Wikipedia, as well as various sources, have given false information about the Romanization Wade-Giles. These sources have wrote that the Hanyu Pinyin -e (as in Tangent /ə/) is written as “o” or “ê” (sometimes even given as “e”), but “o” after velars, while the pinyin -uo is written as “o”, but “uo” after velars and in “shuo”. This is false, and here’s why.
Now it isn’t the first time they have made mistakes. Wikipedia’s article on Latinxua Sin Wenz previously wrote that the pinyin j- q- x- are written as g- k- x-, though the name of the writing has “Sin” for pinyin “xin”, so “there appears to be some interchangeability”. Currently, the article has been somewhat fixed to specify that in deciding whether to use g- k- x- or z- c- s-, one should refer to historical pronunciation. Or, you could actually refer to Chinese varieties which still keep the round-sharp distinction: Hakka, Yue and Min: 曉 is xiao while 小 is siao.
Anyway, back to Wade-Giles.
In modern Mandarin, there is no phonemic distinction (other than for interjections) between -uo and -o. So the medial -u- is inserted even if there historically was never one. The guides claim that “o” is used after non-velars for /ə/, but “uo” after velars “and in shuo”. They have actually overlooked (as the difference of ge and ko is more noticeable when reading Wade-Giles) what is most probably the pronunciation of Mandarin at the time Wade-Giles was developed. Given below are the TCC and my theoretical Wade-Giles Romanization of some characters (without tones):
哥 (ge) - ko - kâ
刻 (ke) - k‘ê - khêk
多 (duo) - to - tâ
過 (guo) - kuo - kwâ
Wade-Giles writes 哥 as “ko”, which is just as the online guides say, and this makes sense too. It is â in TCC, which corresponds to the modern o in Mandarin, except in compounds (and at a later stage, after velars): like for instance, 魔 is “mwâ” in TCC, and “mo” in Mandarin. This all makes sense.
However, it does not make sense that “刻” is written as “k‘o” (note that a forward quote should be used instead of <’>). It did not come from a historical â. What’s more, 哥 is pronounced “ko” (pronunciation given in Eitel) in Cantonese, so “ko” in Mandarin would make sense. However, “刻” is pronounced “hák” in Cantonese. So “k‘o” does not make sense. From 校改國音字典, the pronunciation of “哥” is ㄍㄛ /kɔ/ while that of “刻” is ㄎㄜ /kʰə/ (again, phonemic representation given in Tangent’s analysis, phonetically [kʰɯ̯ʌ~kʰɤ]) in 老國音, which supports my statement. Therefore, “o” is used when it was pronounced /ɔ/, while “ê” is used when it was pronounced /ə/ (can be checked with ease from say the Cantonese pronunciation or from the Kwangwun/廣韻). The statement that “pinyin -e is written as <o> after velars” is false.
Secondly, the guides wrote that “pinyin -uo is written as <o> except after velars, where it is written as <uo>, and in shuo”. This is again, not true, and Wade-Giles reflected the pronunciation at the time. Again, here are four characters for comparison. On the left is the Wade-Giles Romanization and on the right is pinyin.
多: to - duo (TCC tâ)
羅: lo - luo (lâ)
過: kuo - guo (kwâ)
哥: ko - ge (kâ)
說 (as in 說話): shuo - shuo (shiuet)
As you can see, characters like 多 and 羅 did not historically carry the medial -u-, thus Wade-Giles has not written them. (多 is given as ㄉㄛ /tɔ/ in 老國音) By modern Mandarin, phonemic distinction between -o and -uo has been lost, and as previously stated, the medial -u- has been added in front of -o (except after velars, where the -o turns into Tangent /ə/).
As can be seen from the TCC, it just happens so that “過”, “戈” and similar characters were pronounced with the medial (it is “kwo” in Cantonese). Therefore Wade-Giles has written the “u”. It is not an exception for velar initials.
As for 說, as in 說話, its pronunciation in 老國音 is ㄕㄨㄛ [ʂu̯ɔ]. Palatalization is lost after retroflex sibilants, so the /i/ is lost, and following the loss of final stops, the “e” becomes an “o” sound as [u̯ɛ] didn’t exist. Similar cases exist in Hakka, regarding the change of TCC -iue- to -o-. For example: 船 is son in Hakka. In addition, Chao writes “說” as “shot” in General Chinese. But anyway, Wade-Giles did not single out the syllable “shuo”. The character 說 was pronounced “shuo”, and it was thus written this way, and characters that were pronounced [ʂu̯ɔ] (Tangent /ʂwɔ/) and [ʂɔ] are written as <shuo> and <sho> in Wade-Giles respectively, with respect to the pronunciations at the time.
In conclusion, the apparently inconsistencies in Wade-Giles with -o, -uo and -ê are not actually inconsistencies, but how the characters were likely pronounced at the time. As for writing /ɛ/ as “e” and “eh”, it is written as “eh” only syllable-finally, as a word-final <e> gets different pronunciations in different languages (silent in French, for instance). If the ending sound is /ə/ though, it would be written as <ê>, thus the circumflex indicates how it should be pronounced.