How did this one character take on two different pronunciations and meanings? Was it a merger of two different characters?
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In Old Chinese, it is generally thought that some words followed regular morphological alternations (which are preserved in a few places in MSM, but "frozen", i.e., no longer productive). For instance:
Phonetic reconstructions for these alternations generally follow a regular pattern (e.g., voiced/voiceless, 去声/other tone), although specific reconstructions differ.
会 is a somewhat odd example, in that the hui4 reading has displaced the kuai4 reading in almost all places except proper nouns and 会计 = accounting. Branner gives the example from Confucius:
The customary reading of 会 in this passage is hui4, but etymologically, it really should be kuai4.
My references are taken from Branner's excellent essay, "On Early Chinese Morphology and its Intellectual History"
The Middle Chinese readings corresponding to "hui4" and "kuai4" are "hwajH" and "kwajH" in Baxter's notation; the initials being 匣 and 見. Based on this, one would expect the MSM pronunciation of the latter to be "gui4" rather than "kuai4". The common Japanese On reading of かい "kai" (attested earlier as くわい "kwai") is actually a kan-on reading. According to wikipedia, the k- initial in a kan-on reading could correspond to either a 匣 or a 見 MC initial. In any event, the "kai" reading is used in both the 'meeting' and 'accounting' meanings, regardless of which MC reading it corresponds to. Some speculate (e.g., Victor Mair, p31) that the MSM "kuai4" may have been influenced by the Japanese "kai" (i.e., that the etymologically expected "gui4" was displaced by "kuai4" when 会计 was re-borrowed from Japanese to mean "accounting"). It is not the case, however, that the hui4/kuai4 distinction is an artifact of Japanese borrowing--a meaning of "settling accounts" is attested dating back well before the Japanese neologism for accounting (see halfway down random blogpost quoting from dictionary).
It's been said that the word
There are many reasons for why one character have two or more different pronunciations, see here if you want to know more reasons.
In this case, it's mainly because people want to use a different pronunciation to show
There are only two meanings of kuài:
All other meanings should be pronounced as hùi.
For more information about the Heteronym in Chinese, you may read
會 has an entry in 康熙字典 (Kangxi Dictionary):
The bold text are 5 definitions of 會 that is pronounced as kuài (or ㄎㄨㄞˋ).
According to the 1st of the 5 definitions, 會 with this pronunciation is used equivalently with 䯤 (something like a hairpin), which is pronounced as kuài. This might be from the definition of 會 being the sewings in an ancient Chinese hat (弁) used to constraint hair. Of course, these 2 characters and their meanings could have had their influences on each other.
The 2nd and 3rd of the 5 definitions relate to the definition of accountant. I don't quite see its connection with the hairpin definition, but the usage of this word to represent accounting possibly comes from 會's meaning of gathering together, while its pronunciation succeeded the one for another meaning.
For the name of the mountain and a last name, they might had only a verbal pronunciation kuài beforehand and needed a character, and therefore 會 is used.
So in conclusion, the pronunciation of 會 as kuài might comes from its equivalance with 䯤 (because they look alike), and other meanings with this pronunciation exist only because 會 already has a pronunciation of such, and therefore, derived meanings of 會 takes the sound of kuài (such as accounting), as well as sound of kuài takes the character of 會 (such as 會稽山, the mountain).
(answer based on only Kangxi Dictionary and personal deduction)