When searching the Kodansha 大辞泉 by 音読み, for many 音読み entries there are small kanji symbols in a circle such as 漢 or 呉 or 唐. (Unfortunately this dialog box doesn't allow me to put the circle around the kanji.) I'm assuming that these indicate the Chinese dialect or historical era from which the various 音読み came (such as Han, Wu or Tang); is this correct? Also, how would anyone know after all these centuries? Example (again, I apologize for not being able to enclose the characters in circles; I've used parentheses instead): ぶん「分」 [音]：ブン(呉) フン(漢) ブ(慣)
Those Kanji in circles mean that the kanji was incorporated into Japanese from that Chinese dynasty. So yes, 唐 means Tang Dynasty, 漢 means Han Dynasty, and 呉 means Wu Dynasty (4th - 5th century warring states period). The 音読み in Japanese therefore reflects the way the kanji in question were being pronounced during that time in China (Chinese has undergone at least one major vowel shift which is why this is relevant).
慣 is short for 慣用音, which is the generally accepted or popular 音読み of a kanji regardless of whether it's actually rooted in Chinese or not. For example, the 「耗」in 消耗 is read as 「もう」 instead of 「ごう」. Another is the 「緒」in 情緒 which is read 「ちょ」instead of「しょ」.
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