposing them into disjoint cycles? Did you already know how to calculate the product of one permutation followed by another? Quite probably you did; most folk who spend more than a little time on group theory eventually run into “concrete” (ie not quite as abstract) groups such as permutation groups. ....... Did you understand what I meant when I said the group must act transitively on the classes? In this context, that’s the same as saying that, in these societies, everybody has to be related to everybody else.

@EN: Or anyone else. The previous discussion shows that a system of classificatory kinship* and prescriptive marriage has some numbers attached to it that help classify it. I assert without proof that they’re insufficient, in general, to completely specify it. *where FB=F and MZ=M First is the total number of different (kinship) classes or (marriageability) sections or “skins” of each sex in the system. (20 each sex, in my example.) Next are the orders or periods of the ten simple operations on each sex. i. We’ve already talked about the circulum connubium; how many times one could repeat the operations BW or its inverse HZ on the classes of women, or the operations WB or its inverse ZH on the classes of men, before coming back to the same class we started from. (In my example it was 5.) j. Then there’s how many classes of men in a patriline. How many times can one repeat the operations DB=WS (son);or its inverse MH=ZF (father) on the classes of men before returning to the same class? (In my example it was 4). The same number describes how many classes of women are in a patriline. How many times can one repeat the operations BD (a kind of niece) or its inverse FZ (a kind of aunt) on the classes of women before coming back to the same class? (Note a woman’s father’s sister, and her brother’s daughter, are in the same patriline she’s in.) The same number describes the length of a line of mothers-in- law and daughters-in-law. How many times can one consecutively repetitively re-apply the operations HM or its inverse SW to the classes of women before coming back to the starting class? k. There’s also how many classes of women in each matriline, or cycle of mothers-and-daughters. What is the order of the operation BM=FW, and its inverse HD=SZ? In my example, it’s 4. The classes of men in a matriline is the same number. We can repeat the operator MB (maternal uncle) a certain number of times on the set of classes of men; after a certain number (4 in my example) the orbit will always return to the originating class. We’d get the same number for the order of ZS (sister’s son a type of nephew), which is the inverse of MB. Yet another cycle of classes of men is the same size. The orbit of the father-in-law operator WF, and of its inverse the son-in-law operator DH, is the same length. l. Finally, there’s the length of the “rope” or geun, which among classes of men is the order of MF and its inverse DS, and among classes of women is the order of FM or its inverse SD. In one of my examples l = 2; in another l = 3. ——— In an incest-free system (such as all real-life prescriptive marriage systems) only l can be as low as 1. The others, i and j and k, all must be at least 2. ..... If l = 1, then every man is in the same class as his mother’s father MF and as his “daughter’s son” DS. Likewise every woman is a “sister” to her FM (paternal grandmother) and her SD (classificatory “son’s daughters”). So if a man H can marry a woman W, then his mother’s father HMF could have married her father’s mother WFM. That would mean that his mother HM and her father WF could have been full- sister-and-brother. In other words W could be H’s MBD ( mother’s brother’s daughter). Since kinship is classificatory and marriage is prescriptive, that means that H (and his “brothers”) can only marry women [u]classified[/u] as their “mother’s brother’s daughters”, a kind of cross-cousin —— or at least “cross-cousin”. .... If i = 2 then a man’s WBWB and his ZHZH are his “brothers”. A man’s WBW is his “sister”, and his ZHZ is his “wife”. Men have only one term for “brother-in-law” because WB and ZH are classed together. By the same token a woman’s BWBW and her HZHZ are her “sisters”; her BWB is her “husband” and her HZH is her “brother”. She has only one term for sister-in-law; her language doesn’t distinguish between BW and HZ. If i = 3 then a man’s WBWBWB is his “brother”, his WBWBW is his “sister”, his WBWB is his “sister’s husband” (ZH); and his ZHZHZH is his “brother”, his ZHZHZ is his “wife”, his ZHZH is his “wife’s brother” (WB); and his WBW and his ZHZ are each others’ “sisters” From the women’s point of view, her HZHZHZ is her “sister”, her HZHZH is her “brother”, her HZHZ is her “brother’s wife” (BW), her BWBWBW is her “sister”, her BWBWB is her “husband”, her BWBW is her “husband’s sister” (HZ), and her HZH is the “brother” of her BWB. If i >= 4 then the system is of a kind which seems not to have been much published about. .... If j = 2 then every man is in the same class as (is the “brother” of) his FF (father’s father) and as his SS (“sons’ sons”). And every woman is classified with her FFZ (a kind of grand aunt) and her BSD (brother’s son’s daughter, a kind of grandniece). This somewhat limits the possible complexity of the system. If j = 3 then men are classified with their FFF and their SSS, while women are classed with their FFFZ and their BSSD. If j >= 4 maybe the system might be a little more complex and/or interesting. In my example j=4. ..... If k = 2 then men are classified with their MMB “granduncle” and with their ZDS “grandnephew” (“sister’s daughter’s son”). Women are classified with their MM (maternal grandmother) and their DD (“daughters’ daughters”). This might constrain how complicated the whole system can get. If k = 3 then men are the “brothers” of their MMMB and of their ZDDS, a kind of great-granduncle and great-grandnephew. Women are the “sisters” of their MMM and their DDD. If k>=4 the system might be more complex. In my multiracial fantasy examples k=4. ..... If we know both j and k we can deduce something about what relationships are required between a groom and a bride. For instance, suppose j=3 and k=3. Then if H can marry W, his great grandfather HFFF could marry her greatgrandmother WMMM. But that means his grandfather HFF could be the full brother of her grandmother WMM. So we get W=FFZDD. A kind of second-cousin marriage system. But there’s more. His great-grand-uncle HMMMB is classed as his “brother”, and her great-grand-aunt WFFFZ is classed as her “sister”. So if H can marry W, then HMMMB could have been WFFFZ’s husband. So W could be H’s MMMBWBSSD. That’s not a blood relation; it’s a kind of third-cousin- in-law or something. ..... In my Early Adpihi there are 108 classes of each sex. The order i of the circulum connubium is 6; so is the order j = 6 of the patriline-cycle, and the order k=6 of the matriline-cycle. But the order of the “rope” is l=2. Knowing every man is classed with his MFMF and every woman is classed with her FMFM lets us deduce a kind of required third-cousin relationship between groom and bride. If H can marry W, then H’s MFMF could marry W’s FMFM, so H’s MFM could be the sister of W’s FMF. So we have that W must be classifiable as H’s MFMBDSD. ———————————————————— @EN: if you’re interested in group theory, you might be interested in the word problem. What’s a minimal set of relations (or “kinterm homonymies”) to completely specify a classificatory kinship system that would support a prescriptive marriage system? If i=j=k=l=2 we get the Klein 4-group. (if i recall correctly). If i=j=k=l=3 we get the 27-member [s]Bernstein[/s][b]Burnside[/b] 2-generated group with exponent 3. (Or at least we do if we require every non-identity member to have order 3). If i=j=k=l=4 and we require that every member have order at most 4 and make no other restrictions, we could get the 4096-member [s]Bernstein[/s][b]Burnside[/b] 2-generated group with bound 4. I don’t think that’s realistic; I’d need a very large society with some crackerjack yentas to make it work. More likely the group we’d actually end up using would be an image-group or quotient-group of that group; or a subgroup of it, or an image of a subgroup, or a subgroup of an image. For the i =5 system I originally tried to see if I could use the [s]Bernstein[/s][b]Burnside[/b] 2-generated 5-bounded group; its only governing relations are that every element (except the identity) has order 5. But that group has 5^34 members! Asimov’s Galactic Empire didn’t have that many inhabitants! [c][i][size 0.84][edit](Asimov’s Empire had between 1E15 and 1E18 inhabitants. With 40 billion habitable planets and 10 billion inhabitants per planet, the galaxy’s human population couldn’t exceed 4E20 inhabitants.. 5^34 is about 5E23, 1455 times that.)[/edit][/size][/i][/c] So I was pleasantly surprised to make my i =5 j=k=4 l=2 systems! And I did it using permutation group ideas and notations.

@EN: prove to your own satisfaction that the group is commutative if and only if W=MBD. I’ve done it before, and I could probably do it again if you wanted me to. But I think it would be fun for you.

[@]elemtilas,Xhin:[/@] > If i=j=k=l=4 and we require that every member have order at most 4 and make no other restrictions, we could get the 4096-member [s]Bernstein[/s][b]Burnside[/b] 2-generated group with bound 4. I don’t think that’s realistic; I’d need a very large society with some crackerjack yentas to make it work. Elemtilas could probably come up with a great story involving a yenta trying to arrange a decent match for some young person in a society where only 1 out of each 4096 otherwise-eligible single young adults of the opposite sex weren’t too co-eponymous to marry. Xhin could probably make a good game out of such a situation. Other people in this group could do things like that too, I think. But a 4096-class-per-sex PMS/CKS system (with FB=F MZ=M classificatory kinship and prescriptive marriage) would not be a good choice, in my opinion, for a conculture, unless it were going to be [b]the central feature[/b] of the story or game.

I'm familiar with representing and manipulating permutations. I don't have a lot of spare time to try my hand at your proofs/exercises, but thanks for sharing your work!

@chiarizio: You can only tag one person with a tag like that. To do more, use the [``@][/``@] tag: http://gtx0.com/read/the-tag-aka-way-better-post-tagging

> You can only tag one person with a tag like that. > To do more, use the [``@][/``@] tag: > http://gtx0.com/read/the-tag-aka-way-better-post-tagging Oops! If I edit that post, will that work? I’m about to try it. Forgive me if I make a hash of it again!

> @EN: prove to your own satisfaction that the group is commutative if and only if W=MBD. I’ve done it before, and I could probably do it again if you wanted me to. But I think it would be fun for you. > I'm familiar with representing and manipulating permutations. I don't have a lot of spare time to try my hand at your proofs/exercises, but thanks for sharing your work! If you view the group as a permutation group on the classes of men, and you choose any pair of generators and calculate their commutator, you’ll get MBDH or DHMB or WFZS or ZSWF, or something conjugate to one of those. If you view it as a group of permutations on the classes of women, the commutator of any generating pair is FZSW or SWFZ or HMBD or BDHM, or something conjugate to one of those. Setting any of those to the identity permutation yields W=MBD or WF=MB or H=FZS or HM=FZ. That takes care of “only if”. If W=MBD then MBDH is the identity permutation on the set of classes of men. Since, if it’s the identity, then DHMB and WFZS and ZSWF and everything conjugate to one of them is also the identity; any generating pair do commute with each other, so the whole group is commutative. That takes care of the “if” part. Why W=MBD in most* such systems worldwide in real life, surely has nothing to do consciously with commutativity. There must be some other reason it’s most* popular. *(“most” in the sense of “more often than any other single system”; I don’t mean to imply “more often than all other systems put together”.)