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What was the de facto rule by which the imperial title was transmitted in various Chinese dynasties?
What were the rules of succession during the different dynasties[?]
The de facto rule of imperial Chinese succession, like the de jure rule, is that laid out by your source:¹
The first bit about China's five good emperors adopting one another (Huang-ti is the Yellow Emperor or Huangdi, Yü is Yu the Great) may reflect important oral lore—Woo argues it reflected a primitive form of elective monarchy by the early tribes (p. 628)—but it is entirely legendary and can be ignored. The Shang monarchs were certainly real, but we only know their details through much later legends; these point to inheritance by agnatic seniority, with half being succeeded by their brothers and the rest being succeeded variously by sons, nephews, and cousins (p. 631). The bit about the Ch'ing (=Qing) only applied to part of the Qing, and they were Manchu anyway. Kentaro's repeated discussions of the Mongol Empire are similarly aside the point. Han dynasties consistently practiced male primogeniture.
Now, having said that, there was a bit of wiggle room in its actual enforcement. When the Han dynasty got to a place where the Ping Emperor died childless, had no sons, had no brothers, had no uncles, and had no agnate 1st cousins, the chancellor Wang Mang did not just dig out a genealogical book and trust his own fate to the lucky 2nd cousin who happened to be formally next in line. He used their relative powerlessness at court and his control of it to make up an ad hoc rule that filial piety and the spirit of the inheritance laws meant that members of the same generation shouldn't succeed each other, so he could get a malleable young 2nd cousin once-removed instead. He eventually settled on a one-year-old with the excuse that Taoist fortune tellers had chosen him as the most fortunate of the lot. Obviously, he was just making things up as he went and getting away with as much as he could. (Having the dowager empress & her clique on his side against the arrivistes, he eventually became 'acting emperor' and then just overthrew the dynasty altogether on the basis of a mystic's divine vision of Liu Bang, the Han dynasty's founder.)
More often, the fight was about who counted as the eldest son. The Chinese emperors were notionally monogamous despite their harems and Han scholars tied themselves into knots trying to excuse the legend of Shun marrying two sisters. (See further these threads.) Unlike Roman and Christian Europe, the concubine's children weren't held to be meaningless bastards but, theoretically, the eldest son of the Chinese empress came before an older son of one of the concubines (p. 628). When an emperor married his empress while reigning, any sons she already had borne him as a concubine did not succeed to the throne if she bore new sons while empress: the eldest of those new sons succeeded to the throne in place of his older but lower-status brothers (p. 632). However, the emperor was (within poltical limits) able to divorce or demote the empress and promote a concubine, changing whose son would succeed him and whose in-laws would run the country and whose would be slowly tortured to death. Empresses were also able to adopt consorts' children and frequently did so when they proved unable to conceive. Emperors could force such adoptions, so that some Chinese scholars formulated the rule as "from the sons of the empress (嫡) choose the eldest while from sons of the concubines of the emperor (庶) choose the ablest" (p. 628). The Han's Wu Emperor was succeeded by his youngest son because the only son of his empress rose in revolt and he considered his other concubines' sons unlawful and unserious. Yuan Liaofan felt that, in the case of perfectly equal shu, the next emperor should be chosen by lot (p. 629), but I can't think of an example of this actually taking place. On occasion, emperors elevated some of their concubines above the others, giving them almost wifely status; in such cases, the heir would be the eldest son of the empress or, should she have no sons, the eldest of the sons of the first-tier concubines or, should none of them have sons, the ablest of the second-tier concubines (p. 629). The Yuan and the Ming further created two yet-more-elect concubines between the empress and the old first tier. The sons of the "2nd empress" followed the empress's son in precidence, chosen by seniority; the sons of the "3rd empress" followed hers, chosen by seniority; should all three empresses fail to produce a son, the heir was selected from among the sons of the old first-tier concubines, chosen by ability; should all of those fail to produce a son as well, the heir was chosen from among the second-tier concubine's sons by ability (p. 630). The Yongle Emperor followed such rules in his usurpation as he 'accidentally' burnt his nephew to death while trying to 'save' him from 'evil councilors', (probably) changing his mother in the records, and demoting his older brothers to the status of commoners to remove them from consideration. Sometimes an emperor simply said to hell with all of that and chose a younger son over his advisors' strenuous objections because he disliked the older brothers; the bad fate of Jin after its Xian Duke did that for the child of Li Ji was the classical counterexample of what a bad idea that was (p. 631). Rarely a younger son would say to hell with all that and kill his older brother; such an unfilial act should have earned the universal obloquy of the Chinese and made it impossible to rule but Li Shimin famously executed the crown prince Li Jiancheng and is still lionized as one of the greatest Tang emperors. (Official histories would blame the crown prince for having tried to poison his younger brother first.) The various permutations of back-palace scheming you're now imagining form the basis of ⅝ of Chinese historical dramas.
[I]n what fraction of successions did they really apply?
Well above 50%.
First, you seem confused about the idea of usurpation. They (somewhat definitionally) are outside the usual rule of inheritance and mostly don't pretend to be following any secondary one. The members of the royal clan who revolted against Wang Mang didn't rise up in order of seniority or defer to it; they simply killed one another and then, when one leader was left with enough support, started a new line which then proceeded normally. There was no rule any of them claimed besides having the more experienced leader or better army or stronger alliance. The founders of new dynasties like Wang Mang sometimes bothered to create bogus genealogies connecting them somehow to the Yellow Emperor; others didn't bother. Similarly, Cao Pi staged his usurpation as the Xian Emperor's formal request, which he ceremoniously declined three times before accepting. This expedient was used by usurpers whenever possible from then on, starting with Sima Yan tossing out Cao Huan to start the Jin.
Secondly, for an exact percentage of other de facto 'rules', you'd need to be very arbitrary. On each occasion where a younger son succeeded, was it because of his ability or proximity to court at the time of death or was it because of the older son's laziness or mental impairment or the younger son's mother's greater favor or political influence or his father's favor? Can some successions be counted for several categories or just their primary reason? When you have badly gossipy source materials, do you believe them? When two or more sources disagree, which do you go with? You'd also need to have some grad student working on a thesis, because you'd need to justify each arbitrary category you use, explain its limits, and then go through hundreds of transfers of power putting each into a pigeon hole that each of your readers will have some problem with.
¹ "The Rule of Succession to the Throne in China", article 3 of The Chinese Social and Political Science Review, Vol. IX, No. 4 (Oct. 1925), pp. 626–634. Woo Tshung-zuh (胡春澤) is presumably an old Shanghainese form of Hu Chunze (胡春泽).
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As I mentioned in the comments above, the actual succession within each dynasty was very complicated in China, to the point that it might seem as if there were no rule even though there was a definite rule that I describe I will describe under the bar.
For example, in Yuan dynasty's case, very broadly saying, each emperor was elected according to a decision of the Kurultai:
All Great Khans of the Mongol Empire, for example Genghis Khan and Ögedei Khan, were formally elected in a Kurultai; khans of subordinate Mongol states, such as the Golden Horde, were elected by a similar regional Kurultai.
But Genghis Khan himself broke this general rule
Genghis Khan died on 18 August 1227, by which time the Mongol Empire ruled from the Pacific Ocean to the Caspian Sea, an empire twice the size of the Roman Empire or the Muslim Caliphate at their height. Genghis named his third son, the charismatic Ögedei, as his heir.
So both the normal Mongol system and your primogeniture were already broken here (although this was before the formalization of the Yuan in China) and further disputes continued afterwards.
Another example was during the Song dynasty ( 960-1279 ), whose tenth emperor Gaozong
was the ninth son of Emperor Huizong and a younger half-brother of Emperor Qinzong.
Here primogeniture was broken again.
There was a "general rule" of inheritance ( or seizure of power ) or the "requirements" every emperor needed, whatever. According to Wikipedia,
The title of emperor was hereditary, traditionally passed on from father to son in each dynasty. There are also instances where the throne is assumed by a younger brother, should the deceased Emperor have no male offspring. By convention in most dynasties, the eldest son born to the Empress (嫡長子/嫡长子) succeeded to the throne. In some cases when the empress did not bear any children, the emperor would have a child with another of his many wives (all children of the emperor were said also to be the children of the empress, regardless of birth mother). In some dynasties the succession of the empress's eldest son was disputed, and because many emperors had large numbers of progeny, there were wars of succession between rival sons. In an attempt to resolve after-death disputes, the emperor, while still living, often designated a Crown Prince (太子). Even such a clear designation, however, was often thwarted by jealousy and distrust, whether it was the crown prince plotting against the emperor, or brothers plotting against each other. Some emperors, like the Yongzheng Emperor, after abolishing the position of Crown Prince, placed the succession papers in a sealed box, only to be opened and announced after his death.
Unlike, for example, the Japanese monarchy, Chinese political theory allowed for a change in the ruling house. This was based on the concept of the "Mandate of Heaven". The theory behind this was that the Chinese emperor acted as the "Son of Heaven" and held a mandate to rule over everyone else in the world; but only as long as he served the people well. If the quality of rule became questionable because of repeated natural disasters such as flood or famine, or for other reasons, then rebellion was justified. This important concept legitimized the dynastic cycle or the change of dynasties.
This principle made it possible even for peasants to found new dynasties, as happened with the Han and Ming dynasties, and for the establishment of conquest dynasties such as the Mongol-led Yuan dynasty and Manchu-led Qing dynasty. It was moral integrity and benevolent leadership that determined the holder of the "Mandate of Heaven".
There has been only one lawful female reigning Emperor in China, Empress Wu Zetian, who briefly replaced the Tang dynasty with her own Zhou dynasty. Many women, however, did become de facto leaders, usually as empress dowager. Prominent examples include Empress Dowager Lü of the Han dynasty and Empress Liu (Zhenzong) of the Song dynasty and Empress Dowager Cixi of the Qing dynasty.
So it looks as if whoever, whether children or not, might succeed is the under the "Mandate of Heaven" ( including self declaration ) and has the "chance" to be the emperor.
Autocorrect made a grave mistake in the title and someone kindly came along and edited it, as was necessary. But in the process a vital point of the question was lost, namely that I am interested in the rules and what happened de facto and how much of a discrepancy there is
What do you mean by "de facto"? For instance, Liu Bei, who was the founder of Shu Han, was, according to Wiki,
According to the 3rd-century historical text Records of the Three Kingdoms, Liu Bei was born in Zhuo County, Zhuo Commandery, which is in present-day Zhuozhou, Baoding, Hebei. He was a descendant of Liu Zhen, a son of Liu Sheng, who was the ninth son of Emperor Jing and the first King of Zhongshan in Han dynasty. However, Pei Songzhi's 5th-century commentary, based on the Dianlue (典略), said that Liu Bei was a descendant of the Marquis of Linyi (臨邑侯). As the title "Marquis of Linyi" was held by Liu Fu (劉復; grandson of Liu Yan) and later by Liu Fu's son Liu Taotu (劉騊駼), who were also descendants of Emperor Jing, it was possible that Liu Bei descended from this line rather than Liu Zhen's line. Liu Bei's grandfather Liu Xiong (劉雄) and father Liu Hong (劉弘) both served as clerks in the local commandery office.
Virtually nothing is revealed even by the official history of the Three Kingdoms. Another says, Liu Bei was nobody but one of bunches of thugs, and it was a coincidence that his family name was the same as that of the Han's dynasty.
So the "de facto rule" is, like Zhang Jue, who was nobody but a peasant but who led a broad rebellious campaign in the late Han dynasty, whoever had the enough power to go viral against the then dynasty, had the right to become the "founder" or the "successor" of China.
According to the Wiki about Zhang,
Giving himself the title of "Great Teacher" (大賢良師), Zhang Jue led the Yellow Turban Rebellion with his younger brothers Zhang Bao (張寶) and Zhang Liang (張梁) in a campaign called the "Way of Heaven" or "Way of Peace". He and his brothers gave themselves titles: Zhang Bao was the "General of Land" (地公將軍), Zhang Liang was the "General of the People" (人公將軍); and Zhang Jue was the "General of Heaven" (天公將軍).6 The Yellow Turbans claimed to be Taoists, and rebelled against the Han dynasty in response to burdensome taxes, rampant corruption, and famine and flooding, which were seen as indications that the Han emperor had lost the mandate of heaven.
So whether it is because of the famine or the corruption, the emperor himself takes the risk of losing the "Mandate of Heaven", which later becomes the potential cause of the rebellion and the change of the dynasty. Simple enough, eh?