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As for Medieval, this is simpler: medieval is just the name of a "middle period" (medius), between ancient and modern society. Many authors consider that in China, it starts after the Han and ends either with the Tang or with the Song (I'd prefer the Tang). I think it makes sense.
It isn't quite as simple actually. In the traditional Western European historiography established in the Englightenment period, 'ancient history' ends with the fall of the Western Roman Empire, and 'modern history' (literally 'recent history' or 'newer history') begins with the Italian Renaissance or the Protestant Reformation. The period in between (roughly 500-1500) is called the Middle Ages or the Medieval period. For a brief summary of the Enlightenment discourse of modernity and its origins, see
http://en.wikipedia.org/wiki/Modern_TimesI should add here that the neat three-period framework for Western European history has been broken up since the 1980s by the increasingly popular field called Late Antiquity, which concerns history from around 300 to around 700 - roughly from the Tetrarchy of Diocletian to the Islamic conquests of the first caliphs.
If one took the end of the Eastern/Later Han Empire in the civil wars of 190-220 as the end of an 'ancient period', then one would also assume that the subsequent 'medieval period' must end when people in China begin to think that they are in a new era or age, significantly different from and superior to the preceding one. Such a way of thinking could perhaps be found in some sources from the Tang or Song periods, but then one would find it also in some sources for the Ming and Qing periods. Each empire tended to think it was closer to reviving values from the fabled golden age of remote antiquity than the previous empire had been. So how many 'new' and 'modern' ages are there?
The idea that the Song empire represented China's attainment of 'modernity' was first proposed by the Japanese sinologist Naito Konan (1866-1934), whose original name was Naito Torajiro. This 'Naito hypothesis' countered the then-prevalent notion that Chinese modernity began when extensive contact with Europeans (in the form of the Portuguese Empire or, at the latest, the British Empire) was established, by arguing that the society of the Song period was fundamentally different and more 'modern' than that of the Tang - most importantly, the old aristocratic clans were replaced by a new elite produced through the examination system, while the emperor gained more autocratic power and could no longer be deposed by the elite. Naito also argued that from the Song period on, commoners had more rights and more access to political office, the economy was more commercialized and monetized, popular culture became more prominent, and politics diminished in importance compared to literary arts, scholarship, and antiquarian collecting.
Naito's hypothesis had its most marked influence on the historian Miyazaki Ichisada, who openly categorized the period from the end of Eastern Han to the end of Tang as China's 'medieval' period. The idea of the 'Tang-Song transition' has been an influential one outside Japan as well, among 'Western' sinologists who received their training in the late twentieth century. Even today, European and American historians working on Chinese history routinely refer to the Age of Fragmentation as 'early medieval China', while David Graff's book on Chinese warfare from 300-900 was entitled 'Medieval Chinese Warfare'. And yet no one refers to Song, Yuan, or Ming China as 'modern China' - instead, that term is reserved for Chinese history in the past 200 years or so. 'Early modern China' was for a time used to refer to late Ming and early to mid-Qing history (around 1500-1800), on the assumption that this corresponded to the early modern period in Europe, but since the 1990s it has been dropped in favour of the less Eurocentric 'late imperial China'. Meanwhile, 'ancient China' can still be seen referring to any period of Chinese history up to Ming, and the more specific 'early China' is being used to refer to the period up to the end of Eastern Han.
So we have the serious disjuncture between conceptions of 'medieval China' and 'late imperial China' or 'modern China'. What are we to label the nearly six hundred years between 960 and 1500? Some years ago, the term 'Song-Yuan-Ming transition' was proposed and made the title of a book collecting papers presented on a conference. But if Chinese history is going to be seen as a continuous series of transitions (Han-Tang, then Tang-Song, then Song-Yuan-Ming, then Ming-Qing, then Qing-republic), then what is the value of the term 'transition' historiographically?
If it were up to me, I'd just decree that the terms 'medieval' and 'modern' be dropped from the study of Chinese history altogether. Why the need to split time into three periods, when we could more easily use the names of empires for periodization, or coin a term like Age of Fragmentation for periods when there are several competing empires?
One last point: Historiography in China itself does not recognize the concept of a 'medieval' period, since Marxist doctrine has held that there was a single, essentially unchanging 'feudal' period from the Eastern Zhou up to the Opium War (there is a theory that 'sprouts of capitalism' were appearing in the late Ming and then the mid-Qing, but were killed off by the Ming government in the first case and the European imperialists in the second case). However, for the sake of distinction, the Age of Fragmentation and Sui-Tang empires are often referred to as the 'middle-ancient' 中古 period, in relation to the 'upper-ancient' 上古 period (up to the end of the Eastern Han Empire) and the 'recent-ancient' 近古 period from Song up to the Opium War. As for the 'modern period' in Marxist Chinese historiography, it is divided between the 'recent-period' 近代 from the Opium War to the May Fourth Movement of 1919, and the 'present-period' 现代 from 1919 to today.